REGRESSION TO THE MEAN WITH APPLICATIONS TO THE
DESIGN AND ANALYSIS OF EPIDEMIOLOGIC STUDIES
by
Gregory Paul Samsa
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1852T
June 1988
REGRESSION TO THE MEAN WITH APPLICATIONS TO THE
DESIGN AND ANALYSIS OF EPIDEMIOLOGIC STUDIES
by
Gregory Paul Samsa
A Dissertation submitted to the faculty
of The University of North Carolina at
Chapel Hill in partial fulfillment of the
requirements for the degree of Doctor of
Philosophy in the Department of Biostatistics.
Chapel Hill
1988
GREGORY PAUL SAMSA. Regression to the Mean with Applications to
the Design and Analysis of Epidemiologic Studies (Under the direction
of Dr. David Kleinbaum.)
The study of regression to the mean is extended from two-point scalar
designs to two-point vector designs and longitudinal scalar designs.
Definitions, examination of the effects of regression to the mean upon
extreme groups, and analysis recommendations are presented.
When
initial value is measured with error, the usual analysis of covariance
estimator of treatment effects accounts for regression to the mean and
is unbiased unless groups are chosen from separate populations with
different means.
Key words:
regression to the mean, analysis of covariance, errors in
variables, epidemiology.
ii
ACKNOWLEDGEMENTS
The author gratefully acknowledges the financial support of
the Veterans Administration Health Services Research and
Development Training Program and the rapid and professional
typing of Mrs. Delaine Marbry.
iii
Table of Contents
Page
Chapter 1.
A.
B.
C.
D.
Introduction
.
Simul at ion exampl e
.
Goals of the dissertation
Literature review . . . .
Chapter 2.
A.
B.
C.
D.
E.
B.
C.
D.
E.
A.
B.
C.
16
16
19
22
24
30
30
32
36
36
38
43
43
45
50
RTM in Two-point Vector Designs:
Definition and Manifestation of RTM
Introduction
.
1. Initial remarks
.
2. Notation and terminology
.
3. Literature review
. . . . ..
Definition of RTM in Two-point Vector Designs
RTM in Natural Populations ..
1. Introduction
.
2. First simulation example.
3. Second simulation example .
RTM in uncontrolled designs that use truncated
sampl ing
.
Summary and final remarks
Chapter 4.
1
3
8
10
RTM in Two-point Scalar Designs
Introduction
.
1. Overview
.
2. Notation
.
3. Terminology
4. Simulation example
RTM in designs without a treatment intervention
1. Introduction
.
2. RTMG and RTME
.
RTM in designs with treatment and control groups
1. Introduction
.
2. RTME and ANCOVA
.
RTM in designs without control groups
1. Introduction
.
2. External and internal estimators.
Summary and conclusions
.
Chapter 3.
A.
Introduction
52
52
54
56
57
62
62
63
71
78
81
RTM in Two-point Vector Designs Continued:
Regression Analysis when many variables are
subject to RTM
Introduction . . .
Literature review
Recommendations
85
87
91
Chapter 5.
A.
B.
C.
Introduction.....
1. Overvi ew . . . . .
2. Notation and terminology
3. Literature review . . . . . . . . . . . .
Analysis of RTM in longitudinal scalar designs
1. Introduction...............
2. Implementing a general analysis strategy.
a. Simulation example. . . . . . . . . .
b. Analysis of RTM before data collection
Final remarks . . . . . . . . . . . . . . . .
1. Summary . . . . . . . . . . . . . . . . .
2. Comparison between RTM in longitudinal scalar
designs and RTM in two-point scalar designs
Chapter 6.
A.
B.
95
95
96
102
106
106
108
108
123
128
128
130
Examples
Introduction
.
Examples
.
1. Example 1 - Chapel Hill Pediatric Behavior
Study . . . . . .
.
a. Introduction
.
b. RTM and MANCOVA in a truncated two-group
design
.
c. RTM in a single-group truncated design . .
2. Example 2 - Edgecombe County Hypertension Study
a. Introduction
.
b. Accounting for RTM in a regression analysis
c. Conceptual issues involved with interpreting
variables . . . .
.
.
3. Example 3 - Accounting for RTM in a longitudinal
scalar design
.
.
Chapter 7.
A.
B.
C.
RTM in Longitudinal Scalar Designs
133
134
134
134
137
142
148
148
149
154
157
Summary and Final Remarks
Introduction
.
Summary and conclusions
.
Directions for future research
163
165
174
Bibliography
175
Appendix 1 - SAS code for simulation examples.
Appendix 2 - Truncated bivariate normal formulae . . .
Appendix 3 - RTM in the ANCOVA when VI is measured with error
178
180
185
e·
Chapter 1
Introduction
(lA). Introduction:
According to Davis (10), "regression to the mean (RTM) is the
phrase used to identify the phenomenon that a variable that is extreme
on its first measurement will tend to be closer to the center of the
distribution for a later measurement".
RTM is a common phenomenon, as
illustrated by citations from disciplines such as education (4),
genetics (14), psychology (5, 25), epidemiology (10, 11, 15, 27-29, 3233) and business (17), yet RTM appears to be insufficiently understood
(4, 5, 11, 25).
The goal of this dissertation is to provide a general
treatment of RTM beginning with Davis' statement as a point of
departure.
The dissertation is organized into seven relatively self-contained
chapters.
Chapter 2 (RTM in two-point scalar designs), chapter 3 (RTM
in two-point vector designs: definition and manifestation), chapter 4
(RTM in two-point designs: regression analysis when many variables are
subject to RTM) , and chapter 5 (RTM in longitudinal scalar designs) each
discuss RTM in different types of study settings and form the main body
of the text.
two~point
For each type of study design (i.e. two-point scalar,
vector and longitudinal scalar) the treatment includes:
(a)
a definition of RTM;
(b)
a description of the effects of RTM upon groups;
2
(c)
a description of the effects of RTM upon statistical procedures;
and
(d)
a recommended analysis strategy to follow in the presence of RTM.
Chapters 2 and 4 are primarily syntheses of the literature, while
chapters 3 and 5 are almost entirely original in content.
Both chapters
3 and 5 revolve around extended definitions of RTM (i.e. extended to
cover more general study designs), and each of these chapters uses
simulation examples for illumination.
Chapter 6 illustrates the
application of the results of the dissertation to the design and
analysis of three epidemiologic studies, namely:
(a)
a two-point vector study of the effects of psychological
counseling upon mildly disturbed children;
(b)
a two-point vector study of the determinants of change in blood
~ •
pressure; and
(c)
a longitudinal scalar study of diastolic blood pressure.
Apart from this introduction, chapter 1 includes a simple simulation
example, a statement of the goals of the dissertation, and a
nonnotational literature review.
Chapter 7 contains a summary and
concluding remarks.
The following simulation example provides a simple illustration of
RTM.
3
(la). Simulation example
The purpose of this simulation example is to illustrate RTM in a
simple study setting, namely a two-point pretest-posttest design.
The
simulation depicts both the effects of RTM upon individuals and the
effects of RTM upon groups, but no explicit notational distinction is
made between these two manifestations of RTM until" section 10.
There,
regression effects upon individuals are denoted by RTMG while regression
results upon groups are denoted by RTME.
The simulation was implemented
using SAS's pseudorandom normal generator NORMAL (30), as described in
appendix 1.
To begin, let Y1 and Y2 denote scalar measurements of diastolic
blood pressure (OBP) that are taken on the same 10000 individuals at the
time points t1 and t2 (t1<t2)' respectively.
As a working definition,
say that RTM holds if and only if 0<8<1, where 8 denotes the slope
coefficient of the regression of Y2 on Y1. Note that when 0<8<1,
individuals that have extreme Y1s will tend to have less extreme Y2s. In
~
order to generate a data set with 0<8<1, let the observed OBP at time
tj' namely Yj' be modeled by Yj-YT+E j (j-1,2), where
Yj = observed OBP;
YT - true OBP (unobserved), which is assumed to remain stable from
t1 to t2; and
Ej • random error (unobserved), which includes both measurement
error and short-term fluctuations about the true YT'
In addition, assume that the superpopulation distribution of the YTs is
N(80,100) (i.e. normal with mean 80 and variance 100) and that the
distributions of the E1 and the E2 are each N(0,100). Further, assume
that YT, E1, and E2 are all pairwise independent. The above assumptions
4
imply that (Y1'Y2)-N(80,80,200,200, .5) (i.e. bivariate normal with mean
vector (80,80), variances (200,200) and correlation .5), which in turn
implies that 6=.5.
(a)
These simulation specifications are chosen so that:
the sample size is sufficiently large so as to illustrate RTM
effects with a high degree of precision; and
(b)
S is large enough to be realistic (2), yet small enough to clearly
illustrate RTM (i.e. since the magnitude of RTM increases as
decreases).
Note that the "errors in variables" (EIV) component of the present
model (i.e. the true values and random error) is only used in order to
create a natural example.
In particular, the condition for RTM, namely
0<8<1, always holds for the EIV model (at least as formulated here), but
is not unique to that model.
Also, the condition of bivariate normality
may be loosened, as in Das (9).
For concreteness, the first five individuals in the simulated data
set are listed in table 1 below.
Note that only VI' Y2' and D-Y 2-Y 1 are
actually observed by the researcher.
Table 1
First five individuals of the simulated data set
Individual
1
2
3
4
5
YT
E1
76.526
89.138
79.229
67.322
75.988
-4.153
7.657
-1.199
6.696
5.238
Y1
72.373
96.794
78.031
74.018
81. 227
E2
14.842
-2.944
20.634
-10.682
10.836
Y2
91.369
86.194
99.863
56.641
86.824
D
18.996
-10.600
21.833
-17.377
5.597
The means and standard deviations for the entire population are listed
in table 2 below.
In addition, the observed regression of Y2 on Y1 is
•
5
A
A
Y2=80.005+.504(Y l -80.086), which is essentially the Y2=80+.5(Y l -80) that
is expected.
Table 2
Observed and expected moments (n=lOOOO)
mean
YT
E1
E2
Y1
Y2
D
80.045
0.041
-0.035
80.086
80.010
-0.075
observed
(std dev)
(10.031)
(10.051)
(10.093 )
(14.183)
(14.333 )
(14.276)
mean
80.000
0.000
0.000
80.000
80.000
0.000
expected
(std dev)
(10.000)
(10.000)
(10.000)
(14.142)
(14.142)
(14.142)
For concreteness, consider the RTM at two different values of Yl ,
namely Yl=110 (which is "extreme" above the population mean of 80) and
Yl =70 (which is "extreme" below the population mean of 80). In the
former case, E(Y 2 IYl=110) = 80.005 + .504 (110-80.006) = 95.082 , which
is considerably less extreme than 110. In the latter case, E(Y 2 IYl=70)
equals 74.962, which is also less extreme than 70.
This expected RTM is
actually observed, at least on average, for the 26 individuals with Yl
near 110 (i.e. in the interval from 109.5 to 110.5) and the 207
individuals with YI near 70 (i.e. in the interval from 69.5 to 70.5), as
illustrated in table 3 below.
Equivalently, the difference score D is
observed to be negatively correlated with the initial value Yl since D
tends to be negative when Yl is large (i.e. YI near 110) and tends to be
positive when Y1 is small (i.e. Y1 near 70). Note that as Y1 becomes
more extreme the effects of RTM upon D become more pronounced. That is,
when Yl -70, which is only 10 units from the population mean, the "RTM
effect" on D is near +6, while when YI-lIO, which is 30 units from the
6
population mean, the "RTM effect" on D is about -12, which is greater in
absolute value.
Table 3
Illustration of regression effects on extreme individuals
VT
E1
E2
VI
V2
D
109.5<V 1<110.5 (n-26)
mean
(std dev)
69.5<V 1<70.5 (n-207)
mean
(std dev)
96.424
13.480
1.269
109.904
97.693
-12.211
75.720
-5.709
0.726
70.011
76.445
6.435
(7.418)
(7.462)
(12.781)
(0.290)
(12.537)
(12.554)
(7.385)
(7.367)
(9.757)
(0.291)
(12.550)
(12.548)
Table 4 below illustrates the effects of RTM upon extreme groups.
Data from the entire sample are compared with data from the subsample
with V1>95. For example, a design that only includes individuals with
V1>95 might be motivated by a desire to isolate the hypertensives in the
population, with hypertension being operationally defined by Vl>95. Note
that an ideal classification scheme would employ VT rather than VI (i.e.
the scheme would be based upon "true hypertensives" instead of "labelled
hypertensives"), but unfortunately VT is not observable in practice.
Overall, the mean difference score 0 is about zero for the entire sample
while
0 is
-11.056 for the extreme group, which illustrates the effects
of RTM.
That is, each individual with V1>95 has a negative expected
difference score because of RTM (i.e. because of the negative
correlation between D and VI)' leading to a negative 0 overall. In
contrast, the entire sample balances individuals with positive expected
difference scores against those with negative expected difference
scores, leading to no RTM overall.
~ ~
7
Table 4
Comparison of entire sample with truncated subsample
Y1>95 (n=1426)
mean
(std dev)
YT
E1
E2
Y1
Y2
0
91.150
11. 192
0.136
102.342
91. 286
-11. 056
(7.573)
(7.545)
(9.976)
(6.223)
(12.475)
(12.531)
all Y1 (n=10000)
mean
(std dev)
80.045
0.041
-0.035
80.086
80.010
-0.075
(10.031)
(10.051)
(10.093)
(14.183)
(14.333)
(14.276)
To summarize:
(a)
the hypothetical study of "labelled hypertensives" illustrates
RTM under a fairly realistic set of assumptions;
(b)
RTM implies that the difference score 0 is negatively correlated
with the initial value VI;
(c)
even though RTM is expected among individuals, random sampling
-
tends to balance individuals with E(O»O and E(O)<O, so that E(O)=O
overall; and
(d)
on the other hand, sampling schemes that select individuals on the
basis of extreme observed values at baseline can also have group
means that are subject to RTM.
Chapter 1 now continues with a discussion of the goals of the
dissertation.
8
Goals of the dissertation
(~).
The overall goal of this dissertation is to provide a unified
discussion of RTM that covers every possible study situation.
This
discussion includes definitions, description of RTM effects, conditions
for identifying RTM, recommendations for analysis, and the elucidation
of relationships between RTM and other statistical concepts.
The
primary benefit of this research is expected to be the conceptual
clarification of an issue that appears to be insufficiently understood
(4, 5, 11, 25).
In attempting to accomplish this purpose, some specific
subgoals include:
(I)
provide definitions of RTM in
{a}
two-point scalar designs {chapter 2};
{b}
two-point vector designs (chapter 3); and
(c)
longitudinal scalar designs (chapter 5);
The definition of RTM in two-point scalar designs uses previously
published results, while the remaining cases involve conceptual
extensions of the principle of RTM.
Each of the extended definitions of
RTM reduces to the definition of RTM in two-point scalar designs as a
special case.
Note that a definition of RTM in longitudinal vector
designs may be obtained by combining (b) and (c) above, so that this
study design is also treated by implication.
{2}
for each study design, describe the effects of RTM, both upon
groups and upon statistical techniques that might be affected by
RTM; and
(3)
recommend analytical methods that adequately account for RTM.
e"
9
As a rule, these recommendations do not require new statistical methods
but are instead based upon a systematic application of recognized
techniques.
An exception to the above is the analysis of RTM in
situations where a treatment effect must be estimated in uncontrolled
designs which recruit extreme individuals.
The systematic discussion of RTM begins in
concludes with a literature review.
~hapter
2.
Chapter 1
10
(lQ).
Literature review:
This section is intended to be a general overview of the RTM
literature.
More detail, as well as specific formulae, is provided in
in chapters 2 through 5, once sufficient notation is developed.
First consider the two-point scalar design with no treatment
intervention, and assume that
and 01- 02- a for simplicity.
~1- ~2- ~
Define RTMG (for regression to the mean in the sense of Galton) to be
the tendency for individuals with extreme values at t1 to become less
extreme at t2'
Say that RTMG holds whenever 0<6<1, where 6 denotes the
slope coefficient of the regression of Y2 on Y1. Then, for example, an
individual with, say, Y1·~+ka, who is "extreme" (i.e. different from~)
by ko units, is expected to have
which is less extreme than is
Y2-~+6ko,
Yl·~+6ko,
only 6ka units from its mean,
since 6<1.
Note that, strictly
speaking, if we allowed -1<6<0 then extreme individuals at tl would also
tend to be closer to the center of the distribution at t2' but variates
that are negatively correlated over time are not of interest here.
biological variates have 6>0.
Most
Often, the "errors in variables" {EIV}
model, which postulates each observed value to be the sum of a "true
value" and a "random error", is used in order to generate data for which
0<6<1, but RTM does not require EIV {13}.
Cochran {6} discusses the EIV
model in detail.
Define RTME {for regression to the mean in the sense of expected
values} to be the tendency for extreme groups at tl {i.e. groups with
Ylf~l}
to have less extreme group means at t2'
In two-point scalar
designs RTME occurs if and only if a group is composed of extreme
individuals (i.e.
Ylr~l)
and, in addition, RTMG is present {10}.
The
~
-
11
RTM literature seldom makes an explicit distinction between RTMG and
RTME, but nevertheless it is often heuristically useful to do so.
This literature review begins by discussing RTMG and RTME in
two-point scalar designs using:
(a)
no treatment intervention;
(b)
a treatment group and a control group; and
(c)
a treatment group only.
Then, RTM in two-point vector designs and RTM in longitudinal scalar
designs are considered.
There is no literature discussing RTM in
longitudinal vector designs.
To begin, the issue of RTM in two-point scalar designs without a
treatment intervention is typically considered in two study settings. In
the first case, suppose that subjects are selected on the basis of, say,
extremely large values of VI.
Such a design is termed a truncated, as
opposed to a random, sampling design.
Parameters and statistics from
truncated designs are denoted by superscripted primes.
Suppose that
~
Y2<Yl (i.e. that the group means are observed to "regress").
Then, it
.. .
is natural to inquire whether or not the observed difference (Y2-Yl) is
likely to have been due to RTME or whether, in addition, the population
mean has changed (i.e.
~21 ~1)'
Davis (10) gives an example of such an
analysis.
The second group of citations discusses the situation where the
researcher wishes to predict, before applying truncated sampling, what
.
the likely effects of RTM will be.
For example, let tl denote the time
of initial screening using truncated sampling and let t2 denote the time
of the baseline measurement.
Here, the choice of truncation rule at t 1
must take into account the RTM effects between tl and t2' that is the
12
tendency for extreme samples to become less extreme upon remeasurement.
~
Shepard (33) notes that prediction of RTME before data collection may be
understood to be a two-step process, whereby the researcher first
predicts what value of VI is likely to occur (e.g. using the truncated
normal formulae of Tallis (34), then calculates the expected RTME
conditional upon that value of VI actually being observed.
Davis (10)
Ederer (11), Gardner (15), McMahan (24) and Goldman (16), among others,
all recommend truncation rules in this situation.
When a treatment intervention occurs between t l and t2' it is
necessary to disentangle RTM effects from treatment effects, in order to
obtain appropriate estimators of the latter.
That is, each treated
individual is still anticipated to undergo, over and above any effects
of treatment, whatever RTMG that would have been anticipated in the
absence of treatment (1).
When a control group is available, the
A
analysis of covariance (ANCaVA) model
A
A
A
D·~+8lVl+~T
(O-V2-Vl' T-l if
treated, T-O if control) accounts for RTM, as discussed by laird (22),
since the control group provides an indication of the likely magnitude
of the RTM (i.e. both RTMG and RTME) in the treatment group.
Here,
since the ANCaYA using VI as a covariate and V2 as a response is
equivalent to the ANCaVA that uses VI as a covariate and the difference
score 0 as a response (22), there is no problem in having VI as both
part of the response function and as a covariate.
The ANCaYA provides
an improvement in precision when compared to the alternative analysis,
namely a two-sample paired t-test, that does not account for RTMG (note
that the ANCaYA accounts for RTMG by including initial value as a
A
covariate since 81' the slope coefficient of VI in the ANCaYA, is the
same parameter that is used to define RTMG).
In addition, laird claims
e-
13
that the ANCOVA estimator of treatment main effect also provides a bias
adjustment relative to the t-test estimator, since the ANCOVA estimator
takes into account the differential RTME expected between the treatment
and control groups when their group means differ at t 1. Neter (26), for
example, illustrates this argument when Y2 is the response variable and
the covariate is general (i.e. the covariate need not be the measurement
of Y at t 1), so long as the covariate is measured without error.
Cochran (6) disputes Laird's claim about the unbiasedness of the usual
ANCOVA estimator when Y1 is measured with error and, in addition, the
treatment and control group means differ at tl. Crager (8), however,
"-
holds that 62 is unbiased. In truth, the usual ANCOVA estimator is
often unbiased even when the covariate is measured with error. Appendix
3 presents the conditions when this is the case.
James (18), Senn (31) and Davis (10) discuss adjusting for RTME in
uncontrolled two-point scalar designs that employ truncated sampling.
For example, in order to estimate the treatment main effect the observed
-"-"
~~
(Y2-Y1) must be adjusted by a term, (Y 2-Y 1), say, that represents the
magnitude of RTME that is expected to occur even in the absence of
treatment.
Unfortunately, the observed data provide insufficient
information in order to solve this problem, and so recourse must be made
to some untestable assumptions.
specification of
~1'
Davis recommends "external"
~~
01' and S, from which (Y2- Y1) is obtained directly.
In the absence of any outside information, James and Senn both recommend
assuming that the functional form of the treatment effect is known and
that all population parameters remain constant, from which the treatment
effect may again be estimated.
Both authors assume a multiplicative
treatment effect, but section 202 argues that accounting for RTME is
14
most appropriate when an additive model holds, and adapts James' method
to this assumption.
The manifestation of RTM in two-point vector designs has not yet
been systematically described.
Campbell (4) observes that
characteristics that are not part of the participant selection criteria
tend to be less subject to RTME than characteristics that are part of
the truncation rule, but he does not provide any theoretical
justification for this observation.
Chapter 3 discusses this topic in
detail, and explains when and why Campbell's observation holds true.
An important issue in two-point vector designs concerns the
appropriate analysis of data sets where many variables are subject to
RTM.
A reading of the educational and psychometric literatures, in
particular, finds criticism of some models as "not accounting for RTM",
although this criticism and its accompanying recommendations are often
made without sufficient clarity.
Chapter 4 discusses a special case of
the above topic, namely where RTM is caused by EIV and where, in
addition, regression methods are applied.
Citations from the
statistical literature on regression when variables are measured with
error include Kupper (21), Bock (3), Kendall (20), Fuller (12) and
Warren (35).
In brief, these authors discuss the effects of RTM upon
the bias and precision of regression coefficients, and recommend
alternative ways to proceed depending upon what additional information
is available.
Few authors discuss RTM in longitudinal scalar designs.
For this
topic, most of the citations mention that, under the EIV model, the
"separation of screening and baseline" (i.e. using t1 for participant
selection and beginning data collection at t2) suffices to control for
e-
15
RTME, but McMahan (24) demonstrates that this control is obtained at a
cost in precision relative to other similar designs.
Nesselrode (25)
and Anderson (1) provide a more thorough discussion of the persistence
of regression effects over time in longitudinal designs.
In addition,
Nesselrode also illustrates some of the difficulties involved with
defining RTM as a longitudinal, as opposed to simply a two-point,
concept.
The proposed definition of RTM in longitudinal designs (see
chapter 5) circumvents these difficulties.
No authors discuss RTM in longitudinal vector designs, but the
extended definitions of RTM in chapters 3 and 5 may be combined in order
to treat this case as well.
Chapter 2 discusses RTM in two-point scalar designs.
Chapter 2
RTM in two-point scalar designs
(2A).
Introduction:
1.
Overview:
This chapter considers RTM in two-point scalar designs, that is
where each individual has one characteristic that is measured at the
same two points in time.
The two-point design is the simplest study
setting that may give rise to RTM, and these results are used to
motivate the treatment of RTM in more complicated study situations.
Chapter 2 includes:
(a)
a simulation example;
(b)
a discussion of RTM in two-point scalar designs that do not
include a treatment intervention;
(c)
a discussion of RTM in two-point scalar designs that include both
a treatment group and a control group;
(d)
a discussion of RTM in two-point scalar designs that do not
include a control group; and
(e)
a summary and conclusions.
Chapter 2 is primarily a synthesis of the RTM literature for
two-point scalar designs.
With the exceptions of section 202 and
appendix 3, all of these results have been previously reported,
although not always within a general and coherent context.
to simplify and focus the presentation:
In order
17
(a)
the discussion of RTM is limited to studies that employ either
one treatment group and no control group, one control group and
no treatment group, or else one treatment group and one control
group.
In particular, multiple treatment and/or control groups
are not considered;
(b)
an explicit distinction is made as to whether the magnitude of
RTM is to be estimated before sampling or during data analysis.
In the latter case, the goal is usually to obtain an estimator
of treatment effect that is not biased by RTM.
As an example of
the former case, a researcher may wish to predict how much a
group of individuals that are chosen because of extreme values
at time tl is likely to "regress" before a treatment is
administered, say, at time t2 ;
(c)
an explicit distinction is made between the effects of RTM upon
individuals and the effects of RTM upon groups;
(d)
the discussion is limited to only two sampling schemes, called
"random" and "truncated" sampling.
"Random" sampling denotes
the usual simple random sampling (which, on average, is not
to consistently produce extreme individuals at t 1).
Truncated sampling, which produces groups that contain extreme
expec~ed
individuals at time t l , denotes a scheme whereby only
individuals that exceed a fixed cutoff point at tl are eligible
for remeasurement.
Note that the "truncated normal formulae" in
appendix 2 are only applicable to truncated sampling as defined
above.
(e)
it is assumed that there are no missing data;
18
(f)
it is assumed that all individuals are measured at the same two
points in time;
(g)
bivariate normality is assumed throughout this chapter.
In
addition, the means and variances of individual measurements are
assumed to be stable over time unless otherwise stated; and
(h)
a simulation example is used in order to
ori~nt
the reader.
The
introduction now continues with a discussion of notation and
terminology.
e-
19
(2A). Introduction:
~.
Notation:
Let (Y 1,Y 2) denote two observations taken on the same individual
at times t 1 and t 2, respectively (tl<t2)' Note that the subscript j
in Yj denotes time and not individual, and also note that individuals
are not explicitly subscripted. All individuals are measured at tr
same two points in time.
(1)
The true regression of Y2 on Y1 is
].I2··1=].I2+ 8(Y 1-].Il),
where ].12.1 denotes the conditional mean of Y2 given Yl'
In practice,
the parameters of (1) are usually unknown and (1) is replaced by
"
Y2-Y 2+8(Y 1-Yl)'
"
where e is the usual least squares estimator of 8 and Y2 denotes the
(2)
predicted value of Y2'
For consistency of presentation, the
discussion in chapter 2 assumes that the regression of Y2 on Y1 is
estimated from the data (i.e. that (2) holds instead of (1) ), unless
mention is made to the contrary.
Sometimes the distribution of Y2
conditional upon Y1 equaling a fixed value, YI say, is of interest, in
which case the notation Y2IYI-yl is employed. There, YI and Y2 both
denote random variables while the lower case YI denotes the
realization of VI'
Sometimes the observed Yj is postulated to be the sum of a "true
value" YTj and a "random error" Ej , in which case the "errors in
variables" (EIV) model is said to hold. It is usually assumed that
true values do not change over time, in which case YTj is replaced by
YT' Here, the "random errors" may represent either measurement errors
(e.g. as might occur in a complicated biological assay) or real but
short-term fluctuations about the true YT, or both.
20
Studies often proceed by randomly assigning individuals into
either treatment or control groups, which are denoted by the
subscripts "T" and "C", respectively (e.g. YIC denotes the sample mean
of the control group at time tIl.
Any treatment intervention is
assumed to occur between tl and t 2. Since "T" is used to denote both
treatment status and true value, treatment status'is placed inside
parentheses whenever there is a possibility of misinterpretation (e.g.
YIT(C) denotes the true value for an individual in the control group
at time tIl.
Truncated sampling is said to occur when only those individuals
with values of VI that are observed to be above a fixed cutoff point
are considered to be eligible for remeasurement.
Denoting the cutoff
point by AI' the truncation rule is to accept individuals with VI>A I ,
or equivalently with YI*>A I *, where Yl*-(Vl-~I)/crI' and Al*-(Al-~I)/al
are the standardized values of VI and Al ,respectively.
Some other
notation that is used for truncated sampling includes
k*-~(AI*)/{I-¢-I(AI*)}' which is the density of the truncated
standardized normal distribution evaluated at the cutoff point AI'
where ;(0) and ¢-l(o) denote the standardized normal density and
distribution functions, respectively.
The proportion of observations
that exceed Al is denoted by s*, where AI*-¢-I(l-s*).
convenience, it is useful to define q*-k*(AI*-k*).
For notational
Parameters and
statistics from truncated designs are indicated by superscripting a
-~
prime (e.g. VIC denotes the sample mean of the control group at time
tl under truncated sampling).
For convenience of exposition, both
here and elsewhere, extreme is taken to mean extremely large (i.e.
above the population mean), unless otherwise noted.
~
-
22
(2A). Introduction:
~
. Term i nology:
The following definitions are used in this chapter.
Random sampling - is a participant selection scheme whereby all
individuals are eligible for selection regardless of the observed
value of YI .
Truncated sampling - is a participant selection scheme whereby only
individuals that are above a fixed truncation point at time t l , say
with YI>A I , are eligible for selection.
Errors in variables (EIVl model - denotes (Y I ,Y 2) where
2
2
(3) Yj-YT+Ej' YT-N(~T,aT)' Ej-N(O,aE)' (j-I,2), and YT' EI and E2 are
all independent, which imply that
(YI'Y2)-N(~'~T,af+af,af+af,af/(af+af)' Unless otherwise stated, the
EIV model means (3) above as opposed to any of its extensions.
The basic EIV model (3) may be extended in a number of ways.
First, the variances of the error term E may differ from tl to t2' in
· h case aE2 1S
. rep 1ace d byaEl
2 an d aE2.
2
wh1C
Th 1S
. phenomenon 1S
. l'1 ke1y
if, for example, Yj denotes the mean of nj observations, with nl#n2'
Secondly, true values need not remain constant over time, in which
case be YT is replaced by YTI and YT2.
Finally, neither the YTs nor
the Ejs need be assumed Gaussian, although this entire dissertation
does so. Das (9) drops the assumption of normality on the YTs although
he retains it for the Ejs.
Population drift - is said to occur when the population mean changes
over time while the population variance remains constant.
For
example, under the extended EIV model YI-YT1+El' Y2-YT2+ E2' YT2 -YTI+ c ,
population drift occurs when c-O.
e-
23
Regression to the mean (RTM) (working definition) - is a tendency for
variables that are extreme on initial measurement to become less
extreme upon remeasurement.
Two manifestations of RTM, namely RTMG and RTME are considered
here, where the distinction between RTMG and RTME explicitly
recognizes whether the "variables" in the above working definition
refer to individual data values (i.e. RTMG) or to group means (i.e.
RTME).
RTMG (for RTM in the sense of Galton) - is a RTM of individual data
~
values.
RTMG occurs when 0<6<1.
A
In chapter 2, 6 is always assumed to lie between 0 and 1 unless
explicit mention is made to the contrary.
RTME (for RTM of expected values) - is a RTM of group means.
That is,
RTME is a negative bias in YZ-Y1 as an estimator of u2- u 1' which is
caused by both RTMG and the presence of extreme individuals.
The introduction now continues with a simulation example.
24
(2A).
Introduction:
1. Simulation examole:
The goals of this simulation example are:
(a)
to orient the reader to the notation and terminology of
this chapter; and
(b)
to illustrate how RTMG and RTME can affect tRe estimation
of treatment effects.
The simulation example in chapter 1 depicted a two-point scalar
study without a treatment intervention.
Although this terminology was
not used at the time, the EIV model was assumed, and both RTMG and
RTME were illustrated.
To recapitulate, RTMG was present because
A
6-.504, which lies between 0 and 1.
Therefore, at least on average,
individuals who were extreme at t 1 tended to become less extreme at
t2' RTME was illustrated for truncated sampling but not for random
sampling since Y2-Y1--.075 (which was within sampling error of zero)
,#
while Y2-Y1-·11.056.
,#
,.
That is, for truncated sampling Y2-Y1 was a
biased estimator ofU2-U1 (there, U2-U1-0), and this bias is termed
RTME.
The primary intent of the simulation example in chapter 1 was to
demonstrate that extreme samples do not tend to remain stable when
individuals are subject to RTMG, which fact has implications for both
study design and data analysis.
A basic implication for study design
is that, to compensate for RTMG, some "overkill" is required in order
to assure that a group will remain "sufficiently extreme" over time.
A basic implication for data analysis is that an adjustment for the
effects of RTM must be made when analyzing change over time "in order
to get the correct answer".
~ .
25
Having illustrated that RTM can affect two-point scalar studies
without treatment intervention, the simulation is now expanded to
illustrate the necessity for accounting for RTM when assessing the
effects of a treatment.
Suppose that, apart from the 10000
individuals that were analyzed in chapter 1, another 10000 persons are
simulated from the same superpopulation (i.e. Yj=YT+Ej' YT-N(80,100),
Ej -N(0,100), with YT, E1 and E2 all independent). These new
individuals are given a treatment intervention between times t 1 and
t 2, say a drug that is intended to lower diastolic blood pressure.
Further suppose that, instead of lowering diastolic blood pressure,
the drug actually acts to increase diastolic blood pressure by 2 units
in each individual.
Thus, Y1-YT+E 1 while Y2-YT+E2+2. The 10000
individuals from the simulation in chapter 1 are retained as a control
group.
To begin, assume that the control group is not available and
that the researcher is unaware of the effects of RTM.
Table 1 below
presents group means for both random and truncated sampling designs.
First suppose that random sampling is employed.
Then, the natural
estimator of treatment main effect, namely DT' equals 2.000, which is
well within sampling error of the correct value of 2.
In contrast
-'
with the result for random sampling, DT equals -9.650 for truncated
sampling, which illustrates the effects of RTME.
Note that the naive
researcher incorrectly determines that the main effect of the
treatment intervention is to reduce diastolic blood pressure by almost
ten units on average instead of correctly concluding that the
treatment increases diastolic blood pressure by two units.
In sum, a
RTME adjustment is required when estimating treatment main effects in
26
uncontrolled designs that employ truncated sampling but is not
required in uncontrolled designs that utilize random sampling.
Table 1
Treatment group means for random and truncated sampling designs
random sampling (n=10000)
YT
Y1
Y2
0
truncated sampling
with Y1>95 (n=1426)
mean
(std dev)
mean
(std dev)
79.938
80.051
82.051
2.000
(9.959)
(9.826)
(9.896)
(13.996)
90.901
102.251
92.601
-9.650
(7.544)
(6.233)
(12.555)
(12.377)
Next suppose that the researcher is also interested in
determining whether or not the treatment has similar effects on
individuals with differing diastolic blood pressures, that is he
wishes to determine whether or not a treatment by Y1 interaction is
present. So, he regresses 0 on Yl' as in table 2 below, and then
tests whether or not 61, the slope coefficient of Yl' is zero. This
test is significant for both sampling schemes, and so the naive
researcher concludes that the drug is most effective in treating
extreme hypertensives, which is not in fact the case.
To summarize,
this example illustrates that an adjustment for RTMG is required when
analyzing treatment by Y1 interaction in uncontrolled studies,
regardless of whether random or truncated sampling is used.
Table 2
Observed regressions of 0 on Y1 for treatment group,
random and truncatea sampling
Truncated
Random
"
0-1.971-.501(Y I -80.051)
"
O-.173-.443(Y I -80.051)
e'
27
Now suppose that both treatment and control groups are available
to the researcher.
The control group can now be used in order to
assess the likely RTM in the treatment group.
In turn, this
assessment allows the average change score in the treatment group,
DT=Y2T- Y1T), to be adjusted by the additional
namely 0T (i.e.
experience in the control group, in order to account for the RTM that
is expected to occur in the treatment group even in the absence of an
intervention. Now, information from the treatment and control groups
may be combined into an "adjusted" estimator of treatment main effect
in two conceptually distinct ways, namely via the difference of
differences (DT-Dc ) or via the analysis of covariance (ANCOVA), as
illustrated in table 3 below. In table 3A the difference of
differences is 2.075 for random sampling and 1.406 for truncated
sampling, both of which are within sampling error of the correct value
of 2.
Table 38 illustrates a covariance analysis, which begins by
comparing the separate regressions of 0 on VI in the treatment and
control groups in order to formally assess the magnitude of any
treatment by VI interaction.
This test is statistically
nonsignificant for both sampling schemes. Accordingly, the ANCOVA
model
A
(4)
A
A
A
0-Bo+6IVI+62T (T 1 for treatment, T-O for control)
3
is fit to both data sets, yielding 2.058 and 1.362 as the estimates of
treatment main effect for random and truncated sampling, respectively,
both of which are within sampling error of the correct value of 2.
To
summarize, in this simulation example the two methods of using the
control group to "adjust" DT for the effects of RTM, namely the
difference of differences and the ANCOVA, give similar point estimates
28
of the treatment main effect.
In general, this relationship holds
-
-
unless YIT and Y1C differ substantially. Section 2C discusses
technical points about how best to estimate treatment main effects
when a control group is available, but for now the main point is that
either method of utilizing the control group goes a long way towards
eliminating the previously illustrated problems tnat are caused by
RTME.
Table 3
Comparison of treatment and control groups
Table 3A
Group means
Random sampling
Truncated sampling
(n-l0000 treatment,
n=10000 control)
(n-1426 treatment,
n-1456 control)
Y2
Treatment 80.051 82.051
Control
80.086 80.010
Y1
adjusted estimator of
treatment main effect
o
2.000
-.075
Y2
0
Y1
102.251 92.601 -9.650
102.342 91. 286 -11. 056
2.075
1.406
Table 38
Regression lines
Treatment
Control
'"
0-42.077-.501Y
1
0-39.642-. 504Y 1
'"
ANCOVA
'"
0-41.887-.498Yl+2.058T
'"
0-39.053-.481Y
1+1.362T
0-35.635-.443Y 1
0-41. 945-. 518Y1
The remainder of chapter 2 expands upon the ideas illustrated by
this simulation.
Section 28 discusses the mechanism by which RTMG
induces RTME in designs without a treatment intervention.
Section 2C
discusses how best to use the control group in order to estimate
treatment effects in controlled designs.
There, the recommended
29
estimators of treatment effects all account for RTME.
Section 20
discusses the shortcomings of uncontrolled studies and suggests a new
method of adjusting for RTME in this design.
the results of this chapter.
Section 2E summarizes
30
(28).
RTM in designs without a treatment intervention:
1.
Introduction:
This section discusses the relationship between RTMG and RTME in
designs without treatment intervention.
To summarize the results,
RTMG, which occurs when O<B<l, induces RTME in groups that are extreme
at tl (i.e. groups with Y1'Ul)'
The magnitude of expected RTMG for
any individual is (S-1)(Y1-Ul)' while the expected RTME for any group
is (S-1)(Y1-Ul)'
In order to estimate RTME before sampling, the
population parameters sand ul must be specified and the expected
value of VI must be obtained from knowledge of the sampling rule and
the population parameters
~l
and 01'
Then, the resulting values of
ul' Sand Yl are inserted into the above formula for RTME.
In
truncated sampling, the expected value of Yl is ul+k*ol'
The results of this section are used in two ways.
First, an
~
.
understanding of the consequences of RTM in the absence of treatment
is fundamental to describing the role of RTM in more complicated
designs involving treatment intervention.
Secondly, many designs
separate the initial screening, which uses truncated sampling, and the
actual beginning of the study, which occurs sometime later.
For
example, letting tl denote the time of screening, t z denote the formal
beginning of the study, and t3 denote the end of the trial, response
might be calculated as Y3-YZ'
Here, it is often of interest to
predict, before data collection begins, how much a group of
individuals that are to be chosen because of extreme values at tl is
likely to regress by t z'
For this purpose, regardless of the later
design of the study, the time from initial screening to the formal
•
31
beginning of the study may be analyzed as a two-point design without
treatment intervention.
Section 28 now continues with a discussion of RTM in designs
without a treatment intervention.
32
(28).
RTM in designs without a treatment intervention:
z.
RTMG and RTME:
])
8
Diagram 1: RTMG in individuals
First suppose that the regression of V2 on VI is known and that
RTMG is present (i.e. 0<8<1). Diagram lA illustrates that extreme
individuals at tl(i.e. those with V1>UI' say) are expected to be less
extreme at t2' since individuals that are extreme by (V 1-UI) units at
t 1 are only expected to be extreme by 8(VI-UI) units at t2' and
0<8<1. Diagram 18 graphs the regression of the difference score D on
VI' namely
...
(5) O-(U2- UI)+(8-1)(YI-uI)'
Now, D contains two components.
The first component, namely (u2-ul)'
depends upon the magnitude of the population drift, and is the same
for every member of the population.
The second component, namely
(8-1)(Y I - UI)' represents the effects of RTMG upon the expected
...
difference score D. (B-l)(V1-ul) is negative when VI>UI and positive
when Y1<UI' illustrating that a negative correlation between
difference scores and initial value occurs when RTMG is present.
The
magnitude of this "RTMG effect upon individuals" increases as VI
becomes more extreme and also as 8 decreases from one towards zero.
~ .
33
When w1=w2' an individual's difference score is expected to be nonzero
unless either 6=1 (i.e. no RTMG) or Y1=W1 (i.e. the individual is not
extreme).
Now consider a group of extreme individuals, say with mean
Y1*>W1' From (5), it follows that the expected mean difference score
for the group, 0 say, is not (~2-~1) but instead is
A
(6)
rr=(~2-~I)+(8-1)(YI*-~1)'
as reported by Shepard (33).
That is, the expected RTME is obtained
by simply averaging the RTMG effects upon individuals.
When
8'~1'
and
are unknown, as is usually the case, these
~2
parameters are replaced by estimatots.
Here, the usual least squares
A
estimator of 8, namely 8, is unbiased regardless of whether random
sampling or truncated sampling is employed (see appendix 2).
provides an unbiased estimator of
~I
Y1
in random sampling designs. In
truncated designs, Shepard recommends retaining the YI values for all
individuals, not just those that exceed AI' in order to estimate ~1'
In the absence of this information, the method of moments may be used
to estimate
(7)
That is (see appendix 2),
E(Y1)=~l+alk* and E(si 2)=aI 2(I+q*)
~l'
are solved for
(8)
~l
(and also for 01' a nuisance parameter here) by
~I·Y~-k*{S~2/(1+q*)}I/2 and 0I=[si 2/(I+q*)]1/2.
Note that the method of moments does not always produce efficient
estimates, and that alternative methods of solving equations (7) for
~I
might be considered.
Also note that (7) only includes information
that is collected at t i . Thus, solutions of (7) are also applicable
to designs that involve a treatment intervention between tl and t2'
34
As an example of the above calculation, suppose that only 2.5%
of screened individuals exceed the cutoff point A1=13.92 and that
sf=.488, Y1=14.672, and 6-.75 are observed. Then, s*=.025 and
~
".-
A
A~=¢-t1-.02S) = 1.96. Since ~(1.96)=.OS84 and 1-0 (1.96) = .025 it
follows that k*=.0584j.025 = 2.336.
and 1+q*-.122.
estimate of
The estimate of 01 is thus (.488/.122) - 2.
is 14.672-(2)(2.336)
~1
Therefore, q*=2.336(1.96-2.336)
=
The
10 , and so the estimated RTME
is (.75-1)(14.672-10) • -1.168.
Now suppose that it is of interest to predict the magnitude of
RTME before, instead of after, truncated sampling.
For example, this
goal might be motivated by a desire to ensure that the truncated sample
will probably remain "sufficiently extreme" at time t2'
Predicting the
magnitude of RTME before sampling requires a three-step process, namely
Step 1:
Guess 6 and
"
to be 6 and
"
~1;
-'"
Step 2:
(9)
~1
Find E(V 1), which requires that 01 also be guessed, where
E(V1)·~1+0Ik*(see appendix 2); and
Step 3:
Insert 6,
"
~1'
'"
-'"
and E(Yl) into (6) to obtain a predicted RTME of
"
(6-1)(E(Yl)-~I) .
Regardless of the expected value of VI' the value of VI that is actually
observed should be inserted into (6) when calculating a post-sampling
RTME adjustment.
To summarize, RTME is present if and only if RTMG is present and,
in addition, the group is extreme (i.e. Vl;Ul)'
The expected RTME is
(a)
(6-1)(V 1-Ul) , calculated after sampling, 6 and
(b)
(6-1)(Vl-~I)' calculated after sampling, 6 and Ul estimated; or
(c)
(8- 1) (Y 1- ~ 1) ,
"
~1
known; or
"
calculated before sampling, 6,
depending upon the circumstances.
~1'
and 01 guessed,
"
-
Under random sampling, 6 and Y1 are
35
unbiased estimators of 6 and
unbiased estimator of 6.
~l'
For truncated sampling, 3 remains an
In order to estimate
~l'
one design option
involves obtaining the mean of all sampled individuals, not just those
that meet the study selection criteria.
In the absence of this
information, the method of moments may be used in order to estimate
from the observed truncated moments.
applying truncated sampling,
~
..
the prediction of VI is VI
~l
~l
When predicting RTME before
must also be specified, in which case
a~I+k*vI'
Section 2C discusses RTM in
designs that include both treatment and control groups.
36
(2C).
RTM in studies with treatment and control groups:
1.
Introduction:
This section discusses the relationship between RTM and the
analysis of covariance in two-point scalar designs with a single
treatment group and a single control group.
The main conclusion is that
ANCOVA essentially accounts for RTM, although the data analyst must bear
in mind the differences between this situation and the usual ANCOVA
application.
Here, in particular:
(a)
the covariate VI is often measured with error; and
(b)
the response 0-V 2-V I contains the same variable, namely VI' that is
used as the covariate.
Apart from elucidating the relationship between RTM and ANCOVA in
designs with treatment and control groups, the main benefit of this
section involves developing a standard of comparison with which to
illustrate some of the difficulties encountered when analyzing RTM in
uncontrolled designs (see section 20).
In addition, the discussion
reiterates two points about the analysis of covariance that do not
appear to have been fully appreciated in the RTM literature.
In
particular, consideration of the relationship between RTME adjustment
and the formula for the ANCOVA estimator of treatment main effect
suggests that the analysis of RTME should, strictly speaking, only take
place when an additive model holds, which fact has implications for the
discussion of external estimators in section 202. Also, some authors
postulate an ANCOVA model where VI is measured with without error and YZ
is measured with error, even though V2 is merely a remeasurement of the
same characteristic as VI obtained at a different time point.
Section
4It
37
2C2 and appendix 3 both discuss the implications of incorrectly assuming
that YI is measured without error when, in fact, the EIV model applies.
Section 2C2 begins by discussing the relationship between RTME and
the analysis of covariance.
38
(2C).
RTM in studies with treatment and control groups:
~.
RTME and ANCOVA:
D
II.~: J~.
( Er- Dc) - A (~T- Y,cJ = (Y~.,.- yz~) - (~;+I)(~T- Y,J
Diagram 2
Illustration that the ANCaVA accounts for RTM
According to Laird (22), the additive ANCaVA model
(10)
D-eo+6IYI+~T
(T-I for treatment, T-O for control),
is a preferred method of analyzing two-point designs with treatment and
control groups when randomization (or its equivalent) holds and when, in
addition, the treatment effect is additive.
Note that 61 denotes the
pooled slope coefficient of the regressions of D on YI' not of the
regressions of Y2 on YI (i.e.81 -8-1).
Laird shows that the ANCOVA on 0
is essentially equivalent to the ANCOVA model
since the regression
(12)
Y2-~+8(YI-~I)
impl ies
A
(13)
D-(~-~I)+(8-I)(Yl-uI)'
and vice versa.
Now the ANCaVA estimator of treatment main effect from either (10) or
(11) is
39
-
'"
-
-
(14) B2=(Y 2T -Y 2C )- (Y1T-Y 1C )'
which may be rearranged in order to emphasize its relationship with
RTME.
For example,
(15) B2=(Y2T-Y2C)-(YIT-YIC)-{(8-1)(YIT-~1)-(B-1)(YlC-~1))'
where (Y2T-Y2C) is the estimator of treatment main effect that ignores
-
-
the time t 1 data, (YIT-Y 1C ) is the difference between group means at tIl
"
(S-l)(YIT-~I) is the expected RTME in the treatment group and
"
(S-I)(YIC-~I)
282).
is the expected RTME in the control group (see section
Considering the terms
"
(8-1)(YIT-~I)
and
"
-
(8-1)(YIC-~I)'
(i.e. the
RTME adjustments), unless either YIT or YIC is extreme, as for example
occurs in truncated sampling designs, the RTME adjustments are very
small.
When YIT and YIC are both extreme but similar, as for example is
expected in truncated designs that employ randomization, the RTME
adjustments in the treatment and control groups are large but similar.
When treatment and control group means differ substantially, as is the
case in the example in section Blof chapter 6, the RTME adjustments
differ substantially.
The analysis of covariance is often used in order to adjust for
the effects of unequal distribution of the covariate.
Again rearranging
(14) ,
(16)
"
"
82·{(Y2rYlT)-(8-1)(Y1T-~I)}-{(Y2C-YIC)-(8-1)(YIC-~I)}'
"
which illustrates that 82
compares a RTME-adjusted treatment group mean
(i.e.
"
-
(8-1)(YIT-~I)
group mean.
is the RTME adjustment) with a RTME-adjusted control
The difference between the RTME adjustment in the treatment
group and the RTME adjustment in the control group is
(17) (S-l)(YlT-VIC).
40
~
Oiagram 2 above provides a graphical illustration of equation (16).
There, the differential RTME caused by unequal group means at t I is
accounted for by making the assessment of treatment main effect at the
same value of YI . At any YI' the difference between the two regression
lines is just 62,
~
Laird (22) compares the ANCOVA model (10) with a model that does
not account for RTM, namely the two-sample paired t-test model
A A* A*
(IS) 0-60+62T
(T-l for treatment, T=O for control),
which has
(19) 62-(Y2T-VIT}-(Y2C-VIC}
(i.e. the difference of differences) as its estimator of treatment main
~
effect.
Now 62 (i.e. (14) ) and the t-test estimator (19) differ by
(17), that is the differential amount of RTME expected between the
~
treatment and control groups, which leads Laird to conclude that the
ANCOVA estimator is unbiased while the t-test estimator is biased, where
(17) denotes the magnitude of the bias (i.e. (17) is the penalty in bias
that is caused by not accounting for RTM in the t-test model (IS) ).
A
Apart from its alleged unbiasedness, the ANCaVA estimator 82 is more
A*
A
precise than the t-test estimator, 62 say, since 82 removes the effect
of the correlation between VI and 0 when calculating its standard error.
To summarize Laird, then, the ANCOVA accounts for RTMG by including Y1
as a covairate. This leads to a gain in precision relative to the
alternative estimator which does not account for RTMG.
In addition, the
ANCOVA estimator accounts for the RTME in both the treatment and control
groups while the t-test estimator does not, which implies that the
ANCOVA estimator is also preferable on the grounds of unbiasedness.
The
41
magnitude of the bias of ~~, however, is small unless VIT and VIC differ
substantially.
The preceding discussion serves to recast the usual rationale for
applying the analysis of covariance (e.g. Neter (26)) into RTM terms.
Note, however, that the present situation differs from the usual ANCOVA
application in two important respects.
First, trre response variable 0
contains the covariate VI' although Laird demonstrates that this makes
no essential difference in the results. Secondly, the covariate VI is
often measured with error, which violates one of the assumptions of the
usual ANCaVA.
That is, since the ANCaVA assumes that the response V2 is
measured with error it is natural to assume that VI' which is merely a
measurement of the same characteristic at another time point, should be
measured with error also.
Cochran (6) and Crager (8) discuss the
implications of applying the usual ANcaVA model when VI is actually
measured with error.
Extending this work, appendix 3 demonstrates that
A
82 is a biased estimator of the treatment main effect when the treatment
and control groups are taken from populations which have different
means.
In this case, even when VIT-V IC '
A
is the appropriate estimator to use, where
~IT
A
and
~IC
denote estimates
of the (nontruncated) means of the populations from which the treatment
and control groups are chosen.
In all other cases, including
nonrandomized allocation from the same population where V1T'V IC in the
sample, 62 is unbiased. Note that under random sampling from both
populations, (20) reduces to (19), the difference of differences.
To recapitulate, the ANCaVA accounts for RTM by including VI as a
covariate, and this inclusion accomplishes a gain in precision.
The
42
ANCOVA also accomplishes an elimination of bias relative to the
two-sample paired t-test when YI is measured without error.
where YI is measured with error is discussed in appendix 3.
The case
The
~
conclusion is that 62 remains unbiased unless the treatment and control
groups are taken from populations with different means, in which case
(20) is the correct estimator to use.
Section 20 treats RTM in uncontrolled designs that employ
truncated sampling, using the controlled design discussed in this
section as a benchmark.
4Ia
43
(20).
RTM in designs without control groups:
1.
Introduction:
Section 20 discusses the analysis of RTM in designs without
control groups.
Recall that for designs with treatment and control groups, the
analysis of covariance follows a two-step process, namely:
Step 1:
Using the observed regression of V2 on YI in the control group
for comparison, perform an analysis of treatment by YI
interaction.
If this interaction is determined to be either
nonsignificant or unimportant then proceed to step 2a,
otherwise proceed to step 2b.
Step 2a: Estimate the treatment main effect (as in section 2C2).
Here,
the ANCaVA estimator of treatment main effect automatically
takes RTME into account, as explained in section 2C2.
Step 2b: Estimate treatment effects at various values of VI'
As
discussed in section 2C2, the analysis of RTME is not
relevant to this case since an RTME adjustment should only be
performed for additive models (i.e. where treatment main
effects are estimated).
When studies do not contain an appropriate control group, either
by design or by accident, then it is still necessary to perform an
analogue of steps I and 2 above, even though the comparison group of
step I above is no longer based upon observed data.
data analyst has two options.
At this point, the
In the first case, he makes no
assumptions about the nature of the treatment effect, and external
information is used in order to guess what the regression of Y2 on YI
would have been had a control group been used. Based upon this guess,
44
an informal assessment of interaction is performed.
If this assessment
~
indicates that the treatment effect is additive, then (8) is used in
order to produce a RTME adjustment for the estimator of treatment main
effect.
This method is termed "external" estimation.
Alternatively,
instead of specifying the population parameters it might be assumed that
the treatment effect is of a specified form, in which case (8) may again
be used in order to obtain an RTME adjustment.
"internal" estimation.
This method is termed
Based upon the intimate relationship between
estimating treatment main effects and adjusting for RTME, which was
illustrated in section 2e, we recommend that internal estimation only be
applied when an additive model holds, and adapt existing methodology to
this case.
Regardless of whether internal or external estimation is
attempted, at least one untestable assumption is required in order to
proceed, and this is perhaps the major difficulty facing the analysis of
uncontrolled studies.
The discussion now proceeds to the consideration of internal and
external estimators.
section 20.
Truncated sampling is assumed for the remainder of
~
45
(20).
RTM in designs without control groups:
z.
External and internal estimators:
In order to apply the method of external estimators, take S, ul
and 01' which are obtained from another (i.e. an "external") source and
then adjust the observed (Y2T- YIT) to become
(21) (YiT-YIT)-(6-1)~lk*,
the RTME-adjusted external estimator of treatment main effect.
A
Here,
A
(6-1)0Ik* is simply the "pre-sampling" estimator of RTME from section
28.
Note that
a function of
~1
~1
is actually required in the above formula since k* is
and 01.
While there is no airtight way to determine
whether or not the RTME adjustment is appropriate (i.e. since there is
not way to simultaneously test whether or not all of the external
parameter estimates are reasonable), some partial consistency checks are
available.
(a)
In particular,
the observed values of
vi
and s1 may be compared with their
expected values, which are obtained from the external parameter
estimates using equation (8); and
....
(b)
6 may be compared wi th 6 .
When 6 and 6 differ substantially then it remains for the researcher to
....
decide whether 6 has been misspecified or else whether a treatment by Y1
interaction exists (in which case the treatment main effect may not be
of primary importance).
When external estimators of
~1
and 6 are available but 01 is
unspecified then the formula
(22) (Y;T-Y~T)-(B-l)(Y~T-Gl)
may still be used as an RTME-adjusted external estimator of the
treatment main effect.
As compared to formula (21), however, (22) does
46
not allow consistency check (a) above to be applied, which is the
4It
penalty for being unable to specify an estimate of 01'
Note that comparisons (a) and (b) above should be implemented
informally (i.e. not as statistical significance tests), since their
primary goal is the assessment of the appropriateness of model
assumptions.
Also, note that comparison (a) does' not require that the
A
specification of S be correct.
When external parameter estimates are unavailable or unreliable,
then the statistician has no recourse but to use the observed data in
combination with an additional assumption about the form of the
treatment effect when attempting to estimate the RTME in the treatment
group.
This procedure is termed "internal" estimation, since the
parameter estimates S,
~1
and 01 are generated from data that are
entirely internal to the study.
This dissertation argues that an
adjustment for RTME (or perhaps more pertinently, an analysis of
treatment main effects) is most appropriate when an additive model
A~
holds.
In this case, 6 is an unbiased estimator of S (see appendix 2),
and equations (7) may be solved for
~l
using (8), resulting in an
RTME-adjusted estimator of treatment main effect of
(23) (Y;T-Y;T}-(B~-l}(Y~T-~l)' where ~1·Y~-k*{sif(1+q*)}1/2.
The above formula is an original contribution to the RTM literature.
Note that internal estimation is a reasonable alternative to external
estimation (and, in fact, may be preferable since 6 is estimated instead
of specified), so long as the researcher is quite certain that effect of
treatment is actually additive.
The RTM literature contains two articles on estimating RTME in
studies that employ truncated sampling without a control group, namely
~
47
James (18) and Senn (31), where Senn corrects many of James' formulae
and, in addition, uses a superior method of solving James' equations.
Both authors, however, use a multiplicative model of treatment effects,
namely assuming that B(T)=Y6(C)' where 6(T) denotes the slope of the
regression of V2 on VI in the treatment group, 6(C) denotes the
(unobserved) slope of the regression of V2 on VI in the absence of
treatment, and y denotes the multiplicative treatment effect. To apply
James' method, one must carry out the following steps:
A*
(a) estimate Al from the proportion of individuals that were retained
(b)
(c)
by the truncated sampling rule;
A A*
*
calculate k*-;(A 1)/{I-¢(A 1)};
A
A*
calculate l+q*-I+{k*(A 1-k*)};
(e)
calculate ~1-s;l(l+q*);
A A2
calculate °1-(°1)1/2;
(f)
calculate ~1-V;-k*~I;
(g)
calculate
(h)
calculate ~-8/~, the estimate of the multiplicative treatment
(d)
8-{B~2(I+q*)+I-(s;J;I)}1/2; and
effect.
As an example of the above calculations, in the simulation 14.26%
of individuals fulfilled the exclusion criterion of V1>95 and were
treated.
6 =.577.
These individuals had sl-6.233, V1-102.251, s2-12.555 and
Performing the above calculations:
(a)
~i-¢-l (1-.1426) • 1.07
(b)
k*-.2251/.1426
A
=
1.5785
A
(c)
1+q*-I+(1.5785(1.07-1.5785))· .2173
(d)
;Y-(6.233) /.2173
(e)
~1·216.90971/2 = 14.0325 ;
=
216.9097
43
(f)
~1=102.251-(1.5785)(14.0325) =
80.1007 ;
(g)
~(.557f (.2173)+1-(12.555) /216.909)} = .5106
(h)
)=.557/.5106" 1.0909 .
and
Thus, James' method estimates that a small multiplicative treatment
effect was observed when in fact the treatment effect was additive.
However, James' method does provide good estimates of
Applying equation (23) instead of James' method,
~1'
01' and o·
~1-80.1007
and
A
01-14.0325 from equation (8) and so the estimate of the additive
treatment effect is (102.251-92.601)-.557(102.251-80.1007) _ -2.6877 ,
which estimates that the drug adds between 2 and 3 units to each
individual's diastolic blood pressure (recall that the simulation
specification was for exactly 2 units to be added for each individual).
To summarize, both proposed methods of accounting for RTME in
uncontrolled designs proceed by estimating
RTME adjustment discussed in section 2B.
-'
(B-1)(V1-~1)'
which is the
The external method of
estimation simply specifies the population values of B and
other information.
~I
based upon
The advantages of this method are that it is
partially self-checking and that it can also be applied to studies with
a treatment by VI interaction.
The self-checking feature is available
because specifying 01 in addition to ul allows the researcher to predict
the observed
v{ and
s{ using the truncated normal formulae. Similarly,
A,-
if the treatment effect is truly additive then the observed value of B
is expected to equal the specified value of B.
When Band B are very
different then the researcher must decide whether the specified value of
Bwas incorrect or, alternatively, whether the treatment effect is not
truly additive.
A disadvantage of the external method is that the
researcher may not always be able to specify the required parameters.
49
The internal method of parameter estimation only utilizes observed data
~
in generating
and S, but at the cost of having to specify the form of
ul
the treatment effect.
Also, the self-checking features of the external
method are not available for internal parameter estimation.
Strictly
speaking, the internal method should only be applied when an additive
~~
model holds, in which case the observed 8 is an unbiased estimator
of 6.
As before, equation (8) of section 28 is used in order to obtain
~
ul from VI and sl'
Chapter 2 finishes with a summary and conclusions.
50
(2E).
Summary and Conclusions:
The most fundamental situation for the analysis of RTM involves
two-point scalar designs without treatment intervention (section 2B).
There, RTME requires both:
(a)
RTMG (i.e. 0<8<1, where 6 denotes the slope coefficient of the
(b)
regression of Y2 on Yl ); and
extreme individuals at time t l .
The RTMG effect on any individual, that is the degree to which that
individual is expected to be less extreme upon remeasurement, is
which increases as both Yl becomes more extreme and also
as 8 decreases from one to zero. The RTME effect on any group of
(S-I)(Yl-~I)'
-*
individuals, say with -Yl --*
Yl' is (8-1)(Yl-~I). In practice, 8 and ~1
must usually be estimated. Even in truncated designs, the observed 8
is an unbiased estimator of 8.
In
trunc~ted
designs,
~1
may be
estimated from the observed Yl and sl using the method of moments.
In designs with treatment and control groups, the ANCaVA accounts
for RTM by including VI as a covariate, which always accomplishes a gain
in precision in comparison with the two-sample paired t-test.
The
ANCaVA also accomplishes an elimination in bias relative to the t-test
point estimator of treatment effect when
VIT~YIC'
and when, in addition,
YI is not measured with error. When YI is measured with error, 62
remains unbiased unless the treatment and control groups are taken from
populations with different means (see appendix 3).
A
'"
A
In this case,
,.
32-(I-S 1)(UIT-UIC) is the correct estimator to use.
When a control group is unavailable and truncated sampling is
_""
_I
employed, the observed (Y2T- YIT) must be adjusted by an estimate of
51
~
(~I)(YIT-uI)'
which is the magnitude of the RTME that is expected to
occur even in the absence of treatment.
6 and ul may be estimated
either externally or internally, although both of these methods require
the use of untestable assumptions.
Provided that the researcher is
confident about knowing the population parameters, the external method
is to be preferred because it is partially self-checking.
The internal
method was adapted to accommodate an additive assumption about the form
of the treatment effect.
Chapter 3 extends the discussion of RTM to include two-point
vector designs.
Chapter 3
RTM in two-point vector designs:
Definition and manifestation of RTM
(3A).
Introduction:
1.
Initial remarks:
Chapter 3 concerns the definition and manifestation of RTM in
two-point vector designs, that is where more than one measurement is
taken on each individual at the time points tl and t 2. As is the case
for two-point scalar designs, RTM is often encountered as a potential
bias of an estimator of treatment effect caused by the presence of
extreme individuals at tl'
absence of treatment.
Similarly, RTM can also be present in the
That is, under certain conditions both
individuals and groups that have extreme values still tend to
"regress" regardless of their treatment status.
This chapter has two
primary goals, namely:
(I)
to extend the definition of RTM in two-point scalar designs to
include two-point vector designs (see section 38); and
(2)
for two-point vector designs, to describe the effects of RTME
upon extreme groups (see section 3C).
These results apply both
to natural populations (i.e. no treatment intervention) and to
treatment groups.
For uncontrolled designs utilizing truncated
sampling, the results are used to recoRll1end an "external"
estimator of treatment main effect that accounts for RTME.
53
The effects of RTME upon extreme groups are illustrated by simulation
examples.
Chapter 4 discusses another issue concerning two-point vector
designs, that being how to analyze data when many components are
subject to RTM.
The present chapter does not discuss analysis issues.
but rather focuses upon describing RTM effects on extreme groups.
Chapter 3 now continues with a discussion of notation and
terminology.
54
(3A).
Introduction:
z.
Notation and terminology:
Let Yjk (j=l,Z , k=l,Z, ... ,K) denote the observed measurement on
the kth characteristic of an individual at time tj'
These k
characteristics are termed the components of the vector Yj . For each
individual, the vector Y-(Y1,Y Z) is observed, and for simplicity it is
assumed both that all individuals are measured at the same two time
points and that there are no missing data.
Sometimes, the observed
values of Yjk are assumed to be composed of the sum of a true value,
namely YjkT' and an error term, namely Ejk' in which case the errors
in variables (EIV) model is said to hold.
In practice, the error term
Ejk often represents measurement error, but Ejk can also denote, among
others, the difference between a true but unmeasurable quantity and a
surrogate measure, or even a "random shock" (i.e. a real but transient
effect, such as the number of hours of sleep in the night before
taking a test of alertness).
As in chapter 2, this chapter continues to distinguish between
RTMG and RTME, albeit now on a component by component basis.
example, RTME2 denotes RTME in the second component.
For
Also, as in
chapter 2, a notational distinction is made between parameters and
statistics that are calculated using truncated sampling designs and
those that are not.
For example, Bk denotes the population regression
~
coefficient for the regression of Y2k on Ylk' and Bk denotes the
observed slope coefficient from a nontruncated design (i.e. Bk
~
estimates Bk)'
Similarly, B~ denotes the "subpopulation" regression
coefficient of the regression of YZk on Y1k' where only "extreme"
individuals are included in the subpopulation. 8k is estimated by Bk'
~
A
~
55
Here, the "extreme" individuals might be extreme on component k or,
alternatively, the truncation rule might be based upon another
component or components.
Note that Sk need not equal Sk' as was the
case for two-point scalar designs.
For uncontrolled studies employing
-* -*
truncated sampling, (Ylk'Y2k) denotes the first moments that would
have been expected in the absence of treatment.
Ok denotes the
difference score for component k, namely Y2k -Y 1k.
Chapter 3 now continues with a literature review.
56
(3A).
Introduction:
J.
Literature review:
To date, the RTM literature does not contain either:
(a)
a general definition of RTM in two-point vector designs; or
(b)
a systematic description of the effects of RTME upon two-point
vector designs using truncated sampling (note that for
uncontrolled truncated designs, this description of RTME effects
is required in order to apply the "external" method of adjusting
for RTME).
Campbell (4) makes the empirical observation that components that are
part of the truncation rule seem to be subject to RTME more often than
components which are not included in the truncation rule, but he does
not provide any explanation for this observation.
Section 3C explains
when and why Campbell's observation is correct, and attempts to
provide a relatively complete characterization of the effects of RTME
upon two-point vector designs that employ truncated sampling.
In this
regard, Tallis' (34) multivariate normal moment formulae are useful,
although these quickly become unwieldy as the number of components
increases (especially as the number of components in the truncation
rule increases), and nowadays it is often simpler to proceed by
simulation instead of obtaining closed-form solutions for the expected
RTME.
Section 3C adopts a simulation approach.
Chapter 3 continues with the development of an extended
definition of RTM for two-point vector designs.
e-
57
(3B).
Definition of RTM in two-point vector designs:
The principle of RTM states that variables that are extreme on
initial measurement tend to become less extreme upon remeasurement.
Before an extended definition of RTM that is based upon this principle
can be developed t however t two preliminary decisions must be made,
namely:
(1)
to which variables (i.e. components) does this principle apply?
and;
(2)
once the above set of variables is specified t will the principle
of RTM be applied one component at a time (i.e. repeatedly in a
univariate sense) or t alternativelYt will it be applied once
(i.e. in a multivariate sense) to the entire set of components?
Neither of the above two decisions is required in order to define RTM
in two-point scalar designs t since in that case there is only one
component that is potentially subject to RTM.
In this chapter t all continuous variables (i.e. regardless of
whether they are used as predictors t responses, or covariates) are
considered to be potentially subject to RTM.
Note that the set of all
continuous variables may include some that are measured at t 1 only.
The effects of RTM on such components are not directly observable
(i.e. since a post-measurement is not available), but nevertheless are
of interest since RTM can affect analyses that utilize these
variables. This dissertation does not consider the effects of RTM upon
categorical variables that are formed by recoding continuous variables
(e.g. hypertension-2 if DBP>120 t -I if 95<DBP<-120 t -0 if DBP<-95).
Standard methods of analyzing categorical variables that are subject
to misclassification (e.g. (6)) may be applied in this situation.
58
Given that the vector of all continuous components is of
interest, it still remains to decide whether to apply a multivariate
or a univariate definition of RTM.
That is, a multivariate definition
of RTM essentially claims that "individuals that are extreme at t I , in
the sense of multivariate distance, tend to be less extreme, in the
sense of multivariate distance, at t2".
In contrast, a univariate
definition of RTM essentially claims that a population is subject to
RTM if, for any component, k say, RTMk is present".
We choose to
apply the principle of RTM in a univariate rather than a multivariate
sense for the reasons discussed below.
Perhaps the most compelling reason for choosing a univariate,
rather than a multivariate, definition of RTM is that it is extremely
difficult to find a definition of multivariate extremity that is
adequate for the purpose of defining RTME.
Here, the choice involves
K
using either a "nondirectional" metric (e.g. kE I (Yjk-~jk)2), which
determines how many units the point Yj is away from ~j' or a
K
"directional" metric (e.g. kEI(Yjk-~jk»' which determines how far, on
average, the components Yjk are above or below their respective means.
In either case, basing the multivariate assessment of RTME upon a
single quantity (i.e. a multivariate measure of distance) can be
potentially misleading whenever components move in different
directions over time.
To illustrate this difficulty, consider the
following example:
let
~I-~-(O,O)
and let
~he
directional and nondirectional
metrics be k;I(Yjk-~jk) and k;IIYjk-~jkl, respectively.
Suppose
an individual is initially observed to have YI-(I,-I) and that
the goal is to determine whether or not he evidences RTM between
e-
59
tI and tZ'
First suppose that the directional metric is used,
and that the second observation is (.5,-.5).
Now, using the
directional metric, the distance from VI to the mean wI=(O,O) is
zero (i.e. (1-0)+(-1-0)=0 ), the distance from Vz to
Wz
is zero,
(i.e. (.5-0)+(-.5-0)-0 ), and so RTM is not observed (i.e. since
the distance from VI to the population mean" equals the distance
from V2 to the population mean).
Even so, however, both
components Yj1 and YjZ are closer to the center of their
distributions at t2 (i.e. in the inivariate sense).
In
addition, if component 2 is dropped from consideration, then RTM
is now observed (i.e. since the distance from Y1 to ~1 is now
1-0-1 and the distance from YZ to ~2 is now .5-0-.5, which is
less).
The same difficulty occurs when using a nondirectional
For example, let Y1-(1,-1) and Y2-(1.5,-.S) . Then,
the nondirectional distances from Y1 to ~1 and Y2 to ~2 are both
2
2 (i.e. ~-1- 1(1-0)1+1(-1-0)1-2 ), and no RTM is observed (i.e.
metric.
since both distances equal 2) even though component 1 moves away
the mean and component 2 moves toward the mean.
To summarize this example, then, both the directional and the
nondirectional metrics allow inconsistencies where all components
change (i.e. regress and/or egress) but no overall change is observed
using the multivariate assessment of distance.
This illustrates that
applying a multivariate assessment of RTME necessitates specifying
which components are involved in the metric at each stage of the
analysis, since adding or deleting components can potentially change
the results.
For this reason, a univariate, rather than a
multivariate, distance measure is used in the definition of RTME.
60
After considering the above discussion, then, RTM in two-point
vector designs is defined as follows:
Def: Suppose that there are k continuous components.
RTMG is defined
to be present if and only if any of these k components are
subject to RTMGk in the univariate sense (i.e. O<Sk<l).
Def: Suppose that there are k continuous components.
RTME is defined
to be present if and only if any of these k components are
subject to RTMEk in the univariate sense.
Here, RTMEk requires
A
both RTMGk in the data (i.e.
O<~<l)
and the presence of extreme
individuals at t 1.
Notes:
(1)
When only one continuous component is present, the extended
definition of RTM reduces to the usual definition of RTM in
~ -
two-point scalar designs.
(2)
In two-point scalar designs using truncated sampling, the
~
expected value of 8k is 8k' the population (i.e. non-truncated)
regression coefficient.
This relationship does not necessarily
hold for two-point vector designs.
Thus, it is important to
A
note that the above Bk refers to the slope coefficient that is
A
A
~
observed in the data (i.e. denoted by 8k or 8k' depending on the
sampling design), and not necessarily the population regression
coefficient.
(3)
The "extreme" in the above definition of RTME refers to "observed
values exceeding true values" and not necessarily to "observed
values exceeding the population mean".
While these two
operationalizations of "extremity" produce equivalent results in
the two-point scalar case (e.g. since large VI implies positive
61
error, on average), such is not the case for two-point vector
designs, as is illustrated in the simulation example which
follows.
62
(3C).
RTM in natural populations:
1.
Introduction:
Section 3C contains two sets of simulation examples which
describe the effects of RTME upon truncated sampling designs.
use the EIV model with two components.
Both
One goal is to illustrate some
of the differences between the form of RTM effects in two-point vector
designs as compared with two-point scalar designs.
The simulation in
section 3C2 is fairly limited in scope, and concentrates on providing
a counterexample to the intuitively plausible notion that, under the
EIV model, extreme groups are always expected to manifest RTME.
Note
that this claim is always true for two-point scalar designs, but is
not always true for two-point vector designs.
Apart from illustrating
the above, the simulations in section 3C3 describe RTM effects
(especially RTME effects) over a wide range of conditions.
There,
~
-
four conditions are varied, namely the correlation among the error
terms at t 1, the correlation among the true values at t 1, the
truncation rule for the first component, and the truncation rule for
the second component.
In section 3C2, the correlation among the true
values is held constant, and the other conditions are only varied over
a limited range.
Section 3C continues with the first set of simulation examples.
•
63
(3C).
RTM in natural populations:
Z. First simulation example:
The overall goal of this simulation is to illustrate that in
two-point vector designs extreme groups need not always be expected to
manifest RTME.
Because the ultimate assessment of RTM postulates
expected behavior in the absence of treatment, no treatment
intervention is applied.
As was the case for two-point scalar
designs, the conclusions from this example are applicable to more
general study designs, such as those involving a treatment
intervention.
The simulation specifications are as follows:
This simulation depicts the measurement of two components
at two time points for a sample of 60000 individuals.
model is assumed throughout.
The EIV
In order to illustrate the effects
of the correlation structure of the error terms upon RTME, the
60000 individuals in the sample are divided into three groups,
labelled A, B, and C, of 20000 persons apiece.
These three
groups originate from the same superpopulation except for the
effects of the structure of the errors.
In particular, for all
three groups:
(1)
Yjk-YlkT+Ejk, (j-l,2 , k-l,2) , where Yjk denotes the
observed measurement on the kth component at time tj; Y1kT
denotes the true (but unobserved) value; and Ejk denotes the
random (also unobserved) error term;
(2)
true values for individuals are assumed to remain unchanged
from t 1 to t2;
(3) true values are assumed to be drawn from a superpopulation
with Y11T-N(80,100), YI2 T-N(130,100), and correlation matrix
64
1 .95]
[ .95 1 ; and
each Ejk is assumed to be normally distributed with mean a
and variance 100 (j=I,2 , k=I,2). Here, (1) above is the EIV
(4)
assumption, (2) is made for convenience, and (3) and (4) are
made for concreteness.
In particular, the true values in (3)
are assumed to be highly correlated in order to generate
components which have at least moderate correlations among the
observed values.
In turn, these correlations assure that, under
truncated sampling, even components that are not included in the
participant selection criteria (i.e. the truncation rules) tend
to have extreme observed values at t 1 (i.e. in the sense of
exceeding the population mean).
In addition to (1) through (4) above, in group A it is
additionally assumed that error terms are uncorrelated both
within time points and between time points (i.e. the correlation
matrix of
6o 0~
[o 0
~1 ~]0
0
1
).
In addition to (1) through (4) above, in group B it is
also assumed that errors are perfectly correlated at any each
time point, but are uncorrelated between time points (i.e. the
correlation matrix
11 11 00 00]
[oo 00 11 11
).
65
Group C represents the middle ground between these two
extreme correlation structures.
For that group, in addition to
(1) through (4) above, it is also assumed that the correlation
matrix of
(E11,E12,E21,E22) is
[~5
.5
1
0
0
0
0
1
.5
~5]
Groups A and B illustrate two extreme kinds of correlation
structure, namely uncorrelated errors and perfectly correlated
errors, with group C falling in between.
Each set of
correlations is biologically plausible, recognizing that groups
A and B represent extreme cases.
For example, an assumption
about uncorrelated errors might be reasonable to employ when the
EjkS represent actual measurement errors and, in addition, when
measurements on separate components are taken using unrelated
instruments.
On the other hand, highly correlated errors, as
exemplified by group B, might occur, for example, when the EjkS
represent "shocks" (i.e. real but transient effects), and the
same forces that affect the shock for one component also affect
the shock for other components.
Or, for another example,
measurement errors on different components might be highly
correlated if the measurements on these components are taken at
the same time using the same instrument.
For each group, three participant selection rules are
considered. Rule 1 includes everyone (i.e. random sampling).
Rules 2 and 3 identify individuals on the basis of extremely
large values of one or more components at t1 (i.e. truncated
66
sampling).
In particular, rule 2 states that only individuals
with Y11 >95 are eligible for resampling, while rule 3 requires
both Y11 >95 and Y12 >150. Comparisons between rules 2 and 3 for
component 2 illustrate the effect on RTME of including that
component in the participant selection criteria.
Similarly,
comparisons between rules 1 and 2 illustrate the same effect for
component 1.
The vector of mean difference scores is the
response of interest.
Table 1 below describes the results of the simulation.
Table 1
Simulation results
Table lA
Means (and standard deviations)
RG
1
1
1
2
2
2
3
3
3
n
A 20000
B 20000
C 20000
A 2985
B 3908
C 3253
A
693
B 2511
C
904
Yll
80.10 (14.23)
80 . 10 (17. 41 )
80 . 10 (15. 08 )
102.36 ( 6.16)
104.77 ( 8.00)
103.06 ( 6.60)
104.36 ( 6.76)
108.40 ( 7.72)
106.64 ( 7.71)
R=Rulei G=Group
Y21
79.97
79.90
79.94
91.00
88.30
90.16
96.79
89.36
95.36
(14.18)
(17.41)
(15.07)
(12.52)
(16.38)
(13.65)
(11.87)
(16.49)
(13.08)
Y12
130.08 (14.17)
130. 12 (17.41)
130.08 (13.24)
140.75 (12.75)
154.32 ( 8.43)
144.14 (10.10)
157.27 (11.53)
158.72 ( 7.27)
156.36 ( 5.37)
Y22
130.02
129.97
130.03
140.85
137.86
139.89
147.18
139.68
145.99
(14.17)
(17.41)
(13.25)
(12.67)
(16.51)
(11. 83)
(11.53)
(16.44)
(11.01)
e·
67
Table 1B
Regression results
Component 1
R
G
1
1
1
2
2
2
3
A
B
C
A
3
3
B
C
A
B
C
Intercept
40.053
53.113
44.342
35.577
50.586
39.354
57.809
46.698
57.311
Component 2
Intercept
Slope
( .496)
( .546)
( .516)
(3.678)
(3.387)
(3.638)
(6.829)
(4.555)
(5.904)
.498
.334
.444
.541
.360
.493
.374
.394
.357
( .006)
( .007)
( .006)
( .036)
(.032)
(.035)
(.065)
(.042)
( .055)
64.789
86.390
55.499
86.172
75.525
66.575
103.053
81.730
78.366
Slope
( .801)
( .875)
( .759)
{ 2.366)
( 4.738)
( 2.673)
(10.931)
( 7.080)
(10.437)
.501
.335
.573
.388
.404
.509
.281
.365
.432
(.006)
( .007)
(.006)
(.017)
(.031)
(.018)
( .069)
(.045)
( .067)
R=Rule; G=Group
Table lC
Correlations
R
G
P11 °12
1
1
1
2
2
2
3
3
3
A
.500
.335
.445
.266
.176
.238
.213
.184
.210
B
C
A
B
C
A
B
C
P11 °21
.477
.984
.701
.241
.929
.422
.133
.917
.340
P11 °22
.478
.318
.482
.262
.164
.267
.267
.121
.191
P12 °21
.475
.319
.480
.341
.147
.314
.154
.150
.166
P12022
P21022
.479
.984
.698
.356
.982
.641
.273
.983
.613
.501
.334
.572
.391
.206
.434
.151
.161
.211
R=Rule; G=Group
The results of the simulation are presented in table 1 above.
For concreteness, the reader may wish to think of component 1 as
representing, say, diastolic blood pressure (DBP), and component 2 as
representing, say, systolic blood pressure (SBP).
First examining the
correlations (see table lC), note that all of the correlations among
the observed values are positive, even for group A, because of the
positive correlations among the true values.
Also note that the
observed correlation matrices differ according to the sampling rule,
often noticeably so.
This, in turn, causes the observed slope
68
coefficients (see table 18) to differ from rule to rule.
That is,
~
table 18 illustrates that, for two-point vector designs, the magnitude
of RTMG depends upon the truncation rule.
In addition, a comparison
of slope coefficients between models illustrates that the magnitude of
RTMG also depends upon the correlation structure among the error
terms.
Now examining the means (see table lA), note that RTME is not
observed for rule 1 (i.e. random sampling) for any of the three
groups. This is as expected, given that the population mean vector
does not change from tl to t z and since the random sampling results
are expected to mirror those of the population.
Considering rule Z (i.e. truncated sampling using DBP only),
note that DBP evidences about the same magnitude of RTMEI for each of
the three groups (i.e. group means drop from about 103 at tl to about
90 at tZ)'
This is in agreement with the two-point scalar results,
since a truncation rule that is based upon DBP alone will lead to
observed DBP values at tl exceeding true values, which in turn leads
to RTMEI.
Any correlations between DBP and SBP at time tl are
irrelevant for evaluating RTMEl, and so the two-point scalar results
apply directly. The correlations between DBP and SBP do matter when
analyzing RTME2, however.
Note that group A evidences no RTME2 for
-~
-~
SBP (i.e. DA is about zero), group C shows moderate RTME2 (i.e. DC is
-'
about -5), and group B shows large RTME2 (i.e. DB is about -16), even
though the SBP means at tl exceed
of the groups.
~1
by about the same amount for each
Thus, group A illustrates the apparently
counterintuitive situation whereby component 2 evidences RTMG in the
population, RTMG in the sample (see table IB), exceeds its population
~-
69
mean, and yet does not manifest RTME. The explanation for this result
lies in the effects of the correlations among the true values and
among the error terms (especially the latter) upon the mean error term
at t 1. That is, for all three groups, large Y11 implies both E11 >0
and Y11T >80. Because of the high correlations among the true values,
large YIlT implies large Y12T' which implies that both observed and
true SBP values at t1 are large.
error terms.
Now consider the contribution of the
In group A, the fact that
E11 >0
since these error terms are uncorrelated.
has no effect upon
E12 ,
Similarly, in group C the
E12 s are expected to be moderately positive, and in group B the E12 s
are expected to be highly positive. The RTME2 in SBP is caused by the
E12 s, which explains the above result.
Considering rule 3, each group evidences considerable RTME for
both DBP and SBP, which is as anticipated since both DBP and SBP are
part of the inclusion criteria in rule 3.
Group B shows considerably
more RTME than the others, which is also as expected since the perfect
correlations between the error terms essentially means that the
selection criteria is based upon one, rather than two, components (as
is instead the case for the other two groups).
That is, for groups A
and C (A especially), it is more likely that an individual with large
values on both components actually has large true values (recall that
the true values are assumed to be very strongly correlated) and
moderate error instead of moderate true values and large error (i.e.
since one "extreme event", that is an extreme true value, is more
likely than two "extreme events", that is two large errors).
Since
errors tend to be more moderate in groups A and C, less RTME is
observed for these groups.
70
To summarize the results of the simulation, in the EIV model
RTME depends upon the mean error term at t I , and not necessarily upon
either RTMGk or having observed values exceed the population mean. In
turn, for any component, the mean error term depends upon both the
truncation rule (especially, which components are included in the
truncation rule) and also upon the correlations among the error terms
in the population. Section 3C3 presents results describing the amount
of RTME that is expected for various population parameters and
truncation rules.
e-
71
(3C).
RTM in natural populations:
~.
Second simulation example:
The primary goal of this simulation example is to describe RTM
(especially RTME) effects under a wide variety of conditions.
As
before, two components are measured, and an EIV model, namely
Yjk =Y 1kT+Ejk(j=1,2 , k=1,2) is posited. For eacn component k, true
values are assumed to remain unchanged over time. At each time point,
however, the components of the true values (e.g. YIlT and Y12T ) may be
correlated, and this correlation is denoted by Pr. Similarly, at each
time point errors may be correlated, with this correlation denoted by
PE' Normality is assumed, and time t l errors are assumed to be both
independent of time t2 errors and also independent of true values.
Both true values and errors are standardized to have mean zero and
variance one.
Thus, Ylt-N(O,O,l,l,PT) and Ej-N(O,O,l,lPE)'
The
sample size is 100000, chosen to be large enough to illustrate actual
effects with a high degree of precision. Implementation details are
described in appendix 1.
In order to obtain a variety of conditions, a factorial design
is employed, where the four factors to be varied are:
(a)
P'f has levels 0, .25, .50, .60, .70, .80, .90, and 1
(b)
PE has levels 0, .25, .50, .60, .70, .80, .90, and 1
(c)
the truncation rule for the first component has levels
Yll>-~
(i.e. no truncation), Yll >1.5, Yll >2.25, Yl l >3.0, and Yll >3.75;
and
(d)
the truncation rule for the second component has levels
Y12>-~
(i.e. no truncation), Y12 >1.5, Y12>2.25, Y1 2>3.0, and Y12>3.75.
Note that truncating at 1.5 amounts to taking observations that
72
exceed the population mean by about one standard deviation,
truncating at 3.0 amounts to taking observations that exceed the
population mean by about two standard deviations, and so on.
Because of space limitations, selective but representative
results are reported here.
To recapitulate, this simulation example describes RTM
(especially RTME) effects over a wide variety of conditions.
These
conditions are represented by points in the above four-factor design.
Although the design is not all-inclusive, the levels of the above four
factors are intended to cover most of the range of biologically
plausible situations for, say, blood pressure (e.g. diastolic blood
pressure=component one, systolic blood pressure-component 2).
Only
nonnegative correlations (i.e. both for 0T and 0E ) are considered,
~
and measurement errors are always assumed to be uncorrelated over
time, but these assumptions seem to be reasonable for many biological
variates, such as blood pressure measurements.
The simulation results
follow.
Highlighting results of interest:
(1)
without exception (i.e. regardless of the values of PT' 0E' or
the details of the truncation rule), performing truncated
sampling causes the observed variance-covariance matrix,
L~
say,
to differ from the population variance-covariance matrix L.
These data are not illustrated here (but, for example, see table
Ie
of the previous section).
The practical implication of this
result applies to planning studies (e.g. power analysis), in
that one should be aware that the expected variance-covariance
.
73
matrix might not be identical to the population
variance-covariance matrix;
considering designs that truncate on Yll only:
(a) the RTM for the first component (i.e both RTMGl and RTMEl) is
(2)
exactly as in the two-point scalar case (and thus is not
A
illustrated here) . That is Y2l=6 l Y11 . .As the truncation
-"
rule becomes more severe VII becomes more extreme, and thus
the absolute magnitude of RTMEI increases (i .e. 1011
increases), but 61 does not change.
..
The truncated 61 is
expected to equal the population (i.e. the nontruncated) 61'
(b)
the RTM for the second component primarily depends upon
also depends upon PT and the truncation rule.
~,
but
First
considering RTMG2, RTMG2 is observed in all designs (i.e. as
02 is negatively correlated with Y12 )· 62 increases as PE
increases, but this increase is relatively small unless errors
are highly correlated.
62 decreases as PT increases.
below illustrates how 62 varies with 0E and Pr·
Table 2
82 also
increases as the truncation rule becomes more severe, which
relationship is not illustrated in table 2.
Table 2
Slope coefficients (standard errors) for component 2
(V 11 >1.5, Y12>-00)
or=O
PE=O
OE=·5
°E=·7
°E=1
.503
.537
.571
.654
(.007)
(.007)
(.007)
(.008)
or=·5
.474 ( .007)
.508 ( .008)
.548 (.008)
.668 (.009)
(n-14746)
or=·7
.444
.471
.511
.651
(.007)
(.008)
(.009)
(.011)
Pf=l
.365
.354
.373
.506
(.008)
(.010)
(.011)
(.016)
74
Now considering RTME, which is usually of most importance, the
magnitude of RTME2 increases as 0E increases, from no RTME2
when 0E=O to a maximum when 0E=I.
See table 3 below, which
gives RTME2 for some of the points in the "simulation grid".
As illustrated in the previous section, when 0E=O, RTME2 is
"'~
not present even when 82<1 and, in additfon,
Yi2>~12'
Holding
PE constant, the absolute magnitude of the RTME2 does not vary
as PT varies, but does increase as the truncation rule becomes
more severe.
All else being equal, the absolute magnitude of
-~
RTME2 increases as Y12 becomes more extreme.
In turn, Y12
becomes more extreme as the truncation rule increases in
severity (at least so long as 0T and 0T do not both equal
zero), 0T increases, or 0E increases.
To explain, assuming
nonzero correlations, truncation on Y11 implies that E1l>0 and
-,
that Yll>Y 11T ' the more so as the truncation rule becomes more
-~
severe.
Correlations among the true values imply that the
observed Y12 is high, and correlations among the errors imply
that E12 >0 (which also tends to inflate YI2)' RTME2 ;s caused
-~
-~
by EI2 >0.
4It -
75
Table 3
Group means for component 2
( y 1J >1. 5, y 12 >-00) (n=14746 )
(standar errors range from .010-.012)
Y22
.000
.017
°E=O
.556
.015
0E=·5
°E=·7 .778 .013
OE=l 1.110 -.000
Y12
Y12
.559
1.145
1.337
1.669
Y22
Y12
.575
.574
.571
.559
.783
1.338
1. 561
1.893
Y22
Y12
.799 1.119
.797 . 1.675
.795 1.897
.782 2.229
Y22
1.135
1.134
1.131
1.119
(Y >3.0, Y12 >_00) (n=1741)
(standar a1 errors range from .029-.036)
Y12
°E=O -.006
°E=·5 .870
°E=·7 1.220
1. 748
°E=1
(3)
Y22
.009
.021
.024
.023
Y12
.870
1. 747
2.097
2.625
Y22
.886
.898
.901
.900
Y12
1.222
2.098
2.448
2.976
Y22
1.237
1.249
1.252
1.252
Y12
1.780
1.780
2.977
3.504
Y22
1. 766
1.777
1. 781
1.780
Now considering the results when both Y11 and Y12 are included in
the truncation rule, in general the RTM (i.e. both RTMGl and
RTMEl) for component 1 does not change much when another
component is added to the truncation rule, and so is not
illustrated here. VII is slightly larger when participant
selection is also based upon Y12. Y11 also increases when 0E
increases t 0T increases t and as the truncation rule becomes more
severe.
Overall t however t the effects of adding Y12 to the
truncation rule are much stronger on RTME2 than upon RTMEI.
fact t component 2 now acts alot like component It as is
illustrated in table 4 below.
That ;St now RTME2 is always
apparent t even when 0E=O and oT=O.
In
76
Table 4
Group means for component 2
(Y >1.5, Y >1.5) (n=2165-14746)
(sta~rd err~ range from .010-.028)
Y11
°E=O 2.222
PE=·5 2.311
PE=·7 2.340
PE=l 2.374
Y22
1.142
.969
.910
.827
Y12
2.317
2.390
2.470
2.400
Y22
1. 410
1. 232
1.167
1.051
Y11
2.341
2.403
2.407
2.395
Y12
1.502
1..307
1.230
1.121
Y11
2.378
2.401
2.389
2.229
Y12
1.602
1.379
1.297
1.119
The fact that RTME2 is always observed when Y12 is part of the
truncation rule but is only sometimes observed when Y12 is not
supports Campbell's (4) observation to this effect.
The explanation
lies in noting that E12>0 causes RTME2.
Under EIV, when Y12 is
included in the truncation rule then E12 always exceeds zero.
However, when Y12 is not part of the truncation rule then the value of
E12
depends both upon Ell and upon the correlation between Ell and
E12 ·
Summarizing the simulation results, under the EIV model
truncated sampling affects both the expected variance-covariance
matrix of the observed values as well as the anticipated slope
coefficients for the regression of Y2 on Y1. Researchers should bear
these facts in mind when using the above quantities during the
planning stage of experiments (e.g. as inputs into a power analysis).
For any component, RTMGk plus large observed values is no longer a
sufficient condition for expecting RTMEk.
Instead, the condition is
large observed values plus positive measurement error.
On average,
positive measurement error may be obtained by either including a
component in the truncation rule or else via Qf>O (i.e. a positive
correlation between the measurement errors for the component in
e-
77
question and a component that is included in the truncation rule).
For components that are not included in the truncation rule, RTMEk
depends most strongly upon PE' but also depends upon PT and the
severity of the truncation rule.
For most practical purposes, the
effects of other components can be ignored when considering components
that are included in the truncation rule.
Overall, similar conclusions should apply to situations with
more that two components, even when a number of these components are
included in the truncation rule.
Chapter 3 continues with a discussion of how to adjust for RTME
in uncontrolled designs that employ truncated sampling.
78
(3D).
RTME in uncontrolled studies that use truncated sampling:
This section discusses how to account for RTME in uncontrolled
designs that use truncated sampling.
The primary goal is to obtain
estimators of the treatment main effect vector (i.e. the vector that
•
consists of the main effect of treatment upon each component in the
response function).
For simplicity, assume that
population is stable (i.e.
~1=~2tLl=L2).
~he
underlying
As was the case for
two-point designs, in truncated sampling studies estimating the
treatment main effect vector requires that each of the observed mean
difference scores (Y2k-Ylk) be adjusted by a term, (V;k-V;k) say,
which is an estimate of the RTMEk that is expected even in the absence
of treatment.
-* -"'*
Ylk) may be
Now, in the two-point scalar design (Y2k-
estimated either:
(a)
externally (i.e. the observed data are not used in the estimation
process); or
(b)
internally (i.e. the observed data are used in the estimation
process while outside information is not).
Owing to the additional complexities of the two-point vector case,
however, only an "external" method of estimation is recommended here.
As before, some strong and essentially untestable assumptions are
required in order to estimate the magnitude of RTME in this design,
which serves to illustrate the cost of not including an appropriate
control group.
First considering external estimation, if L and
~
(i.e. the
first and second moments of the nontruncated data in the absence of
treatment) can be specified then the truncated normal formulae (34)
may be applied in order to obtain the expectation of the RTME that
~
-
79
would have been observed even in the absence of treatment.
-* -*
~
estimated treatment main effect is (Y2-YI)-(Y2-YI)'
observed values of B~ and
The
Here, the
YI may be used as an informal check on the
external estimates of wand L.
That is, the truncated normal formulae
-~
predict not only the magnitude of the RTME but also Band YI . A
considerable discrepancy between the observed YI And the expected YI
indicates that either L or w is misspecified. Similarly, a
-~
discrepancy between the observed and predicted slope coefficients
either indicates a misspecification of B or else a nonadditive
treatment effect (note that it is impossible to test which of these is
the case).
It is also impossible to test whether the assumption that
wT=w2 is true, since
wZ"~
is confounded with the treatment effect.
To explain why an internal estimation is not recommended here,
begin by recalling the internal method proposed for the two-point
scalar design.
There, (B-I)(YI-WI) is the expected RTME, where wI is
estimated by the method of moments and B is estimated directly from
the data via an assumption of additivity.
As is illustrated in
section 3C, however, (Bk-I)(Ylk-Wlk) is not the estimated RTMEk in the
two-point vector case, and so a direct extension of the above
procedure can not be developed. What is reqUired, instead, is an
estimate, for each component, of the mean error term at time tl
(assuming the EIV model).
Since this model assumes that errors are
uncorrelated over time, then the mean error term at tl also equals the
expected RTMEk.
However, it is a very difficult matter to estimate
the mean Elk on the basis of observed data alone, because of the
proliferation of nuisance parameters.
procedure is not recommended.
Thus, an internal estimation
80
To summarize, because of the differences between the
manifestations of RTM in two-point scalar designs and two-point vector
designs, an extension of the internal method of estimating RTME can
not be developed for the two-point vector design.
method is recommended.
In this case, care should be exercised in
oraer to ensure that the population
stable.
Thus, the external
characteristi~s
are known and
In addition, the partial checking of assumptions described
above should be pursued.
The best solution, of course, is not to
employ uncontrolled designs although it must be recognized that this
design option is sometimes the only alternative.
Section 3E contains a summary and concluding remarks.
•
81
(3E).
Summary and final remarks:
This chapter's discussion of RTM in two-point vector designs
follows a similar format as that of chapter 2.
That is, a definition
of RTM is proposed, then the effects of RTM in natural populations
(i.e. no treatment intervention) are described, then an adjustment for
expected RTME in uncontrolled truncated designs is recommended.
Extending the definition of RTM from the two-point scalar case
to the two-point vector case requires choosing between a
"multivariate" definition of RTM (e.g. "individuals that are extreme,
in the sense of multivariate distance, tend to be less extreme, in the
sense of multivariate distance, upon remeasurement") and applying the
"univariate" definition of RTM (i.e. from chapter 2) on a component by
component basis.
Since the multivariate definition leads to apparent
contradictions, RTM is defined to hold if and only if any of the
continuous components are subject to RTM in a univariate sense.
distinction between RTMG and RTME is maintained.
The
As illustrated in
the simulation examples, defining RTME in the two-point vector case
requires that the concept of "extreme YI " be clarified. Under the EIV
model assumed throughout this chapter, "extreme YI " requires that both
Ylk>fllk and Elk>O, which is not necessarily implied by
recapitulate, the extended definitions of RTM are:
Ylk>~lk.
To
82
Def: Suppose that there are k continuous components.
RTMG is defined
to be present if and only if any of these k components are
subject to RTMGk in the univariate sense (i.e. O<Sk<I).
RTME is
defined to be present if and only if any of the k components are
subject to RTMEk in the univariate sense.
both RTMGk in the data (i.e.
individuals (i.e. EIk>O and
O<~<I)
YIk>~Ik)
Here, RTMEk requires
and the presence of extreme
at tl.
The simulation examples in section 3C describe the effects of
RTM in truncated sampling designs.
For components that are part of
the truncation rule, RTM effects are similar to those encountered in
the two-point scalar case.
In fact, when truncating on a single
component, these effects are identical since the other components are
irrelevant to this manifestation of RTM.
Now considering components
that are not part of the truncation rule, various RTME effects are
possible, ranging from no RTME to extreme RTME, depending primarily
upon the value of PE (i.e. the correlation of the error terms at tIl,
but also depending upon PI (i.e. the correlation among the true values
at tI) and the severity of the truncation rule.
Surprisingly, in a
model with correlated true values and uncorrelated errors, a component
that is not part of the truncation rule is expected to show large
observed values at tI' RTMG in the data, and yet not manifest RTME.
A~
In comparison to two-point scalar designs, the truncated Bk need not
have an expectation of Sk' the (nontruncated) population regression
coefficient, which fact should be borne in mind when planning studies
(e.g. in power analyses).
In the uncontrolled truncated design, only an "external"
estimation procedure is recommended for obtaining a RTME adjustment to
~ •
83
the observed treatment main effect.
This is in contrast to the
two-point scalar case, where both external and internal methods are
recommended.
Here, internal estimation was not pursued, in part
because of the proliferation of nuisance parameters.
As in the
two-point scalar case, while the external method requires untestable
assumptions (e.g. that the population parameters "do not change over
time), it is at least partially self-checking.
Chapter 4 now considers the effects of having many components
that are subject to RTM upon regression techniques that are used to
analyze two-point vector data.
Chapter 4
RTM in two-point vector designs continued:
Regression analysis when many variables ar~ subject to RTM
(4A).
Introduction:
This chapter continues the discussion of RTM in two-point vector
designs by examining some of the effects of RTM upon multivariate
analyses when many variables are subject to RTM.
Now, the above topic
is too general for the present circumstances, and must be limited.
In
particular, only regression methods are considered here, and also RTM
is always assumed to be a consequence of the EIV assumption.
All
continuous variables (i.e. predictors, covariates, and response) may
be subject to RTM.
Unless otherwise noted, the discussion applies
equally well to both random and truncated designs.
Note that the
error term in the EIV model need not necessarily represent measurement
error, and thus the EIV model is more generally applicable than it
might first appear. For example, in Galton's (14) analysis of the
regression of son's height upon father's height, the error term
represents the combined effect of a large number of random, but real,
genetic factors.
Since rare genetic events are not likely to be
repeated, RTM occurs.
Recognizing that RTM is, in practice, often
caused by EIV allows insights from the literature on "regression when
variables are measured with error" to be applied to the present
problem.
~ ~
85
For concreteness, consider the fol ,owing situation.
A
researcher wishes to determine which variables are predictive of a
drop in diastolic blood pressure (DBP).
Apart from the response
variable (i.e. DBP 2-DBP 1) and initial DBP (i.e. DBP 1), the researcher
measures various variables at tl such as age, gender, social class,
stress level, nutrition status and cholesterol, some of which may be
subject to RTM. The research question asks whether any of the above
variables predict later change in DBP.
Or, alternatively, the
regressor variables might also be measured at t2' in which case the
research questions could also include examining the relationship
between changes in the regressor variables and changes in DBP.
In
either case, the researcher recognizes that, even in the absence of
any other factors, the most extreme initial DBPs will tend to have the
greatest drops, and wishes to avoid spurious conclusions caused by
this RTM (i.e. RTMG) in DBP.
In addition, the researcher is concerned
about what effects the RTM in the regressor variables will have upon
the regression analysis.
Example 2 of chapter 6 discusses the effects
of RTM upon the analysis of the Edgecombe County Hypertension Study
(James (19), one of whose components loosely follows the above design.
The goals of this chapter are:
(1)
to describe the effects of including variables that are subject
to RTM (i.e. variables satisfying the EIV assumption) upon
regression analyses; and
(2)
based upon the above results, to recommend a simple analysis
strategy for this study situation.
Note that this analysis
strategy codifies existing theory, and no new mathematical
results are developed.
86
Chapter 4 contains this introduction, a literature review, and a
description of a recommended analysis strategy.
Example 2 of chapter
6 illustrates the application of this analysis strategy.
Chapter 4 now continues with a literature review.
87
(48).
Literature review:
Few authors directly discuss the effects of RTM upon regression
analysis, although RTM is often mentioned within other contexts.
In
particular, the errors in variables (EIV) literature provides useful
insights, since it is assumed throughout this chapter that RTM is
caused by the presence of error at t1.
This literature review
considers two separate issues, namely:
(1)
given a regression equation, what is the effect of having some
variables measured with error? For the present purposes, it is
equivalent to ask what is the effect of having some variables
subject to RTM? ; and
(2)
how should the regression equation be formulated when variables
are measured with error (i.e. subject to RTM)?
Cochran (6), Kupper (21), Bock (3), Kendall (20), Fuller (12),
and Warren (35) all consider some of the ways that EIV affects
regression analyses.
This topic is also discussed at length in the
econometric and psychometric literatures.
Each of the above authors
assumes that RTM is caused by EIV (and, in fact, is primarily
concerned with the effects of the errors in variables assumption
rather than RTM, per se).
Although these authors consider slightly
different aspects of the effects of RTM upon regression analysis,
their results are in essential agreement.
(1)
To summarize:
in the absence of any other effects, the slope coefficient of a
regressor that is measured with error tends to be dampened.
That is, the slope coefficient of the observed regressor (i.e.
true value plus error) is smaller in absolute value that the
slope coefficient that would have been observed were the true
88
values known.
Note that the former term is sometimes called the
structural regression coefficient while the latter term is
sometimes called the functional regression coefficient;
(2)
the above dampening tends to be more extreme when the model
contains many, rather than one, regressors;
(3)
even variables that are measured without error may have incorrect
slope coefficients when they are correlated (in the population)
with variables that are measured with error;
(4)
the relationship in (3) above is complicated, but is particularly
worrisome because the direction of the bias may be either
positive or negative;
(5)
as a consequence of (3) and (4) above, covariates that are
measured with error may be most dangerous, because of their
unpredictable and possibly large effects on the slope
coefficients of variables of greater interest;
(6)
the effect of having the response variable measured with error is
similar to that of having the predictors measured with error, in
that, on average, the bias tends to be toward the null; and
(7)
apart from biasing parameter estimates, observing variables that
are measured with error has adverse effects upon the precision
of observed slope coefficients.
Now considering recommendations about how to proceed in the
presence of RTM, Fuller (12) describes a method of adjusting the above
biased slope coefficients, provided that the various components of
variance (i.e. the variances of the true values and of the error
terms) are either known or estimated.
to
at
need be specified.
In fact, only the ratio of
01
Note that, at present, only large-sample
89
properties of this method have been examined.
Kendall (20) and Warren
(35) discuss the difference in interpretation between adjusted and
unadjusted coefficients.
Cochran (6) and Bock (3) provide examples of
how to estimate variance components in some designs.
In the absence
of this information, Kendall discusses some situations where reducing
the level of measurement provides relief.
In particular,
categorization of the variables that are measured with error is
recommended under some circumstances.
Alternatively, variables might
be replaced with ranks when the observed data are spaced widely enough
so that one can be confident of the rankings despite the effects of
the error terms. Conover (7) claims that replacing observed data with
ranks is often a surprisingly robust and efficient procedure for many
statistical techniques.
Rosner (29), following examples from the psychometric
literature, recommends replacing any difference score that is subject
to RTM by a residual, where the residual is obtained from the
regression of time t 2 values on time t 1 values. While intuitively
appealing, this method, at least in its simplest form, may ignore a
key multivariate dimension of the situation by considering separate
univariate regressions in order to account for the RTM in each
difference score.
Note that including initial value as a covariate
when accounting for the RTM in the response variable is equivalent to
Rosner's suggestion so long as this initial value is uncorrelated with
the other regressors.
To recapitulate, in two-point vector designs RTM causes problems
that are not necessarily limited to the variables that are measured
with error.
An adjustment procedure is available when all variance
90
components are known.
recommendations.
Otherwise, the literature only contains ad hoc
The literature does not recommend any coding schemes
(i.e. any way to define the regression model) that completely avoids
the effects of RTM when variance components are unknown.
91
(4C).
Recommendations:
Perhaps the most important recommendation is a general one,
namely to be aware that using variables that are subject to RTM in a
regression analysis may lead to incorrect results.
The second most
important recommendation is to appeal to the large (albeit incomplete)
literature on regression when variables are measured with error for
gUidance, since in epidemiologic practice it is reasonable to assume
that RTM in two-point vector designs is most often caused by EIV.
Based upon the literature, a general outline of the steps to follow
when analyzing two-point vector designs when some variables are
subject to RTM is:
(1)
separate variables into predictors, covariates, and response;
(2)
determine which variables are directly subject to RTM;
(3)
of the remaining variables, determine which of these are
correlated with the variables that are subject to RTM.
These
variables are "contaminated" in the sense that RTM in (2) above
may also affect the parameter estimates in (3); and
(4)
if all variance components are known or can be estimated, perform
a sensitivity analysis (i.e. determine how sensitive the results
are to variables being measured with error) using the method of
Fuller.
That is, obtain adjusted and unadjusted parameter
estimates (and variance-covariance matrices) and compare.
Even
if the variance components can be specified, consider dropping
any regressors, especially covariates, that are measured with
extreme error since such variables probably do more harm than
good.
Now, the adjusted coefficients fall into two groups,
namely coefficients of regressors that are subject to RTM and
92
coefficients of contaminated regressors.
Clearly, the
contaminated slope coefficients should be adjusted.
For the
regressors that are measured with error, determine which
relationship is of greater scientific interest, namely the
regression of response on true values (which is estimated by the
adjusted coefficients) or the regression of response on observed
values (which is estimated by the unadjusted coefficients).
If
true values are of greater substantive interest, then retain the
adjusted coefficients for these regressors also.
Note that
adjustments need not be made when the regression model is used
for prediction only (12), that is when the values of the
parameter estimates are not of interest.
In view of the above difficulties in using variables that are
measured with error in regression analyses, the major design
recommendation is to avoid such variables whenever possible,
especially when these variables are covariates.
If variables that are
subject to RTM must be used, only use those with variance components
that are known or can be estimated (e.g. from a pilot study) so that
the proper adjustment procedure can be applied.
Sometimes, variance
components can be estimated from study data when an appropriate degree
of replication is present.
To recapitulate, the researcher should avoid placing his
conclusions in jeopardy by using variables that are EIV (and, thus,
subject to RTM) when variance components estimates are unavailable.
Special care should be taken with covariates, since RTM in these
variables leads to unpredictable results.
RTM in predictor and
response variables tends to bias results towards the null.
~.
93
Example 2 of chapter 6 illustrates the calculations required in
order to adjust coefficients that are measured with error.
Chapter 5 discusses RTM in longitudinal designs.
Chapter 5
RTM in longitudinal designs
(5A).
Introduction:
1.
Overview:
Chapter 5 discusses RTM in longitudinal scalar designs, that is
where one characteristic is measured on each individual at more than
two points in time.
Few authors discuss RTM in these designs, and
although some special cases have been treated no one has covered this
topic in any generality.
In particular, no one has proposed a general
definition of RTM that takes into account the special features of
longitudinal designs.
Nor has anyone proposed a comprehensive
strategy about how to analyze longitudinal data that are subject to
RTM.
(a)
Specifically, then, the goals of this chapter are to:
develop an extended definition of RTM that includes longitudinal
scalar designs;
(b)
illustrate, by example, the general applicability of this
definition;
(c)
provide recommendations as to how to account for RTM in the
analysis of longitudinal scalar data; and
(d)
contrast RTM in longitudinal scalar designs with RTM in two-point
scalar designs, and discuss to what extent RTM is a "two-point
concept".
This chapter contains an introduction, a discussion of the
analysis of RTM in longitudinal scalar designs, and a summary and
~
A
95
conclusions. Apart from this overview, the introduction includes a
discussion of notation and terminology and a literature review.
The
literature review is very sparse due to a paucity of published
research on RTM in longitudinal designs.
For the same reason, the
section on notation and terminology is rather lengthy, in order to
provide adequate means to express the new concepts that are developed.
In particular, this section includes a discussion of an extended
definition of RTM that applies to longitudinal designs.
The main body
of the chapter includes discussion of analysis recommendations,
implementation issues, and a simulation example.
Example 3 of chapter
6 provides another example of the application of the results of this
chapter.
Chapter 5 ends with a summary and conclusions section, which
includes a comparison of the manifestations and effects of RTM between
two-point scalar and longitudinal scalar designs.
The introduction now continues with a discussion of notation and
terminology.
96
(SA).
Introduction:
l.
Notation and terminology:
Throughout this chapter Yj , j=I,2, ... ,p denotes the value of the
scalar Y at the time t j . The capital Y denotes a random variable
while the lower case y denotes its realization (e.g. YI =2 means that
the random variable Y has taken the value 2; YI=YI means that the
random variable YI has taken the value YI)'
assumed that:
For simplicity, it is
(a)
at each time point Yj is scalar instead of vector;
(b)
all individuals are measured at the same points in time;
(c)
there are no missing data; and
(d)
the time points (tl,t2, ... ,t p) are equally spaced (e.g. so that
there is no loss in generality in replacing (t l 't2' .... 't p) with
(l,2, ... ,p).
Longitudinal designs are often used to assess the long-term
effects of a treatment intervention.
Unless otherwise stated, it is
assumed that any such treatment is initially administered between t l
and t2' The treatment might, for example, consist of a one-time dose
of a drug or, alternatively, might consist of a drug that is
administered continuously from t l to t p ' In either case, interest
lies not just in the initial response to treatment (e.g. Y2-Y I ), but
also in the response at the later time points (t3, ... ,t p)'
Longitudinal designs are often analyzed by obtaining a
measurement (e.g. an estimated slope coefficient) at the level of the
individual and then amalgamating individual results within groups.
Individuals are sometimes explicitly subscripted using the subscript
(i).
For example, Yj(i) sometimes replaces Yj' B (i) denotes the
e.
97
observed slope coefficient of the regression of Y on time for
individual (i) (i.e. this regression uses the points
«1'Y 1(i}},(2'Y 2(i}}, .. ,(P,Y p(i}) as inputs).
When only time points
from t a to t b (i.e. instead of tl to t p) are used in the above
regression then Bab(i} denotes the observed slope coefficient. B
denotes the average individual slope coefficient (note that B is not
equivalent to the slope coefficient of a line fit to the points
«I'YI),(2'Y2) .. )). The simulation example uses a superscript notation
to distinguish within-group averages.
For example, S(2} denotes the
average slope coefficient for individuals in group 2, while
YI -.5(Y 2+Y3)(3) denotes the average value of YI -.5(Y 2+Y3) for all
individuals in group 3.
Some of the terminology that is used in this chapter is listed
below.
Response - An individual's observed response (or response
vector) is defined to be the vector (YI'Y 2' .... ,Y p).
Expected response - Suppose that an individual has already been
observed at t l (i.e. YI is known to be YI; Yl-YI' say). Then that
individual's expected response, conditional upon Y1-YI' is
(YI'~2.1' .....
'
~p.l)·
Note that the extended definition of RTM
compares the patterns of expected response (i.e. the ETPs) of
individuals with varying values of YI.
Expected time path (ETP) - The ETP is a plot of expected
response (on the y-axis) against time (on the x-axis).
Response function - For an individual, the response function
f(Ya'
'yb) , (a>-l,b<-p) , is defined to be a scalar function of
(ya'
yb).
For simplicity, it is usually assumed that all time
98
points from t a to t b are included in the response function. The
rationale for not including (Y1' ... 'Ya-l) in the response function is
that the effects of RTM can sometimes be reduced or eliminated by
dropping the first few time points from the response function.
In
practice, the data analyst might choose to use Yb instead of Yp as the
final time point if, for example, many missing values are present
after tb.
Note that response functions are calculated at the level of
the individual, then individual response functions are amalgamated
within groups as "average response functions" (see below).
Section B
of this chapter discusses how to choose response functions that are
unaffected by RTM or, barring this, how to analyze response functions
in a manner that appropriately accounts for RTM.
Here, response
functions are often used in order to assess treatment effects.
Average response function - An average response function for a
group is defined to be a scalar function that is obtained by averaging
the response functions of the group's members.
For example, if the
response function of interest is 8(i)' then the average response for
group 2, say, is denoted by 8(2).
In the sense of this definition,
the average response function need not necessarily be the mean of the
individual response functions.
For example, the median of the 8(i)s
is also an average response function.
Summary response measure - A summary response measure is defined
to be a scalar function that uses summary statistics (e.g. group
means) as inputs.
For example, both Yl /Y 2 and the slope coefficient
of a line fit to the points ( (I,Y l ), (2,Y 2), ... ) are summary
response measures.
Note that this chapter concentrates on average
response functions instead of summary response measures.
~ •
99
Dependence upon initial value - The response function
f(Ya, .. ,Y b) is said to "depend" upon initial value if and only if the
shape of the ETP from t a to tb varies as YI varies. Note that this
dependence is only checked in the interval from t a to tb' that is in
the interval of points used in the response function.
RTM is defined
to be a form of dependence upon initial value.
RTMab (definition) - Suppose that the response function is a
function of (Ya""'Yb) , (a>=I,b<=p).
Then individuals are defined
to be subject to RTM from t a to tb' denoted by RTMab, if and only if
the response function f(Ya, ... ,Y b) depends upon initial value in such
a way that the ETPs of individuals with varying values of Yl become
closer together (i.e. converge) in the interval from t a to tb'
RTMab (discussion) - Note that the above definition of RTMab,
which is an addition to the RTM literature, does not depend upon the
specification of the exact form of the response function, but does
depend upon the particular time points that are used as inputs to the
response function.
Also note that the same population might be
subject to RTMab but not RTMcd (a,c and/or
b~d),
depending upon the
choice of the time points that are used in the respective response
functions.
To be precise, we define "converge" as follows.
Denote the
elements of the expected response rector (VI' 2 l' 3 1"") by
(gl,g2"")'
Consider two individuals with expected response vectors
(91,92, ... ,gp) and (gi,g2, ... ,gp), respectively, and suppose that the
former individual is more extreme at tl'
Create the sequence
(sa, ... ,sb)=(lg:-ual-19a-ual, ... ,lgb-Ubl-lgb-Ubl).
Say that response
does not depend upon initial value if and only if sa=",=sb'
Say that
100
the ETPs coverge if and only if
inequality is strict.
only if
sa~... ~sb'
s~ ... ~sb'
where at least one
Similarly, say that the ETPs diverge if and
where at least one inequality is strict.
Note that
response may depend upon initial value in ways other than covergence
or divergence.
Also note that, while one may formally assess
convengence in any event, it is probably irrelevant to do so in
situations where individuals have ETPs of different shapes.
Also,
when the OJ'S are not all identical (j=l, ... p), then the alternative
metric
(Iga-~a)/oal-Ig:-~a)/oal, ... )
might be considered.
The choice
between these two metrics is exemplified by the deterministic
autoregressive process model
Yj=~8{Yj-1-~)'
0<8<1.
The first metric
holds that RTM occurs (i.e., because extreme observation converge) but
the second metric does not (i.e., because extreme observations do not
change status in terms of standard deviation units).
response functions are seldom scaled by
OJ'
Since individual
we prefer the first metric
in general, recognizing the above difficulties.
As is the case for two-point scalar designs, the ultimate focus
of this chapter addresses the effects of RTM upon groups.
Even so,
however, the assessment of RTM is performed at the level of the
individual, since individual response functions are used as the basic
unit of analysis.
Note that, in contrast to two-point scalar designs,
the "individual perspective" (i.e. RTMG) and the "group perspective"
(i.e. RTME) may yield different results.
For example, in the
two-point scalar design with treatment and control groups the ANCOVA
estimator of treatment main effect, which is a function of group means
that accounts for RTME, is equivalent to the estimator that is
obtained by adjusting each individual's difference score for RTMG.
••
~
101
Considering longitudinal designs, however, for example the average
A
response function B need not equal the slope of the regression line
fit to the points ((l,yl)' .... ), which is a summary response measure.
To recapitulate, the "RTM of group means" and the "RTM of individual
values" need not yield equivalent results in longitudinal designs.
Even though the ultimate focus of the analysis
i~
upon groups, RTM is
understood to be the "RTM of individual data values", since response
functions are actually calculated at the level of the individual (this
is the level at which the effects of RTM are manifested), and only
then combined over group.
RTM-induced bias - For any individual, the RTM-induced bias in a
response function is defined to be the difference between that
person's expected response function and the expected response function
for a person with Y1=UI.
For a group, the RTM-induced bias in an
average response function is defined to be the average RTM-induced
bias of that group's members.
Note that the RTM effects on an average
response function need not be limited to an RTM-induced bias, but
might also include an RTM-induced loss of precision as well (e.g. see
chapter 2).
The introduction now continues with a literature review.
102
(SA).
Introduction:
1.
Literature review:
RTM is typically formulated as a two-point concept (i.e. using
the slope of the regression of Y2 on VI)' A few authors, mostly in
the educational and psychometric literatures, discuss the application
of RTM ideas to longitudinal designs but no one-provides a complete
treatment or even a general definition that takes into account the
special features of longitudinal data.
Thus, one of the goals of this
chapter is to develop and apply an extended definition of RTM that
includes longitudinal scalar designs.
Nesselrode (25) probably provides the most thorough discussion
of RTM in longitudinal scalar designs to date.
He uses the expected
response vector, in conjunction with a series of two-point definitions
of RTM (e.g. RTM from t 1 to t2' RTM from t 2 to t3' etc.) in order to
illustrate some of the differences between RTM in two-point scalar
designs and RTM in longitudinal designs.
Generally speaking,
two-point scalar designs:
(a)
usually use difference scores or their equivalent to quantify
change; and
(b)
are circumscribed in their ability to differentiate between
models by the paucity of time points.
In contrast, longitudinal scalar designs:
(c)
allow a variety of options for quantifying change (i.e. allow a
variety of possible response functions); and
(d)
allow a variety of mathematical models to be considered.
When a treatment intervention is employed, point (d) above may be
expanded by noting that longitudinal scalar designs:
4It .
103
(dl) allow the postulation of a variety of models of what would have
occurred for treated individuals even in the absence of the
treatment intervention; and
(d2) allow the consideration of a number of possible forms of the
treatment effect, if any.
Thus, any proposed definition of RTM in longitudinal scalar designs
must apply to a much wider range of models, response functions, and
(if applicable) treatment effects than must the definition of RTM in
two-point scalar designs.
In turn, this suggests that a definition of
RTM in longitudinal scalar designs may be more difficult to develop
than was the definition of RTM in two-point scalar designs.
In contrasting RTM in two-point designs with RTM in longitudinal
designs, Nesselrode presents a series of models which are
indistinguishable between t 1 and t 2 but have different RTM effects
after t 2. In addition, he distinguishes between RTM effects caused by
measurement error (as illustrated in the simulation example) and RTM
effects caused by trends among true values (e.g. the autoregressive
parameter
e in example 3 of chapter 6). Nesselrode also stresses the
difference between following an individual that is extreme at tl and,
instead, choosing persons because of extreme values at later time
points.
For example, under the ElY model an individual that is
selected because of an extreme value at tl is likely to regress
between tl and t2' but is not expected to regress between t2 and t3
even though he will probably remain somewhat extreme at t2'
In
contrast, persons that are selected because of extreme values at t2
are expected to regress between t2 and t 3 . Nesselrode did not
describe a comprehensive analysis strategy for dealing with RTM in
104
longitudinal designs, but did comment upon the importance of
developing one.
Most of the biostatistical and epidemiological literature on RTM
in longitudinal designs concentrates on a particular special case,
that being the "separation of participant selection and baseline" as a
remedy for RTM.
For example, Davis (10), Ederer (11), and others
postulate the basic EIV model Yj=YT+Ej' and consider the situation
where study participants are chosen on the basis of having extreme
Y1s, say YI large. Under these assumptions, RTM affects response
functions that include YI but does not affect response functions that
do not include YI . In particular, a recommended study design utilizes
t I entirely for recruiting extreme individuals (i.e. "participant
while the assessment of response begins at t2 (i.e. t 2 is
used as a "baseline
thus IIseparating participant selection and
selection
ll
)
ll
baseline
ll
),
•
To give an example of this literature, McMahan (24) assumes the
basic EIV model Yj=YT+Ej' j-I,2,3, and compares two three-point
designs, both of which use Y3-Y 2 as a response function. The first
design completely separates participant selection, which takes place
at t I (i.e. only individuals with large Yls are selected), and the
baseline t 2. Under the EIV model, such a design is not subject to
RTM.
The second design uses both t I and t2 for screening (i.e. only
individuals with large YIs and large Y2s are eligible for
participation), and so is subject to RTM since t2 is used for both
screening and baseline.
Thus, the first design is the superior one
under the criterion of E(Y 3-Y 2) since E(Y 3-Y 2)=0 for each individual
in design one while E(Y3-Y2) depends upon initial value in design two.
e
105
However, McMahan also demonstrates that Var(Y3- Y2) is larger for the
first design than for the second, which illustrates that the latter
design is the superior one on the basis of precision.
Therefore,
McMahan concludes that the design option of using Y1 as the sole
criterion for participant selection involves a tradeoff between bias
(which is caused by RTM) and precision. Davis (10) considers similar
issues when (Y 1,Y 2,Y 3) is multivariate normal and not necessarily EIV.
To recapituate, most of the literature on RTM in longitudinal
scalar designs is devoted to the special case of the separation of
participant selection and baseline as a potential remedy for RTM under
EIV.
This dissertation considers a broader topic, namely that of
developing and applying an extended definition of RTM to longitudinal
scalar designs.
Application of this extended definition of RTM is
part of a general strategy for the analysis of longitudinal scalar
data that are subject to RTM.
106
(58).
Analysis of RTM in longitudinal scalar designs:
1.
Introduction:
Recall that RTMab occurs whenever the response function
f(Y a ,. "Yb) depends upon initial value in such a way that the ETPs of
individuals with different Yis converge in the interval from t a to t b.
When YI is extreme, then this tendency for the ETPs of individuals
with different Yis to converge amounts to claiming that later
measurements are expected to become less extreme, as per the principle
of RTM.
Note that there exist forms of dependence between response
and initial value that are not manifested as RTM.
For example, in the
hypertension literature the "tracking hypothesis" holds that a
predisposition towards hypertension causes individuals with somewhat
elevated blood pressures, say, as young adults, to evidence extremely
high blood pressures later in life.
~
Under this hypothesis, ETPs
diverge over time instead of converge, which illustrates a dependence
between YI and the shape of the ETP that is not manifested as RTM.
RTM affects the analysis of longitudinal scalar designs in two
main ways, both of which are illustrated in the simulation example
which follows.
First, even in the absence of a treatment
intervention, the presence of RTM gives misleading information about
the shape of individuals' time paths (i.e. misleading information
about the nature of the response).
Secondly, RTM can affect both the
bias and precision of average response functions (e.g. causing an
inappropriate or inefficient assessment of treatment effects).
The
overall analysis strategy recommended here, which is illustrated in
the simulation, involves first applying the extended definition of RTM
in order to determine whether or not RTM occurs for the design taken
107
as a whole.
If so, then ETPs are examined in order to determine which
set of time points might reduce or eliminate the effects of RTM.
For
example, the simulation illustrates a case where dropping VI from the
response function serves to entirely eliminate the effects of RTM.
Once a set of time points is tentatively entertained, the form of the
treatment effect is analyzed in order to determine whether or not an
estimator of treatment effect can be developed that is a function of
this set of time points.
Finally, if RTM effect remain, then initial
value is included in the model of response.
Note that this proposed
analysis strategy applies equally well to forms of dependence between
response and initial value that are not manifested as RTM.
This section continues with a simulation example, followed by an
expanded discussion of analysis strategy and implementation issues.
108
(5B).
Analysis of RTM in longitudinal scalar designs:
Z. Implementing a general analysis strategy:
~.
Simulation example:
The discussion of a general analysis strategy to apply to
longitudinal scalar data that are subject to RTM begins with a
simulation example.
(a)
The goals of this simulation example are to:
orient the reader to the notation and terminology of this
chapter;
(b)
illustrate RTM in a longitudinal scalar design; and
(c)
illustrate the relationship between RTM and the estimation of
treatment effects in a longitudinal scalar design.
Both random and truncated sampling designs are considered.
To begin, suppose that a clinical trial of two antihypertensive
drugs proceeds by randomly allocating a population of 15000
individuals into three groups of 5000 persons apiece.
Everyone is
observed at five equally spaced time points (tl,t2,t3,t4,t5)' which
are sometimes taken to be (1,2,3,4,5) without loss of generality.
Let
yjk)denote the diastolic blood pressure (DBP) measurement at time tj
(j=I,2,3,4,S) for an individual in group k (k-l,2,3).
Sometimes the
DBP at time tj is denoted by the shorthand Yj (i.e. without the
superscript), in which case this notation applies to all individuals
regardless of group.
Assume that the EIV model holds in each group, that is Yj=YT+E j ,
where YTj denotes the true value at time tj and Ej denotes the random
error term at time t j
.
For each group, YTI is assumed to be normally
distributed with mean 80 and variance 100.
Each Ej is assumed to be
normally distributed with mean 0 and variance 100, independent of the
109
other error terms and independent of the true values.
For any person,
true values are assumed to remain unchanged over time (i.e. equal to
YTI ) except for the effects of the treatment intervention, if any.
Group I is the control group. There, Y3l2YTI+Ej (j=1,2,3,4,S).
Note that every control person's true values are assumed to remain
unchanged over time.
Each person in group 2 receives one dose of drug A between t 1
and t2' and it is assumed that this treatment has an immediate and
permanent effect on true DBP, the magnitude of which is denoted by T.
y~2~YTl+El while y~2~(YTI-T)+Ej' (j=2,3,4,S).
Y~~~YTI if j=1 and y~~tYTI-T if j=2,3,4,S. Drug A is
That is,
Equivalently,
assumed to
affect some persons more than others, and so each member of group 2 is
randomly assigned a value of T that is taken from a uniform
distribution over the interval (0,20).
The mean value of T,
~T
say,
which is one of the quantities that the researcher wishes to estimate,
turned out to be 9.98, which is essentially as expected.
Instead of a single dose, as was the case for group 2, each
person in group 3 is dosed nonstop with drug B beginning immediately
after tl' and it is assumed that the effects of this regimen upon true
DBP are the same in each time period and that these effects
accumulate.
That is, y~2lYTI+O(j-I)+Ej' where 0 denotes the magnitude
of drug B's effect per unit time period.
Drug B is also assumed to
affect some persons more than others, and so each member of group 3 is
randomly assigned a value of 0 taken from a uniform distribution over
the interval (0,5). The mean value of 0,
~o
say, which is another of
the quantities which the researcher wishes to estimate, turned out to
be 2.49, which is essentially as predicted.
110
To summarize, the researcher wishes to estimate Wo and wT ' whose
values for this realization of the simulation are known to the
omniscient statistician but not to the researcher.
The implementation
details for the simulation are discussed in appendix 1.
The
discussion now proceeds to examine the effects of RTM upon the
estimation of Wo and W for both random and truncated designs, both
T
with and without a control group.
Truncated designs assume Y1>95.
Table 1 below restates the simulation specifications.
III
Table 1
Simulation Specifications
Observed values
tl
t3
t4
ts
Yn +E 3
Yn +E 4
Yn +Es
Yn - nE 3
Yn -28+E3
Yn- HE 4
Yn-HEs
Yn- 30+E4
Yn -4 0+E 5
t2
Group 1
Yn +E 1
2
Yn+ E1
Yn +E 2
Yn-nE 2
3
Yn+E l
Yn- 8+E 2
True values
Group 1
Yn
Yn
2
Yn
Yn
Yn
Yn -T Yn-T
Yn
Yn- o Yn -28 Yn -38 Yn- 48
Expected response with an individual with
3
Yn-T
Group 1
ll+ka
~+8ka
~+8ka
2
~+ka
~+8ka
~+8ka-~
3
ll+ka
ll+8 ka
ll+8ka-2~8
80, a2
=
Note:
~
Yn
Yn -T
=
200, 8 ,. .5
~+8ka
~+8ka
T
~+8ka-~
Yl"~+ka
T
~+8ka-3~8
~+8ka-~
T
ll+8ka-4~0
112
O=9ro up I
0=3,.0 up 2')(:31"0 up .3
v
II)
~
Q..
10
"
"
u
Q,:
0
•
~:
~
;.
~
Q.
><
~
?
*.
H.
U
ill
U'
.,
rim.
u
U
t~
is
Time
Figure IA
ETPs for all groups
(no treatment effects)
Figure 18
ETPs for all groups
(treatment effects included)
Recall that the first step in the recommended analysis strategy
involves checking for the expected existence of RTM.
Since any RTM is
expected to affect individuals regardless of their later treatment
status, this assessment does not include treatment effects.
For the
same reason, the overall assessment of RTM is performed in the same
manner regardless of the design (i.e. random or truncated, control
group or no control group), and so is only illustrated once here.
Figure IA above, which is abstracted from table 1, plots expected
response against time in the absence of treatment for individuals with
different values of VI' VI·~+kcr say, where ~.80, cr 2=200 and 8=.5 here.
That is, the expected response for a person in group 1 (who remains
untreated throughout the study) is (VI'
(~ks,
~+8kcr,
u+8kcr,
~+8kcr,
~+8kcr),
~2.I' ~3.I' ~4.I' ~5.I)·
which is plotted above. For
individuals with different values of VI (i.e. different values of k),
the ETPs converge from tl to t2' and so RTM (i.e. RTMI5) is present
for the design as a whole.
As discussed above, this same RTM is
4It
113
. expected in groups 2 and 3.
three groups.
Figure IB above plots the ETPs for all
Note that group l's plots represent the effects of RTM
only while the plots for groups 2 and 3 combine both RTM effects and
treatment effects.
Since RTM is in fact expected to be present, the second step in
the recommended analysis strategy is performed.
That is, the form of
the RTM effect is analyzed in the hopes of obtaining a set of time
points which, when used in the response function, reduce or eliminate
the effects of RTM.
Here, since the ETPs in figure lA are parallel
from t 2 to ts, this set of time points is (t 2,t 3,t 4,t S)' Thus, any
response function that includes Yl is affected by RTM while any
response function that does not include Yl is not.
After deciding to tentatively entertain (Y 2'Y 3 'Y 4'YS) as the
inputs to the response function, the next step in the recommended
analysis strategy involves examining the ETPs in each group in order
to determine whether or not a function of these time points can be
developed that actually provides an appropriate estimator of the
treatment effect parameter.
This step accomplishes the link between
the previous assessment of RTM, which does not consider treatment
effects, and an analysis of treatment effects that properly takes RTM
into account.
As discussed below, when considering group 3 it turns
out that there are many such functions of (Y 2'Y 3 'Y 4 'Y S) that are
unbiased estimators of u5' but that there are no unbiased estimators
of u using (Y 2'Y 3 'Y 4'YS) from group 2 only. This illustrates that
T
the proper choice of a response function depends not just upon an
analysis of RTM but also upon the form of the treatment effect.
In
addition, this illustrates that it is not always possible to eliminate
114
the effects of RTM
t~rough
the choice of a response function, even
when portions of the ETPs are parallel (i.e. not subject to RTM).
First considering the estimation of We' the expected response
vector for a person in group 3 with YI=w+ka is
(w+ka,w+Ska-we,w+Ska-2we,w+Ska-3we,w+Ska-4We) , which is plotted in
figure 18 above.
Considering this vector, it turns out that many
functions of (Y 2'Y 3'Y 4 'Y S) have an expected value of we' so that RTM
in fact can be accounted for by the expedient of not including YI in
the response function.
For example, both 1/3 (Y 2-Y S) and -62S (i.e.
the negative of the average slope coefficient of the individual
regressions of Y on time, where these regressions do not include the
data point (1, YI(i»' have expectations of we (e.g. the expected
value of I/3(Y 2-YS) for an individual with Y1=u+ka is
I/3{(w+Ska-we)-(w+Ska-4we)}=we). Table 2A below illustrates that both
of these estimators of we perform well in the simulation.
That is,
these estimators are near 2.49 in group 3.
Similarly, table 28
-(3)-(1)
illustrates that the controlled comparisons (e.g. B2S- S2S) are also
near 2.49, which is as expected since all response functions
f(Y 2'Y 3 'Y4'YS) have an expectation of zero in the control group.
Note
that a control group is not required in order to estimate We' even
when truncated sampling is employed, so long as it is known that the
population mean of Y does not change over time. To recapitulate, a
control group is not required to adjust for RTM since the above
response functions account for the expected effects of RTM upon each
member of the population, instead of relying on RTM effects to
possibly average to zero.
Of course, without a control group any
4It
116
Table 2A
Group Means and Mean Response Functions
All Individuals
Group 1
Observed data:
n
5000
YI
Y2
Y3
Y4
Y5
R1
R2
R3
'"
Estimators of ~f: -S2
/3(Y 2- V5)
Estimators of
~T:
Group 2
5000
5000
80.16
79.96
80.09
80.03
80.08
(.20)
(.20)
(.20)
(.20)
(.20)
79.95
70.27
70.20
70.14
70.15
(.20)
(.. 22)
(.22)
(.22)
(.22)
.20
.08
.12
(.20)
(.20)
(.16)
9.69
9.81
9.77
( .22)
( .22)
( .18)
.03
-.04
(.06)
( .07)
Only Individuals With Y1>95
Group 1
Group 2
Observed data:
Estimators of
n
~
Y1
Y2
Y3
Y4
Y5
102.17
90.84
91.10
90.75
90.85
:
Rl
RZ
R3
'"
Estimators of ~~: -8 25
/3(Y 2-Y 5)
T
745
Group 3
80.08
77 .47
74.87
72.62
70.09
( .20)
(.26)
(.27)
( .29)
(.22)
2.44
2.46
( .07)
( .07)
Group 3
745
745
(.22) 102.26
(.44) 81. 57
(.46) 81.42
(.45 ) 81.89
(.44) 81.41
(.22) 102.16
(.51) 89.46
(.53) 86.72
(.52) 85.35
(.53 ) 82.26
11.33
11.32
11.28
(.45)
(.44)
( .36)
(.51)
(.53)
(.41 )
- .03
-.00
( .16)
( .17)
20.69
20.85
20.69
2.30
2.40
(.22)
(.46)
(.65 )
(.78)
(.53)
(.20)
(.19)
e
117
Table 28
Controlled comparisons
All individuals
Estimators of
~:
T
R
9.49
9.73
(.30)
(.30)
R3
9.65
(.24)
R1
2
2.41
2.50
(.10)
(.10)
Only individuals with Y >95
9.37
(.68)
9.53
(.69)
(.52)
9.41
2.26
2.40
Legend:
(.25)
(.25)
Mean and (standard error) are presented.
R1=Y 1-Y 2
R2=Y 1-Y 5
R3=Yl-·25(Y2+ Y3+ Y4+ Y5)
Controlled comparisons subtract response function in
control group from response function in group 2 or 3,
as appropriate.
118
Now considering the estimation of
~
L
, the expected response
vector for any individual in group 2 with fixed YI , say YI=~+ko, is
(w+ko,~+8ko-~ ,~+8ko-~ ,~+8ko-~ ,~+Bko-~).
Figure 18 above plots
L
1
1
1
this expected response vector against time for various values of YI .
Note that the expected change between t i and t 2 for someone in group 2
with
YI=~+ka
is
(I-S)ka+~l'
which represents a confounding of RTM
effects (i.e. S) and treatment effects (i.e.
Also note that the
~).
T
expected value of (Y 2'Y 3 'Y 4 'Y S)' conditional upon
YI=~+ka,
(Y2'Y3'Y4'YS)·(~+Ska-~T,~+Bka-~T,~+Bka-~T,~+Ska-~T)
,
each Yj (j=2,3,4,S).
is
the same for
Since Y is not expected to change after tl' no
function that uses only (Y 2'Y 3'Y 4 'Y S) can have an expectation of ~T'
Insofar as RTM is concerned, then, this example is not different from
the two-point scalar case except that a more precise post-measurement
may be obtained from (Y 2'Y 3 'Y 4'Y S) in comparison to using Y2 only.
Continuing to focus upon the estimation of ~T' let RI =Y I -Y 2,
R2=Y I -Y S' and R3=YI-.2S(Y2+Y3+Y4+YS) for notational convenience. Note
that RI , R2 and R3 each compare a pre-dosing score to a post-dosing
score, as would be the case for a two-point design.
As expected from
the results of chapter 2, then, each of the controlled comparisons
-(2)
R
I - R-(l)
I ' R-(2)
2 - R-(l)
2 ' an d -R(2)
3 - -R(l)
3 are un b'lase d es t'lma t ors 0 f ~1' as
illustrated in table 28 above (i.e. all of these estimators are near
9.98 for both the random and truncated designs).
For the nontruncated
design, these estimators are unbiased because the individual RTM
effects within each group are expected to average to zero leaving, for
example, E(Y 3 )=0 and E(Y 3 )=~. For the truncated design,
1
randomized allocation implies that the RTM effects for group 2,
119
although nonzero, are about the same as the RTM effects for the
control group (e.g. E(Yi(2))=(1-B)ka+~T and
E(V;(3))=(I-B)ka), and therefore cancel.
As is also discussed in
chapter 2, even when RTM effects cancel, leaving no bias in the
average response function, the inclusion of YI as a covariate involves
a gain in precision when estimating ~T' As an illustration, the
standard error of 82 from the ANCOVA (using R3 as the response and YI
as the covariate) on the controlled truncated design is .49, which is
less than .S2, the standard error of R;(2)_R;(I) from the same design.
Now considering the uncontrolled designs, table 2A illustrates that,
as is the case for the two-point design, the confounding of treatment
effects with RTM effects implies that there is no unbiased estimator
of
~
in the uncontrolled truncated design.
of RI(2),
R~(2),
There, for example, each
and R;(3) are near 20 instead of 9.98 .
As a final comment on the estimation of
~T
in the simulation
example, note that the design option of separating participant
selection and baseline avoids the above confounding of RTM effects
with treatment effects.
As an illustration, now suppose that drug A
is administered to group 2 between t2 and t3 instead of between t l and
t 2. Then, the expected response vector for an individual in group 2
with
Yl=~ka
is
(~ka,~+8kcr,~+8kcr-~ ,~+8kcr-~ ,~+8kcr-~T)'
T
T
As before,
RTM effects are manifested between tl and t2' but now the treatment
effects occur between t2 and t3'
For notation, let
R4=Y2-1/3(Y3+Y4+YS) and RS·{,S-(Yl -,S(Y 1 -80))+.SY2} - 1/3(Y 3+Y 4+Y S)'
R4, which is the analogue of R3 in the previous case, is now an
unbiased estimator of
~
T
in all designs, including the uncontrolled
truncated design, as illustrated in table 3 below.
The general
120
analysis strategy recommended in this chapter would employ R4, or some
other function of (Y 2'Y3'Y 4'Y S)' as an estimator of ~T that
automatically accounts for RTM.
Note, however, that if the precise
magnitude of the RTM between t 1 and t 2 is known or can be estimated
(e.g. from a control group or from external sources), then this
information may be used in order to include Y1 i~ the response
function and, thus, obtain a more precise estimator of ~.
For
T
example, both Y1-.S(Y I -80) and Y2 are unbiased estimators of Y2T' and
so RS' which averages these two quantities, might be considered as an
alternative estimator of
Table 3 below illustrates that RS is
somewhat more precise than R4. Note that Yl-.5(YI-80) is simply the
~.
T
RTMG-adjusted value of Y2 discussed in chapter 2.
The above
adaptation of the recommended analysis strategy, which is in the
spirit of McMahan (24), requires that 8 be correctly guessed or
A
estimated (here 8-.5), which is an additional risk that only seems to
be worth considering when a~ is very large relative to
ai.
Table 3
Treatment between t2 and t3
Means (standard errors)
Group 1 random Group 1 truncated Group 3 random Group 3 truncated
Response (observed, see previous paragraph for notation)
R4 -.11 (.16)
-.06 (.41)
10.09 (.18)
9.56
R5 -.05 (.12)
.06 (.30)
9.95 (.14)
9.56
True value (unobserved)
0
0
Controlled comparison
random sampling
R4
RS
10.20
10.00
(.24)
(.19)
(.47)
(.38)
9.57
9.98
Controlled comparison
truncated sampling
9.62
9.49
(.62)
(.48)
121
To summarize the results of the simulation example, analysis of
the ETPs of individuals without treatment intervention suggested a set
of time points that, when used in the response function, are
unaffected by RTM.
In other words, for these time points, individual
response is not correlated with initial value and there are no RTM
effects on either the bias or the precision of estimators of treatment
effect.
In particular, even estimators from uncontrolled truncated
designs are adequate, at least when it may be assumed that the
underlying population mean does not change over time.
After deciding
upon a set of time points to tentatively entertain as potential inputs
to the response function, an analysis of the form of the treatment
effect determined that many unbiased estimators of
from these points, but that no such estimators of
developed.
RTM biased the estimators of
~T
~6
~T
could be created
could be
in the uncontrolled
truncated design and affected the precision of estimators of
designs.
~T
in all
In any case, when RTM effects remain, then information about
initial value should be used when modeling response.
This
accomplishes both a gain in precision and a bias adjustment, if
necessary.
The proof of the claims about the precision gain and
possible bias adjustment (i.e. when V1T1VIC) proceeds exactly as in
the two-point case (see section 2C), but with the more general
response function obtainable from a longitudinal design.
When the
study design includes both treatment and control groups, then
information from the control group allows the specification of any
RTM-induced relationship between VI and the expected response
function. For uncontrolled designs, outside information is required,
again as in the two-point case (see section 20), in order to specify
122
the nature of the RTM-induced relationship between initial value and
expected response (e.g. it is necessary to specify 6=.5 and
~=80
in
order to obtain the more precise R5 above). Once this relationship is
specified, then one proceeds to include VI as a covariate as before.
Section 58 now continues with a discussion of some of the RTM
issues that may be considered before, rather than after, data
collection.
In particular, assessment of the form of RTM effects is
illustrated, where this assessment might be part of the planning of
the study design (e.g. so as to avoid the design-induced problems
discussed in the estimation of
~L
above).
Finally, a simple method of
predicting the magnitude of RTM-induced bias of estimators of
treatment effect is presented.
•
123
(58).
Analysis of RTM in longitudinal scalar designs:
l.
Implementing a general analysis strategy:
(Q). Analysis of RTM before data collection:
\~-
/
(A)
Time
(f)
Time..
"'--~
.
(Co)
Tim.
.
(0)
,./me.
Figure 2
ETPs for some different models
Before data collection, the assessment of RTM begins by
determining the sets of time points for which RTM is present and the
sets of time points for which RTM is absent.
This is accomplished by
plotting the expected response, in the absence of treatment, for
individuals with varying values of YI . Figure 2 above illustrates
these plots for four models. Figure 2A plots ETPs from the EIV model
in the simulation example.
Clearly, the set of time points for which
RTM is absent is very large, namely all of the time points except for
YI, which illustrates why the EIV model is perhaps the simplest case
to consider.
Figure 2B plots the ETPs for an extended EIV model,
namely Yj=YTj+Ej' where the true values at tj are assumed to vary,
124
say, in response to a factor that affects everyone equally.
Again,
RTMlb (b>=2) is present but not RTMab (a>l, b>a), exactly as for the
simulation.
Figure 28 illustrates that the definition of RTM implies
that "extremely large Y1s tend to become less extreme relative to the
population mean at later time points", not necessarily that "extremely
large YlS tend to decrease over time".
Figures
and 2D plot ETPs
~C
for models that are subject to RTMab for every a and b (a<b).
2C uses the autoregressive stochastic process model
O<e<l.
Figure
Yj=~+e(Yj_1-~)+Ej'
There, for example, the expected response vector for an
individual with Y1=~ko is (~ko,~eko,~+e2ko,... ) . Although RTM is
expected to be unrelenting for this individual, its magnitude is
expected to decrease as time passes. In particular, the rate of
decrease is rapid if eis small, which in turn implies that dropping
the first few time points from the proposed response function will
greatly reduce the effects of RTM.
However, dropping these time
points also produces less extreme data.
The above autoregressive
model, which is discussed in example 3 of chapter 6, might plausibly
be applied to biological processes that have an "automatic regulatory
mechanism", the strength of which is represented bye, that takes
effect whenever the individual begins to evidence extreme values.
Figure 2D illustrates another case where RTMab exists regardless of
the values of a and b (a<b).
Suppose that, in the absence of
reatment, the functional relationship between a rat's weight in grams
(y) and its calendar age (x) is y=250{1-exp(x)}.
Thus, rats tend to
gain weight as they age, albeit more slowly, up to 250 grams. However,
suppose that a study that examines the effects of various treatments
upon weight gain (i.e. weight at end of study minus weight at time of
125
treatment) includes rats of different calendar ages at the time of
treatment.
Then, figure 2D illustrates that the relationship between
weight gain and calendar age is manifested as RTM (i.e. older rats are
in~tially
heavier but will tend to gain less weight over time).
Here,
including calendar age as a covariate is one potential solution to the
difficulties caused by RTM.
Another solution is to reparameterize the
response scale from Y to logit(Y), if it is biologically reasonable to
do so.
Suppose that it is impossible to create a response function
where individuals are unaffected by RTM.
For example, in the
simulation this is the case for estimators of
~T.
As illustrated in
the simulation, the effects of RTM are always manifested as a loss of
precision in estimators of treatment effects.
In addition, depending
upon the design, RTM may also induce a bias in estimators of treatment
effects. The magnitude of the bias in the average response function
that is attributable to RTM may be obtained by averaging the bias
estimates of the individual response functions.
After sampling, this
calculation requires both:
(a)
knowing or estimating the parameters in the model of response in
the absence of treatment; and
(b)
deciding upon the response function.
When estimating the bias before sampling instead of after data
collection, the sampling distribution of YI is additionally required.
As an example of the above calculations, suppose that in the
simulation example it is known that, in the absence of treatment,
Yj=YT+Ej'
2
YT-N(~,oT)'
2 , and the YTs and the Ejs are al 1
Ej-N(O,oE)
independent, but that of and o~ are not known exactly except that both
126
parameters exceed zero.
Then the previous section illustrates that
RTM is present for any response function that includes Yl . Now
suppose that 8=af/{af+a~) is either known or estimated (i.e. from the
contro1 group) to equal .5, f.l=80, that the response funct ion- is
R3=Yl-·2S{Y2+Y3+Y4+YS), and that the average value of Yl in group 2 is
100. Then, the RTM-induced bias of R3 is 10, which is obtained as
follows:
Recall that the RTM effect for any individual, say with Yl=v+ka,
is the difference between the expected response for that person,
namely (f.l+ka)-.2S{{f.l+Ska)+{f.l+Ska)+{f.l+Bka)+{f.l+Ska)) = {l-S)ka,
and the expected response for a nonextreme individual (i.e. a
person with Ylsf.l), namely f.l-.2S{f.l+f.l+f.l+f.l) •
o.
The RTM bias for
any group of m individuals, say, is just the average of the RTM
m
I
effects on these individuals, namely M- ~=I{{l-B)kma), which
m
equals M-I~.1{.5){20)=10 here.
Note that km usually varies from
person to person, depending upon the value of YI .
When the RTM bias is to be estimated before sampling then the
values of Yl are unknown, and so the sampling distribution of Yl must
be used in the above calculation. In that case, the estimated RTM
bias is J[f{YI)r{Y1)]dy, where f{Y 1) is the density function of Yl and
r{Y l ) is the RTM effect on any individual with that value of Yl .
To summarize, when the functional form of the model in the
absence of treatment is known, then the RTM effect on any individual
is obtained by subtracting the expected response function for that
individual from the expected response function for an individual with
Yl=f.l.
These individual bias adjustments are then averaged in order to
obtain the appropriate bias adjustments for the average response
~
127
function of any group.
Before sampling, the same calculation applies,
but the sampling distribution of VI must be used in order to predict
the likely values of VI and, hence, the likely magnitude of individual
RTM effects.
When the functional form of the model in the absence of
treatment intervention is unknown (or the functional form is known but
the exact parameter values are unknown), then information form either
a control group or else from outside sources may be used in order to
obtain appropriate estimators.
Chapter 5 continues with a summary and conclusions.
128
(5C).
Final remarks:
1. Summary:
To summarize chapter 5, RTM, which is defined to be a special
case of a relationship between expected response and initial value,
holds whenever the ETPs for individuals with varying values of VI
converge over time.
This extended definition of RTM reduces to the
usual definition of RTM when the study design includes only two,
instead of many, time points
Here, the time points of primary
interest are the set of time points that are used in the response
function.
Thus, for example, the same study might evidence RTM for
one response function but be free from RTM when an alternative
response function is employed.
The existence of RTM is assessed
assuming the absence of any treatment effects.
Chapter 5 assumes that the usual analysis of longitudinal
designs involves first estimating response functions at the level of
the individual (e.g. using individuals' estimated slope coefficients),
then amalgamating individual results within groups.
Since the
individual is the basic unit of analysis, the ultimate assessment of
RTM is performed at the level of the individual.
For any individual,
the RTM-induced bias is defined to be the difference between the
expected value of that individual's response function and the expected
value of the response function for an individual with
Yl·~l'
The
effects of RTM upon groups are obtained by averaging RTM effects upon
individuals.
When individual RTM effects do not cancel, RTM is said
to induce a bias in the average response function.
Even when
individual RTM effects cancel, RTM induces a loss of precision in the
average response function.
As is the case for two-point scalar
~
129
designs, including Yl in the model of response accomplishes both a
gain in precision and a bias adjustment, if necessary.
RTM effects
can be predicted before sampling provided that the sampling
distribution of Y1 is used in place of the observed distribution of
Y1·
An analysis strategy for dealing with RTM i'n longitudinal
designs was recommended and illustrated using a simulation example.
First, RTM is assessed for the design taken as a whole.
If RTM is
present, then ETP plots are examined in order to determine what set of
time points, if any, eliminates or at least reduces the effects of
RTM.
Once a set of time points is tentatively entertained, the
anticipated form of the treatment effect is examined in order to
determine whether or not an appropriate response function can be
obtained from this set of time points.
If some RTM effects on
individuals still remain, then Y1 is included in the model of
response. This analysis strategy applies equally well to forms of
dependence between initial value and expected response that are not
manifested as RTM.
Chapter 5 concludes with a comparison between the analysis of
RTM in longitudinal scalar designs and the analysis of RTM in
two-point scalar designs.
130
(SC).
Final remarks:
2.
Comparison between RTM in longitudinal scalar designs and
RTM in two-point scalar designs:
The proposed definition of RTM in longitudinal scalar designs
includes the usual definition of RTM in two-point scalar designs as a
special case.
That is, for two-point scalar
converge when 0<8<1.
d~signs,
the ETPs
Even so, however, it remains of interest to
compare the manifestations and analysis of RTM between these two types
of study design.
To this end, some of the similarities between RTM in
two-point and longitudinal scalar designs include:
(I)
Both definitions of RTM describe a relationship between initial
value and the pattern of later measurements or, equivalently,
between initial value and expected response.
For both designs,
an assessment of RTM is part of a larger examination of which
variables affect response;
(2)
In both definitions, the relationship between initial value and
expected response, which is expressed through the ETP, takes the
form that initially extreme individuals tend to become less
extreme during the period of time that response is measured; and
(3)
Even though RTM exists in many data sets, the relationship
between VI and expected response is seldom of primary interest.
For example, in a clinical trial, RTM is essentially a nuisance
when the overall goal is to estimate the effects of a treatment
regimen.
The analysis recommendation of including VI as a
covariate in order to account for the effects of RTM upon both
the bias and the precision of average response functions is
appropriate for both types of study design.
4It
131
In contrast, some of the differences between RTM in longitudinal and
two-point scalar studies include:
(1)
For two-point designs, a relationship between initial value and
ETP will, in practice, almost always be manifested as RTM.
is not the case for longitudinal designs.
This
In other words, the
concepts of RTM and a relationship between "initial value and
response are distinct for longitudinal designs while they are
practically inseparable for two-point designs.
In longitudinal
designs, the latter concept seems to be more fundamental than
RTM since a relationship between response and initial value that
is not manifested as RTM must still be adequately accounted for
in the data analysis;
(2)
Two-point designs almost always use difference scores or their
equivalent as the response function.
In contrast, VI is not
always part of the response function for longitudinal designs.
More generally, the analysis option of dropping some data points
from the response function so that the remaining data points are
not subject to RTM is possible for longitudinal designs but not
for two-point designs; and
(3)
In two-point designs, performing RTM (i.e. RTMG) adjustments upon
individuals and performing RTM (i.e. RTME) adjustments upon
group means typically leads to identical results.
For example,
the ANCOVA estimator of treatment main effect, which is a
function of group means that accounts for RTME, is equivalent to
first performing RTMG adjustments upon individuals and then
averaging RTMG-adjusted individual responses. For longitudinal
designs, however, the "individual perspective" and the "group
132
perspective" often yield different results, and the "individual
perspective" seems to be of more fundamental importance.
The dissertation now continues with chapter 6, which consists of
a set of examples illustrating the application of the theory that was
expounded in the previous four chapters.
Chapter 6
Examples
(6A).
Introduction:
This chapter contains three examples of applications and
extensions of the results of the dissertation.
The first example,
which involves assessing the effectiveness of pediatric counseling,
primarily illustrates the results of chapters 2 and 3.
For a
two-point vector design with truncated sampling, the magnitude of
RTME, as well as the magnitude of RTMG-induced misclassification, is
estimated.
The effects of RTME upon testing an a prior hypothesis
within a MANCOVA model are also described.
The second example, whose
two-point vector design is abstracted from James' (19) study of
hypertension among Edgecombe County blacks, illustrates the use of
regression analysis when many variables are measured with error and
are, thus, subject to RTM (see chapter 4). Finally, the third example
extends the discussion of RTM in longitudinal scalar designs (see
chapter 5) to include a model with unrelenting RTM, namely an
autoregressive stochastic process model.
The general analysis
strategy proposed in chapter 5 is applied, and some of the
modifications required by the presence of unrelenting RTM are
discussed.
Chapter 6 now continues with a discussion of RTME in the Chapel
Hill Pediatrics Behavioral Study (Martin (23)).
134
(6B).
Examples:
1. Example 1 - Chapel Hill Pediatrics Behavior Study:
~.
Introduction:
One of the components of the Chapel Hill Pediatrics Behavior
Study (Martin (23)) involves using a two-point vector design in order
to test whether or not pediatric psychologists can provide effective
short-term treatment for mild behavioral disorders.
The actual design
amounts to truncated sampling with a control group, and thus might
potentially be affected by RTM.
The proposed analysis involves using
a multivariate analysis of covariance (MANCOVA) model in order to test
a predetermined null hypothesis, namely that, after accounting for the
effects of age and sex, psychological counseling does not affect
either of two behavioral outcomes measured on each child.
selection issues are not discussed here.
Variable
e-
In section c, the control
group is ignored, in order to illustrate how RTM affects a truncated
design with a single group.
The study population consists of all children, aged two to seven
years, that have a well-child visit during the recruitment period.
At
the time of the first such visit (i.e. tl)' the attending physician
evaluates whether or not the child is suffering from some kind of mild
behavioral disorder that might be amenable to short-term psychological
treatment (e.g. biters might benefit, psychotics would not), in which
case the child is referred to the team of pediatric psychologists that
is associated with Chapel Hill Pediatrics.
In conjunction with the
physician's evaluation, two indices of the child's emotional health
are prepared from the parents' report, namely the Eyeberg Child
Behavior Inventory Intensity Score (VI) and the Eyeberg Child Behavior
135
Problem Score (Y2)'
Changes in these two indices are used as the
YI is a summated score of 36 Likert-type items,
each scored from one ("never") to seven ("always"). For example, one
response variables.
of the items is "teases or provokes other children".
Y2 is a summated
score of zero-one responses to the same 36 items, this time measuring
whether or not the particular item is interpreted as a "problem" by
the parents.
YI and Y2 were observed to be approximately normally
distributed in a sample of similar children, and clinical cut-offs of
YI =I27 and Y2=II have been recommended. As might be expected, YI and
Y2 tend to be positively correlated. Although YI I and Yl 2 were not
used in the physician's assessment, it may be assumed that every child
that is recommended for counseling has scored high on YII , and
probably high on Yl 2 as well.
The subpopulation of interest involves all children with a high
emotional health index (i.e. high YII and possibly high Y12 ) at tI'
that is at the time of initial screening. Note that some of these
children are assigned to the treatment group (i.e. psychological
counseling) and some are not, and that treatment assignment is not
,
strictly random.
,
In particular, YIT>Y1C' since relatively more of the
"extremely disturbed" children are assigned to the treatment group.
Each child with a high emotional health index is remeasured
approximately three months after the initial clinic visit, with the
vector of difference scores O-(OI,02)-(Y21-Y1l'Y22-Y12) being the
response of interest.
The analytical goal is to assess the effects of
treatment upon response after accounting for any effects of age and
gender.
Since a modified version of truncated sampling is applied in
order to determine eligibility, it is of interest to determine the
136
likely effects of RTM not just upon the subpopulation as a whole, but
in particular upon the assessment of treatment effects.
137
(6B).
Examples:
1.
Example 1 - Chapel Hill Pediatrics Behavior Study:
Q.
RTM and MANCOVA in a two-group truncated design:
Recall that the researcher wishes to assess the main effect of
treatment upon the vector of difference scores 0=(01'02)' while
controlling for the effects of age and gender.
Recognizing that the
treatment group is expected to suffer differentially more RTME than
the control group (i.e. since it is more extreme at t 1), she is
concerned whether this differential RTME will bias the estimator of
treatment effect, namely (6 31 , 632 ), in the model
(1)
(Ql); (~OI
+~11 age +~21 gender +~31 trt)
02 : 632 +612 age +622 gender +632 trt
To answer, recall the results of chapter 2 (especially section
20) and appendix 3 which describe the relationship between RTME and
the ANCOVA for two-point scalar designs.
There, when Y1T1Y1C'
including initial value as a covariate was required in order to obtain
an unbiased estimator of the treatment main effect.
In addition, the
above result continues to hold, even though the covariate is typically
measured with error, unless the treatment and control groups are
sampled from separate populations with different means (note that
treatment and control groups are obtained from the same population
here).
The present situation differs from the simple two-point scalar
case in that:
(a)
a multivariate, rather than a univariate, response is modeled;
and
138
(b)
the additional control variables age and gender are included.
Nevertheless, an examination of the proof in appendix 3 suggests
that the above result also applies here.
That is, the MANCOVA
model
(2) (QI)~(~OI +~II age +~2I gender +~3I trt +~4IVII +§SI VI2\
\02 ~\602 +6 12 age +6 22 gender +632 trt +642 VII +6 S2 VI2 )
provides an unbiased estimator of the treatment main effect vector
(~I'~2)
while model (1) does not.
In order to illustrate the above, models (1) and (2) were fit to
a simulated data set using the design of the Chapel Hill Behavioral
Study. As described below, the results of the physician's clinical
judgment in assigning children to treatment are translated into a
truncation rule that is based upon the observed values of VII.
In
addition, the covariates age and gender are assigned to individuals in
such a way that age is unrelated to initial value, treatment status,
or response.
Gender is positively correlated with both initial value
and treatment status.
Thus, the simulated data set illustrates a
fairly wide range of covariate behavior.
Using these covariates as
representative of the sort of covariates that would be encountered in
actual practice, the estimates of the treatment main effect vector are
within sampling error of the correct values using model (2), while
model (1)'s estimates are clearly biased.
Thus, we conclude that
"including initial value appropriately accounts for RTME when
assessing treatment effect in the Chapel Hill Behavioral Study".
In order to develop the simulation example, suppose that V
follows the EIV model, i.e. Vjk-VjkT+Ejk' where
139
(a)
in the absence of treatment, the true value vector (i.e.
(YIIT'YI2T)) remains unchanged over time;
(b)
(c)
the error terms Elk and E2k are uncorrelated over time;
true values and error terms are uncorrelated;
(d)
at each time point, the correlation between the two components of
(e)
the true value vector (i.e. YIlT and YI2T ) 'is PT;
at each time point, the correlation between the two components of
the error vector (i.e. Ejl and Ej2) is PE;
(f)
each of the error trems is normally distributed with mean 0 and
.
2
varlance 0Ek;
(g)
across individuals, the distribution of true values is normal
with mean ~lkTM and variance 0ikT for males and with mean ~lkTF
and variance oi~T for females; and
(h)
specifically, ~IITM=110, ~IITF=117, ~12TM-8, ~12TF=9, 011T=100,
222
012T=2, 0El=100, 0E2=2, PT=·8, and PE-· 5 .
In addition to the above, suppose that age, which is coded as an
indicator variable for simplicity, and gender values are randomly and
independently assigned to individuals (e.g. the probability of having
age=O and gender=O is .25).
The treatment is postulated to reduce
true values at t2 by 5 units for Yj1 and by .7 units for Yj2'
Finally, assume that the participant selection rule amounts to
dropping all children with Y11 <127, placing all children with Yll~135
into the treatment group, and randomly assigning (with probability
1/2) all children with 127<Y<135 into either treatment or control
groups.
Thus, we assume that the physician bases his decision on the
child's behavior only (i.e. Yl1 ) and not upon the parent's perception
of the child's behavior (i.e. YI2 ). All extremely disturbed (i.e.
140
Y11~135)
children are recommended for treatment, while moderately
disturbed (i.e. 1272Yll<135) children might or might not be
recommended for treatment.
Those children with little or no
disturbance (i.e. Y11 <127) are never recommended for treatment and are
not of interest. We assume that 25000 children are screened, which is
considerably more than are screened in the Chapel Hill Pediatrics
Behavioral Study.
The large size of this simulation example is
intended to illustrate the RTM effects on the MANCOVA that will be
applied to the Chapel Hill Pediatrics Behavioral data with a high
degree of precision.
Applying the above assumptions, 4409 (i.e. 17.6%) of the
screened children have
Yll~127,
and are thus included in the study.
Of these 4409 children, 3126 are assigned to treatment and 1283 are
assigned to the control group.
-~
-,
The group means at tl are
-,
-~
(Y 11 T'Y 12 T)=(136.77,10.72) and (Y 11C 'YI2C)-(130.69,10.19),
respectively. As expected, the treatment group is more extreme than
is the control group at tl'
In order to illustrate that including initial value adequately
accounts for the effects of RTME on the MANCOVA assessment of
treatment effect in this design, models (1) and (2) were fit to the
simulated data set.
The results are presented in table 2 below.
For
model (2), the estimates of the treatment main effect vector are
within sampling error of the correct values of (-5,-.7), while model
(1)'s estimates are clearly biased.
Note that age is not
significantly correlated with either component of 0, while, on the
other hand, gender is (these data are not illustrated here).
In
addition, gender is positively correlated with treatment status (these
141
data are also not shown).
Thus, the supposition that "including
initial value appropriately accounts for RTME when assessing treatment
effects" has been illustrated in a data set that contains, in addition
to initial values and an indicator of treatment status, a set of
control variables, some of which are statistically significant and
some of which are not.
In addition, these control variables had
various correlations with the indicator of treatment status.
Thus, it
is natural to assume that "including initial value appropriately
accounts for RTME when assessing treatment effects" regardless of the
nature of the covariates (i.e. regardless of whether the covariates
are continuous or discrete and also regardless of their correlations
with either response or treatment status) and also regardless of
whether the response is univariate or multivariate.
Table 2
Model-fitting results
Intercept
Age
Gender
Y1
Y2
Treatment
Component 1
Model 1
est. (s.e.)
Component 2
Model 1
est. (s.e.)
-9.819 (.476)
.133 (.378)
2.706 (.406)
-.757 (.075)
.021 (.059)
.186 (.064)
.
-8.081 (.416)
.
- .885 ( .065)
Component 1
Model 2
est. (s.e.)
53.960
-.757
3.139
-.544
.700
-5.164
(3.993)
(.075)
( .397)
(.032)
(.112)
(.443)
Component 2
Model 2
est. (s.e.)
1.958
.034
.390
.019
-.529
-.730
(.576)
( .053)
(.057)
(.005)
(.016)
( .064)
The Chapel Hill Pediatrics Behavioral Study example continues
with an assessment of the magnitude of RTME and misclassification in a
single group design.
142
(6B).
Examples:
1.
Example 1 - Chapel Hill Pediatrics Behavior Study:
£.
RTM in a single-group truncated design:
Continuing to discuss the Chapel Hill Pediatrics Behavior Study,
now suppose that only one group is available, and that the primary
goal is to estimate the amount of RTME that is -likely to occur in this
group between t 1 and t 2. One might want to obtain this expected RTME
either:
(a)
when a treatment intervention occurs between tl and t2' the
estimated RTME is required in order to apply the "external"
method of parameter estimation.
That is, an unbiased estimator
of the treatment effect vector is obtained by subtracting the
expected RTME from the observed mean difference score; or
(b)
even when no treatment is applied between tl and t 2, it is often
of interest to predict how much a sample that is chosen because
of extreme values at tl will regress (i.e. RTME) by t2'
For
this application, interest might also center upon predicted
"misclassification", that is predicting what percentage of
individuals that exceed a clinical cut-off point at tl will no
longer exceed the cut-off point at t2'
When the design
specifies that study participants must exceed the clinical
cut-off at both tl and t 2, then this analysis of "RTM-induced
misclassification" also amounts to predicting the amount of
attrition that will be caused by RTM.
The 3126 children from the treatment group in the previous
section comprise the group of interest here, and it is now assumed
that the study design includes three, rather than two, time points.
~
143
In particular, t1 is used for participant selection only, t z is used
for the baseline measurement, and t 3 is used for the final
measurement. Response is calculated as Y3-Y z. The primary concern is
how extreme the sample is likely to be at t z , recognizing that RTM
will likely cause the sample to be less extreme at tz than at t 1. Two
aspects of this phenomenon are considered, namely the change in group
means and the degree of RTM-induced misclassification.
Finally, some
methods for obtaining the inputs to this analysis are described.
To begin, one must specify three things in order to predict RTME
in a two-point vector design (note that here, only the two time points
t l and t2 are of interest), namely:
(a)
the mean vector;
(b)
the variance-covariance matrix of the multivariate normal vector
(c)
of observed values (YI,Y Z); and
the truncation rule.
Then, the above information is utilized (i.e. either via simulation or
through the truncated multivariate normal formulae (34) in order to
obtain the estimated RTME.
Usually, specification of the truncation
rule is straightforward, although such is not the case here.
In
particular, since the present truncation rule (which was obtained by
translating the physician's decision criteria into a rule that is
based on YII only) includes an element of randomized allocation, the
truncated normal formulae do not apply directly and so we proceed by
simulation.
The parameter estimates from the previous section (e.g.
0T=.8) are used in this example as well.
A discussion about some
possible ways to obtain the parameter estimates in (a) and (b) above
follows later.
144
Table IA below illustrates the RTME that is expected for this
design.
Both components are expected to regress, that is component I
is expected to drop by about II units between t l and t 2, while
component 2 is expected to drop by almost one unit. This exemplifies
the results of chapter 3, which state that, regardless of the values
of PI and PE' component I is expected to evidence RTME by virtue of
being included in the truncation rule.
Component 2, which is not
included in the participant selection criteria, is expected to
evidence RTME only because PE-O.
Table IB below illustrates the magnitude of misclassification
that is caused by RTME.
Overall, only 1439 (46.0%) of the 3126
children that were extreme at tl (i.e.
Yll~127)
are also extreme (i.e.
'21>127) at t2' Even under a relaxed definition of extremity at t 2
(i.e. Y21~127 or Y22~11), only 1674 (53.6%) of the children are
extreme at t2'
Thus, RTM ;s expected to induce considerable
misclassification in the Chapel Hill Pediatrics Behavior Study data
set.
Table lA
Group means
mean
(s.e.)
Yl1 136.77 (.13)
Y12 10.72 (.03)
Y21 125.81 (.23)
Y22 9.92 (.04)
145
Table 18
Classification results
time t 1
n percent
1361 ( 43.5%)
1765 ( 56.5%)
o ( 0.0%)
o ( 0.0%)
3126 (100.0%)
3126 (100.0%)
time t2
n percent
682
757
235
1452
1439
1674
(21.8%)
(24.2%)
( 7.5%)
(46.4%)
(46.0%)
(53.6%)
The above analysis of the effects of RTM on the one-group
truncated design required information about the underlying (i.e.
nontruncated) population parameters in order to proceed.
A discussion
of some ways to obtain the required parameter estimates follows:
When it is applicable, the most direct method of obtaining the
required estimates of the first two moments of the nontruncated
population is by simply observing the population (i.e. in the
absence of any treatment intervention) at two distinct time
points.
For example, Davis (10) does this in a two-point scalar
design.
Where possible, it is advantageous, both in terms of
cost and time, to obtain estimates from the published
literature.
Unfortunately, in the literature Yj often
represents not a single observation but rather the mean, say of
nj observations, possibly taken over Vj visits.
Unless one's
proposed design exactly matches a design that has been reported
in the literature, using the observed moments of (Y 1'Y2) will
not suffice, because the variance-covariance matrix of (Y 1'Y2)
depends upon how each Yj is generated (i.e. how many
observations nj are taken at each visit Vj).
146
When the EIV model holds, many researchers estimate the variance
components (i.e. a~, a~, PT' and PE', then combine these
variance components estimates into predicted moments of the
observed (Y I , Y2). For example, in the two-point scalar case,
2
2
assuming We=O and estimating ~T' aT' and aE allows the
researcher to specify that
2 2 2 2 2
2 2
(Y1'Y2)-N(~T'~T,aT+aE,aT+aE,aT/(aT+aE))'
Variance components
estimates require repeated observations on at least some
individuals at each time point.
for example
Yab=~+aa+8b
Considering any component, if,
then the variance components are given
by the well-known formulae (e.g. Rosner (29)),
~=LL(Yab-Ya)2/A(B-I) and
a~max(O,L(YA-~)2/A-LL(Yab-Ya)2/AB(B-I).
Given a set of
repeated observations on the same individual, the correlations
or
and PE may be similarly estimated (e.g. Cochran (6)).
Even
in the absence of any guidance from the literature, in some
cases the researcher may be willing to specify some of the
variance components based upon a prior considerations.
In
Jarticular, when Ej represents measurement error, then it is
sometimes biologically reasonable to assume that PE is
approximately zero.
Recall that, under truncated sampling, when
PE-O then RTME is only expected for those components which are
part of the truncation rule, even when all components are
extreme (i.e.
YIk>~lk)
at tl (see chapter 3).
Thus, when PE=O
the consideration of RTME may be limited to the relatively few
components that are used in selecting extreme individuals.
~
147
To recapitulate, RTME effects were described for a two-point
vector design utilizing truncated sampling.
These effects are of
independent interest in:
(a)
uncontrolled designs (i.e. in order to apply the external method
of RTME adjustment); and
(b)
designs that separate participant selection and baseline.
In a design with treatment and control groups, including initial value
as a covariate sufficed to avoid a RTME-induced bias in an estimator
of the treatment main effect.
Chapter 6 continues with an example of a regression analysis
when many variables are subject to RTM.
148
(68).
Examples:
l.
Example 2 - Edgecombe County Hypertension study:
~.
Introduction:
One component of the Edgecombe County Hypertension Study (19)
involves assessing the determinants of blood pressure change in
Edgecombe County blacks.
The response variable i's change in diastolic
blood pressure (i.e. D8P 2-DBP 1). Predictor variables include
sociodemographic characteristics (e.g. age), psychosocial measures
(e.g. an anger score), and environmental factors (e.g. diet).
Both
the response variable and some of the predictor variables are measured
with error and are, thus, subject to RTM.
Section (b) illustrates
both:
(1)
some possible effects of RTM upon a regression analysis using
~
this data set; and
(2)
Fuller's (12) method of accounting for these RTM effects by
adjusting the parameter estimates of the regression model.
Recall that, apart from adjusting the appropriate parameter estimates,
the analysis strategy recommended in chapter 4 also includes applying
substantive criteria in order to determine how variables should be
entered into the above adjustment procedure (i.e. for variables that
are measured with error, it must determined whether "true values or
observed values are more important").
Section (c) illustrates making
this determination for some of the variables in the Edgecombe County
Hypertension Study, whose design and research hypotheses are greatly
simplified for the purposes of the present discussion.
Section (b) now continues with an illustration of accounting for
the effects of RTM upon a regression analysis.
149
(68).
Examples:
£. Example 2 - Edgecombe County Hypertension study:
Q.
Accounting for RTM in a regression analysis:
Recall that the Chapel Hill Pediatrics Behavioral Study (see
example 1) illustrates a regression analysis where the response
variable variable is subject to RTM while the predictor variables are
not.
The present example expands this discussion to include the case
where both predictor and response variables are subject to RTM, and
not just the response variable only.
As is easily demonstrated, when
both predictor and response variables are subject to RTM merely
including initial value as a covariate does not suffice to account for
the effects of RTM, and more specialized techniques must be applied
instead.
In particular, Fuller's (12) method of adjusting the
observed slope coefficients is described.
For simplicity, we assume
that the response variable Y and the three predictor variables, which
have the generic labels Xl' X2 and X3, are all subject to RTM. Note
that Fuller's method may also be applied to data sets where some,
rather than all, variables are subject to RTM, as described below.
Also note that, in practice, Fuller's method also requires adjusting
the standard errors of the observed slope coefficients (i.e. and not
just the point estimates), which is not illustrated here.
To begin, suppose that each of Y, Xl' X2 and X3 are modeled as a
sum of a true value and an error. For example, Y=YT+EYT' with
YT-N(O'alO) and EYT-N(O,a~o)'
a€o=
a~l= a~2= a~3=1/9.
Let aYo=aYl=aY2=al3=1, and let
Then, each Y, Xl' X2 and X3 have Gaussian
distributions with mean zero and variance 10/9. The ratio of error
150
variance to total variance is .1 for each variable.
Finally, suppose
that the correlation matrix of the
=
[
.~
.8
.7
the correlation matrix of the errors is
.9
1
.8
.6
.7
.5
.6
.5
1
.5
.5
1
o1
01 00 0]
0
0
0
1
[0 0 0 01
and that errors and true values are independent.
Two hundred individuals are simulated from the above
superpopulation.
For these individuals, the unobservable regression
A
of Y on XT (i.e. Y on true values) is Y•. 594XIT+.348X2T+.222X3T'
Instead of the above, the observed regression of Y onX
observed values) is Y=.552XI+.3IIX2+.238X3'
(i.e. Y on
Note that this regression
is similar to the regression of Y on XT (i.e. because of the moderate
magnitude of the error terms), and that the coefficients of Xl and X2
are dampened. The coefficient of X3 increases slightly, which may be
an artifact of chance or may, instead, illustrate Cochran's (6) claim
that observed regression coefficients can behave unpredictably when
more than one variable is measured with error.
In any event, let us
describe Fuller's adjustment procedure and examine its effects on this
data set.
For notation, let k=(one plus the number of predictors), n-the
number of observations, D=diag(crXl,crX2,crX3)' E-diag(Al,A2,A3)' where
Aj denotes the ratio of error variance to total variance for predictor
Xj' and let A-OED.
Note that Aj equals zero for any variable that is
measured without error.
Also note that E need not be diagonal, so
~
151
that this method applies equally well to situations with correlated
errors among the predictors.
The adjustment proceeds by replacing the usual least-squares
B=[n1x~Xj1[nX~Y] by s=[n1x~X - (1-k/n)Aj1[n1X~Y] , so long as the
matrix [n1X~X_ (l-k/n)A] is invertible and well-conditioned.
In the present application, n=200, k=4,
D=diag(1.024,1.039,1.014), E-diag(.l,.l,.l) ,
[n1X~X]= [1.044
.601
.481
.601 .481]
1.075 .409
.409 1.023
and so 8 -(.606,.301,.227).
, [n1X~Y]=
.878]
.764
[ .637
That is, the adjusted regression is
Y=.606X 1+.302X 2+.227X 3, which is closer to the actual regression of Y
on XT than was the observed regression of Y on X.
Note that, in
~
essence, this method operates by reducing the diagonal elements of X X
(i.e. the variance estimates of the observed predictors) to estimates
of the variances of the XTs.
Now, the above example illustrates a case where the recommended
adjustment procedure performs well.
That is, on average the adjusted
slope coefficients are both close to the actual coefficients and also
closer to the actual coefficients than are the observed coefficients.
2
Now consider exactly the same situation as before, except that 0E2=9
instead of 1/9, so that A2=.9 instead of .1.
now measured with extreme error.
That is, component 2 is
Following the same process as
before, the true, observed, and adjusted regressions are
true:
Y=.594X1T+.348X2T+.222X3T
A
observed:
Y-.316X1+.041X2+.465X3
adjusted:
Y=.304X 1+.391X 2+.342X 3
A
152
Comparing the true and observed regressions, note that including a
predictor that is measured with extreme error in the regression model
(i.e X2) causes all of the regression coefficients to be severely
affected, not just the component of the variable that is measured with
extreme error.
Also note that while the adjustment procedure did
produce an improved estimate of the regression of' Y on X, the
estimated slope for component 1, which is not the variable that is
measured with extreme error, is still qUite far from its actual value.
Although not illustrated here, further simulation indicated that the
results of the adjustment procedure are also very sensitive to the
specification of E and 0 when a predictor is measured with extreme
error, which suggests that Fuller's method may not be robust when some
predictors are measured with extreme error.
To recapitulate the discussion of Fuller's adjustment method,
these simulations indicate that RTM affects observed slope
coefficients, usually to dampen theffi but not always.
Such effects can
be considerable when even one regressor is measured with extreme
error.
In this case, slope coefficients of other variables may be
seriously affected as well. Adjusted slope coefficients tend to be
closer to the actual regression coefficients than do the observed
slope coefficients, but it is worrisome that the fit is not better
when variables are measured with extreme error.
In this case, one
cannot even be confident about the point estimates of the coefficients
of the variables that are not measured with extreme error.
Thus, it
seems likely that Kupper's (21) assertion that covariables that are
measured with extreme error are most dangerous because of their
unpredictable effects on other coefficients might well be strengthened
~
153
by adding "and, in addition, because they short-circuit the usual
adjustment strategy".
Note that Fuller's adjustment method is based
upon asymptotic theory, and so its application in small to moderate
sized samples remains open to investigation.
Applying the above results to the Edgecombe County Hypertension
Study, two implications are to drop any variables-that are measured
with extreme error and to apply Fuller's adjustment method to the
remaining regression.
Note that a preliminary step is required in
order to apply this adjustment method, namely to separate the
regressor variables into those that are measured without error (i.e.
no RTM and Aj=O), those that are measured with error but for which
observed values are more important than true values (i.e. for these
variables, we retain the observed values and set Aj=O), and variables
that are measured with error for which the true values are of more
substantive importance than observed values.
Section (c) illustrates
this determination for some of the variables in the Edgecombe County
Hypertension Study.
154
(68).
Examples:
z.
Example 2 - Edgecombe County Hypertension study:
£.
Conceptual issues involved with interpreting variables:
Recall that the previous example assumes that the EIV model
holds for each variable (e.g. Xj=XTj+Ej' where Ej denotes the random
error term).
Note that when a variable is measured without error then
a~j=o. Also note that Ej need not necessarily represent measurement
error, and thus the EIV model is more generally applicable than it
might first appear.
For example, in Galton's (14) analysis of the
regression of son's height upon father's height, the error term
represents the combined effects of a large number of random, but real,
genetic factors.
Similarly, Ej might also represent a "random shock",
that is a real but transient effect, an example of which is the number
of hours of sleep in the night before a test of alertness.
2
that aEj>O.
Suppose
The goal of this section is to discuss when the
researcher might wish to explicitly account for the RTM caused by
a~j>O (i.e. by setting Aj.a~j/(a~j+atj) in Fuller's method), that is
to estimate the regression coefficient of XT rather than the
regression coefficient of X, and when, alternatively, he might wish to
ignore this RTM and direct inference toward observed, rather than
true, values.
Note that, throughout this section, it is assumed that
the magnitude of the regression coefficients is of importance, and
that the analytical goal is not simply prediction.
One might as well
use observed values when the only goal is prediction.
To begin, consider the case where Ej represents measurement
error, for example final-digit preference in blood pressure
measurements.
Here, true values are usually more important that
~
155
observed values since, for example, a scientist is more likely to be
interested in the biological implications of one's actual cholesterol
level than in some erroneous measurement of the same.
An exception
might be the case where the research question specifically pertains to
observed variables, for example "can a cheap but error-prone clinical
measurement of cholesterol be used to predict lat'er heart disease?"
(i.e. rather than, "does cholesterol level predict later heart
disease?").
When observed values are retained, note that regression
coefficients are not strictly comparable across studies with different
2
0EjS
(for example, when X denotes the mean of a number of measurements
then varying this number affects
oi j ),
although this might not make
much of a difference unless some variables are measured with
considerable error.
Now suppose that Ej represents within-person variability.
For
example, both a person's daily caloric intake and nightly hours of
sleep tend to vary, often considerably.
Here, the primary question is
which quantity is more important: the individual's long-term mean
va1ue or the present val ue.
For example, daily caloric intake is
usually intended to be a measure
i nd i vidual' s mean cal ori c intake.
(albiet a poor one)
of the
On the other hand, the number of
hours of sleep on the night before taking a test of alertness (i.e.
the present val ue) is probably of more importance than the long-term
mean, since "sleepy is sleepy" regardless of one's normal habits.
Finally, in an example such as Galton's where the Ejs represent
real but random effects, it is more likely that observed values are of
interest.
That is, a person's actual genetic composition is what
156
affects response, regardless of what composition is expected
considering one's set of parents.
To apply the above approach to some variables from the Edgecombe
County Hypertension study, age and education are clearly measured
without error (at least assuming that respondents are knowledgeable
and forthright).
For EIV variables, true values are probably always
more important than observed values.
For example, this holds true for
systolic blood pressure because Ej represents measurement error.
In
addition, this holds true for variables that are subject to
within-person variablility (e.g. dietary variables, anger score, and
so on), because individuals' long-term mean values are the focus of
inference.
Note that in some analyses variables such as the anger
score are dichotomized, in which case regression methods when
variables are subject to misclassification, rather than Fuller's
adjustment method, should be applied.
Chapter 6 now continues with an example of unrelenting RTM in a
longitudinal design.
157
(68).
Examples:
~.
Example 3 - Accounting for RTM in a longitudinal scalar
design:
The simulation example in section B of chapter 5 illustrates the
application of an analysis strategy to a longitudinal data set that
follows the EIV model and, thus, is subject to RTM.
To recapitulate,
the steps in the recommended analysis strategy are:
(1)
ignoring any treatment effects, assess whether RTM (i.e. RTMG) is
present by plotting ETPs for individuals with different values
of VI (recall that the ETP plots expected response, in the
absence of treatment, against time);
(2)
from the above plots, attempt to obtain a set of time points for
which the effects of RTM are minimized or eliminated;
(3)
for this set of time points, attempt to find an unbiased
estimator of the treatment effect parameter; and
(4)
if steps 2 and 3 are not fully successful, attempt to account for
RTM by including initial value in the model of response.
As illustrated in chapter 5, this analysis strategy works well for the
EIV model, in part because the effects of RTM conclude by t2.
Note
that one of the implications is that the set of time points in step 2
above is very large, which in turn suggests that an unbiased estimator
of the treatment effect parameter can usually be found in step 3.
The present example is intended to complement the example in
section 5B.
transient.
Here, the effects of RTM are unrelenting instead of
As a consequence, step 2 above cannot be fully
accomplished, and thus step 4 must be attempted.
That is, the form of
the data analysis differs from that suggested for the EIV model.
158
Taken together, the overall goal of these two examples is neither to
exhaust all of the ways by which RTM may be manifested in longitudinal
designs, nor to illustrate all of the statistical techniques that may
be applied to longitudinal data sets that are subject to RTM.
Rather,
the goals are:
(a)
to illustrate two typical manifestations of'RTM in longitudinal
designs; and also
(b)
to illustrate that the general strategy developed in chapter 5
may be applied in qUite dissimilar applications.
In particular,
it is noteworthy to elucidate some of the differences in the
implementation of the same overall approach caused by the
presence of unrelenting, as opposed to transient, RTM.
In the same spirit, the following data analyses are not recommended as
being either complete or optimal, but rather are proposed as an
illustration of simple methods of accounting for RTM in this
particular application.
To begin, let Vj' j=I,2, ... ,J , denote the standardized
diastolic blood pressure for an individual at time tj (i.e.
standardized to have mean zero).
Assume that Vj follows the
autoregressive model Vj -8V j _1+E j , j-I,2 ... ,J , 0<8<1, where VO-N(O,I),
each Ej-N(O,I), with YO' E1, ... ,f , all independent. Here, e
J
represents the effects of a "biological regulatory mechanism" that
serves to dampen extreme OBPs toward the mean, while Ej represents a
real but stochastic effect (e.g. stress level at tj).
An uncontrolled
truncated design, which tends to exacerbate any problems caused by
RTM, is utilized.
The researcher wishes to account for RTM in the
159
data analysis, in particular obtaining an unbiased estimator of the
treatment effect parameter that accounts for the effects of RTM.
In order to develop an appropriate estimator of the treatment
effect, let us apply the recommended analysis strategy.
the ETPs for individuals in the absence of treatment.
Step 1 plots
For any
individual with fixed VI' say VI=ka (where k indexes the number of
standard deviations by which VI exceeds the mean of zero) the expected
response is (ka,eka,e 2ka,e 3ka, ... ). Figure 1 below plots expected
response against time for individuals with various values of VI.
Since these ETPs become closer together at every pair of time points,
RTM is unrelenting.
Time.
Figure I - ETPs
Step 2 in the recommended analysis strategy involves examining
the plots in figure 1 above in order to obtain a set of time points
for which the effects of RTM are reduced or eliminated.
Since RTM is
unrelenting, however, the effects of RTM are never eliminated.
Thus,
step 3, which is usually easy to accomplish in the presence of
transient RTM (see chapter 5) is bypassed here in favor of step 4.
That is, initial value must somehow be included in the model of
response in order to account for RTM.
In other words, the RTM
parameter e must be estimated "internally".
not available for internally estimating
Since a control group is
e, then, at least two time
points must be observed prior to treatment in order to estimate e and
160
at least one time point must be observed after treatment in order to
estimate the treatment effect.
..
For example, suppose that the design includes J=3 time points.
Let the treatment be administered immediately after t 2. Then, e may
be estimated by taking all of the points (Y I (i)'Y 2(i)) (i.e. one such
point per individual) and regressing Y2 on YI . The intercept of this
regression should be zero with the slope being e. Treatment effects
are estimated by comparing this regression with a similar regression
of Y3 on Y2' For example, when the treatment acts to reduce each
person's DBP by T units above and beyond the effects of e, then the
intercept of this latter regression should be
-T
(with slope e).
Similarly, when the treatment acts to reinforce the automatic
regulatory mechanism by changing e to
e*=e~(O<~<I),
then the
regression of Y3 on Y2 should have a slope of e* instead of e.
Table 3 below illustrates the above method of estimating the
treatment effect.
25000 individuals are simulated from the previously
described superpopulation, with all 2330 persons with YI >I.5 being
retained. The group means are YI =2.00, Y2-0.99, and Y3=0.50, which
_.#
-'
-"'"
illustrate the unrelenting RTM among persons that are extreme at t l .
In the absence of treatment, the intercept estimates are about zero
and the slope estimates are about .50, so that the RTM parameter e is
adequately estimated.
Similarly, the estimates of the treatment
effect parameters are near the simulated values of T=-.7 (i.e. since
the intercept in column 3 is near -.7) and
in column 4 is near .50*.50=.25).
~-.50
(i.e. since the slope
~
161
Table 3
Parameter estimates
Y{ on Y2
Y~ on YI
(no reatment) (no reatment)
Intercept
Slope
-.022 (.097)
.507 (.'047)
.026 (.029)
.479 (.020)
Y3 on Y2
(1=-.7)
Y3 on Y2
(E,;=.50)
-.674 (.029)
.479 (.020)
.026 (.029)
.229 (.020)
Now suppose that many time points are available.
Then, these
additional time points allow model parameters to be estimated within,
rather than across, individuals.
For example, if the treatment is
administered immediately after tQ(I<Q<J), then two regressions may be
calculated for each individual, the first using the data points
(Yk(i)'Yk+I(i)) k=I,
,Q-I and the second using the data points
(Yk(i)'Yk+I(i)) k=Q,
,J-I.
Alternatively, more specialized
techniques for dealing with autoregressive models may be used to
estimate these parameters.
In any event, estimates of treatment
effect (e.g. 8(i)/8(i)) may be calculated within individual and then
averaged.
Estimation of treatment effects within individuals, which
is also illustrated in section 58, is usually superior whenever either
the autoregressive parameter 8 or else the treatment parameter (;,
say) differs markedly from individual to individual.
Note that this
approach, which can, in theory, be implemented with as few as three
time points (e.g. here, Y2(i)/Y I (i) and Y3(i)/Y2(i) could provide
estimates of 8 and 8* respectively), generally becomes preferable as
the number of time points increases from three to "many".
Where the
cutoff point lies depends upon the model employed, the magnitude of
individual differences, the degree of assurance that the data analyst
has in the form of the model, and so on.
162
To summarize, in the autoregressive models described above, RTM,
which is caused by a biological mechanism that acts upon extreme
values, is expected between every pair of time points .. Even though
this unrelenting RTM is in contrast to the transient RTM observed when
selecting extreme individuals under the EIV model, the recommended
analysis strategy was still applied successfully "(i.e. by skipping to
step 4).
The basic analytical decision, which primarily depends upon
the number of available time points, requires choosing between a
"within-individual" analysis and a "between-individual" analysis,
either of which can estimate both RTM effects and treatment effects.
Chapter 7 continues with a summary and conclusions.
Chapter 7
Summary and final remarks
(7A).
Introduction:
The dissertation now concludes with a nontechnical summary and
some final remarks.
In order to make this chapter relatively
self-contained, the notation and terminology that are used here is
reviewed below.
Yjk' j=I,2, ... ,J , k=I, ... ,K , denotes the observed values,
where j indexes time and k indexes component.
The component k is
suppressed for scalar designs.
The EIV model has Yjk=YjkT+Ejk' where Yjk denotes the observed
value, YjkT denotes the true value, and Ejk denotes the random error.
Under the EIV model, PE denotes the correlation between the
components of the error term (e.g. Ell and EI2 ), and PT denotes the
correlation between the components of the true value (e.g. YIlT and
YI2T )·
8k denotes the slope of the regression of Y2k on Ylk.
Dk denotes the difference score Y2k-Ylk.
A truncated design only chooses individuals with YI above a
certain cutoff point (denoted by AI). Parameters and statistics from
truncated designs are denoted by superscripted primes.
k* is a function of the standardized cut-off point in truncated
* namely
sampling (i.e. AI)'
k*=~(AI*)/{I-¢(AI*)}.
q*=k*(AI*-k*).
164
RTM (regression to the mean) is a tendency for variables
that are observed to be extreme to become less extreme upon
remeasurement.
RTM has two manifestations, namely RTMG and
RTME, of which RTMG is the more fundamental.
RTMG, which is a tendency for extreme individuals to regress to
the group mean (i.e. to become less extreme upon "remeasurement) ,
occurs when 0<8<1.
RTME is a tendency for extreme groups to have mean values that
regress.
Equivalently, RTME is a negative bias in Y2- Y1 as an
estimator of
~2-~1
that is caused by both RTMG and the presence of
extreme individuals at t 1.
ETP (expected time path) plots, for an individual,
(Y1'~2.1"'~J.I)
on the y-axis versus time on the x-axis.
Here, it is
assumed that YI has already been observed.
In longitudinal designs, the response function f(Ya, .... ,Y b)
a~l
, b<J, a<b, is a scalar function that is used to assess, for
example, response to treatment.
.-ok --It
In uncontrolled truncated designs, (Y 2-YI) denotes the RTME that
is expected even in the absence of treatment.
ETPk, RTMGk, RTMEk, and 8k denote the ETP, RTMG, RTME, and 8 for
component k.
165
(7B).
Summary and conclusions:
This dissertation explicitly considers RTM in two-point scalar,
two-point vector, and longitudinal scalar designs.
Since the proposed
definition of RTM in vector designs amounts to repeatedly applying the
scalar definition of RTM, then the additional topic of RTM in
longitudinal vector designs is also treated by implication.
For each
design, the coverage begins with a definition of RTM, followed by a
description of the effects of RTM upon extreme samples, followed by a
discussion of the effects that RTM may have on statistical procedures,
finally concluding with a proposed analysis strategy to follow when
data are subject to RTM.
Note that most of the present RTM literature
treats the two-point scalar case, and that the definition and analysis
of RTM in two-point vector and longitudinal designs is, in the main,
an original extension of this literature.
For each study design, a definition of RTM that satisfies Davis'
(IO) "RTM principle", namely that "RTM is the phrase used to identify
the phenomenon that a variable that is extreme on its first
measurement will tend to be closer to the center of the distribution
for a later measurement", is proposed.
Note that the definitions of
RTM in the more complicated designs reduce to the basic definition of
RTM in the two-point scalar design as a special case.
For example,
the proposed definition of RTM in the two-point vector design is
equivalent to that of RTM in the two-point scalar design when the
number of components equals one instead of k.
Two definitions of RTM in two-point scalar designs are proposed,
depending upon whether the "variables" in Davis' RTM principle refers
to individual measurements or to group means.
RTM of individual
166
measurements, denoted by RTMG (for RTM in the sense of Galton), is
'"
defined to occur whenever 0<6<1.
One direct implication of RTMG is
that the difference score Y2- Yl is negatively correlated with the
initial value Y1. RTM of group means, denoted by RTME (for RTM of
expected values), is defined to occur whenever Y2-Yl is a negatively
biased estimator of u2- uI' where this bias is caused by both RTMG and
the presence of extreme values at tl'
The RTM literature seldom makes
an explicit distinction between RTMG and RTME, although it is often
heuristically useful to do so.
In two-point vector designs, RTMG and RTME are defined to occur
whenever, respectively, RTMGk and RTMEk occur for any component k.
This "component by component" definition of RTM is preferable to a
"multivariate" definition of RTM, since any multivariate definition of
RTM can lead to inconsistencies.
For example, a multivariate
extension of Davis' RTM principle might state that individuals that
are extreme, in the sense of multivariate distance, at t 1, tend to be
less extreme, in the sense of multivariate distance, at t2' An
example of the inconsistencies that this alternative definition allows
"
occurs when some components regress to the mean (i.e. O<8k<I)
while
A
other components egress from the mean (i.e. 6k>I), leaving no overall
difference between the multivariate distances (YI-Ul) and (Y2-u2) and,
thus, no RTM.
For longitudinal scalar designs (and thus, by implication,
longitudinal vector designs as well), only RTMG is defined.
This is
because most statistical techniques for treating longitudinal data
obtain response functions at the level of the individual, then
amalgamate these results within groups.
Thus, even though the
167
ultimate focus is usually upon groups, RTM (i.e. RTMG) is accounted
for at the level of the individual since individual response functions
are used as the basic unit of analysis.
Note that for longitudinal
designs it is necessary to specify whether response functions are to
be initially calculated at the level of the individual and then
combined over group (as is assumed here) or whether, alternatively,
data points are to be combined into group means, say, before analysis.
This distinction is in contrast to the analysis of two-point designs,
where the "individual perspective" (i.e. RTMG) and the "group
perspective" (i.e. RTME) typically yield equivalent results.
For
example, in the two-point scalar design with a treatment and a control
group, the ANCaVA estimator of treatment main effect, which is a
function of group means that accounts for RTME, is equivalent to the
estimator that is obtained by adjusting each individual's difference
score for the expected effects of RTMG.
In longitudinal scalar
designs, individuals are defined to be subject to RTM from t a to tb'
denoted by RTMab, if and only if the response function f(Ya, .... 'Yb)
depends upon initial value in such a way that the ETPs of individuals
with varying values of YI become closer together in the interval from
t a to tb. Note that RTM is only assessed over the interval of time
points that are used in the response function, which might be a subset
of the time points that are actually observed.
After defining RTM, the effects of RTM upon single-group
truncated designs were considered.
When the group of interest is
merely followed over time and not subjected to a treatment
intervention, the primary application of these results is in the
two-point case.
Note that many designs (even those that include more
168
time points and/or a later treatment intervention) separate
participant selection (i .e. at t 1) and baseline measurement (i.e. at
t 2) and thus can be considered as a two-point design without treatment
intervention over (tl,t2)'
When participants are selected on the
basis of extreme values at t 1 (i.e. truncated sampling), interest
often centers on determining how extreme such ~ sample is likely to
remain at t2' that is after the initial effects of RTM have been
manifested.
In this application, a closely related problem is that of
"RTM-induced misclassification", that is in determining what
percentage of persons that exceed a cut-off point at tl are likely to
exceed the cut-off at t2'
When the single group receives a treatment
intervention (i.e. an uncontrolled truncated design), interest centers
on obtaining an estimator of treatment effect that accounts for the
effects of RTM.
For any fixed value of YI , note that treated
individuals are expected to manifest the same magnitude of RTM as
would untreated individuals.
In the two-point scalar case, the expected RTME for a truncated
,#
design is
(6-I)(VI-~I)'
This result follows from the fact that the
RTMG for any individual is
(6-I)(YI-~I)'
Before sampling, the
expected value of VI is ~+k*crI' so the expected RTME, before tI' is
A
6k*cri' Conveniently, E(6'#)-6, so that the observed slope coefficient,
even under truncated sampling, estimates the population regression
parameter.
Note that this result does not necessarily hold in the
two-point vector case.
For uncontrolled truncated designs, an unbiased (i.e. adjusted
-*-*
for RTME) estimator of the treatment main effect is (Y2-Yl)-(Y2-YI)'
-
-,#
where (Y-*2-Y-*I ) denotes the change in group means that would have been
4It
169
caused by RTME even in the absence of treatment intervention.
is obtained by inserting estimates of S and
above.
~1
into
-* -*
(Y2Yl)
(6-1)(Yl-~l)
These estimates may be either obtained externally (i.e. from
outside sources), in which case some partial checking of assumptions
is possible, or internally (i.e. based on observed data only), in
which case an additional assumption about the form of the treatment
effect is required.
Since a RTME adjustment is closely related to
estimating a treatment main effect, the internal method should only be
applied when an additive model holds.
Assuming additivity, the
'"
observed 8' is an unbiased estimator of 8, and the method of moments
may be used in order to obtain
the truncated
v; and ~;
~1
(and 01' a nuisance parameter) from
via the formula ~1=Y~-k*{s'i/(1+q*)}1/2.
In the two-point vector case, the effects of RTME upon extreme
groups (e.g. which are produced by truncated sampling) have not been
systematically described until now.
Assuming that the EIV model holds
for all components, the main result is that components that are part
of the truncation rule are always expected to manifest RTME, whereas
components that are not part of the truncation rule mayor not
manifest RTME.
In fact, a component that is not included in the
truncation rule may be extreme at t 1 (i.e. Y1k>ulk)' have RTMG both in
'"
the population (i.e. O<Sk<I) and in the data (i.e. O<Bk<l), and yet
not manifest RTME. For components that are not included in the
truncation rule, the primary determinant of the magnitude of RTME is
PE' although PT and the details of the truncation rule also playa
role.
In particular, when PE=O no RTME is expected.
The explanation
for this initially counterintuitve result is as follows: RTM only
occurs when extreme components also have positive error (i.e. E1k>O)
170
at t 1. That is, since E(E 2k )=O regardless of the observed Ylk' such
extreme components regress on average. Now, components that are part
of the truncation rule tend to have both extreme true values (i.e.
-~
Ylk>~lk)
and positive error (i.e. E1k >O), and, thus, RTME. When true
values among components are correlated, this correlation implies that
the true values of the other components are also extreme and, thus,
that observed values are extreme also.
requires positive error at ti'
However, RTM additionally
When PE>O, the positive error in the
components that are included in the truncation rule induces positive
error in all other components.
However, when PEzO, such is not the
case and RTME is not expected.
In uncontrolled truncated designs, the internal method of
parameter estimation can not be extended to adjust for RTME, in part
because of the proliferation of nuisance parameters in the vector
case.
Therefore, the external method, which is again partially
self-checking, is recommended.
In longitudinal designs, the primary focus of describing the
effects of RTM was upon whether the effects of RTM are transient (i.e.
in which case the ETPs are parallel for some set of time points) or
permanent (i.e. in which case the ETPs are not).
the effects of RTM are transient.
Under the EIV model,
That is, suppose that an individual
is chosen because of an extreme value at tI' say Y1-u+ka, where k
indexes the number of standard deviations by which YI exceeds the
population mean.
Then the ETP is
(~+kcr, ~+Bka, ~+Bko, ....
), where
At t I , an individual that is chosen to have YI=~+kcr is
most likely to have a moderately extreme true value (i.e. u+Bka) and
B=(df/(dt+or).
positive measurement error (i.e. (I-S)ko.
This positive measurement
•
171
error is not likely to be repeated, on average, so that the effects of
RTM end by t 2. Comparing the EIV model to the autoregressive model
Yj=u+e(Yj_l-~)+Ej,O<e<l, we find that the ETP of an individual with
YI=u+ka is (~+ka, ~+eka, ~+e2ka, ... ), and so unrelenting RTM occurs.
Transient and unrelenting RTM effects require somewhat different
analytical treatment, as is discussed later.
After describing the effects of RTM in one-group designs, the
effects of RTM upon statistical procedures were discussed, and
analysis strategies were recommended.
In two-point scalar designs,
the main question involves whether the usual ANCOVA appropriately
accounts for RTM when treatment and control group means differ at t l .
Now the ANCOVA model D=Bo+BIY+B2T "accounts for RTMG" by including
A
A
A
A
initial value as a covariate (recall that an equivalent way of
describing RTMG is to state that difference scores are negatively
correlated with initial value).
treatment effect, namely
~,
In addition, the ANCOVA estimator of
may be rewritten as a function of group
means that explicitly accounts for the RTME in each group.
However,
the usual ANCOVA model assumes that the covariate YI is measured
without error, which is seldom true in RTM applications.
We examined
A
the effects of violating this assumption, and found that 62 is biased
only when treatment and control groups are obtained from separate
populations with different means. Otherwise, the usual ANCOVA holds
without change.
Using the Chapel Hill Pediatric Behavioral Study (23)
as an example, we posited that this same relationship holds true in
the two-point vector case (i.e. for a MANCOVA with a multivariate
response factor and general covariates), and demonstrated the above by
example.
172
Probably the most practically important two-point vector case
concerns the effects of RTM in analyses where a number of variables
are subject to RTM.
This is an overly general topic, and we chose to
limit the discussion to RTM that is caused by EIV (generally speaking,
RTM is most often caused by EIV in practice) and also to regression
methods. Then, results form the literature on regression with errors
in variables (e.g. (3, 12, 20, 21, 35»
may be applied directly.
Noting some results of interest, slope coefficients of variables that
are measured with error tend to be dampened, as is also true in the
univariate case.
However, and most worrisome, is the fact that the
slope coefficients of variables that are measured without error may
also be affected in unpredictable ways, so long as these variables are
correlated with variables that are measured with error.
The
recommended policy is to perform a "sensitivity analysis", that is to
determine how much the results change when the fact that variables are
measured with error is explicitly taken into account.
This analysis
requires that the components of variance (i.e. the variances of the
true values and of the error terms) are known or can be estimated.
Clearly, the "contaminated" coefficients of those variables that are
not measured with error should be adjusted, while the researcher may
or may not choose to adjust the remaining coefficients. That is, the
researcher must decide whether the observed values or the true values
are of more fundamental scientific interest.
This analysis was
illustrated using the design of the Edgecombe County Hypertension
Study (19).
RTM can also affect the bias and precision of parameter
estimates obtained from longitudinal designs.
An analysis strategy
~
173
was proposed whereby the ETPs are first examined in order to determine
what set of time points, if any, eliminates or at least reduces the
effects of RTM. Once a set of time points is tentatively entertained,
the anticipated form of the treatment effect is analyzed in order to
determine whether or not an appropriate response function can be
obtained from this set of time points.
If
som~
RTM effects on
individuals remain, then Y is included in the model of response.
1
This analysis strategy applies equally well to forms of dependence
between initial value and response that are not manifested as RTM.
addition, the analysis strategy applies, albiet with some
modification, to both transient and unrelenting RTM.
In
174
(7C).
Directions for future research:
For general study situations, RTM has been defined and its
effects described.
However, normality was assumed throughout, and
these results might also be developed for the non-normal case.
The
effects of RTM (which are usually assumed to be caused by EIV) upon
regression models have been described.
These results might be
extended to other statistical techniques (e.g. logistic regression)
and also extended to include RTM that is not necessarily caused by
EIV.
In addition, the behavior of Fuller's adjustment procedure might
be examined for small to moderate sized samples with various
correlation structures, error variances, and so on.
The discussion of
RTM in longitudinal designs is somewhat speculative, in that RTM is
usually thought to be a two-point concept.
In particular, the
discussion of analysis strategies to pursue in longitudinal designs
that are subject to RTM might be fleshed out considerably.
Also, the
proposed definition of RTM in longitudinal designs applies to most,
but not all, longitudinal models.
Considering the above topics,
though, it appears that the approach used in this dissertation might
perform well in developing such extensions.
~
175
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5.
Clarke DB, Clarke AM and Brown RI
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176
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Hotelling H Review of "The triumph of mediocrity in business" by
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pp463-465 1933.
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James S Work in progress
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McMahan CA Regression toward the mean in a two-stage selection
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25.
Nesselrode JR, Stigler SM, and Balton PB Regression toward the
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Neter J and Wasserman W Applied linear statistical models
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Irwin Co.
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Rosner B The analysis of longitudinal data in epidemiologic
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178
Appendix 1
SAS code for simulation examples:
The overall logic of the SAS code, which is illustrated via a
portion of the code used to create the simulation example in appendix
3, is essentially identical for each simulation, namely:
(a)
through I repititions of a "do loop", where I is the sample size,
create the members of the sample;
(b)
within each pass of the "do loop";
(1)
take the next element from a stream of standard normal
variates, which are created through the pseudonormal
generator "NORMAL" (30);
(2)
transform the standard normal variates to normal variates
with an appropriate transformation.
In the present
example, the transformation consists of multiplying by the
standard deviation and adding the mean.
In the
simulations for chapter 3, various correlation matrices
were obtained by multiplying by the appropriate Cholesky
decomposition, obtained through the function "HALF" (30);
(3)
create new variables by adding "true values" and "error
terms", where appropriate;
(c)
where applicable (e.g. truncated designs), choose a subset of the
simulated sample; and
(d)
finally, run descriptive procedures (e.g. PROC UNIVARIATE and
PROC CORR (30) in order to check the above code.
The SAS code used to create table 28 of appendix 3 follows.
~
179
DATA TEMPI;
DO 1=1 TO 100000;
YIT=80+10*NORMAL(1);
El=10*NORMAL(2);
Y2T=YlT;
E2=10*NORMAL(3);
Y1=YlT+El;
Y2=YlT+E2;
GROUP=O;
OUTPUT;
DROP I;
END;
DATA TEMP2;
DO 1=1 TO 100000;
Y1T=100+lO*NORMAL(4);
E1=10*NORMAL(S);
Y2T=Y1T+2;
E2=10*NORMAL(6);
YI=YlT+El;
Y2=YlT+E2 j
GROUP-I;
OUTPUT;
DROP I;
END;
DATA TEMP3;
SET TEMPI;
IF Yl>100;
DATA TEMP4;
SET TEMP2;
IF YI>IOOj
DATA TEMPS;
SET TEMP2;
IF YI>120;
DATA TEMP6;
SET TEMPI TEMP2;
DATA TEMP7;
SET TEMP3 TEMP4;
DATA TEMP8;
SET TEMP3 TEMPS;
PROC MEANS DATA-TEMPI; PROC MEANS DATA-TEMP2; PROC MEANS DATA=TEMP3;
PROC MEANS DATA-TEMP4j PROC MEANS DATA-TEMPSj
PROC REG DATA-TEMP6j
MODEL Y2=YI GROUPj PROC REG DATA=TEMP7j
MODEL Y2=YI GROUPj PROC REG DATA-TEMP8j
MODEL Y2-YI GROUP;
180
Appendix Z
Truncated bivariate normal formulae:
Let (X,Y) denote a standardized bivariate normal random variable
(i.e. (X,Y)-N(O,O,I,I,p)) and truncated at (X=A,Y=B), that is retain
only those individuals for which X>A and Y>B.
Notes:
(a)
In chapter 2, (Y 1'Y Z) replaces (X,Y). Here, X and Yare used in
order to reduce the number of subscripts.
(b)
For most designs, truncation only occurs at time t l , which
represents the special case A=A, B=-oo.
Let,
PA,B=S(A,B)=I-~(A,B)=Pr{X>A,Y>B),
where
standard normal distribution function,
S(o).l-~(o),
the standard normal density function.
Also, let
~(Al)
~(o)denotes
the
and ¢(o) denotes
~(A,B,p)
and
denote bivariate and univariate normal distribution functions,
respectively.
For notational convenience, let
:\= (1_p2 )-1/2 ,
C=:\(B-fA), and
D=:\(A- ~) .
Weiler (36) shows that the first two moments of the standardized
bivariate normal (X,Y) truncated at (A*,B*) are
(1)
PA*,B*E{Y)-¢{B*)S(D)+P¢(A*)S(C)
(2)
PA*,B*E{X)=¢{A*)S{C)+P¢(B*)S(D)
(3)
(4)
PA*,B*E(y2)=B*¢(B*)S(D)+p2¢{A*)S{C)+p{l-p2)~{A*,B*,p)+PA*,B*
PA*, B*E (X 2)=A* ¢{A*)S (C )+p2 ¢(B*) S(D)+p( 1- p2) ~(A*, B*, p)+P A*, B*
(5)
PA*,B*E(XY)=p{A*¢(A*)S(C)+B*¢(B*)S(D)+PA*,B*(I-p2)~(A*,B*,p)}
~
181
When truncating at X=A* only, B*=-oo, and so
(6)
S(D)=O,¢(B*)=O,B*¢(B*)=O,S(C)=l,¢(A*,B*,p)=O
Thus,
(7 )
E(X) =¢( A*)/ S(A*) =k*, say
(8)
E(Y)=~(X)=pk*
E(X 2)=1+A*k*
(10) E(y 2)=I+p 2A*k*
(9)
(11) E(XY)=p(I+A*k*)
Letting q*=k*(A*-k*) for convenience, then
(12) V(X)=I+q*
(13) V(Y)=I+p2q*
(14) I-V(X)=E(X){E(X)-A*}
(15) 1- V( Y) =p2 (l- V(X) }
(16) p(X*, Y*) =p{ (l+q*)/ (1+ p2q*)} 1/2
Still truncating at X=A*, let (xo,YO) N(~,~,a2,a2,p), that is
(xo,YO) is bivariate normal (i.e. not standardized) with
axo=oyo=a. Then,
(17)
X°:ll~+aX
(18) YO=u+aY
Since
(19)
E(XO)=~+aE(X)
(20)
E(YO)=~+aE(Y)
(21) V(XO)=a 2V(X)
(22) V(YO)=a 2V(Y)
(23) p(xo,YO)=o(X,Y)
~Xo=~yo=~
and
182
it
follows that
(24) E(XO)=]..l+ok*
(25) E(YO)=]..l+ok*
(26) V(xo)=o2(1+q*)
(27) V(yo)=o2(1- p2q*)
(28) p(XO, yO)=p{(1+q*)/(l+p2 q*)}l/2
Still truncating at XO=A (i.e. X=A*), now let
(X", Y..) -N (lJX' lJy, oX, 0y, p) . Then,
(29) X"=lJX+oXX
(30) Y"=lJy+Oyy
and so
(31) E(X ")=lJX+oXk*
(32) E(y")=lJy+oyk*
(33) V(X ")=ai(l+q*)
(34) V(Y ")=~(1+p2q*)
•
(35) p( X.. , y..)=p{ (1+q*/ (1 +p2 q*)} 1/2
Since S(Y",X")=p(Y",X")Oy";0X" it follows that S(y",X")=S(y,X),
that is that the slopes of the regression of Y on X are the identical
in the truncated and the nontruncated cases.
Also,
(36) E(Y ..-X..). (lJy- lJX) +k* (pay - oX)
Equation (36) above illustrates
(a)
when lJX1lJY, the estimator of treatment main effect should include
a correction for lJy"-lJX" as well as a correction for RTME; and
(b)
when 0X,Oy , the expected RTME includes a term k*(poy-oX)rather
than k*(S-l).
When 0X,Oy, the use of (S-l)instead of (poy-oX)
provides an additional source of error.
183
Gardner (15) derives results for the EIV model X=XT+E, where X
denotes the observed value, XT denotes the true value, and E denotes
an error term. At present, the truncation is at (A,B).
u=(XT'YT'X,y)~
Now the vector
variance-covariance matrix
(37)0=
0
0
Io( a+ 1\)
1\ =
has mean
(wX,Wy,WX'~Y)~
and
I ' where
0[0
oia
0
0
0p2
2
0p
where XT-N(wx,oi) and Ej-N(O,O]).
Now the vector u may be standardized to
(38) we
-1
01
W-[wl,w2,w3,w4']~
by
(u-v)
0 .... 0
-1
op
-1
€p
where v-(wX,Wy,WX,Wy)
222
and €i -oi +oi .
Weiler's results may then be applied to w, resulting in
(using a slightly more general notation)
(39)
-1
E(wl)-PA,B[Pi3~(A)S(C)+Pi4~(B)S(D)],
where Pij is the ijth element of R, the correlation matrix of w. In
the above case,
184
1 pOl/El
1
02/ E2
po l 0 2/E 1E l
1
Thus, it follows that the expectations for the nonstandardized u
are
(41) E(XIX>A,Y>B)=U1+(E1/PA,B){
¢(A)S(C)+Pol02/E1E2¢(B)S(D)}
(42) E(XTIX>A,Y>B)=Ul+(01/PA,B){01/El
¢(A)S(C)+P02/E2
¢(B)S(D))
(43) E(YIX>A,Y>B)=Ul+(El/PA,B){pol/E1E2 ¢(A)S(C)+
¢(B)S(D)}
(44) E(YTIX>A,Y>B)=Ul+(01/PA,B){P0102/El ¢(A)S(C)+02/E2
¢(B)S(D))
For single-stage truncation (i.e. at X-A only),
(45) E(XIX>A)=U1+k*E1
(46) E(YT/X>A)=U1+k*(oI/El)
(47)
E(YIX>A)=~+k*(
pOI 02/ El)
(48) E(YTIX>A)zU2+k*(pol02/Q).
185
Appendix 3
RTM in the ANCOVA when
YI
is measured with error:
According to Cochran (6), YI being measured with error has the
following effects on the ANCOVA (using Y2 as the response and YI as
the covariate):
(a)
the precision of all parameter estimates decreases relative to
the precision of the parameter estimates that could have been
obtained were the true values known;
A
(b)
RTMG is present, since 61' the slope coefficient of the
regression of the observed Y2 on the observed YI , is expected to
lie between zero and one; and
(c)
in nonrandomized designs, B2 may be a biased estimator of the
treatment main effect.
This appendix is intended to clarify point (c) above, in
A
particular by specifying the exact circumstances for which 62 is a
biased estimator of the treatment main effect.
The same assumptions are used as in section 2C, although with a
slightly different notation in order to reduce the number of
To wit, let X=time tI observed value, Y=time t 2 observed
value, X*·time tI true value, Y*=time t 2 true value, O·time tI error,
subscripts.
and E=time t2 error.
Assume that the regression of y* on X* in the
absence of treatment is Y*=X* (i.e. that true values do not change
over time), so that y* may be replaced with X*.
The subscripts "T"
and "C" are used to distinguish treatment and control groups, and T
is used to identify the (additive) treatment effect.
For the control
*
*
group, it is assumed that Xc=Xc+O
c and Yc=Xc+E c (note that x* replaces
* and YT=XT+T+ET
*
Y*), while it is assumed that XT=XT+OT
in the treatment
186
group.
°and E are assumed to be independent, with distributions
X*,
X*-N(w,af), D-N(O,a~), and E-N(O,a~).
A.A.,..
The ANCOVA model is
A.
Y2=!?o+81Y1+82T (T=l for treatment and T=O for control).
Using the observed values, the ANCOVA estimator of treatment
main effect is
"
-
-
In terms of true values and errors, (1) may be rewritten as
(2)
"
- -* -
~=(XT+T+ErXc-Ec)
"-* - -*-
- 81 (XT+DrXc-D c )'
We want to calculate E(E1). For simplicity, the sampling error of 81,
which is of order n- 2 (6), is ignored in the following discussion.
To begin, note that for any given X, X' say, that E(EIX=X')=O,
even if X is extreme.
(3)
Also, conditional upon X equalling X',
E(XjX=X )=l.I+B1(X'-w) and
(4) E(0 IX=X ,) =(1- 131) (X ' - W) .
Here, (3) follows since the regression of V on X predicts not only
E(VIX')' but also E(Vix') and E(Xix')'
Since D=X-X*, (4) follows.
Thus, for a given X, X' say,
(5)
(6)
E(EIX=X')-O,
-*
E(XIX_X')=W+S1(X'-w), and
(7)
E(Dlx=x')-(1- 61)(X'-U)'
"
Using the above, a general expression for E(62)
is
(8)
..
"""""* """""*
"""""* """""*
-E(S2)=T+E{XT-Xc)-Sl(XT-Xc)-S1(DT-Dc)}'
which may be simplified to
Note that (9) does not necessarily equal T, especially in
nonrandomized designs.
In particular, two cases are of interest.
.
187
Case 1:
wXT=wXC but XT#X C either because of chance or because of
nonrandomized allocation to treatment and control groups (e.g. if
X>w, then allocate 75% of individuals to treatment but if X<w then
allocate 25% of individuals to treatment).
-
Here, conditional upon
-
the observed values of XT and XC,
~-
-
~-
-
(10)
E(XTIXT)=J,.I+ 81 (XT- w) ,
(11 )
E(X CIXC) =J,.I+ 81 (XC].I) ,
-
(12)
-
-
E(DTIX T)=(18 1)(XT-].I), and
- (13 ) E(D CIX c)=(1- 61)(Xc-].I)·
Thus,
In conclusion, the usual ANCOVA estimator is unbiased in case 1.
J.lXT1J.lXC (even if XT=X C) because treated and control
individuals are drawn from different populations. Here,
Case 2:
-
-
conditional upone the observed values of XT and XC,
(15)
(16)
E(XTIXT)=J.lXT+Bl(XT-J.lXT),
E(XCI Xc)=J.lXC+61(XC-J.lXC),
(17)
E(DTIXT)=(I- Bl) (XT-J.lXT) , and
(18)
E(DCIXc)=(I-Bl)(XC-].IXC)'
-
-
-
Thus,
A
(19)
_
_
E(B2)=T+E{I- Bl) (J.lXT- 61(XT-J.lXT)-(J.lXC+61 (XC-J.lXC))
-
-
- 61 (1- 61 ) (XT-J.lXT ) - (XC -].IXC) )}.
Simplifying (19) above,
A
(20)
E(~)=T+(1-Bl)(J.lXrwXc)'
A
as reported by Cochran.
Note that in case 2 62 remains biased
even when XT=X C' Here, 62 should be replaced by the unbiased
(21) 62- (1- 61)( wxr J.lXC) ,
A
A
188
where wXT and wxc are estimates of the actual (i.e. nontruncated)
~
means of the populations from which the treatment and control groups
are chose.
•
Under random sampling from both populations, (21)
reduces to the difference of differences estimator
because XT and Xc estimate llXT and wxc' respectively.
The above conclusions are illustrated by a set of simulation
examples.
Tables lA, 18 and lC below illustrate case 1.
There, a
sample of 200000 individuals is simulated from a superpopulation with
X-N(80,100), D-N{0,100), and E-N{0,100), under the same EIV
assumptions as in the previous theoretical development.
The sample,
which at present does not include a treatment intervention, is
intended to be large enough to illustrate actual trends with a high
degree of assurance.
The sample is disaggregated into deciles, with
each decile containing 20000 persons, based upon the observed values
of X.
Table 1A presents summary statistics for these deciles.
Note
that X· tends to be small in the lower deciles and tends to be large
in the higher deciles.
Table 1A
Means (standard errors) by decile
Decil e
1
2
3
4
5
6
7
8
9
10
X·
67.552
72.551
75.102
77 . 273
79.081
80.846
82.770
84.839
87.400
92.366
0
(.053)
(.053)
(.051)
(.050)
(.050)
(.051)
(.051)
(.050)
( .050)
(.053)
-12.405
-7.428
-4.818
-2.818
-0.887
0.914
2.703
4.749
7.331
12.341
X
(.053) 55.145 (.041)
(.051) 65.132 (.012)
(.051) 70.285 (.009)
(.050) 74.455 (.008)
(.050) 78.193 (.007)
(.051) 81.761 (.007)
(.051) 85.473 (.008)
(.050) 89.588 (.009)
(.051) 94.731 (.012)
(.053) 104.707 (.041)
Y
67.485
72.577
75.040
77 . 160
79.080
80.800
82. 794
84 .843
87.353
92.337
(.088)
(.088)
(.087)
(. 087)
(.087)
(.087)
(. 087)
(. 086 )
(.087)
(.089)
189
Table 18 below illustrates Cochran's formula
(22)
E(BO)=(I-B 1)wl'
which predicts that the observed intercept estimates will be about 40
(i.e. (1-.5)(80)=40 ) in each decile.
(a)
Note that:
these intercepts are expected to differ from zero, which is the
intercept for the regression of y* on X*; and
(b)
these intercepts do not depend upon X, and thus are expected to
be about the same in each decile.
The results of table 18 are essentially as predicted.
Table 18
Slope and intercept estimates (standard errors)
for regression of Y on X , by decile
Decil e
1
2
3
Intercept
40.940
47.770
45.262
40.985
47.691
45.001
36.519
41.916
44.142
42.190
4
5
6
7
8
9
10
Slope
(0.845)
(3.162)
(4.646)
(5.816)
(6.687)
(6.713)
(6.582)
(5.994)
(4.643)
(1. 568)
.484
.380
.426
.485
.400
.439
.540
.479
.456
.480
(.015)
( .049)
(.066)
(.078)
( .086)
(.082)
(.077 )
(.067)
( .049)
(.005)
Table 1C below presents ANCOVA results from three allocation
schemes.
Scheme (a), which represents randomized allocation to
treatment or control group, is implemented by sorting each decile by a
random number and then placing the first 10000 individuals from each
sorted decile into the treatment group.
Here, and throughout this
*-X *+2+ET (i.e. 8 =2.
appendix, all treated individuals have YT
T
2
Schemes (b) and (c) represent nonrandomized allocation rules.
In (b),
75% of the members of the upper deciles and 25% of the members of the
lower deciles are placed in the treatment group.
In (c), all
190
individuals in the upper deciles are placed in the treatment group.
As suggested by both equation (14) and table 1B, however, the fact
that XT>X C in schemes (b) and (c) does not bias the resulting
estimators of 62' since neither the intercept estimate in the
treatment group nor the intercept estimate in the control group are
affected by the magnitude of the observed Xs.
As' expected, then,
~
table 1C shows 82 to be close to the true value of 62=2 in each case.
Table 1C
ANCOVA results for case 1
Scheme(see above)
a
b
c
Intercept
Slope
Treatment
39.885 (.160)
39.872 (.162)
40.217 (.225)
.501 (.002)
.501 (.002)
.496 (.003)
2.004 (.055)
1.977 (.060)
2.127 (.091)
Tables 2A and 28 below illustrate case 2, that is where the
treatment and control groups are chosen from two different
populations. In each case, one sample of 100000 individuals is taken
from each superpopulation.
In table 2A, both superpopulations have
X*-N(80,100) , with everything else as before.
200000 individuals are used in the ANCOVA.
and nT-7821 persons with X>100 are retained.
In scheme (d), all
In (e), only the nC=7855
In (f), only the nC-7855
persons with XC>100 and the nT=192 persons with XT>120 are retained.
Thus, both random and truncated designs are illustrated. In addition,
(f) illustrates the case where XT'XC'
However, equation (20) predicts
that schemes (d), (e) and (f) will all yield the same result, that is
no bias in 62 since
~T=~C'
-
Table 2A below supports this prediction,
even in scheme (f) where XT'X C'
•
191
Table 2A
ANCOVA results for case 2
d
e
f
Scheme(see above)
(~T=~C)
Slope
.501 (.002)
.498 (.018)
.483 (.025)
Intercept
39.865 (0.159)
40.243 (1.888)
41.872 (2.627)
Treatment
2.038 (.058)
2.064 (.195)
1.722 (.993)
In contrast to table 2A, table 28 below illustrates the case
where the treatment and control groups are chosen from populations
with different means.
Here, the control superpopulation has
*
*
XC-N(80,100)
and the treatment superpopulation has XT-N(100,100),
with
everything else as before.
Now, for each of schemes (d), (e) and (f)
A
the expected bias of
~
is 10 (i.e. (1-.5)(100-80)=10 ), illustrated
This bias calculation holds even when XT-X C' and also holds
when XT is the same distance from ~T as Xc is from ~C.
below.
Table 28
ANCOVA results for case 2
Scheme(see above)
d
e
f
(~T1~)
Intercept
Slope
Treatment
39.865 (0.159)
39.909 (0.677)
40.243 (1.888)
.501 (.002)
.501 (.006)
.498 (.018)
12.010 (.067)
11. 965 (.151)
12.100 (.403)
To conclude, even though measurement error in Y (i.e. "X")
A
affects the precision of 82 in all designs, the usual ANCOVA estimator
A
~
is a biased estimator of the treatment main effect only when
treated and control individuals are drawn from different populations
and, in addition, these populations have different means.
When this
A
is the case,
~
is biased even when observed means are identical at
'"
time t 1, and should be replaced by
A
A
A
~-(l-81)(~lr~lC).