A MIXED MODEL ANALYSIS OF TRACKING
FOR INCONSISTENTLY-TIMED LONGITUDINAL DATA
by
Penelope Susan Pekow
Department of Biostatistics, University of
North Carolina at Chapel Hill, NC.
Institute of statistics Mimeo Series No. 1890T
September 1991
A MIXED MODEL ANALYSIS OF TRACKING
FOR INCONSISTENTLY-TIMED LONGITUDINAL DATA
by
Penelope Susan Pekow
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
1991
Advisor
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ABSTRACT
PENELOPE S. PEKOW. A Mixed Model Analysis of Tracking for InconsistentlyTimed Longitudinal Data. (Under the direction of Paul W. Stewart).
Tracking
analysi~
is primarily directed at measuring the extent to which
individuals maintain their relative position for some attribute among a group of peers
over time. A review of the literature shows a wide variety of definitions and measures
of tracking. However most require complete balanced data or do not make use of the
longitudinal structure of the data.
This work extends the tracking indices of McMahan (1981), defining tracking
as maintenance of a constant relative deviation from the population mean over time,
to the mixed model with linear covariance structure. This model can accommodate
data characterized by missing and/or mistimed data as well as deliberately incomplete
designs.
Several tracking indices and their estimators are defined for the mixed model
setting, based upon both observed sets of measurement occasions and selected points
across an interval of interest.
Randomly generated data are then used to explore
some of the properties of the tracking estimators for various designs:
complete
balanced data, inconsistently-timed data, complete data with mistiming, complete
plus incomplete data, and -a linked cross-sectional design.
An example application is given using inconsistently-timed data on the growth
of pulmonary function with height as the time scale.
Some of the effects of the
modeling assumptions on the tracking indices are discussed, along with suggestions for
future directions in tracking analysis.
iii
ACKNOWLEDGMENTS
My special
thanks
to
my
advisor
Dr.
Paul
Stewart, for
help
and
encouragement well above and beyond the call of duty. He was a real friend as well
as an advisor over the course of this project.
My gratitude also goes to my family for giving me the time to complete this
work. Thanks to my husband Francis for giving up his own projects and taking over
all the at home duties, and to my daughters Sela and Leah for putting up with all
those times that Mommy was too busy.
;
iv
TABLE OF CONTENTS
Page
LIST OF TABLES
vii
LIST OF FIGURES
viii
Chapter
1. INTRODUCTION AND LITERATURE REVIEW
1.1 Introduction
'
1.2 Literature Review of Tracking Analyses
1
1
3
1.2.1 Goals of Tracking Analysis
3
1.2.2 Tracking Definitions
4
1.2.3 Tracking as Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Correlation Coefficients as Measures of Tracking
9
1.2.5 Tracking as Maintenance of Extreme Position
11
1.2.6 Individual Tracking Patterns
13
1.2.7 Correlation Between Slope and Intercept
17
1.2.8 Notation for the Growth Curve Model
19
1.2.9 Indices of Growth Separation
20
1.2.10 Indices of Growth Constancy
24
1.2.11 Applications of Tracking Indices
28
1.3 Mixed Effects Model Approaches to Incomplete Longitudinal Data Analysis.. 30
1.3.1 Introduction to the Mixed Effects Model
30
1.3.2 Estimation
31
1.3.3 Hypothesis Testing
34
1.3.4 Example Applications
35
1.4 Statement of the Problem and Outline
36
v
Page
2. DEFINING TRACKING IN THE MIXED MODEL
38
2.1 Choice of Tracking Definition for Unbalanced Data
38
2.2 McMahan's Tracking Index in the Mixed Model
45
2.2.1 Defining Tracking in the Mixed Model
46
2.2.2 Individual Design Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47
2.2.3 A More General Model
52
2.2.4 Indices of Tracking in the Mixed Model
53
2.2.5 Adjusting for Zero Correlation
59
2.2.6 Tracking Indices Based Upon Observed Values
65
2.2.7 Parameter Dependence Upon Measurement Occasions
67
3. ESTIMATION
73
3.1 Estimation in the Mixed Model
3.2 Tracking Index Estimators . . . . . . . . . . . . . . . . . . . .
73
. . . . . . . . . . . . . . 77
3.3 Jackknife Estimation
4. MODELING ISSUES
81
84
4.1 Under- or Over- Fitting a Tracking Model
85
4.2 Including Within-Individual Error in the Tracking Parameters
96
4.3 Exploration of Designs for Measurement Occasions
97
4.4 Disscussion
121
5. AN EXAMPLE
126
5.1 Description of the Data
126
5.2 Data Selection
128
5.3 Model Selection and Fitting
133
5.4 Tracking Estimation
137
vi
Page
5.5 Results
141
5.6 Discussion
143
6. SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH
152
SELECTED BIBLIOGRAPHY
155
vii
LIST OF TABLES
Table 4.1.1: Under-Fitting: Forcing Perfect Tracking
87
Table 4.1.2:
90
Under-Fitting: A Quadratic Model
Table 4.1.3: Over-Fitting Example
Table 4.3.1: Summary of Tracking Estimates for Inconsistently-Timed Data
(Design a)
95
103
Table 4.3.2: Summary of Tracking Estimates for Complete Data (Design b) .... 103
Table 4.3.3: Summary of Tracking Estimates for Complete Data with Mistiming
(Design c)
104
Table 4.3.4: Summary of Tracking Estimates for Complete Data with Additional
Incomplete Data (Design d)
104
Table 4.3.5: Linked Cross-Sectional Example
114
Table 4.3.6: Expected Tracking at Selected Points from Complete Balanced Data
Example
116
Table 5.2.1: Number of Observations Per Child in the Analysis Dataset
132
Table 5.3.1: Results of Model Fitting for FEF 2 5-75 and Height
African-American Boys
135
Table 5.3.2: Results of Model Fitting for FEF2 5-75 and Height
African-American Girls
135
Table 5.3.3: Results of Model Fitting for FVC and Height
African-American Boys
136
Table 5.3.4: Results of Model Fitting for FVC and Height
African-American Girls
136
Table 5.3.5: Tracking Estimates for FEF2 5-75 with respect to Height
African-American Boys
139
Table 5.3.6: Tracking Estimates for FEF2 5-75 with respect to Height
African-American Girls
139
Table 5.3.7: Tracking Estimates for FVC with respect to Height
African-American Boys
140
Table 5.3.8: Tracking Estimates for FVC with respect to Height
African-American Girls
140
viii
LIST OF FIGURES
Figure
1: Illustration of Growth Constancy and Growth Separation
6
Figure
2.1: Visual Impression of Tracking
Figure
2.2: Inconsistently-Timed Data and Visual Impression of Tracking .... 42
Figure
2.3: Defining the Nearest Tracking Contour at the Measurement
Occasions
57
2.4: Defining the Nearest Tracking Contour at the Selected
'Test' Measurement Occasions
58
Figure
39
Figure
4.1.1: Forcing Perfect Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Figure
4.1.2: Example of Under-Fitting
92
Figure
4.3.1: Predicted Lines from an Example with Inconsistently-Timed
Data (Design a)
99
Figure
4.3.2: Predicted LInes from an Example with Complete Balanced
Data (Design b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99
Figure
4.3.3: Predicted Lines from an Example with Mistimed Data
(Design c) • . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . '.' ..... 100
Figure
4.3.4: Predicted Lines from an Example of Complete Plus Incomplete
Data (D.esign d)
101
4.3.5: Comparing Four Designs
Distribution of Pa: Mean Adjusted Observed Tracking
106
Figure
Figure
4.3.6: Comparing Four Designs
Distribution of var(Pai): Variance of Adjusted Individual
observed Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 106
Figure
4.3.7: Comparing Four Designs
Distribution of 1': Mean Adjusted Expected Tracking
107
4.3.8: Comparing Four Designs
Distribution of var( T i): Variance of Adjusted Individual
Expected Tracking
107
4.3.9: Predicted Lines and Observations for Six Individuals
with Inconsistently-Timed Data
109
Figure
Figure
Figure 4.3.10: Comparing Four Designs
Distribution of Pat: Mean Adjusted Observed Tracking
at Test Points
111
ix
Figure 4.3.11: Comparing Four Designs
Distribution of var(Pati): Variance of Adjusted Individual
Observed Tracking at Test Points
111
Figure 4.3.12: Comparing Four Designs
Distribution of Tt: Mean Adjusted Expected Tracking
at Test Points'
112
Figure 4.3.13: Predicted Lines from Linked Cross-Sectional Design
115
Figure 4.3.14: Plots of Pai' Adjusted Individual Tracking Estimates vs. Mean,
Minimum, Maximum and Range in Heights for
Inconsistently-timed Data (Design a)
118
Figure 4.3.15: Plots of T i' Individual Adjusted Expected Tracking Estimates
vs. Mean, Minimum, Maximum and Range in Heights
for Inconsistently-timed Data (Design a)
119
Figure 4.3.16: Plots of Pati' Adjusted Individual Tracking Estimates
at Selected Heights vs. Mean, Minimum, Maximum and
Range in Heights for Inconsistently-timed Data (Design a)
120
Figure
5.2.1: Distribution of Observations across Height
for African-American Girls, Ages 5-13
130
Figure
5.2.2: Distribution of Observations across Height
for African-American Boys, Ages 5-13 . . . . . . . . . . . . . . . . . . . . 131
Figure
5.4.1: Predicted Lines for FEF 2 5-75 vs. Height
for 30 African-American Boys . . . . . . . . . . . . . • . . . . . . . . . . . . 144
Figure
5.4.2: Line Segments Connecting Observations of FEF 25- 75
for 30 African-American Boys
145
5.4.3: Predicted Lines for FEF 25-75 vs. Height
for 26 African-American Girls
146
5.4.4: Line Segments Connecting Observations of FEF 2 5-75
for 26 African-American Girls
147
5.4.5: Predicted Lines for FVC vs. Height
for 30 African-American Boys
148
5.4.6: Line Segments Connecting Observations of FVC
for 30 African-American Boys
149
5.4.7: Predicted Lines for FVC vs. Height
for 26 African-American Girls
150
5.4.8: Line Segments Connecting Observations of FVC
for 26 African-American Girls
151
Figure
Figure
Figure
Figure
Figure
Figure
Chapter 1
INTRODUCTION AND LITERATURE REVIEW
1.1 Introduction
Repeated measurements on individuals taken over time often show patterns
that can be used to describe or evaluate a process. Tracking is a term that has been
used to describe a pattern in which individuals maintain their relative position with
respect to some attribute among a group of peers over time. The concept of tracking
is closely tied to prediction, and it's use in the medical literature indicates that
tracking generally encompasses the idea that knowledge of an individual's past
contains useful information about the individual's present, and also future status.
The term tracking has traditionally been used in the description of growth of physical
characteristics; for example, height and weight are said to track with age throughout
childhood. It has also been used to describe change over time in intelligence testing
and other psychological measurements, as well as in measures of educational
attainment, such as reading levels.
Much attention
has focused
upon the
identification of individuals who do not track 'properly' for further evaluation for
health or developmental problems.
Recent interest has also focused upon assessing
the extent to which risk factors for disease, such as high cholesterol levels or high
blood pressure, track from childhood into adulthood.
This work was motivated by a study of pulmonary function in children with
two or more measurements over time.
The questions of interest are whether the
growth of lungs shows the same pattern for all children as they grow taller, and
2
whether certain children or groups of children with markedly different patterns of lung
growth can be identified.
Lung growth corresponds 'with height more closely than
with age throughout childhood, so that interest lies in whether or not lung growth
tracks with height. Using height as a time scale produces 'inconsistently-timed' data
since it is not feasible to follow children at exact heights or exact height intervals.
A survey of the literature shows no generally agreed upon definition of
tracking.
Specific definitions, and the corresponding methods of evaluation have
varied widely. However the methods seen in the literature are dependent upon having
all individuals measured at the same points in time, or else do not take full advantage
of incomplete longitudinal data.
In any longitudinal study there is usually some
missing and/or mistimed data, and the current methods for analysis of tracking
cannot adequately handle these problems.
There also exist data for which it is
difficult or impossible to obtain measurements at the same point in time, as well as
study designs with deliberate staggering of entry times, such as a linked crosssectional study, where a question of tracking may be of interest.
The first section of the literature review will discuss definitions and analyses of
tracking and related measures.
The second will give a brief review of the use of a
mixed model for the analysis of unbalanced longitudinal data.
•
3
1.2 Literature Review of Tracking Analyses
1.2.1 Goals of Tracking Analysis
The basic goals of tracking analysis may be quite variable, rendering different
definitions and analysis techniques appropriate. Interest may focus upon
-- Describing growth or change:
Does height track with age for this
interval?
-- Evaluating growth or change:
Is individual i growing 'properly'?
-- Predicting future levels:
What is the expected future level of
pulmonary function?
-- Assessing risk or hazard:
Does childhood high blood pressure
imply
high
blood
pressure in
adulthood?
Historically, tracking has been used to describe the growth of physical
characteristics.
Height and weight are said to track with age throughout childhood,
and pediatricians typically plot height and weight of children on standard growth
charts to evaluate their growth. Tanner (1978a,b) and Goldstein (1979) describe the
creation and use of standard growth charts.
These charts provide cross-sectional
standards, and are also used to evaluate a child's growth rate.
Marked departure
from a child's original percentile over time is considered a signal for further evaluation
of the child's growth.
This strategy is based upon the assumption that height and
weight do track with age, and is appropriate in an age range where growth indeed
does track. However healthy children grow at different rates and cross percentile lines
frequently; the amount of percentile crossing depends upon the age range and the type
of measurement (height or weight) under consideration (Smith et al., 1976; Berkey et
al., 1983).
One approach to providing better standards for growth evaluation has
focused upon developing conditional standards which take previous measurements into
4
consideration (Cameron, 1980; Berkey et al., 1983). Another approach to longitudinal
evaluation of growth focuses upon tracking indices, which can provide information on
the degree to which percentile crossings occur.
Recent interest in tracking has also focused upon assessing the extent to which
risk factors for disease, such as high blood pressure or poor lung function, track from
childhood into adulthood.
Here, emphasis is on the value of early measurements for
predicting future levels. This can facilitate identification of individuals at high risk of
developing disease, for early intervention programs. A high degree of tracking would
mean only 1 or 2 measurements would be needed for assessing an individual's at risk
status, while a lesser degree of tracking would indicate need for longer follow-up in
identification of those at risk.
A measure of tracking may also be useful in evaluating the use of a time- (e.g.
age- or height-) adjusted value for comparison of groups in a cross-sectional study.
Adjusting values to some specific age or height for comparison assumes that an
individual's growth is
parall~l
to some population curve, i.e., that an individual
.
maintains a relative deviation from the population mean over time, which is one
definition of tracking seen in the literature (see McMahan, 1981).
1.2.2 Tracking Definitions
Specific definitions of tracking, and their mathematic interpretations have
varied considerably in the literature.
One general definition that appears to have
widespread acceptance is the maintenance of relative position over time within a
group of peers.
The other theme that appears to be consistently important to a
concept of tracking is the idea of prediction.
•
/
5
The phrase 'maintenance of relative position' has been interpreted generally in
two different' ways, which have led to two different types of indices of tracking.
In
the first, maintenance of relative position is taken to mean maintaining rank or order
across time.
The individual at the bottom of the group stays at the bottom, etc.,
although the distance between that individual and the next may change across time.
This is basically a comparison of how individuals perform relative to each other,
without reference to a population distribution. In this definition of tracking, growth
patterns may vary considerably across individuals, but rank order is unchanged.
Maintenance of relative position is also interpreted as maintenance of position
within a population distribution, i.e., maintaining percentile rank, or maintaining a
constant relative deviation from the population mean over time. This is a comparison
of how individuals perform relative to some internal standard.
In a paper discussing what he calls measures of "growth stability", Goldstein
(1981) evaluates tracking in both ways, creating indices of growth separation and of
growth constancy. Growth separation measures the extent to which individual's lines
cross or change rank order.
Growth constancy measures the extent to which
individuals vary about their expected percentile, or about their expected deviation
from the population mean. It is quite possible for curves which are well separated to
vary considerably about their expected deviation, and vice versa, as can be seen in
Figure 1. This picture shows the hypothetical growth curves of three children along
with the population 10th, 50th, and 90th percentile lines.
Child 2 maintains a
constant percentile rank across time. Child 1 varies considerably around her average
percentile ranking, and child 3 grows slowly at first, and then faster than average.
Children 1 and 3 show considerable growth variability--they do not 'track' well in the
sense of maintaining a constant percentile rank.
Yet all three curves are well
separated--they do not intersect in the interval, and so track well in that sense.
6
140
-r-----------------------------,
1
130
2
~
u
z
f-
3
120
I
'-'
W
I
110
100
6
7
8
9
10
11
•
AGE IN YEARS
Child 1:
Figure 1:
Variable growth around average percentile
Child 2:
Maintains approximately constant percentile
Child 3:
Growth is first slow then fast
Illustration of Growth Constancy and Growth Separation
•
7
Thus, perfect growth constancy implies perfect growth separation, but the converse is
not true. It seems clear that these two types of indices measure very different aspects
of growth stability.
Growth constancy is a descriptor of performance relative to a
population distribution, while growth separation describes how individuals perform
relative to each other.
Definitions of tracking found in the literature are represented by those given
below:
1.
Systematic change in repeated measures that facilitates prediction
of future or missing values (Ware and Wu, 1981)
2.
Tendency for each individual's measurements to remain equal to a
fixed percentile of the population distribution which may be
changing. (Clarke, Woolson, and Schrott, 1976)
3.
Tendency for each individual's measurements to remain a
constant relative deviation from the population mean over time.
(McMahan, 1981)
4.
Tendency for the expectations of an individual's measurements to
remain in the same relative order over a specific time interval.
(Foulkes and Davis, 1981)
5.
The consistency of an individual's serial measurements. in a
longitudinal survey. Non-trackers are defined as those who follow
different, usually more complex, growth patterns than trackers.
(Foster, Mohr, and Elston, 1989)
Definition 1 emphasizes the idea of prediction but does not stress the
relationship to others over time.
Definitions 2 and 3 define measures of growth
constancy in Goldstein's sense, while Definition 4 defines tracking as growth
separation.
Definition 5 focuses on comparison of patterns of growth rather than
comparing levels (e.g., rank or percentile) over time.
8
1.2.3 Tracking as Prediction
Some investigations of tracking have focused on assessing the value of early
,
observations for predicting future levels. This is tracking in the sense of Definition 1
given above.
When the emphasis is on prediction in a repeated measures study,
observations on individuals at times t lt t 2 ,
times tl:+ 1 '
... ,_ tl:+ m
... ,
tl:+ m ' are used to predict values at
for future subjects with observations for only the first Ie time
points.
The issue of prediction of future or of missing values in the setting of growth
curve analysis is discussed in Lee and Geisser (1972), Rae (1975; 1987), and Ware
and Wu (1981).
Growth curve analysis provides for the fitting of individual curves,
which take full advantage of the longitudinal nature of the data, rather than
analyzing serial sets of paired measurements.
Many of the goals of growth curve
analysis are closely related to those of tracking analysis.
11
Ware and Wu relate growth curve analysis directly to their concept of
tracking.
"Tracking" to Ware and Wu means staying on a projection of the
individual growth
~urve
-.
predicted by earlier measurements, rather than maintaining
position relative to others. This is a useful concept for ri!lk assessment and prediction
analysis, but it is not necessarily what is typically meant by tracking.
After fitting a polynomial growth curve model, Ware and Wu compute the
predicted mean square error (PMSE) for a sample of size
jN. They then divide their
sample into thirds, successively predicting the final measurement for each third using
prediction equations based on complete data in the remaining two-thirds. The PMSE
is used as a measure of how well their model predicts a final level for new data. They
suggested that their approach and model could be altered to handle individual design
matrices, ~i'
•
9
They compare PMSE's from several models, most importantly demonstrating
that the PMSE's are smallest for models incorporating information on the series of
repeated measures, compared to models that use only a single earlier measurement for
prediction, or the mean of the earlier measurements. This is a clear demonstration of
the greater information provided by the use of individual trajectories for prediction
compared to a single measurement, or the mean of earlier measurements.
Ware and Wu also point out that their growth curve analysis of tracking has
the advantage of providing, in addition to the PMSE, an estimate of the correlation
between intercept and slope. This correlation is another parameter that is discussed
in relation to tracking analysis and patterns of change over time (see section 1.2.7
below).
1.2.4
Correlation Coefficients as Measures of Tracking
One simple measure of tracking that has been used frequently is the
correlation coefficient between successive measurements on an individual.
A high
correlation is considered indicative of high tracking, meaning in this case high
predictive value.
Several studies on the tracking of blood pressure have used this
measure, trying to evaluate the degree to which blood pressure tracks through
childhood into adulthood.
Levine et al. (1978), and Rosner et al. (1977), defined the tracking correlation
as the correlation coefficient between measurements on the same person taken at two
different times.
Separate correlations are computed for each pair of timepoints.
In
effect, this utilizes only the measurements at times t k and t k + 1 , and ignores
measurements at other timepoints. Using the correlation coefficient does not
effectively use all of the information contained in serial measurements, although it
10
does allow the inclusion of all individuals with data at the two timepoints considered,
for each pair of comparisons.
Rosner et ai. explored the effect of the length of the
time interval between measurements of blood pressure, finding that the correlation
•
decreased with increasing interval. Levine et ai. compared the predictive values of the
tracking correlation for different specified age ranges.
Voors et ai. (1979) also evaluated tracking by looking at Pearson correlation
coefficients between mean blood pressures for various time intervals. In addition, they
looked at partial correlations, controlling for some measure of body size, as well as
using the previous year's blood pressure as a way of incorporating more information
on blood pressure history.
Clarke, et ai. (1976) in a study attempting to quantify tracking of blood
pressure, blood lipids and obesity, used data from three surveys, at two year intervals,
of children ages five to eighteen.
group for each survey.
Percentile ranks were computed for each age-sex
Using data on those children with repeated measures, rank
correlations between the surveys were used to measure tracking.
Results were
compared to rank correlations for height and weight (considered examples of
attributes which track) for different intervals and age ranges.
Clarke, et ai. (1978), defined tracking as "the phenomenon of children
maintaining their rank within their age-sex group."
Using data from four large
surveys, they computed age-sex standardized scores for each survey. They evaluated
tracking in height, weight, other measures of body size, and blood lipid levels, by agesex specific Pearson correlation coefficients for all pairs of surveys, using data from
those children in all four surveys. They also computed percentile ranks at each age for
each individual, and compared these across time, since tracking has often been
described in terms of maintenance of percentile over time.
..
11
1.2.5
Tracking as Maintenance of Extreme Position
Sometimes there is interest in individual behavior over time for the whole
range of the distribution, but often interest focuses on the extremes.
For example,
the question of interest may be "Is a child with high blood pressure going to maintain
high blood pressure?" or "Is a child with low pulmonary function going to maintain a
low level into adulthood?"
The text by Remington and Schork (1985) describes
tracking in this context, in which epidemiologists are interested in evaluating the
probability of remaining in the upper quintile of a distribution of some risk factor over
time.
Clarke, et 0/., (1978) in the paper discussed above, divided the population into
quintiles at each age, and computed the probability of 'surviving' or remaining in the
upper quintile, using clinical lifetable
technique~.
They also evaluated the conditional
transition probability of being in the ith quintile (2, 4 or 6 years later) given a start in
the jth quintile at the initial visit, where their emphasis was ·on· the upper quintile.
Voors et 0/. (1979) also discussed tracking in terms of maintaining relative
position within a distribution.
They were most interested in evaluating whether or
not those at the extremes for blood pressure remained extreme.
For those in the
upper and lower deciles at the initial measurement, they computed expected mean
blood pressures one year later, using information on between- and within- child
variability of blood pressure measurements to adjust for regression to the mean. These
were considered the values expected under ideal tracking. A measure of goodness-offit of the observed values was used as a measure of tracking.
For tracking defined as the tendency to remain in the same quantile over time,
Ware and Wu (1981) give an example, for a population grouped in quintiles, of how
the Kappa statistic (see Fleiss, 1971) can be used to measure the agreement of
quintile rank over time. This method can be described as follows:
12
For n subjects divided into k cells (e.g., 5 quintiles) at each of p times, let n ij
the
be
p .
.J
number
n
= "I'i=l
'"'1 En..
lJ
is
t
of
the
p. = -1()
n .. (n .. -1)
1·
p p-1 j=l lJ lJ
P
..
n
= It1 i=l
E Pi.
where
Pe
times
subject
proportion
measures
of
all
quintile
in
observations
concordance
is the average concordance.
k
=i=l
E p~.
1
is
for
the
in
ith
j.
the
Then
jth
quintile,
individual,
.
Then kappa is computed as
IC
and
-pe
= P{-pe
- ,
is the expected concordance if individuals are randomly and
independently distributed into categories. This approach however, does not adjust for
interoccasion variability, which may be high in some measures, such as blood pressure.
Therefore
IC
values near one will not be seen. This technique can be used to focus on
agreement of extreme vs. non-extreme rank over time. Although individual indices of
tracking were not discussed in this paper, it would also be possible to use the Pi. as
individual measures of tracking.
Webber et al. (1983), in a study of the tracking of
car~iovascular disease
risk
factors in children, define tracking as persistence in rank over time, but evaluate
tracking only in the upper' portion of the distribution. Trackers were defined as those
initially at or above the 90th percentile who remained above the 90th percentile for
their age-sex-race group over time.
Discriminant analysis was then used to identify
risk factors associated with 'tracking' as defined above.
The discriminant functions
were then applied to the same data to see how well trackers vs. non- trackers were
identified.
This definition of tracking, while it may be useful for identifying those likely to
maintain extreme levels of a risk factor, does not correspond to tracking as growth
separation or growth constancy, as defined by Goldstein (1981), McMahan (1981), or
Foulkes and Davis (1981) (see section 1.2.10 below).
It seems clear that crossing a
specified level such as the 90th percentile line is not the same as evaluating the
13
amount of change around a line. For example, by this definition a child moving from
percentile 91 to 89 is a non-tracker, while one moving from 99.9 to 90.1 is a tracker
by this definition.
The opposite would be true when defining tracking as growth
separation or growth constancy, where the first child crosses few percentile lines, and
the second many. As noted by Webber et al., those initially the highest among the
high tended to be the trackers, as would be expected under this definition.
1.2.6
Individual Tracking Patterns
Most of the analyses of tracking found in the literature focus upon defining
and measuring tracking for a population. They ask whether or not an attribute, such
as height or blood pressure, tracks with age, and develop population indices for
tracking.
Another approach is to develop a model that identifies individuals as
trackers or non-trackers within a population.
These methods would be more in
keeping with a research goal of identifying individuals at risk of disease later in life.
Goldstein (1981) mentions' the possibility of using individual counts of the number of
other curves crossed to identify individuals whose growth patterns differ markedly
from others (see section 1.2.9). Two papers, by Lauer et al. (1986) and Foster, et al.
(1989) develop methods for identifying individuals with differing growth patterns.
Lauer et al. (1986) present an analysis which focuses upon individual measures
of tracking in a study of blood pressure in childhood. There is no attempt to create
any overall indices describing whether or not blood pressure tracks. Instead, the focus
of the paper is upon using different aspects of tracking to identify particular groups of
children with high risk of developing hypertension later in life.
paper seems to mean following a specific growth pattern.
"Tracking" in this
14
Individual indices were defined as follows:
Using data on 4313 subjects with
three to six measurements over time, all measured variables were transformed into
age-sex specific percentile ranks for each survey year. Let Yij be the age-sex specific
percentile rank for ith measurement, i-I, ..., n j for the ith individual, i=l, ..., N.
For each child, i, a least squares regression line was fit to the transformed data and
then the following were defined:
1) LEVEL - the mean percentile rank across time -- Yi
=
n·
LJ 1C
~Yij
j=l
I
2) TREND - the slope of the regression line, or the change in percentile rank
over time --
and
3) VARIABILITY - the residual standard deviation --
Trend is essentially a directional individual measure of growth separation subjects with low trend have decreasing rank across time; with flat
trend,
approximately the same rank across time; and with high trend, increasing rank across
time.
Variability measures how an individual's ranks vary around the fitted line of
their percentile rankings, and is a standardized measure of growth constancy. Level is
merely an individual's average rank across time, useful in determining overall position
(e.g., extreme or not), as well as in conjunction with the other two indices.
The resulting indices, computed for all children, were then ranked and divided
into quintiles of level, trend and variability for systolic and diastolic blood pressure.
Different groups of children were identified as 'tracking' in various ways. Those with
high level (high rank), flat trend (no change in rank) and low variability were
considered to have consistently high blood pressure, those witli moderate or high level,
15
high trend (increasing rank) and low variability as tracking towards high blood
pressure, and those with high level and high variability as having labile blood
pressure. Other patterns of interest were also defined. Within the identified groups,
associated factors, such as level, trend and variability of various measures of body size
were identified as correlates of specific "tracking" patterns.
Foster, et al. (1989) develop a model for identifying trackers vs. non-trackers
in a population, again without any attempt to define or measure whether the
attribute 'tracks' for the population as a whole. They do, however, assume that the
majority of a population are trackers, meaning that the majority have a more
homogeneous set of curves than the population as a whole.
For their model they assume that the shapes of individual growth curves will
differ, and that individual growth will differ from the population mean. Trackers are
defined in their model to be those who have a simpler growth pattern than nontrackers.
In other words, it· is assumed that the growth curves of trackers can be
described by a polynomial of degree v, which is less than the degree necessary for the
population as a whole.
Non-trackers are those whose growth pattern is more
complex, better described by a polynomial of degree v+q
~
p.
Using these asumptions, for a population of r individuals all measured at the
same n points in time, a principle components analysis is used to find the n
eigenvectors that span the data space. Then Akaike's information criterion (AIC) is
minimized to select the first p eigenvectors:
AIC(~,,)
= -2{log likelihood(~,)} + 2m,
where m" is the number of parameters estimated for the sth family, and ~" is the
maximum likelihood estimate of the m,,-parameter vector.
Thus in determining p,-
goodness-of-fit is balanced with the number of parameters estimated.
16
The vector of observations for the ith individual, y.,
-, can be expressed as
Y
_,"
= G- ..r-'1'
c" + e"
-'1' ,
i
= 1, ..., r
Here £i1' is a px1 vector of individual parameters, and
•
G1'
is an nxp orthonormal
design matrix, the same for all individuals. It is also assumed that ~i1' ..... N(g, lT~!n).
Once p is determined by the AIC(p) criterion,
G1' , £i1"
and iT~ are computed.
To separate the population into trackers and non-trackers,
G1"
£i1" and iT~
are assumed to be fixed, and for each individual, wi ~ P is found by minimizing
The frequency distribution of the wi is evaluated to separate the population into
trackers and non-trackers, defining, for example, v as the modal value of the Wi
distribution.
Trackers are those with w"t<
v, and non-trackers those with w," > v.
-
New estimates of the individual curves are made, and the tracker and non-tracker
mean curves .and variances can be estimated separately.
Evaluation of the method by simulation studies showed that its success is
dependent upon the proportion of non-trackers in the population being small. When
the proportion of non-trackers, or those with more complex curves becomes large, v
becomes large -- everyone is identified as trackers in this case.
The method was applied to data on growth of height over four years, for
children in eight age-sex cohorts.
It appeared to be useful in determining which
children hit adolescent growth spurts early or late. These children were identified as
non-trackers, requiring a polynomial of higher degree in order to fit their curves well.
This method may be useful for identifying individuals with markedly more
complex growth patterns than is typical. It would not, however, identify those whose
growth rate may be flatter or more steep than is typical, but not necessarily more
complex. These individuals could be better identified by a method such as Lauer's.
17
1.2.7 Correlation Between Slope and Intercept
The correlation between slope and intercept is one other parameter related to
the concept of tracking that has been of interest to some investigators.
A positive
correlation between slope, or an average growth velocity, and intercept means relative
order would be maintained with growth or change over time.
For example, it has
been hypothesized that those with relatively high blood pressure initially will have a
higher growth rate for blood pressure, and subsequently even higher blood pressure.
It has also been hypothesized that children with poor lung function will have slower
pulmonary growth rates, and subsequently poor lung function as adults.
The simple estimate of the correlation of slope and intercept, or of the
regression of slope on intercept, based on the observed values, is biased toward zero,
due to the presence of random error in the independent variable (see Blomqvist, 1977;
Rogosa and Willet, 1985; Wu, Ware and Feinleib, 1980; and Neter and Wasserman,
1974).
The bias is enough, in some applications, e.g:, studies of blood pressure, to
make a significant positive correlation appear to be a significant negative one (see
example in Blomqvist, 1977).
Blomqvist, assuming a linear relationship of change over time, and a bivariate
normal distribution for slope and intercept, derived the maximum likelihood (ML)
estimate of the regression coefficient of the true slope on the true initial value, as a
simple adjustment to the sample regression coefficient computed from the estimated
slope and the estimated intercept.
Wu, Ware and Feinleib (1980) recommend the Blomqvist ML estimator, after
comparing the ML estimator to a method (FHG) regressing the slope based on all but
the initial observation, on the initial observation.
The FHG estimate is biased
downward, and is asymptotically less efficient than the ML estimate.
18
Even when growth or change is not strictly linear but is quadratic or cubic
across time, a linear assumption is often adequate for the time intervals considered in
many studies (see Rogosa and Willett, 1985, Wu et al., 1980), so the correlation
between slope and intercept from this simple model can still be informative in tracking
analyses. The slope from the linear regression is also a good estimate of the average
rate of change over an interval (Seigel, 1975) when a higher order model applies.
Rogosa and Willett (1985) discussed a major problem with the interpretation
of the correlation of slope and initial value for growth studies in which subjects are
observed at the same p times.
The correlation is highly dependent on the choice of
the initial time, t 1 • Change that is correlated with initial status at age 10 may not be
correlated with status at age 5, so that the initial time and interval of interest
strongly influence the results.
Rogosa and Willett identified the functional dependance of the correlation
between slope and intercept,
9 the slope.
p(t)"
on time, where e(t) is the .response at time t, and
For any particular set of straight-line growth curves, they identified a
centering point,
and a scaling constant,
to is the point of minimum variance in the response, and
K.
determines the rate
at which the variance in the response increases as the distance,
I t-t"I,
from the
centering point increases. The relationship of t" to the time interval of interest will
determine whether or not a positive correlation is found for slope and initial status,
Le., whether or not 'tracking' is seen.
19
If to is in the interval of interest, more crossing over of lines will be seen, and if
to is 'far' (scale determined by Ie) to the left or right of the interval, fewer lines will be
seen to cross, and a positive or negative correlation, respectively will be found. Thus,
tracking will be seen for a data range where to is outside the interval of interest.
1.2.8 Notation for the Growth Curve Model
The following notation will be used when discussing growth curve models in
the next sections.
For a population of r individuals observed at the same n points in time, and
Yi
the vector of observations on the ith subject, the growth curve model is defined as
Y.=XR.+
e- I.,
-I
-~I
Y.IR.
""" N(XR.,
(72y),
-I~I
-~I-
in which
I!i """
and
N(f!, Q),
independently of
so that
where
y.
-
1
is a vector of the n observations on the ith individual
(nxl)
fJ·
_I
is a vector of unknown individual parameters,
(PXl)
~
is a known constant design matrix
fJ
is a vector of unknown population parameters
(nxp)
(PXl)
(72y
(nxn)
is a positive-definite, symmetric, within-individual covariance
matrix of ~ i'
is a positive definite symmetric covariance matrix of the
individual parameters, fJ_I.,
In some applications the assumptions of homoscedasticity and zero correlation
of within-individual error is made, so that (72y
= (72!.
20
1.2.9
Indices of Growth Separation
Two measures of growth separation, defined earlier as the extent to which
individuals' lines cross or change rank order, have been developed. Goldstein (1981)
creates an index without modeling the data.
Instead, an individual's path is
determined by the line segments connecting their n observations. Foulkes and Davis
(1981) first model the data and define their index base upon the expected curves of
the individuals.
For his index of separation, assuming all i=1, ... , r individuals are measured
at the same j=1, ... , n timepoints, Goldstein counts the number of times (mJ the
line segments connecting the measurements of the ith individual intersect at least
once with the line segments of the other individuals over the n timepoints.
He
defines, for a sample of N individuals, the proportion of pairs of individuals whose
paths do not cross in the interval' [tt, to]:
Thus, Goldstein's index is an estimate of the probability that two randomly
chosen individuals' paths will not cross in the interval [tt, to]. R=O when all possible
pairs cross, and R=1 when no crossings occur.
When the correlations between
occasions are zero, the probability of not crossing for a randomly selected pair of lines,
formed by connecting the successive measurements, is 2
for determining when R indicates real separation.
J-
1
'
which gives a lower bound
It is clear that as the number of
measurements, n, increases, the probability of lines crossing increases. It should also
be noted that the probability of lines crossing increases with the sample size, r, as well
as with the interval length, to -tt.
The standard error of R is computed by a
jackknifing technique, and can be used to construct confidence intervals about R.
.
21
Goldstein's separation index doesn't correct for short-term within-individual
variability. This may not be a problem in a study of tracking of a characteristic such
as height, but may cause considerable difficulty in an investigation of the tracking of
blood pressure or pulmonary function. Foulkes and Davis (1981), in a paper discussed
below, derive an analogous index that adjusts for inter-occasion variability by first
modeling the individual curves to get 'true' values at each point.
Goldstein's m i can be used as measures of separation for each individual, to
identify those who are growing differently, e.g., crossing many more lines than is
typical. However, as Goldstein points out, the individual separation measure tends to
be highly correlated with distance from the population mean-- those at the extremes
tend to cross fewer lines than those in the middle of the pack-- so it must be
interpreted along with information on the expected percentile ranking for that
individual.
Foulkes and. Davis (1981), who do not use the term growth separation,
considered tracking in the context of maintenance of relative rank over a given time
span, stating,
"Perfect tracking occurs when a group of individual growth curves do
not intersect; or equivalently when relative rank within the response
distribution is maintained over time."
For an interval [T H T 2 ], and the curves of any two randomly chosen
individuals, f(t,1! i)' and f(t,l!j)' they define an index of tracking as the probability
that the curves do not cross,
22
An estimator of the index, t(T 1 ,T 2), is the U-statistic for a sample of size N, given
below.
The established theory of U-statistics provides asymptotic normality and
variance estimation. In practice, this method assumes that a vector of observations,
r i = f(~,~ i)
+ ~i
is available, rather than the curve f(t,~ i)' The r i are observed at
the timepoints [t 1 , t 2 ,
••• ,
t n]. f(t,p.) is assumed to be a polynomial in t, the same
- I
degree for all individuals, and
-I
t(T itT 2 )
= (2) i~/,(ri,rj,Tl,T2)
,
where,
if
or
o
and ~'
= (1,t,t 2 , ••• ,t P ).
otherwise,
¢J is an indicator of whether or not the curves of the ith
and jth subjects cross in the interval.
When measurements are made at only two
points in time, this is equivalent to the probability of concordance.
The estimator can also be rewritten as
where
¢J.I is the number of times the curve of the ith individual crosses at least once
with the curves of the other r-l individuals, and could be used like the m·I in
Goldstein's separation index as a measure of how well the ith individual is tracking,
with the same warning as to its confounding with distance from the mean.
23
Foulkes and Davis emphasize the importance of defining the interval of
interest when evaluating tracking, since an attribute which tracks well through early
childhood may not track on into adulthood. They also note that -y(T 1,T 2) decreases
monotonically as the interval length increases. It should also be noted that the degree
of the polynomial affects the index of tracking.
Generally -y(T1,T1) decreases as the
degree of the polynomial increases (see Goldstein, 1981).
As an index of tracking, -y(T l' T 2) is easily interpreted-- it takes values from 0
to 1, and tracking is said to occur when the probability that any two randomly
chosen lines cross is less than ~,or equivalently when -y(T 1 ,T 2 ) ~ ~.
This index measures the same aspect of tracking as Goldstein's R, namely
growth separation.
By first modeling the data, and basing the index on the expected
values rather than the observed, it adjusts for measurement error, which may be
considerable in some applications. However, as noted above, the index will be affected
by the model selected, i.e., the degree of the polynomial.
Tracking will be said to
occur when the variability in the slopes or shapes of the set of curves is much less
than the variability in the intercepts or locations of the curves.
This index could be applied to evaluate tracking in repeated measures with
missing data, as long as the interval of interest, [T 1,T 2]' is covered by all individuals,
i.e., ti,l ~ T l' and ti,n. ~ T 2 for all i. Individual growth curves could be fitted to the
•
data, and the index evaluated over the interval of interest at the measurement points
of interest. For example this could be done for data from a purposefully incomplete
longitudinal design, where all individuals are measured at the first and last occasions,
with varying times of
me~urement in
between. This method would not be applicable
for only partially overlapping data, such as in a linked cross-sectional design, where no
single individual has data spanning the entire interval of interest. This would entail
extrapolation of the growth curves, which often has serious problems.
24
One problem with the index, is that it doesn't matter where within the
interval the lines cross, so long as they cross. Foulkes and Davis note, in response to
a letter in Biometrics (1983), that the absolute maintenance of relative rank
throughout an interval may be too stringent a requirement for tracking, since it
doesn't take into account the length of the interval over which rank is maintained.
1.2.10
Indices of Growth Constancy
Goldstein and McMahan developed measures of growth constancy, defined
earlier as the extent to which individuals vary around their expected percentile or
about their expected deviation from the population mean.
Goldstein measures growth constancy by first standardizing measurements at
each occasion, again assuming all i=1, ... , r individuals are measured at the same
j=l, ... ,n timepoints.
When the measures at each timepoint have mean 0, and
variance 1, the total variance is n(r-1). A measure of constancy for each individual is
2
S.
1
=j=l
En (y..
_
- y ..)
13
2
1
which measures how individuals vary about their own mean level, given by
y·.=aE
1
j=1
Yij' where the y .. are the standardized measures for the ith individual at the
v
jth timepoint~ The proportion of total variance n( r-1) not attributable to individual
variation around their own mean is
r
D = 1 - (r!l)n
}:S~ .
a=1
25
When all inter-occasion correlation is zero, D=~, so a modified index is
D_1
C - -" - 1
- I-A - -
that is, the average of the
1
r
~S2 - f
(r-l)(n-l)~ i -
(2) inter-occasion correlations.
,
This index of constancy
also uses the observed values without modeling across time, and does not correct for
interoccasion :variability.
The same warning applies as for the separation index:
adjusting for interoccasion variability may be more important for some applications
than others. McMahan derives comparable indices, discussed below, which adjust for
interoccasion variability, as well as handling data which are not initially standardized
at the times of measurement.
McMahan (1981) defined tracking in a manner related to the concept of
maintenance of percentile rank over time.
When the distribution of an attribute is
approximately normal at each timepoint, then maintaining percentile rank over time
is equivalent to maintaining a constant relative deviation from the population mean.
In McMahan's definition,
"Tracking occurs when the expected value of the relative deviation from
the population mean remains unchanged over time."
He analyzes tracking in the setting of a growth curve model, defined earlier,
where all individuals are observed at the same n occasions, and
(T2y
=
(T2!.
Perfect tracking is then defined by,
E(Y tJ.·1.8_ t.)
= p.J + K..t (TJ,.
i=l, ..., rand j=l, ..., n
where Pj is the population mean, and (Tj the population standard deviation at time j.
K. i is a constant for the ith individual, and E(K.i)=O, var(K.i)=l.
26
In matrix notation this is
1
1
where ~ is a diagonal matrix with elements ~ll = {?;)1, = {~Q~')1,. Perfect tracking
is
equivalent
to
a
restriction
on ·the
covariance
structure
in
the
model
In other words, the correlation between measurements is perfect, between all times of
measurement:
McMahan proves this is true if and only if Q is of rank 1. He also adds the
restriction that the non-zero elements of
~11
are of the same sign, where 11 is the
eigenvector corresponding to the non-zero eigenvalue of Q.
This added restriction
merely guards against the possibility of a complete reversal in direction of the relative
deviation, from
+lCiD'j
to
-lCiD'j'
which would not be consistent with tracking.
McMahan then defines an index of tracking, as the proportion of the total
variance,
l' ~~l, apart from
within-subject error, that is explained by tracking:
( - -l'~XDX'~l
----
-
--
0'~~l)2
( can also be thought of in terms of the expected sum of squared deviations
from tracking. Defining the deviation from tracking for the ith individual as
11·
-I
and minimizing ~~~i
Il!i
I fJ·
_I = XfJ·
- _ I - XfJ
-_ -
to obtain k i =
k·~l
1--
,
0' ~~lr11'~(~I!i
- )5.1!).
(can then be
seen as one minus the proportion of the total variance that deviates from perfect
tracking,
( =1 -
E(II~v .)
-1-1
l'~~l
27
( will be in the interval [0,1], but since it may be high in situations that do not
correspond to tracking (e.g., ~l)~' diagonal), a modified index is defined:
1'4(~l)~' - 4 2 )41
T _
- l'4(4U4' - 4 2 )41
T
can take negative values, but only when some covariances are negative, which
would not be likely when tracking occurs.
McMahan likens his analysis to a principle components analysis, but here the
first component is chosen by a subject matter consideration.
That is, what
proportion of the variability is in the direction of tracking?
When the variables are standardized by
1'R1
(p
T
and
where
p -
J3.
= 4-1~l)~'4-1
4- 1 ,
the modified indices are:
1'R1
= (-1'1)2 = -~2-'
1'(J3. - IH _ 1'J3.1 -
m
1'(U' - IH - m(m - 1)
is a correlation matrix.
Also,
Tp
= (2)
-1
LPi;
= p,
the
i<i
average pairwise correlation coefficient, corrected for within-subject error.
Tp
is analogous to Goldstein's adjusted measure of growth constancy, C, which
is the average pairwise correlation coefficient without adjustment for within-subject
error.
28
1.2.11
Applications of Tracking Indices
Rogosa and Willett (1983) applied both McMahan's and Foulkes and Davis's
methods to the same dataset with very different results, indeed drawing different
conclusions as to the presence of tracking.
In responding, both McMahan, and
Foulkes and Davis noted that the 2 types of indices will agree when there is close to
perfect tracking.
They also note that the index of separation does not take into
account the proportion of the interval over which ranks are maintained, and that this
is at the heart of the difference.
Dockery et al. (1983) indicate that they applied McMahan's definition of
tracking to data on pulmonary functions of children ages 6 - 11.
From a 7 year
study, using over 40,000 observations from over 13,000 children, they created
population percentile lines for forced vital capacity (FVC) and forced expiratory
volume in the first second (FEV l ), controlling for height, weight, age, race, sex, and
their interactions. To do this, they assumed the multiple observations on a child were
independent, claiming that the degree of underestimation of the standard errors of the
regression coefficients caused by this assumption were small, for this dataset.
They
said they then used residuals from this analysis for the 3,264 children with at least 5
annual visits to analyze tracking by McMahan's method.
It is unclear exactly how
McMahan's method applies to this data. It is also unclear what affect the assumption
of within-child independence has on the subsequent tracking analysis.
Using the tracking of height with age through middle childhood as a standard
of good tracking, they compared the tracking of FVC and FEV 1 with height, based
upon their own population standards, to the tracking of height with age, based on
NCHS standards, for the same children.
They found that FVC and FEV l show
tracking with height that is comparable in strength to the tracking of height with age.
29
They indicate that the strong degree of tracking for the pulmonary functions may be
less readily apparent in graphical comparisons, due to their greater measurement
error, as compared to height.
Hibbert et al. (1990) applied McMahan's index
T
to data on lung function of
healthy children to determine whether lung function tracks with age from middle
childhood through adolescence.
Additionally, they used Kendall's coefficient of
concordance, W, a measure of the degree of association between the rankings at each
of the timepoints, as another measure of tracking. Separate models were fit for boys
and girls in two cohorts each, beginning with mean ages 8.8 and 12.6 years. Each was
followed for 5 years. They found all lung functions to track well for these age groups,
as measured both by McMahan's
T
and Kendall's W.
However, the requirement for complete, balanced data over a 5 year period
meant that they could use measurements from only 226 children divided among 4
groups, out of a total of 568 children followed. A method that could include children
with 3 or 4 measurements over the study period would be advantageous. The authors
also note that lung function is highly correlated with height. A method for analyzing
tracking of lung function with respect to height may be more informative.
30
1.3
Mixed Effects Model Approaches to Incomplete Longitudinal Data Analysis
1.3.1
Introduction to the Mixed Effects Model
The analysis of data by mixed models is covered in texts such as Searle (1971,
variance components models only) and Hocking (1985).
The mixed model contains
both fixed and random effects, and is written as follows.
Assume each individual, i=1, ... , r is observed on n i occasions, for a total of
r
I: n .=N observations.
i=1
Let
Y i be the the vector of observations from the ith subject.
1
Y i is modeled as:
Y.=A
.• +B.d.+e.,
-1
-1-1-1
-1
in which,
i
Va cJ ]
{ ~i
=
[I)9 g]
In
~
2
i
and
is a vector of the n i observations on the i-th observational
unit
•
is a vector of unknown constant 'fixed' population parameters,
(PX1)
A.
- 1
(nixp)
d.
-1
is a known constant design matrix corresponding to the fixed
effects, t, in which ~ =[~~,... ,~W is of rank p,
is a vector of unknown random individual parameters,
(QX1)
B.
-1
(nixq)
is a known constant design matrix corresponding to the
random effects, cJ i ' in which lJ = [l}~, ... ,lJW is of rank q,
is a positive definite symmetric covariance matrix of Y .,
-
1
is a positive definite symmetric covariance matrix of the
random effects, cJ i '
is the within-individual variance,
o
(mXl)
I),
var(~i)
is a vector of variance components, 0"
elements of
plus ~2.
the g=1,... , m-1
31
Further, in many instances, it may be assumed that I} has a linear structure
given by
m-l
I}
where each
G,
=g=l
E D,G"
is a known constant matrix.
Yi has a linear covariance
Thus
structure given by
~i
m
= ViI}V~ + u2!ni =,=1
E D,Gi"
-
2
where G," =B.G,B~
- ,- -, for g=l, ..., m-1, and G_ ,'m= _In,.. and Dm =u •
1.3.2
Estimation
Methods for estimation of parameters for the
mixe~
model with unbalanced
data include maximum likelihood (ML) estimation, restricted maximum likelihood
(REML) procedures, moment estimators, and general linear model or
estimators.
ANOVA
Hocking (1985) discusses estimation by these procedures, and Harville
(1977) gives a comprehensive review of ML and REML procedures, along with
computational techniques. Laird and Ware (1982) discuss the Bayesian approach to
variance component estimation, and the application of the EM algorithm to
estimation.
Under assumptions of normality, the ML estimates of the fixed effects,
given by
i
-
= {tA~t~IA,}-IJtA~t~ly.}
'=1 li=1 - ,1- 1
and the asymptotic variance of
-
1
1
-
1
t is given by
asm. var[fl
=
{ I:
-1 }-1 •
,~~i~i
~i
'
t, are
32
An estimator of the random effects,
9.i'
is derived from an extension to the
Gauss-Markov theorem to cover random effects, and is given as (Harville, 1977; Laird
and Ware, 1982):
a.. = DB .t-:1(y. -I
--I-I
-I
A_ ;~_.
) •
•
For ML estimation of the variance components, closed form solutions to the
likelihood equations are not available. Therefore, iterative techniques must be used.
Under certain regularity conditions the ML estimators have the desirable properties of
being consistent, asymptotically normal, and efficient (Harville, 1977; Magnus, 1978).
However, as Harville noted, mixed models have several problems, including:
1)
computation requires solution of constrained non-linear equations, 2) estimates of the
variance components are not adjusted for the loss in degrees of freedom resulting from
the estimation of the model's fixed effects, and
3) the estimators are derived under
particular distributional assumptions for the parameters, which may be problematic in
some settings.
The particular form of the maximum likelihood estimators for the variance
components depends on the structure assumed for E..
- I
Jennrich and Schluchter
(1985) and Magnus (1978) consider a general covariance structure, where the
elements of
vector
~.
~i
are known functions of the unknown parameters contained in the
The ML estimates are obtained by solving the likelihood equations:
where ~i =
Yi
-
~it.
33
Fairclough and Helms (1984) and Andrade and Helms (1984) explored
maximum likelihood estimation for the general linear mixed model with linear
covariance structure under the assumption of normality. Under these conditions, the
log-likelihood function of the model is
The ML estimates of the fixed effects,
of the variance components,
~
t, are as above, and the ML estimates
are given by
where [()gh] is an m x m matrix with g,hth element the expression in ()gh' and [()g] is
an m x 1 vector with gth element the expression in
give the asymptotic variance and covariance for
asm.
.
var[~]
A
A
-
=
t
Og.
Fairclough and Helms (1984)
and ~ as
{t,
A·E.-1 A. }-1
~
A
~1-I-1
-I
Algorithms such as the Newton-Raphson, the Method of Scoring, and the EM
algorithm can be used to solve the likelihood equations.
Jennrich and Schluchter
(1985) review and compare the efficiency of the computational methods in terms of
rate of convergence, cost per iteration, and sensitivity to starting values. They found
that the EM algorithm is least expensive per iteration, but is most sensitive to a poor
starting value, and may not converge in a reasonable number of iterations.
34
The Newton-Raphson and the Method of Scoring are much more expensive per
iteration, especially the Newton-Raphson, but they always converged much more
quickly than the EM algorithm.
They recommend looking into hybrid algorithms,
perhaps starting with the Method of Scoring, and then switching to the NewtonRaphson
method.
Schluchter has developed software to analyze incomplete
longitudinal data via the mixed model, available in the program BMDP 5V.
Fairclough and Helms (1984) also compare the EM algorithm and the method of
scoring, with similar conclusions to those of Jennrich and Schluchter..
1.3.3
Hypothesis Testing
General results on hypothesis testing for the unbalanced mixed model are
limited. Linear hypotheses concerning the elements of
~,
the fixed effects, and
variance components, can be tested by likelihood ratio test statistics.
~,
the
However the
distributional theory is not well developed, especially for tests concerning the variance
components.
When assumptions are made on the covariance structure, other tests
have been proposed.
Andrade and Helms (1984) developed estimators and test
statistics for linear hypotheses on
covariance structure, Le., where ~.
- I
known.
~.
and
~
under the assumptions of a linear
= B. DB. + tr
,
- I
-
- I
2
In
-
i
= Em DgG. , and
g=1
- Ig
the G. are
- Ig
McCarroll and Helms (1987) developed several tests, including a REML-
based approximate F-test for general linear hypotheses on
~,
in the linear covariance
structure models, which they compare for a few incomplete longitudinal designs.
35
1.3.4
Example Applications
Fairclough and Helms (1984) used the mixed model to analyze data from a
longitudinal study of lung
functi~ns
of children. They assumed the lung functions of
interest were linearly related to height over the range of heights observed.
The
number of observations per child varied widely, as did the heights at which lung
functions were measured, resulting in a highly unbalanced dataset.
Four race-sex
groups were studied, and the modeled fixed effects, <!" were the slopes and intercepts
of the group growth curves, and the random effects,
intercept and slope for each child.
~i'
were the increments to the
They assumed a linear covariance structure and
estimated parameters using maximum likelihood methods.
They tested a variety of
hypotheses on race-sex differences in intercepts, slopes, and child-to-child variability,
using likelihood ratio test statistics and asymptotic chi-square approximations due to
Andrade and Helms (1984).
Laird and Ware (1982) also give an example of the use of a mixed model in
analyzing the effect of air pollutants on lung function development, where, as in
Fairclough and Helms, lungs are said to grow with height, resulting in an
inconsistently-timed dataset.
The fixed effects were the population intercept and
height coefficents, as well as effects of covariates such as pollution exposure, smoking
in the home, etc., and the random effects were the individual deviations.
They did
not assume any special form for the covarince structure, Q, of the random effects, and
used both maximum likelihood and empirical Bayesian approaches in the estimation.
Laird and Ware also use the mixed model to study the short term effects of an air
pollution alert on lung functions of children, as well as to identify the characteristics
of children who are particularly sensitive to pollution.
36
1.4
Statement of the Problem and Outline
The tracking analyses discussed in section 1.2 depend upon having all
individuals measured at the same set of timepoints, or else restrict interest to sets of 2
points at a time for computation of correlations.
This need for complete balanced data has meant that a great deal of data has
been discarded, restricting analyses to those with complete sets of observations (e.g.,
Dockery et al., 1983; Hibbert et al., 1990). It has also meant that a time scale such
as height which will result in inconsistent timing across individuals, could not be used
in a tracking analysis.
This present research will focus on developing an analysis of tracking that can
accomodate inconsistently-timed data, using a mixed effects model with a linear
covariance structure.
This model will also be able to accomodate datasets with
missing and/or mistimed data.
Mixed effects models are particularly well suited for handling data that are
unbalanced.
For analysis· of growth or change over time, mixed models may be
specified in terms of polynomial functions of time, where an individual's covariance
matrix depends upon the pattern of measurements over time.
problems arise for highly unbalanced
inconsistently-timed data.
data,
making
Thus, no special
this model practical for
In addition, the mixed model is based upon the explicit
definition of population and individual characteristics which makes it particularly
attractive for analyzing tracking, where interest focuses upon comparison of the two.
The mixed model is also useful for exploring different variance assumptions
and how they affect resulting indices of tracking without necessarily imposing a
37
structure on the variance which is too restrictive.
Tracking indices will be strongly
affected by the variance structure imposed by the choice of random effects,
~i'
and
by the choice of linear structure imposed upon I), given by the (;g.
The linear covariance structure can be very general, imposing no particular
pattern on the covariance by letting the i,jth and j,ith elements of the (; 9 matrices
equal 1, and all others equal zero.
where I)
For example, for a quadratic model with time,
6
=pEI Og(;g, the G- g matrices can be specified as
o
o
o
o
o
1
o
o
o
o
o
1
o
o
o
o
o
1
Other models for the covariance structure, such as autoregressive models, can
handle unbalanced data, but may impose a correlation structure which is too strict for
an analysis where the tracking indices are functions of the estimated variance
parameters.
In Chapter 2, the choice of definition for tracking in the mixed model is
discussed.
Tracking is then defined for this model, and several indices are proposed.
In Chapter 3, the estimation methodology for the tracking indices is developed.
Chapter 4 focuses on evaluation of the indices and how they are affected by the model
assumptions, along with comparing their use for several differerent designs for
measurement occasions. Chapter 5 presents an empirical application to illustrate the
methods. Chapter 6 provides a final discussion and suggestions for further research.
Chapter 2
DEFINING TRACKING IN THE MIXED MODEL
2.1 Choice of Tracking Definition for Unbalanced Data
One appropriate definition of tracking for unbalanced data is the one given by
McMahan (1981):
Tracking occurs when the expected value of the relative deviation from the
population mean remains unchanged over time.
This definition has several advantages. It effectively represents tracking very
simply, and this idea of maintaining a constant relative position within a population
distribution is the really the classic idea of tracking, as opposed to the idea of
projection along an individual trajectory.
This definition of tracking also coincides most closely with a visual impression
of tracking.
The index developed by Foulkes and Davis defines perfect tracking as
lines not crossing in a specified interval, but doesn't take into account the proportion
of the interval over which the lines remain in rank order. For example, consider the
graphs in Figure 2.1.
In both graphs the proportion of curves crossing within the
interval is identical, and would give identical results for the index of tracking of
Foulkes and Davis. But the visual impression of tracking is much stronger for 2.1a,
where the curves retain their rank order over most of the interval, and only cross at
the ends. An index of growth constancy rather than growth separation would better
distinguish the two situations pictured, resulting in a higher index of growth
constancy for the first.
39
Fi ure 2.1a:
150
All
140
130
~
u
120
~
5:<.:>
110
Gj
J:
100
90
80
9
7
5
AGE
Figure 2.1b:
Crossing in the middle of the interval
150
140
130
~
u
z
120
l-
I
'-'
LU
I
110
100
90
80
L.---I------L-
....L-
...L.-
.l.-
.L.----..J
5 7 9
AGE
Figure 2.1: Visual Impression of Tracking
All the lines cross in both 2.1a and 2.1b, but the visual impression
of tracking is much greater for 2.1a, where all crossing is early
in the interval.
40
What does one expect to see when visually evaluating a graph of a group of
growth curves for evidence of tracking? There are two main elements: 1) the curves
tend not to cross much over the course of the interval, and 2) there is a general
impression that the curves that are initially at the upper end of the distribution stay
there, and those at the lower end stay there.
In a sense, what is needed is an index of tracking that coincides more closely
with a visual impression of tracking.
Growth constancy, as in the example above,
seems to do this. Also, a high level of growth constancy implies a high level of growth
separation, but the converse is not true. It therefore seems that growth constancy is
the more appropriate measure to focus upon.
Tracking defined as growth constancy also has the advantage of lending itself
to analysis and interpretation as a function of the variance parameters of growth
curve models, which are frequently used to model such data.
The one drawback to
this definition is the requirement that the response be' normally distributed at each
measurement occasion, which may not be appropriate for response measurements
which are known to have a non-normal distribution.
This concept of tracking also has several advantages for use with unbalanced
data.
It is relatively simple to define in a mixed model setting which can handle
highly unbalanced data, and it does not necessarily require the projection of data over
the range of interest for all individuals.
When data are highly unbalanced it is
difficult, if not impossible, to compare individuals at a set of specified timepoints.
Rank order or percentile rank for an individual cannot be defined at any point
without first extrapolating or projecting data to the specified points. In other words,
some kind of modeling asumptions must be employed for inconsistently-timed data.
41
When tracking is defined as the proportion of growth curves not crossing over
a specified interval, as in the method of Foulkes and Davis, tracking can be estimated
for unbalanced data, as long as the data for all individuals span the interval of
interest.
However, when individuals have been measured over only partially
overlapping intervals this can lead to wide variation in results, depending upon the
assumptions and model used in projecting the data.
An example which illustrates the problems of defining and modeling tracking
for unbalanced data can be given using data on pulmonary function and height.
Figure 2.2a is a plot of log(FEF 2 5-7S) vs height, where the 2 measurements on each of
62 boys are connected by line segments. In this picture there is a visual impression of
tracking -- there is some crossing over of lines, but those at the bottom tend to stay
there, and those at the top tend to remain there, too.
In Figure 2.2b, the line
segments have been extrapolated to cover the full height range of interest for this
study.
It is. clear that most of the lines cross. within the inter:val, so that resulting
tracking indices will be low, and do not correspond with the initial visual impression
of tracking in Figure 2.2a.
This simple extrapolation of the lines also projects data
into regions where it is never seen, especially the very low projections for the low end
of the height range.
While it may not be obvious from Figure 2.2a, there is a
correlation between the initial height measurement and the slope of the line segment - the steeper slopes tend to come from the boys who were taller at the initial
measurement occasion.
When this correlation between slope and initial height is
taken into account by allowing a quadratic population curve, as illustrated in Figure
2.2c, the proportion of curves crossing is smaller than in Figure 2.2b, resulting in a
greater visual impression of tracking.
42
Figure 2.20
LOG(FEF25-75) AND HEIGHT
2.2
2
1.8
1.6
1.4
y:;-
"
1.2
I
I/)
N
b
I.J..
8
...J
0.8
0.6
0.4
0.2
0
-0.2
110
130
150
170
190
HEIGHT IN eM
Figure 2.2:
Extrapolation of Data and Visual Impression of Tracking
Inconsistently-timed data gives a visual impression of tracking in 2.2a.
In 2.2b, following, the extrapolated lines appear to cross more than
they do in 2.2c, where an added population growth acceleration has
been included in the extrapolation.
43
FIGURE 2.2b
EXTRAPOLATED LOG(FEF25-75) AND HEIGHT
4
3
-
2
Lt'l
,...,
I
Lt'l
('oj
lL.
L&J
lL.
'-'
<::J
0
...J
0
-1
-2
110
130
150
170
HEIGHT IN eM
FIGURE 2.2c
Extrapolated Curves, after adding population growth acceleration
4
-2
.
~--L-_--...L_
110
_- ' - - _ - - " -_ _- ' - - _ - - ' -_ _" - - _ - - L - _ - - - J L . . - - . J
130
150
HEIGHT IN CM
170
190
44
What may be less obvious, is the effect this differen t modeling has on an index
comparable to McMahan's r. In the first case (Fig. 2.2b), the individual lines are not
constrained to parallel any population trend, and an index such as r, measuring the
proportion of variance in the direction of tracking may be small. In the second case
(Fig. 2.2c), the individual curves are all constrained to follow the same quadratic
trend as the population curve, e.g., some of the variation is forced into the direction
of tracking, as seen both in the visual impression, as it would be in resulting tracking
estimates. This issue is discussed further in Chapter 4.
Tracking as the maintenance of constant
relative deviation from
the
population mean is, strictly speaking, a measure of growth constancy as defined by
Goldstein. Other concepts of tracking, such as growth separation, may also be useful
to consider in identifying individuals who are out of step, e.g., crossing many more
lines than others. While direct comparison of curves requires extrapolation over the
data .range of interest., growth separation may also be seen as a comparison of the
variance of the intercepts and variance of the slopes or shapes of a group of curves.
In a mixed model these pa.rameters are elements of {>, the covariance matrix of the
random effects. For example, consider a model for linear growth over time, individual
i measured at times til' ..., tin.'
with random effects for intercept and slope, i.e.,
I
and
(Jl is the variance of the intercepts across individuals, and
(J2
the variance of the
slopes. When the intercept is defined at the left endpoint of the data range, and the
variance of the intercepts is much greater than the variance of the slopes, i.e., (Jl
(J2'
>>
then the curves will be well separated, and tracking, in the sense of growth
separation, is seen.
45
In this special case of linear growth, the correlation of slope and intercept may
also be a parameter of particular interest when discussing tracking issues. This also
can be derived from I), as PIS
2.2
= ~.
~Ol02
McMahan's Tracking Index in the Mixed Model
The growth curve model used by McMahan, described in section 1.2.10 is a
special case of a mixed effects model with the added constraints that the design
matrix for the random effects must be the same as the design matrix for the fixed
effects, and must be identical for all individuals. The model can be rewritten in mixed
model notation as follows:
Assuming all i=l, ..., r individuals are observed on the same n occasions,
McMahan's model for the ith subject is
with
and
so that
Now l~t f!i =
f! +
~i' i.e., an individual's curve is equal to the population
mean curve plus individual deviations from the mean. Thus,
and
so that
~i .....
N(g, I).
46
Or by letting
Ii
=
t, the resulting mixed model is:
with
independent of
~i -
N(g, J),
so that
2.2.1
Defining Tracking in the Mixed Model
Following McMahan, perfect tracking is said to occur when an individual
maintains a constant relative deviation from the population mean,
~t,
aside from
random measurement error, i.e., when
in which "i is a constant for the ith individual, E("i)=O, var("i)=l, and ~ is a
1
diagonal matrix with (~}ll = (~J)~')~. Le., the 11th element of ~ is the population
standard deviation at the lth measurement occasion. McMahan's index is applicable
to evaluate tracking -- this is merely a restatement of his model.
Following
McMahan, this definition of tracking is equivalent to a restriction on the covariance
structure:
47
This, McMahan shows, occurs if and only if Q is of rank 1, with the added
restriction that the non-zero elements of
~2'1
are of the same sign, where
eigenvector corresponding to the one non-zero eigenvalue of Q.
2'1 is the
This restriction
guards against the possibility of a complete reversal in direction of the relative
deviation from
+"i
to
-"i'
which satisfies the above relationship, but would not
represent tracking.
2.2.2
Individual Design Matrices
The next step is to generalize the model to allow for individual design
matrices,
with
~i
~i'
as follows: For an individual observed at points til'...,tin ., let
I
and 4i as above, so that
Tracking may be defined in this model as above:
tracking occurs when an
individual maintains a constant relative deviation from the population mean.
This
should be true regardless of the measurement occasions, and can be stated as:
1
in which 4i is a diagonal matrix with (~)Il = (~iQ~:)~. i.e., the 11th element of 4i is
the population standard deviation at the lth measurement occasion for the ith
subject, 1=1, ..., ni' for all i=l, ..., r individuals.
48
Theorem
2..:.l: Tracking defined as a constant relative deviation from the population
mean is equivalent to a restriction on the covariance structure:
A.DA~
- 1 - - 1 = ~ol1'~
_1
1..
£I22f:
The proof follows directly from McMahan.
Proof
~:
1.
A.DA~
= var(~.)~.l1'~.
= ~.11'~
..
-1--1
I
-1-- -I
-1---1
2.
Proof
$:
Let (A.d.).
be an arbitrary element of A.d.
with non-zero variance, and let
-I-I)
-I-I
denote the vector of remaining elements of A
od o.
(A.d.).,
-I-I)
-1-1
If ~il)~~ =~in'~i' then (rearranging ~i' ~i so jth element is last)
A.do)'J
- I -I )
Var(A.d.) = Va{(
- I-I
(A.d.).
- 1- I
=A'DA~=[
-
I -
-
I
)
( ~.).,.,
- I ))
(~
( ~.) .. ,
- I JJ
(~
-
.).,.
))
I
.) ..
- I
JJ
Then Var[(A.d.)
·,I(A.d.).J
= (~.)
.,.,-(~.)
.,.(~ .)~~(~.)
.. , .
-I-I)
-I-I)
-I))
- I ) ) -IJJ
-IJJ
49
Now,
2
tT il
(sym.)
~.11'~.
-t-- -t
=
2
tTin.
t
=
so that (4i)jJ'
("i-1 x "i-I)
2
tT in.
t
as
4 i ll'4 i
above, with the jth row and column removed,
and
Thus,
(~.) .,.,-(a.) .,.(~ .)~!(~.) ..,
-tJJ
-tJJ -IJJ -IJJ
=
1
-
tT~.
IJ
50
Rewriting (A.d.).
as the mean plus a relative deviation:
- I-I J
( A.d.).
-I-IJ
= (A.O)
+
-1-
It·fr··
IIJ
.
=
It·fr ..•
IIJ
(A.).,.,
-IJ3[
] + (A)
= E[(A.d.).,
0-]-1
-I-IJ
. , . It·fr··IIJ
-
I
JJ
(A.).,.,
= E[(A.d.)
.,] + -dijJJ
-I-IJ
[It.fr ..
]1
IIJ-
= E[(A.d.).,]
+1t.(A.).,.,l
-I-IJ
1-IJ3That is, ~iQ~~ = ~in'~i implies a constant relative deviation for all j=l, ..., n
Theorem
U:
i.
The condition for perfect tracking refers to Q: Perfect tracking occurs
if and only if Q is of rank 1.
4i n' ~i is of rank 1 since the vector 1 is of rank 1 (product has minimal
rank), so that for
~i
of full rank, Q must be of rank 1.
51
Using the eigenvalue, eigenvector decomposition, there exists an orthogonal
matrix
r
so that
l) =
and diagonal matrix ij such that
Therefore,
r'l)r
= ij. If
l) is of rank 1 only h ll i=
0,
hllI11~·
~il)~~
=
hll~iI11~~~
1
=
1
(h~1~iI1)(h~1~iI1)'·
!
Thus, the 11th element of (~il)~~)2, (¢.i)ll = Ihll~iI1111 ' so that
1
! has elements t,=
1 if
(h~1~iI1)' >
-1 if (h~1~iI1)'
and
!
It,l = 1 if (hfl~iI1)' =
If all non-zero elements of
to It,l for zero elements, then! =
0
1
<
0
o.
~i11
are of the same sign, and this sign is assigned
±!, so that ~il)~~ = ¢.ill'¢.i .
Thus the conditions for perfect tracking for McMahan's model hold, even
when allowing for individual design matrices.
52
2.2.3
A More General Model
So far it has been assumed that the design matrix for the fixed and random
effects are identical, Le., that for the model
Allowing
~i
Yi =
~it
+
lJi9i
+
~i' ~i
= lJi for all
and lJi to differ gives much more flexibility in modeling the
variance structure, although this can greatly complicate the interpretation of resulting
tracking parameters. This will be discussed in more detail in Chapter 4. Typically,
the columns of lJi will be a subset of the columns of
~i'
or vice versa, although other
structures are possible.
For this model,
Y i "'" N(~it, lJi1)lJ: + 0'21). Tracking, defined as above as
maintenance of a constant relative deviation from the population mean can be stated
as:
1
in which ~i is a diagonal matrix with (A}II - (~i1)~:)~, Le., the 11th element of ~i is
the population standard deviation at the Ith measurement occasion for the ith
subject, 1=1, .•., ni' for all i=1, ..., r individuals.
Theorem
2.:.3.:
This statement of tracking is equivalent to a restriction on the
covariance matrix structure:
B.DB~
= A.U'A
..
-1--1
-1---1
Theorem 2.4: The condition for perfect tracking is on 1): Perfect tracking occurs if
and only if l) is of rank 1.
53
Proofs of these statements hold as in the previous section, substituting
A..
-
I
~i
for
Thus, tracking as maintenance of a constant relative deviation from the
population mean has been defined for the mixed model situation.
2.2.4 Indices of Tracking in the Mixed Model
As described in Chapter 1, section 1.2.10, McMahan defines an index of
tracking as
which can also be written as
,=
where
~i
1
E(v~v
.)
-1-1
1'441
is the vector of deviations from tracking for the ith individual:
v·IQ·
-I~I
=
(AQ.
- AQ)
-~I
-~
k.~I,
1--
and k i is the average relative deviation from the population mean for the expected
values of the ith individual. k.I is obtained by minimizing v~v·I{J·,
resulting in k.I
-1-1.1
=
A modified index is also defined, adjusting for a situation that could give high
values for' but does not correspond to tracking. This occurs when ~l)~' is diagonal.
The modified index,
T,
is defined as:
54
where
In mixed mod~l notation, where
as above, noting only that
~i
Yi
= ~t + ~~i
+ ~i' the indices are defined
and k i may be rewritten as
v·ld.
-I -I
= Ad.
--I
- k.a1,
1--
and
For the model
Yi =
~it
+
!Ji~i
several indices of tracking are proposed.
+
~i'
allowing for highly unbalanced data,
McMahan's tracking index is defined as a
proportion of the total variance that is in the direction of tracking.
In the model
defined above, it is not as simple to think in terms of a total .variance for the system,
but the model does give an estimate of I), the covariance matrix of the individual
deviations from the population parameters, and the condition for perfect tracking is
on I).
Based on I), the model allows for the estimation of the population standard
deviation at any timepoint across the range of the data, i.e., the elements (a i ), of
where
4i
4i,
is a diagonal matrix of the square roots of the diagonal elements of !JiI)!J~ .
Thus a tracking index can be based upon the estimates of I), and the resulting
estimates of the population standard deviation at any timepoints of interest.
Two types of indices are proposed. The first type is based upon the observed
measurement occasions and (weighted) averages of these across individuals, numbers
(2) and (3) below. The second type is defined by first adjusting to a selected set of
'test' points for each individual and then computing the indices, as in 1) and 4) below.
One version, denoted by , for each type, is the expected tracking, based upon
E(~~~i)' the expected squared deviations from perfect tracking.
by p, is based upon the observed squared deviations, ~ ~~ i'
The other, denoted
55
The following indices are proposed:
(1)
(t
l' 4tl}tQIJ{4 t l - 1
=
0' 4t4 t !)2
where l}t is an appropriate design
-
matrix containing a set of 'test' timepoints across the interval of interest.
The number and spacing of the points would be dependent upon the
average number and spacing of the observed measurement occasions for
the individuals in the dataset, or else could be a pre-specified set of points
across the range of interest.
It would also be possible to choose several
sets of test points and to either average or otherwise evaluate. the
distribution of the resulting set of (t's.
E(v~v. )
l' ;.'~1. , where l} i is the design. ma:trix for the
1
-1-1-
random effects for the ith individual. This is the expected tracking for an
individual with the same set of measurement occasions as the ith
individual. An overall index could be defined by a weighted average of the
r
(i' (=
1
f
L
i=1
for all i,
r
w i (·, where
1
(b)
n·
N'
L
i=1
w.=I, and the weights, w., could be either (a)
1
1
which weights for the number of measurement
occasions for i, (c) depend upon the proportion of the interval of interest
spanned by the observations of i, or (d) some combination of (b) and (c).
(3) Pi = 1 -
v~v.
,-1-1
1
.!l
..!l.1
-1-1-
,the observed tracking for the ith individual, based upon
v·ld.
-1 -1
and
k.1
=
B.d.
- k 1..!l.I,
-1-1
-1-
= (1'.!l ..!l.1)-11'.!l.B.d ..
-
-1-1-
-
-1-1-1
•
56
k i is the average deviation from the population mean of the expected
values for the ith individual, at the observed measurement occasions. v.
is
-1
the nix! vector of deviations of the individual's expected values at the n i
measurement occasions from the individual's nearest tracking contour.
These are illustrated in Figure 2.3.
The Pi can be used as individual measures of tracking, and can be
evaluated relative to the expected tracking for an individual with the same
set of measurement occasions,
'i' defined in (2) above.
A weighted average of the Pi' p =
r
E w.p., can
i=1
1
be defined for a measure of
1
overall observed tracking, with weights defined as for', above.
(4)
Pti = 1
v'.v '
,-tl -tl ,the 'test' tracking for the ith individual, based upon
! '~h4t!
vt,ld.
= Btd,
- kt·~tl,
-1-1
--1
1--
and
kti is the average deviation from the population mean of the expected
values for the ith individual, at the test set of points.
~ti
is the ntxl
vector of deviations of the individual's expected values at the nt selected
measurement occasions from the individual's nearest tracking contour.
These are illustrated in Figure 2.4.
As for the Pi' a weighted average of the individual measures can be used
as an overall measure, defined as Pt
r
=i=l
E w .Pt', with the weights defined as
1
above.
1
57
•
Mean
---------
o
o
Observed data points
Individual predicted curve
-
Line of average deviation from mean
(closest line of perfect tracking)
- Deviation from treKking
at observed measurement occasions
Figure 2.3:
Defining the nearest tracking contour at the
measurement occasions.
58
o
Mean
,/'//t /t(/'"
1,/,//'
O--T
~:j::--------t-
_____________________
to
o
Observed data points
Individual predicted curve
-
Line of average deviation from mean
(closest line of perfect tracking)
projection of predicted line
Deviation from tracking
at selected 'test' measurement occasions
Figure 2.4:
Defining the nearest tracking contour at the
selected 'test' measurement occasions.
59
2.2.5
Adjusting for Zero Correlation
Ideally a tracking index, such as (t or (i' should have high values representing
tracking, with (t
=1
for perfect tracking, and (t
=0
for no tracking.
simplicity, (t will be used, though the same results apply for (i' substituting
tit
(For
tli
for
in the discussion below.)
To define a modified index with the desirable properties:
Let
"It
be the value of
(t
when tltQtI~ is diagonal. This is the case of zero correlation
between measurements made at the timepoints represented in
l'4- t (tit Qtlf)4-tl
0' 4- t 4-tl)2
tit.
When diagonal,
l'4- t (4-:)4- t l
0' 4- t 4- t l)2
1'4-:1
0' 4-:1)2 Since this situation represents zero correlation between measurements, it is
one possible definition of 'no tracking.' Following McMahan, an adjusted index of
tracking can be defined as:
This
rt
~as
the desired properties:
= 0 when tltQtI~ is diagonal:
and rt
= 1 when tracking is perfect:
60
Perfect tracking implies !\tl)!\~
= ~t!1'~t
(Theorem 2.3), so that
_ l' ~t(~t!1'~t)~t1
(1' ~t~t1)2
_ (1' ~t'~h1)(1'~t~t1) - 1
(1'~ ~ 1)2
-,
-
Thus, r t
= 11 --
""t
""t
- t - t-
= l.
rt can take negative values when ·the covariances between some timepoints are
negative. This is a consequence of defining 'no tracking' as zero correlation between
measurement occasions. While it is easy to define perfect tracking, departures from
tracking can occur in many ways. Another definition of 'no tracking' could be regular
shifts between positive an,d negative correlations.
negative value for rt as defined above.
This situation would result in a
This seems acceptable, since it is not a
.situation that represents tracking.
The modified index can also be written as
which is another way of seeing the modification of Ct as an adjustment for the case of
zero correlation between measurements at the selected timepoints.
matrix for the times
represented in
Vt
is simply ~~ for
The covariance
zero correlation.
61
t3tl)t3{
This quantity is subtracted from the covariance matrix
from
the
covariance
under
perfect
tracking,
~tll'~t'
in the numerator, and
in
the
denominator.
Equivalence is demonstrated below:
1'~tCt3tl)t3{)~tl/0'~t~tl)2 - 1'~:1/0'~:1)2
1 1'~:1/0'~~l)2
l' ~t(t3tl)t3D~tl - l' ~:1
0' ~:1)2 - l' ~:1
l' ~t(t3tl)t3{)~tl - l' ~t(~n~tl
0'~t~tl)2 - 1'~t(~:)~tl
T
i
can be defined, as
Tt
above, substituting
t3i
for
t3 t. and
~i for ~t.
T
can
then be defined as a weighted average of the T i , as ( was, of the (i.
The adjusted versions of the tracking indices,
Tt
or
T
i
,
are of particular
interest since they can easily be shown to be a weighted average of all pairwise
correlations:
62
where·
(sym.)
2
(fin.
I
o
o
o
o
o
Evaluating the numerator:
l'A.(B.DB~-A~)A.l
=
-I -1- -I
-I
-1-
-
·0
o
(sym.)
o
(f.
In·I
63
Evaluating the denominator:
o
o
(sym.)
o
Thus,
(T.
In·I
T·I -
This index of tracking can be seen as a weighted average of correlations
between all pairs of measurement points j,k, weighting by the variances at j and k.
This makes some intuitive sense, in that when tracking is high, we might expect order
to be maintained at those times. of measurement where the spread among individuals
is greater, but that more crossing might occur where the variablility among
individuals is small.
'i
is less easily interpreted as a function of variances and correlations. In the
same manner as above, it can be shown that
64
One advantage of the first 2 indices, (t and (, and their adjusted versions
and
T
proposed above, is that they can easily be shown to generalize to McMahan's
indices, ( and
T,
in the special case where
To show that
(t
= ( and
Tt
=
square roots of the diagonal elements of
tJ i
T,
= ~i = ~ for all i:
choosing
tJt
~,
as
gives
tJtl)tJ{ = ~l)~', or 4 t
4t
= 4.
containing the
Thus,
!'4tmtl)tJ{-4~)4t!
!'4t(4t!!' 4t-4~)4t!
and
_ !'4(~l)~'-42)4! _
- !'4(4!~'4-42)4! -
= /'
To show that ,
/'. , " and
4,
Tt
T·
=T
when B.
-I
T
= A. = A for all i, note that ~. =
-I
-
the diagonal matrix of square roots of the diagonal elements of tJil)tJ~
Thus,
so that any weighted average, (, of the (i=( would be equal to (,
and
T·I
!'4i(tJil)tJ~-4~)4i!
= !'4
(4 n'4 -4~)4i!
i
i
i
- !'4(~l)~'-42)4!_T
-!'4(4n'4-4 2)4! and thus
T
=
T.
.
-I
= ~l)~'.
65
Adjusted versions of the individual observed indices, can also be defined:
Pai
= p.-w.
_'__
I
I-wi'
and
with adjustment factors wi and
"It
defined as above.
The average adjusted indices
would then be
r
Pa =
E w·Pa·,
i=l '
r
Pat = i=1'
E w'Pat', .
and
2.2.6 Tracking Indices based upon Observed Values
Thus far the tracking indices discussed have been based upon the expected
values for individuals, and measure how well the expectations track, rather than how
well the observed values track.
To do this would be the same as including within-
individual variability, 0'2, in determining the contour lines for tracking. It is possible
to define tracking indices similar to those defined above that take into account the
within-individual variability.
In such a case, the covariance structure between
observed values for an individual is ~il)~~
+ 0'21n.,,
rather than ~il)~~ , which is the
covariance between the expected values. Thus, the following indices for tracking of
observed values may he defined:
66
and
where
~t
is a diagonal matrix containing the square roots of the diagonal elements of
Similarly,
,- and
r-
are then defined as weighted averages
of'i
and
ri :
r
,- = E w·e,
i=1 • •
and
r
-= i=1 w.r.,
• •
r
£J
~
with the weig"hts, Wi defined as for' earlier.
Similarly, versions of the individual tracking measures, p., and Pa " and their
•
•
averages, P, and Pa, can be defined which include within-individual variability in the
formulation. For the 'observed' indices these are:
67
. -,
,
p~-w~
l-w~'
Pat -
p
p~
and
k~
,=
-
•
.
w.p. ,
i=l ' ,
=
r
E w.p·.,
i=1 ' a,
.~)
-, d.
-, =- (Y-,.-A-,-
and
-,
r
~
=.f..J
v~1
where
~~
.
,
-
k~~ ~1,
,-,-
(l'~~~~l)-11'~~B.d
-,-,-,-,-,..
-
.
and w; are defined as above.
Individual indices based upon the test points cannot be defined in a similar·
manner, since there are no observed values at the test points.
The observed
deviations from tracking are based upon the difference between the observed values
and the population mean,
Yi-~it,
and their difference from the expected tracking
level. With no observations at the test points, this difference cannot be defined.
2.2.7
Parameter Dependence on Measurement Occasions
While many articles in the literature have made it a point to note that
tracking is interval dependent so that specifying a range of interest for the data is
crucial, the dependence of tracking on particular measurement occasions has not been
made explicit. In this section we assume that Q is fixed; Le., that Q is not a function
of HiH~l' the times of observations.
68
In section 2.2.2 above it was shown that the condition for perfect tracking is
on I), the covariance matrix of the random effects.
tracking, (., and
,
T.,
The parameters of degree of
however are actually functions of B
.D B~, the covariance matrix of
-'--I
the individuals' expections at the measurement occasions j=l,... , n i . The parameter,
or index of tracking, depends upon the actual observation points, and not only the
range covered by the data. It is possible, however, to define situations in which the
parameter is independent of the particular observation points.
Theorem 2&:
.fIQQf:
T
-
1 -
l' ~l(llll)ll~-~n~ll
l' ~l(~ll1'~l-~n~ll'
Substituting these in
Tl
above gives:
69
When the correlation between all points is constant, then
T 1 =T2
the measurement occasions. Letting Pijk=P, for all i=l,... , rj
regardless of
j,k=l,... , n i
•
= p.
Thus,
T 1 =T2=P,
regardless of the number and spacing of the observations.
When the variance is not constant across measurement occasions, then for
and
:e2' with :ell):e{ '#
correlation, p.
:e21):e~
:el
, the tracking index is still the interoccasion
Ci or Ct , which include the sum of the squared variances,
would not be equal when the variances differ across time, even when the
interoccasion correlations are all p.
It is important to note that it may be impossible to find a I) matrix that will
satisfy this situation.
If in fact the correlation is constant between all
possible pairs of measurement occasions, then another model rather than the
mixed effects model, that will estimate the single correlation parameter is
more appropriate.
The mixed effects model, however, does a good job of
finding the I) that will best approximate this, for the measurement occasions
represented in the data.
There are two special cases worth noting.
70
~
1: "Perfect Tracking."
Q
When
42l!'42'
is of rank 1,
:l:hQtJ{ = 4lH'41>
so that
Thus,
T2=1.
Tl=T2,
so that
even though
Tl=l,
:tJlQ:tJ{
and tJ2QtJ~
=1= tJ2QtJ~.
=
This
occurs when the correlations between all measurement occasions are one, but
the variance is not constant across measurement occasions.
~ ~:
"Zero Correlation between observed measurements occasions."
Zero correlation between measurement occasions means that
and also, :tJ2QtJ~
=
tJlQtJ{ =
4~,
4~. Again, this can occur when the correlation between
any two observations is always zero, but the v~riances differ, so that 4~ =1=
4~ when
There
tJl =l=tJ2'
are
also
In this case
other
Tl =0=T2'
possible
structures
for
the
covariance
between
measurements that, when coupled with special sampling schemes such as equidistant
measurements, would yield tracking parameters that are independent of the exact
measurement occasions. An example would be an AR(l) structure. But any cases for
which this is true are all of stationary processes.
Such cases are better fit by other
models rather than random effects or mixed effects models.
However, most of the
examples of tracking analyses in the literature deal with real time processes, where
not only the distance between measurements, but also the exact measurement
occasions are important.
For example, it is not expected that the growth of height
with age, or lung function with height is the same between ages 6 and 8 as between 8
and 10, so that a mixed effects growth model might be appropriate for this type of
data.
It is also important for a tracking analysis not to impose a particular
correlation structure on the data that in effect imposes a particular tracking
structure.
71
This dependence of the tracking parameter on the exact measurement
occasions is problematic.
It means that even for studies with complete, balanced
data, if the measurement occasions differ between studies, then the underlying
parameters estimated from the studies are different. Great variation in results across
studies could have as much to do with differing sampling schemes across time as with
differences in the individuals included in the studies.
However, the difference between 2 studies where the sampling across time is
identical, one with complete data, and the second including individuals with data
missing at one or more points, is a different issue. Although not all possible pairwise
correlations will be included in the formulation of
Ti
for an individual missing some
-data, the estimate of Q and the within individual error should be improved by
avoiding the loss of information from excluding individuals with incomplete follow-up.
This loss of data can often be extreme, as in the work of Hibbert et al. (1990) where
less than half of the children followed in their study were included in the final
analyses.
T
t could be defined as the tracking parameter for the complete set of
observation occasions, and would be equal to
T
data. When some individuals are missing data,
Ti
T
when all individuals have complete
would be a weighted average of the
's produced by the different patterns of missing data, weighting by the number of
individuals with that pattern.
The
Ti
computed from data patterns of individuals
with missing values would be missing some components of all possible pairwise
correlations.
The analysis, in other words, would be weighted towards correlations
between those occasions with more observations.
When considering the problem of inconsistently-timed data, the mixed effects
model may be a good compromise choice for modeling an unknown covariance
structure.
However this
poses some problems for
an
analysis of tracking.
•
72
For a fixed
Q,
is of rank 1.
it is not possible that tJiQtJ~ =
tJjQtJj
when
tJi#tJ j , except
Murray and Helms (1990) show this for the case of
tJ i
when
Q
containing an
intercept and covariate.
But with no fixed determination of where to sample across an interval, this
model is at least a good first approximation.
The Q matrix estimated should be a
good compromise so that the average of all estimates of tracking,
r
T=L: T.,
i=l
different observation patterns, as well as the variability among the
information on tracking across the interval.
Ti
for the r
1
should give
This will not hold up well when the
observations for many individuals are close together and do not span the interval of
interest.
When data are merely mistimed, the Q estimated from the mixed model
should be a good approximation across the different patterns of observations across
the interval of interest.
Examples to illustrate the above points concerning missing data, mistimed
data, and inconsistently timed data will be given in Chapter 4.
Chapter 3
ESTIMATION
The tracking indices proposed in Chapter 2 are functions of the variance
components of the Mixed Effects Model, so that the estimators of the indices will be
functions of the mixed model variance estimators.
The first section of this chapter
gives a brief review of estimation for the Mixed Effects Model.
In the second section, tracking estimators are defined and their properties
discussed. In addition, jackknife estimators of the tracking indices are derived so that
confidence intervals can be computed for the degree of tracking.
3.1
Estimation in the Mixed Effects Model
The generalized Mixed Model is Yi -
Chapter 1.
~it
+
lli~i
+
~i'
defined as in
To review:
Assume each individual, i=l, ... , r is observed on n i occasions, for a total of
r
E n .=N observations.
i=1
Let
Yi
be the the vector of observations from the ith subject.
I
Y i is
modele~
as:
y.
= A- 1.•-I
in which,
+ B.d. + e.,
-I-I
-I
74
and
is a vector of the n·I observations on the i-th observational
unit
•
is a vector of unknown constant 'fixed' population parameters,
(PX1)
A.
is a known constant design matrix corresponding to the fixed
effects, t, in which ~ =[~~,... ,~Q' is of rank p,
- 1
(ni XP)
is a vector of unknown random individual parameters,
d.
-1
(QX1)
B.
- I
is a known constant design matrix corresponding to the
random effects, ~i' in which :D = m~,
,:DQ' is of rank q,
E.
is a positive definite symmetric covariance matrix of Y i'
...
(niXQ)
- I
(niXn i )
is a positive definite symmetric covariance matrix of the
random effects, ~i'
D
(QXQ)
is the within-individual variance,
var(~i)
is a vector of variance components, 9
elements of Q, plus CT 2 •
"
9
(mXl)
the g=l,... , m-l
. Further, it may be assumed that Q has a linear structure given by
m-l
Q =
where each
G,
E
g=l
9I
G"
is a known constant matrix.
Thus
Yi
structure given by
E·
-I
= B.DB~ +
-1--1
CT
2
I- n I.
m
="
LJ (J,G_
1=1
I•
"
has a linear covariance
75
Under these assumptions the following maximum likelihood (ML) estimators
are defined:
For the fixed effects,
~
A
t,
.-1 y. ),
= ~
~ A·E.-1 A· )-1~
~ A·E.
·=-1 -
and for the variance components
.=-1 -
•
I - I
-
I
€ = [° 1 ,
•••,
I - I
-
I
Om-I' .,.2], assuming a linear covariance
structure,
•
so that
mel.
I} =
E O",G"
g=l
and
The ML estimators of the variance components are not adjusted for loss in the
degrees of freedom resulting from the estimation of the model's fixed effects, so that
they are not unbiased estimators.
Fairclough and Helms (1984) discussed another
problem with estimation of the variance components. Neither the method of scoring
nor the EM algorithm constrains
Q to
be positive definite.
In practice, they found
non-positive definite estimates of I} to occur when I} was nearly singular, or when the
within-subject variability,
B.DB~.
- 1- - I
.,.2In i ,
was large relative to the between subject variability,
76
This is of great concern in tracking analyses, where the tracking parameters
are functions of the variance. Perfect tracking implies perfect correlation:
so that 1) may indeed be close to singular for data that tracks very well.
In this
situation non-positive definite estimates of 1) are quite likely. One tactic suggested to
handle this problem was to adjust the model so that 1) is non-singular. For example,
in a model with fixed and random effects for intercept and slope, if the variance of the
slopes is close to zero, a model with no random effect for slope could be fitted.
However this in effect forces the slopes to be parallel -- in other words, forces perfect
tracking.
The ,same situation can arise with a quadratic model. If the variance of
the quadratic term is close to zero, a model with random effects for the constant and
linear terms only can be fitted.
However, this again forces the individual curves to
follow some aspect of the population curve, with resulting higher estimates for
tracking. This is discussed further, and illustrated with some examples in Chapter 4.
Another tactic suggested was to forego the linear covariance structure and
.
use 1) =
r.., .
E cjicji JD
•
place of 1) =
i=l
m-l.
E (JgGg·
This may be a more reasonable approach
pi
for a tracking analysis.
When the within-subject variability,
CT 2 I n .,
-
is large relative to the between
I
subject variability, ~i1)~~, along with causing difficulties in estimation, it may be
reasonable to question whether it is appropriate to say that an attribute tracks. This
might occur, for example in a study of the tracking of blood pressure with age.
this case it may be better to rethink the analysis.
In
This issue was touched upon in
section 2.2.6, where tracking parameters that are not adjusted for within individual
data are defined.
77
3.2. Tracking Index Estimators
The following estimators of the tracking indices can be defined:
,
A
A'
A
1.:1.(B.DB.).:1.1
-I -1--1 -I(1''&.'&.1)2
'
-1-1-
A'
,
-
i -
and
where *i is the diagonal matrix containing the diagonal elements of l.liQl.l~ , and
1f.&~1
A
Wi
__
,-
= 0' *~!)2·
The overall index estimators are then defined as
r
, =i=1
Ew.'.,
A
A
I
and
A
T
=
I
r
~
A
.l.JW.T.,
i=1
where the wi are weights defined as either
I
I
1
a) f for all i, or
b)
n·
N'
i=l, ..., r.
't and Tt can be defined as 'i and T i above, substituting l.lt' the design
matrix for the random effects for the set of test points, for l.li'
An estimator for the observed tracking for the ith individual is defined as
p.= 1 A
I
where
v. =
and
k.1
I
B.a.
- k 1..&.l,
-I-I
-1-
= (1 f .:1 ..:1.1)
A
-
A
-1-1-
-1
f
A
A
1- .:1.B·d
..
-1-1-1
i/V.
I
I
A'
1.:1 ..:1.1
- I - 1fA
78
The overall observed tracking estimator is
.
p. = L",w.p.,
r
~
i=l
I
I
with the weights, wi' defined as above.
Similarly, an estimator for the individual tracking at the set of test points is
defined as
where
and
The mean of the individual tracking estimates at the test points is
r
Pt = i=l
Ew,pt"
I
I
with the weights, w.,
defined
as above. It is interesting to note that with the weights
I
,
defined as ~ for all i~dividuals, this becomes
Some algebraic manipulation can show that this is similar to the estimator for (t,
defined as (t
However, the terms in
~t
ii~.iit·, the
i=l
I
_ r.. ,
•
of Pt are based upon Q=E 9i 9 i rather than upon Q
i=l
m-l.
numerator
I
•
=pE l (JgG g , while the 4 t
are the
•
79
same in both estimators, as the square roots of the diagonal elements of
cannot quite be written neatly in the same form as
't,
lhl):tJf·
Pt
but one basic difference
between the two is the partial use of different estimators for I).
In practice, the
estimates tend to be fairly close.
The estimators of the individual indices adjusted for zero correlation at the
measurement occasions and at the test points respectively are:
p.-w.
Pai
= --wi
I.',
'
and
The mean of the individual adjusted
indices are simply
r
Pa
=i=1
E woPao,
'
,
and
Estimators for the indices that do not adjust for within individual error are:
and
i-.
••
... -w.
' ,
- l-w~'
,
where
is
the
diagonal
matrix
containing
the
diagonal
elements
of
80
The overall index estimators are then defined as
•
C
r
1
t· =
and
•
= i=l
Ew·C,
1
r
E w.t~,
i=l
1
where the wi are weights defined as either
a)
1
Jfor all i, or
b)
n·
N'
i=l, ..., r.
For the indices at selected points, the estimators are:
and
=
.
;;.. .
1 -Wt
.• ,
.. t -"It
For the ith individual, the observed tracking indices based upon observed
values rather than expected values are:
where
and
v~1
= (Y- I·-A- 1.~)
-
k~1
= (1'.&~.&~1)-11'.&~B.a ..
-
-1-1-
wi is defined as above.
-
k~.&
~1,
1-1-
-1-1-1
81
The overall observed tracking estimators are
p.
r
=i=1
~ w.p~ ,
I
pi
and
I
r
=i=1
~ w.p·. ,
I
al
with the weights, wi' defined as above.
The ML estimators from the mixed effects model are asymptotically normal
with asymptotic variance given in section 1.3.2. The expected tracking estimators (i
and
T.,
I
as differentiable functions of these estimators, are ML estimators and will also
have asymptotic normal distributions.
However, beyond the work of deriving the
variance of the estimators, it would be another matter to determine when the sample
size is large enough for the asymptotic results to hold.
It is quite likely that
extremely large sample sizes would be necessary, since the tracking estimators are
functions of the p(p+1)/2 variance components, where p is the number of random
effects estimated. Instead, a resampling technique, the jackknife, is developed for this
model, which is effective as well as being computationally feasible. In the next section
jackknifing estimation is described and developed for the tracking estimators.
3.3 Jackknife Estimation
Jackknifing is a technique that can be used to get an estimate of the variance
of the tracking parameters.
By dividing the sample into subsets and repeatedly
computing the parameter, with each subset deleted in turn, an estimate of the
variability among .these estimators can be computed.
confidence limits for the tracking estimators.
This can be used to compute
82
Briefly, the technique can be described as follows:
Divide the sample into n groups of k observations, where nk=N, the total
number of observations.
Letting f
t
denote the estimator of
Tt
using all N
observations, and f t.-i the estimator when the ith observation is deleted, the jackknife
estimator of T t can then be defined as
1 n _
Tt
= Ii i=1
E
T
t .t. ,
where
If the diagonal elements of
4- t are positive and finite, then
,.." N(O,I)
as n becomes large. Thus the variance of Tt is estimated by
n
...1...1 E
n- i=l
{T t i-Tt)2, and
•
confidence intervals on the index of tracking may be computed.
In many applications k=l, so that one observation is deleted at a time and a
total of N estimates of Tt,i are computed. However this may become computationally
time-consuming,
especially
when
an
iterative
computation of Tt,i requires the re-estimation of
proocedure is
necessary.
Q for each group of size k
The
deleted, so
that the EM algorithm or the scoring procedure must be run a total of n times, where
n is the number of individuals.
Another approach is to re-estimate
Q using
defining
_
D .
- -t
r-l.. ,
=';=1-'-'
E d.d.,
with the ith individual deleted, i = 1, ..., r.
the random effects estimators, ~i'
83
However, this doesn't take into account the change in the fixed effects that
result from the deletion of an observation, or group of observations. This may be a
reasonable approach, however, when the number of individuals is very large, so that
continued re-estimation using an iterative algorithm becomes too costly on the
computer.
For the examples and applications considered in the next chapters, N is
considered to be the total number of individuals, N:=r, and one individual is deleted at
a time, so a total of r procedures are run after the initial full data model. The final
parameter estimates, ~ and
€from the full data model, are used as the starting points
for each run. This should require only a very few iterations at each run, since deletion
of a single individual should not have a large effect, in most cases.
Another tactic, when the number of individuals is large, would be to delete
groups of individuals at a time, keeping the number of groups, n, large, since the
asymptotic· normality depends upon this.
However, this has some problem for
inconsistently-timed data, since it would be possible to delete individuals with similar
observation occasions as a group, making the variability larger.
Iteration time for the EM algorithm seems to increase more rapidly with the
number of observations per individual than with the number of individuals (this is
based upon general impression, rather than a formal study). So the appropriate tactic
for jackknifing will depend as much upon the number of observations per individual as
the number of individuals in the study.
Programs for jackknife estimation have been developed, and the technique is
applied to the data example presented in Chapter 5.
Chapter 4
MODELING ISSUES
In this chapter some of the issues involved with the modeling of data for
tracking analysis are explored.
In the first section some of the effects of under- or
over- fitting a model on the resulting tracking parameters are investigated. The next
section discusses the effects of not including within-individual error in the formulation
of a tracking index, and in section 4.3 some exploration of the effects of the spacing of
observations across an interval of interest is undertaken.
Randomly generated data are used to illustrate and investigate some of the
characteristics of tracking estimates in the mixed model.
These are not intended to
be formal simulations in any sense, but are merely a few examples to explore the.
effects of different models and different designs for measurement occasions.
In the examples that follow, the data were generated based upon the model:
1
where
IJ.,
is a vector of randomly generated standard normal deviates, and ( )2 is the
Cholesky root. The standard normal deviates were generated using the SAS function
RANNOR.
The fixed effects,
~,
number of subjects (50 in most examples), and
number of observations per subject (3--5) were chosen to resemble data used in
preliminary analyses of log(FEF 2 5-7S) with height in centimeters.
individual error,
(1"2,
The within-
was made very small in most examples, since this decreases the
computer time rather drastically. Details of the particular parameters used are given
in the relevant section.
85
4.1 Under- or Over- Fitting a Tracking Model
The tracking indices, as functions of the covariance matrix of the random
effects of a mixed model, are particularly sensitive to the appropriate specification of
the random effects.
In the discussion that follows, under-fitting means fitting a
polynomial model with too few parameters, i.e., of not high enough order; and
ove7'-
fitting means attempting to fit too many parameters, i.e., a model of too high an
order.
Under-fitting the random effects can have an especially large effect on the
tracking indices. By allowing less variability, some tracking is forced into the model.
That is, by forcing individual curves to follow some aspect of the population curve,
the individual curves are made more similar and resulting tracking estimates can be
greatly inflated.
For example, constant, linear and quadratic fixed effects could define the
population intercept, slope and growth acceleration with time, while constant and
linear random effects would allow individual deviations in slope and intercept, but
assume the same growth acceleration for all individuals.
This model may be
appropriate in some settings, but in others could over-estimate the degree of tracking.
For the case of constant and linear fixed effects and only constant deviations
the inflation of the tracking estimate is particularly clear. In this case Q consists of
the variance of the constant deviations only, and is, of course, of rank 1. This implies
perfect tracking. An example follows.
Data for example 4.1.1 were generated to have normally distributed intercept
and slope deviations for 50 individuals, with three observations per individual, based
on parameters given in Table 4.1.1. The 'heights' were generated to have a uniform
distribution across the interval (0,50), using the SAS function RANUNI to represent
heights ranging from 100 to 150 centimeters.
The covariance structure was
deliberately chosen to give poor tracking for the set of test points, spread at 25
86
centimeters across the interval of interest.
This can be seen in the resulting low
tracking estimates for the adjusted indices, Pat and f t , for the set of test points listed
in Table 4.1.1.
Two models were fit to the data:
(a) the correct model, with fixed
and random effects for intercept and slope, and (b) a model with random effects for
intercept only.
Details of the models are given in Table 4.1.1.
Figure 4.1.1a
illustrates the individual predicted lines from the model including individual deviations
for slope and intercept.
In model 4.1.1b the individual lines are allowed to have
different intercepts, but are forced to parallel the population curve.
In this case no
lines can cross, and individuals must maintain a constant deviation from the
population mean, so of course tracking is perfect, as illustrated in Figure 4.1.1b.
87
TABLE 4.1.1
UNDER-FITTING: FORCING PERFECT TRACKING
Parameters used for Randomly Generated Data
~=[.2171
-
D-
.020J
.015
[
-.0006 ]
.000045
-.0006
MODELS
Model 4.1.1a:
i1
ModeI4.1.1b:
Yi
ht ]
= 11 ht
i2
[ 1 ht
[ 4>int
J+
4>slope
i3
Parameter Estimates
i=[
i=[
Model 4.1.1a:
ModeI4.1.1b:
-
.2169 ]
.0200
• [ .0140
D-.00058
-
.1866]
.0213
0=[-0142J
·.00058 ]
.000047
0- 2 =.000011
0- 2 =.00806
Tracking Estimates
~ 2Jl
Based 2Jl observed heights
Model
4,1.1a
4.1.1b
P
.829
1
C
Pa
.688
1
f
.835
1
Pt
.685
1
100,125,150 gn
.644
1
Ct
Pat
.194
1
.642
1
f
t
,189
1
88
FIGURE 4.1.1a
Predicted Lines from Model with Random Effects for Slope and Intercept
1.6.----------------------------,
1.5
1.4
1.3
1.2
1. 1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Ol---.t:.---------------------------l
-0. 1 L-..-10.L..0------l----1..J.2-0-----L..----14L..0-----1---J
Height in cm
FIGURE 4.1.1b
Predicted Lines from Model with Random Effects for Intercept Only
1.5 . . . . - - - - - - - - - - - - - - ' - - - - ' - - - - - - - - - - - - - - - - - ,
1.4
1.3
1.2
1. 1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Of--__...."e....------------------------j
-0. 1 '----10'-0-----'-----1......2-0-----'-----1....L.- - - - - - L . . . - - - - - J
40
Height
In
cm
Figure 4.1.1: Forcing Perfect Tracking
4. 1. 1a show predicted lines from the full model, and 4.1. 1b from the
model with random effects for intercept only.
89
Tracking estimates will be greater when individuals are forced to follow some
aspect of the population curve. This is less obvious but just as true for a model with
constant, linear and quadratic fixed effects, and only constant and linear random
effects.
This is case is illustrated in Example 4.1.2.
Data were generated to have
constant, linear and quadratic and fixed and random effects, for 50 individuals all
observed at the same 5 points: 100, 112.5, 125, 137.5, and 150 centimeters.
models were fit to the data:
Four
(a) the full model with constant, linear and quadratic
fixed and random effects, (b) a model with constant, linear and quadratic fixed effects
and only constant and linear random effects, (c) a model with constant and linear
fixed and random effects, and (d) a model with constant and linear fixed effects, and
constant, linear and quadratic random effects.
Models (a) and (d) contain the
random effects for the quadratic term, and models (b) and (c) do not.
The
parameters used to generate the data, along with the models fit and resulting
parameter and tracking estimates are given in Table 4.1.2.
It is clear that leaving the random effects for the quadratic term out of the
model results in an increase in the tracking estimates. Depending upon the size of the
quadratic term as well as its correlation with the other terms, the increase in tracking
can be small, or rather dramatic. In this example, the models with only constant and
linear random effects, (b) and (c) showed increases in the adjusted indices from
Pa=f=.374 in the full model (a), to
Pa~.56
and f=.49 for models (b) and (c).
Predicted lines from the 4 models are given in Figure 4.1.2 so that a visual impression
of the degree of tracking can be evaluated.
-,
90
TABLE 4.1.2
UNDER-FITTING A QUADRATIC CURVE
Parameters used for Randomly Generated Data
t
=
.217J
.020
1)
= [ .. 099629 -.002956 .0000197]
[ 0000
.0004383 -.0000072
'sym.
1.438E-7
MODELS
4.1.2a:
1
t
t~I4Jll
: : : 4J2
[ 1 .,. t~ 4J a
Yi =
1
t~
4.1.2b:
Yi - [
:
+
+ e·
-I
q2
= .00001
91
TABLE 4.1.2 continued
UNDERFITTING A QUADRATIC CURVE
Parameter Estimates
Model 4.1.2a:
t = .[.217~
.0200
Q =[
sym.
0000
t
Model 4.1.2b:
~
.2173~
= .0170
0001
Model 4.1.2c:
.099413 -.002991 .0000136]
.0004535 -.0000077
~- = [.1710J
.0237
"If
Model 4.1.2d: . ~ = [.2170]
.0200
-
Q = [.09407
-.002095 ]
.0000737
0'2 = .01733
-.002062 ]
.0000724
0'2 = .01940
.099412 -.002991 .0000135]
.0004535 -.0000078
sym.
1.702E-7
Tracking Estimates
Model
P
Pa
,
T
4.1.2a
4.1.2b
4.1.2c
4.1.2d
.503
.659
.667
.482
.374
.559
.570
.345
.503
.607
.609
.482
.374
.493
.495
.345
~ 2l!
0'2
= .0000104
1.592E-7
D = [.09530
Q= [
•
observed heights: 100,112.5,125,137.5,150 em
0'2
= .0000104
92
2.2
2
Figure 4.1.2a:
Quadratic Data and Model
.---------~--------------------,
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
oH~~~~~---==:::::::::="""~------i
-0.2
-0.4
- 0.6 l.--J10L.-O----JL.-----J12L.-O----JL.-----14l..-0:-------L....-----l
Height in cm
Figure 4.1.2b:
Underfitting
2.--------------::.----------------'---.....,
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
O.~ U2~~~::::::::====------J
-0.2
-0.4
- 0.6 L..--J0L.-0---....!----1...L.2-0---~---14.l-0------L....-----l
1
height in cm
Figure 4.1.2: Example of Underfitting
4.1.20 shows the full model. 4.1.2b shows the predicted lines from a model
with a quadratic population curve, and slope and intercept deviations.
Individual curves are forced to follow the population growth acceleration.
93
Figure 4.1.2c:
Underfitting
2.--------~~---=-=---------=:-:------~
1.8
1.6
1.4
1.2
1
0.8
0.6
O.4LJ~~=-------J
o
0.2
-0.2
-0.4
- 0.6
0-0---....L-------:1~2-=-0---..l...------:-14-:-0:::------l.-----...l
L--l
1
height in em
Figure 4.1.2d:
Quadratic Random Effects
2.2....---------=--------------------,
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
OH~~~~~___==~~-----___l
-0.2
-0.4
- 0.6 1.---l0~0----L----1....1..2-0---...I..----1.J....4-0- - - . . 1 - - . - - - '
1
height in em
Figure 4.1.2 continued: 4.2.1c shows predicted lines from a model
with no quadratic terms, and 4.1.2d from the model with quadratic
random effects, but a forced zero population quadratic term.
.
94
The next example is an attempt to see what happens when a model is overfitted.
For example 4.1.3, data were generated to have linear fixed and random
effects for 50 individuals, each observed at the same 4 heights. Parameters are listed
in Table 4.1.3. Three models were fitted to the data: 4.1.3a) the 'correct' model with
linear fixed and random effects, 4.1.3b) a model with linear fixed effects and quadratic
random effects, and 4.1.3c) a model with quadratic fixed and random effects.
One clear indication that a model may be over-fit is the failure to converge in
a 'reasonable' number of iterations.
Model 4.1.3a converged in 4 iterations while
model 4.1.3b took 215 iterations to converge. Model 4.1.3c failed to converge in 250
iterations, and was not pursued further.
The interesting result for tracking
estimation, however, is the very close agreement in the tracking estimates for models
(a) and (b). The 'extra' variance components that were estimated were all very close
to zero, and had no real impact on the tracking estimates. In addition, the estimates
of the within individual error,
0'2,
changed very littl~ with the additional .terms in the
model. This may be one way of determining when a polynomial tracking model has
been properly fit:
a large increase in the number of iterations required to fit the
model, without any real change in the tracking estimates, or any decrease in the
within-individual error would indicate overfitting.
Of course there are many other
ways to over-fit a model, but failure to converge in a reasonable number of iterations
is often a good clue that something is wrong.
Table 4.1.3.
Results for this example are given in
95
TABLE 4.1.3
OVER-FITTING EXAMPLE
Parameters used for Randomly Generated Data
~- = [.217J
.020
Q
=
.015 -.0006]
[ -.0006 .000045
(T2
= .00001
.
4.1.3a:
Parameter Estimates
4.1.3a:
4.1.3b:
~- = [.2170]
.0200
~- = [.2166]
.0200
f>
=[
-
_
.0120 -.000495l
. .0000423J
0'2
= .00000985
-.000485 -1.53E-7]
.0000414 9.19E-9
[.0119
Q=
0'2 = .00000958
3.056E-12
Tracking Estimates
~ 2!!.
Model
4.1.3a
4.1.3b
observed heights: 100,112.5,125,137.5,150 cm
Pa
.662
.663
.433
.434
f
.661
.663
.433
.433
96
4.2 Including Within-Individual Error in the Tracking Parameters
The tracking indices considered thus far, like the ones defined by McMahan
(1981), evaluate how well the expected values track -- that is, how well the expected
values for individuals maintain a constant relative deviation from the population
expected values.
But it may not be reasonable to say that an attribute tracks well
when the within-individual variability is large relative to the between individual
variability. In such a case, the expectations may track well, but it would still not be
easy to predict an individual's future level relative to others based upon a few earlier
measurements.
In section 2.2.6 a set of indices was defined based upon the covariance matrix
of the observations, E.=
B.DB~+(7'2In.,
rather than upon the covariance matrix of
-I
-1--1
I
the individual expectations. When the within-indvidual error is small, the difference
between the resulting tracking indices is small, but as (7'2 increases relative to the
variance of the intercepts or relative to the variance at the observed measurement
occasions, the tracking indices based upon
:eil):e~.
~i
decrease relative to indices based upon
This difference can be important to the interpretation of how well an
attribute tracks.
A good example is found in the next chapter, with the tracking of the
pulmonary functions FVC and FEF 2 5-75 with respect to height. For FEF 25_75 , the
within-individual variability is about the same size as the variability of the intercepts
for both boys and girls.
This attribute exhibits near perfect tracking when the
expectations are evaluated, but only poor to moderate tracking when withinindividual error is included.
For FVC for the girls, the within-individual error is
about four times the size of the variability of the intercepts. While the expectations
are found to track very well, when (7'2 is included in the indices, the conclusion is that
FVC does not track with respect to height.
97
The issue of adjustment for
,,2
also comes into play when considering the
effect of smoothing caused by modeling assumptions, such as assuming a normal
distribution across the indivdual's slopes (discussed further below), or when a model
has been poorly fitted (e.g., underfitted).
variability is assigned to
u2 ,
In the case of underfitting, more of the
which is then adjusted 'out' before tracking is estimated.
This can be seen in example 4.1.2, where the estimates of
u2 go from -.00001 when
the 'correct' model is used, to -.018 when the random effects are underfit.
4.3
Exploration of Designs for Measurement Occasions
This work has been aimed at developing an analysis for tracking of
inconsistently-timed observations using the mixed effects model. So far the evaluation
of the indices has focused upon some of the
effe~ts
of the mixed model structure. This
section looks at the variability introduced into the model when different designs for
the timing of the observations are considered.
First, a comparison of the tracking estimates from the different designs will be
made, and then some evaluation of the differences among the indices estimated from
the same design will be considered.
For this set of examples, data were randomly generated using the same seed
for each design, with the following parameters:
~=[.2171
-
.020J
Q=[
.015
-.0003
-.0003 ]
.000045
,,2=.00001
98
For each example, data were generated for 50 individuals, with 5 observations
per person, except for the last, which is explained below. The following designs for
measurement occasions were considered:
Design a:
Inconsistently-timed data -- observations were generated to have a uniform
distribution over the interval (0,50), representing heights from 100 to 150
centimeters , using the SAS function RANUNI.
Design b: Complete balanced data -- observations were generated for each individual
at 12.5 centimeter intervals from 0 to 50, representing data from 100 to
150 cm in height.
Design c: Mistimed data -- some 'noise' was added to heights that were evenly
spaced at 12.5 centimeters from 0 to 50, by adding a uniform random
variable on (-2,2). Thus the initial height was between -2 and 2, the
second between 10.5 and 14.5, etc.
Design d: Complete data for Q!l individuals. with additional individuals with some
missing ~ -- data were generated as for design b, with the addition of
15 individuals with 4 of the 5 observations selected at random, and 10
with 3 of the 5 selected at random.
The last design. was chosen as a way of evaluating the effect of missing data in
a complete design.
Rather than randomly deleting some observations from the
original 50 individuals, an additional 25 individuals were included with only 3 or 4 of
the 5 observations.
This method was selected since it is more often the case, in
practice, that a complete design is intended, but the analysis carried out only on those
individuals with complete data.
This comparison looks at the effect of including
additional individuals with incomplete data.
Figures 4.3.1--4.3.4 show the predicted lines from a single example of each of
the four designs considered.
99
1.9
r-----..:...--------------------------,
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
01----=-"'::::::::.----------------------1
- O. 1 L-..-lOLO------l.-----12LO-------L-----14.l.-0-----:..L----l
Height in em
Figure 4.3.1:
Predicted lines from an examples with inconsistently-timed data.
2.2.---------------------------,
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
Ol-~:::::::::::::=-----------------___l
- 0 . 2 '---::-"10':-:0=-------'-----:-12~0:-------L...----1-L..4-0----L.....--l
Height
Figure 4.3.2:
In
em
Predicted lines from an example of complete, balanced data.
100
2. 2 ~
:..-Fi:..:::g!..:::u~re=--4.:..:•.:::3.:..:.3:..:a:.::.-:P~r:....:e:.:d:.:.i c.:..t:.:e:..:d~lin~e.:..s=---:.f:..-.ro:;,..m..:....:......_m_i s.: . . t:;,. m
i _e_d_d_at_a:...-_ _--,
2.1
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1. 1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
O. L----I.-_ _~~_.....L..__ _
d
.l.....:_--L...--.J
. . L __ _..l___ _..l___ _
90
110
130
150
Height in em
1. 7
1.6
Figure 4.3.3b:
Mistimed data. showing predicted values
.....--~--=-------:--------------------__,
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
OL---l------J..------L-_---JI....-_ _. l - -_ _...L-_ _...I-_ _-l...-----'
90
110
130
150
Height in em
Figure 4.3.3: Predicted lines from an example with mistimed data.
4.3.30 shows all 50 lines. while 4.3.3b shows 10 lines along with predicted
values, to give an idea of the degree of mistiming allowed.
101
2.2...------------------------------,
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
01-----1I!!:::::=-------------------------1
- 0.2 L-.....I10'-0----L-----1-1..2-0---....L...----1..L40-------I-----J
Height in cm
Figure 4.3.4: Predicted lines from an example of complete plus
additional incomplete data.
102
Due to limitiations on computer equipment along with time considerations, a
half dozen examples of each design were run, and the results are compared below.
For each example run, eight tracking indices were estimated. These are four indices
based upon observed measurement occasions: p, the average across all individuals of
the observed tracking; (, the average across all individuals of the expected tracking;
and their adjusted versions, Pa, and
selected set of heights.
The other four indices are based upon a
T.
In this case the heights used matched the complete design:
100, 112.5, 125, 137.5, and 150 cm.
These indices are Pt' the average observed
tracking at the test heights; (t, the expected tracking at the test heights; and their
adjusted versions, Pat' and Tt, respectively.
Tables 4.3.1--4.3.4 report the results of the six runs for each of the designs.
Note that for design b, complete balanced data, there is no difference between the
tracking for the observed heights and selected heights, since the heights were identical
for all individuals, and were the same as the selected heights. Therefore only one set
is reported.
For the indices that are averaged across individuals, the variances of the
individual estimates were also computed.
A) = 1(
f.. (A A)2 .
F or examp1e, var( Pi
l \~ P .-p
r- )j=1
'
These can give an indication of how the design for measurement occasions affects the
variability across individuals, as well as determining which of the indices are most
affected by the design. The distribution of the variances of the individual indices for
the six runs for each design are also summarized in Tables 4.3.1--4.3.4, which follows.
103
TABLE 4.3.1
Summary OT Tracking Estimates Tor Inconsistently-Timed
Data (Design a)
Tracking
Index
Minimum
Range
Maximum
Mean
Std Dev
,r
pa
0.853
0.799
0.857
0.804
0.915
0.883
0.913
0.881
0.0624
0.0843
0.0563
0.077
0.882
0.838
0.886
0.844
0.0236
0.0325
0.0199
0.0269
var(Pi)
var(Pai)
var ('i)
0.0156
0.03
0.0027
0.0058
0.0859
0.166
0.0076
0.0149
0.0703
0.136
0.0049
0.009
0.0402
0.0765
0.0046
0.0093
0.027
0.0511
0.0018
0.0036
0.771
0.665
0.770
0.665
0.868
0.815
0.867
0.815
0.097
0.150
0.097
0.150
0.815
0.736
0.815
0.736
0.035
0.0522
0.0351
0.0523
0.0261
0.0508
0.0632
0.126
0.0371
0.075
0.0479
0.098
0.0141
0.0297
P
yarer i)
var(pu)
var(Pati)
TABLE 4.3.2
Summary OT Tracking Estimates Tor Complete Data
(design b)
•
Tracking
Index
Minimum
Maximum
Range
Mean
Std Dev
--------------------------------------------------P
pa
,r
0.835
0.762.
0.835
0.762
0.905
0.859
0.905
0.859
var(Pi)
var(Pai)
0.009
0.019
0.0342
0.0714
0.0701
0.097
0.0702
0.097
0.0252
0.0525
0.869
0.808
0.869
0.808
0.0271
0.039
0.0271
0.039
0.0205
0.0439
0.0102
0.0214
---------------------------------------------------
104
TABLE 4.3.3
Summary of Tracking Estimates for Complete Data with
Mistiming (design c)
Tracking
Index
Range
Mean
Std Dev
Minimum
Maximum
T
0.840
0.769
0.843
0.773
0.905
0.858
0.906
0.860
var(Pi)
var(Pai)
var«i)
var(Ti)
0.0097
0.02
517E-7
0.0001
0.035
0.0728
0.0002
0.0003
0.0254
0.0529
0.0001
0.0002
0.0208
0.0443
0.0001
0.0002
0.01
0.021
426E-7
0.0001
0.842
0.772
0.842
0.772
0.907
0.862
0.907
0.862
0.0658
0.0902
0.0658
0.0902
0.869
0.809
0.869
0.809
0.0255
0.0368
0.0256
0.0369
0.0085
0.0189
0.0326
0.0676
P
pa
(
var(pti)
var(Pati)
0.869
0.065
0.0887 0.808
0.0638 . 0.869
0.0876 0.808
0.024
0.0487
0.0208
0.0446
0.0254
0.0363
0.0251
0.0363
0.0101
0.0213
TABLE 4.3.4
Summary of Tracking Estimates for Complete Data with
Additional Incomplete Data (design d)
Tracking
Index
Minimum
Maximum
Range
Mean
Std Dev
T
0.818
0.711
0.820
0.719
0.912
0.853
0.905
0.844
0.0937
0.142
0.0858
0.124
0.861
0.771
0.859
0.769
0.0358
0.0586
0.0343
0.0573
var(Pi)
var(Pai)
var(C i )
var(T i)
0.009
0.0305
0.001
0.0034
0.0513
0.204
0.0027
0.0291
0.0424
0.174
0.0016
0.0257
0.0299
0.108
0.0018
0.0117
0.0151
0.0669
0.0007
0.0092
Pt
Pat
Ct
Tt
0.815
0.735
0.815
0.735
0.904
0.856
0.904
0.856
0.0885
0.121
0.0885
0.121
0.860
0.794
0.860
0.793
0.0332
0.0464
0.0332
0.0464
var(pti)
var (Pati)
0.0118
0.0263
0.0482
0.099
0.0364
0.0729
0.0265
0.0571
0.0136
0.0276
P
pa
(
105
More direct comparisons of the estimates resulting from the different designs
are given in Figures 4.3.5--4.3.8.
These plots compare the distributions of the
adjusted tracking estimates from the six runs of each design.
Since the tracking indices are functions of the measurement occasions rather
than the range covered by the data, the complete balanced data will be used as a
standard for tracking at the specified heights, against which the other designs are
compared.
The focus for the comparisons will be upon the adjusted indices, since
these have a more clear interpretation as weighted functions of the interoccasion
correlations, as well as having clearly defined values for perfect tracking and the
absence of tracking (defined as zero correlation among values at the given
measurement occasions).
Looking first at Pa in Figure 4.3.5, it is clear that' the estimates from the
inconsistently-timed data are a bit high relative to the complete data, and the
estimates from the complete plus incomplete data are lower and more variable.
Figure 4.3.6 compares the variability of the individual adjusted indices, var(Pai)'
across the 6 runs. It is clear that the individual indices are more variable for the first
and last design. The same can be said of the expected tracking, f, shown in Figure
4.3.7, and the variance of the individual indices, var( f i)' in Figure 4.3.8. f tends to
be high and more variable from the inconsistently-timed data and low and more
variable from the complete plus incomplete data.
This is easy to explain when thinking of the indices as weighted averages of all
pairs of interoccasion correlations. For the inconsistently-timed data, it is possible for
individuals to have clumps of data closely related in time (or height). The individual
predicted values at these points close in time would be more highly correlated than
pairs of distant observations. It is also possible to have a clump of observations and
one
distant
one
(e.g.,
observations
at
100,
102,
103,
105,
and
148
cm).
106
~
1r _
Pa=rE Pai
i=l
I
1 +
I
I
I
0.9 +
I
I
I
0.8 +
I
I
I
0.7 +
Design
+-----+
I + I
+-----+
+-----+
+-----+
I
I
I
*--+--*
+-----+
I
*--+--*
+-----+
I
I
+-----+
I
I
+
I
I
+-----+
I
----------+-----------+-----------+-----------+--a b c
d
Inconsistently
Timed Data
Complete
Data
Mistimed
Data
Complete +
Incomplete
FIGURE 4.3.5:
Comparing Four Designs
Distribution ofpa: Mean Adjusted Observed Tracking
var(Pai)
I
0.3 +
I
I
I
0.2 +
I
I
I
0.1 +
I
I
I
o
+
Design
I
I
I
I
I
+-----+
+-----+
I
+
I
I
+-----+
*--+--*
+-----+
I
I
I
I
*--+--*
+-----+
I
----------+-----------+-----------+-----------+--a b c
d
Inconsistently
Timed Data
Complete
Data
Mistimed
Data
+
Complete
Incomplete
FIGURE 4.3.6: Comparing Four Designs
Distribution of var(p~i): Variance of Adjusted Individual Observed Tracking
107
1r _
T=rE
T·
i=l '
A
I
1 +
I
I
I
0.9 +
I
I
I
0.8 +
I
I
I
0.7 +
+-----+
*--+--*
+-----+
I
+-----+
+-----+
+-----+
I
I
*--+--*
+-----+
I
I
I
*--+--*
I
I
I
+-----+
+-----+
----------+-----------+-----------+-----------+--a b c
Design
+
I
I
I
Inconsistently
Timed Data
Complete
Data
Mistimed
Data
d
Complete +
Incomplete
FIGURE 4.3.7:
Comparing Four Designs
Distribution of f: Mean Adjusted Expected Tracking
I
o
0.03 +
I
I
I
I
I
I
I
+--+--+
0.02 +
I
I
I
0.01 +
I
I
I
o+
Design
I
+-----+
I + I
I
I
+-----+
I
*--+--*
*--+--*
----------+-----------+-----------+-----------+--a b c
Inconsistently
Timed Data
Complete
Data
Mistimed
Data
d
Complete +
Incomplete
FIGURE 4.3.8:
Comparing Four Designs
Distribution of var( Ti ): Variance of Adjusted Individual Expected Tracking
108
In this case the individual index would be an average of 6 pairs of high correlations
among the closely spaced observations, and 4 low correlations, between the distant
observation and the others.
In general, it seems that the clumping of data has the
largest effect, resulting in a higher value for the tracking indices, as well as more
variability across individuals as the pattern of measurement occasions varies widely.
The other factor which can lead to more variability across individuals as well
as higher indices is the interval over which the individuals have data.
Looking at
Figure 4.3.2, the predicted lines for the complete data, it appears that most of the
crossing of lines occurs in the first half of the height range, between 100 and 125 cm.
Individuals with data spanning this range only, might be expected to have lower
individual indices; however, this is offset by the smaller distance between observations
and resulting higher correlation between expected values.
Those with observations
only in the upper half of the height range tend to have extremely high indices since
they have observations that are closely related in time as well as actually maintaining
a constant deviation from the population mean, within that limited range. These are
illustrated in Figure 4.3.9. This graph, from the first run of the inconsistently-timed
data, shows the predicted lines and observations, along with expected tracking for
those measurement occasions, for a few individuals. The highest expected tracking is
for those individuals with observations restricted to the upper half of the range of
interest.
For'the incomplete data, when the individual index is based upon only 3 or 4
of the 5 measurement occasions, the proportion of distant pairs of observations is
greater while no clumping of observations is possible. The additional individuals with
incomplete data tend to have lower indices than those with complete data. There is
more variability across individuals, and lower values for
Pa,
depending upon which 3
or 4 of the 5 observations are included. This becomes clear when comparing the mean
1.4 I r - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - ,
tau=.771
1.3
1.2
1. 1
tau=.974
tau=.975
0.9
0.8
tau=.995
tau=.740
tau=.729
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
100
120
Height
140
In
em
Figure 4.3.9: Predicted lines and observations for six individuals with inconsistently-timed data.
The expected tracking (tau) for each set of measurement occasions is included.
Note that the high values are for the shorter intervals.
....
o
co
110
for the individuals with complete data to those with incomplete data.
For example,
for the first run of complete+incomplete data, Pa=.867 for those with complete data,
while the mean for individuals with incomplete data is Pa=.771.
Comparing the results of the mistimed data (design c) to the complete
balanced data (design b), for the adjusted indices, Pa and f, in Figures 4.3.10--4.3.12,
it appears that there is no difference between the mistimed and complete data (at
least to 3 significant digits), even when evaluating variability of the individual indices.
This is not an unreasonable result -- no clumping of measurement occasions is
possible, and all individuals span the full range of interest. Of course, the degree of
mistiming in this example is fairly small, and as the mistiming increases, the data
could come to look more like inconsistently-timed data. However the small range for
mistiming. tried in this example seemed reasonable, as it represents ±2 units in -12
unit intervals, which could represent being 2 months early or late for an annual visit
in a time scale is measured in months.
When evaluating the indices based upon a set of test points, a somewhat
different picture appears.
Within each design, Pat and f t were the same to 3
significant figures for each run, so that the discussion will focus on Pat. It should be
noted, however, that the expected tracking, f
t,
is defined for a set of measurement
occasions, so that for a set of test points, there are no individual indices, but only a
single value associated with that set of times or heights.
The results for tracking estimated at a set of test points from the
inconsistently-timed data (design a) were consistently lower than the estimates from
the other designs, as seen in Figure 4.3.10. The individual adjusted indices were also
considerably more variables, as seen in Figure 4.3.11.
So it seems that projecting
data to the test points, and then estimating tracking may be problematic.
inconsistently-timed data, where individuals cover the whole time range of interest
111
Pat
I
0.9 +
I
I
I
+-----+
I
*--+--*
0.8 +
I
I
I
*-----*
Design
+-----+
+-----+
I
I
*--+--*
+-----+
I
I + I
+-----+
0.7 +
I
I
I
0.6 +
I
+-----+
I
I
+
I
I
*-----*
+-----+
I
I
----------+-----------+-----------+-----------+--a b c
Inconsistently
Timed Data
Complete
Data
Mistimed
Data
d
Complete +
Incomplete
FIGURE 4.3.10:
Comparing Four Designs
Distribution of Pat: Mean Adjusted Observed Tracking at Test Points
var(Pati)
0.15
0.1
I
+
I
I
I
+
I
I
I
0.05 +
I
I
I
o +
Design
*-----*
I
I
I
I
+
I
I
+-----+
I
I
+-----+
+-----+
+-----+
I
I
I
+-----+
*--+--*
I
+-----+
*--+--*
I
+
I
*-----*
+-----+
I
----------+-----------+-----------+-----------+--a b c
d
Inconsistently
Timed Data
Complete
Data
Mistimed
Data
+
Complete
Incomplete
FIGURE 4.3. 11 :
Comparing Four Designs
Distribution of var(Pati): Variance of Adjusted Individual Observed Tracking at Test Points
112
0.9
I
+
I
I
I
0.8 +
I
I
I
0.7 +
I
I
I
+-----+
I
I
+-----+
I
+-----+
I
I
I
*--+--*
+-----+
I
+
I
+-----+
+
I
I
+-----+
I
I
+-----+
I
0.6 +
Design
----------+-----------+-----------+-----------+--a b c
Inconsistently
Timed Data
Complete
Data
Mistimed
Data
d
+
Complete
Incomplete
FIGURE 4. 3 • 12 :
Comparing Four Designs
Distribution of Tt: Expected Adjusted Tracking at Test Points
/
113
should not have this severe a problem. The mistimed data again has results almost
identical to the complete data, and the complete plus incomplete data also has a
distribution that is more similar to the complete data, though the individual indices
are more variable. While the mean for Pat for the complete plus incomplete data is
the same as for the complete data, the median is lower. The runs with lower values
for Pat resulted when there were more individuals with only the first 3 or last 3 of the
5 observations. In other words, the lower estimates are associated with projection or
extrapolation of the data across the interval to the test points, as with the
inconsistently-timed data.
To investigate the effect of projecting data across the interval of interest, one
other design was considered, using the same parameters as in the first four designs,
given above.
A single example of a linked cross-sectional design was run, where 5
groups of 10 individuals were each observed 3 times in contiguous sub-intervals. The
first group were observed at heights of 100, 105, and 110 cm, the second at 110, 115,
and 120 cm, and so on, up to the fifth group with observations at 140, 145, and 150
em.
Paarameter estimates and tracking results are given in Table 4.3.5.
The
predicted lines are given in Figure 4.3.13.
The estimates for the indices based upon observed heights are all extremely
high. This is a result of both the short intervals considered and the smaller distance
between points (5 cm in this example vs. 12.5 cm for the complete balanced data).
The results for the indices based upon test points are right on target for these data.
They are higher than the maximum among the 6 runs for the inconsistently-timed
data, and are right within range for the complete, balanced data.
So this type of
design may be useful for estimating overall, or attribute, tracking for selected points
across an interval, if not for individual tracking. One reason that this design may do
better than the inconsistently-timed data, is that there is balance across the whole
114
interval of interest. The individual tracking estimates depend upon the sub-interval,
and in this linked cross-sectional design all sub-intervals are equally represented,
which is not likely to be the case for inconsistently-timed data.
includes the mean adjusted individual tracking,
Pa
Pat
and
Table 4.3.5 also
for each sub-interval, so
that the effect of each interval can be seen. The test points for
Pat
considered were
still 100, 112.5, 125, 137.5, and 150 cm for comparability with the earlier examples.
TABLE 4.3.5
LINKED CROSS-SECTIONAL EXAMPLE
Parameter Estimates
~- = [.2170]
.0200
Based
Q!!
f>
=[
-
iT 2 = .00000996
Tracking Estimates
observed heights
Based
Pa
.981
.0136 -.00031l
.0000349J
Q!!
T
.971
.977
.966
100.112.5,125,137.5,150 £!!!.
Pat
.849
.825
Tracking Estimates
Based on Individuals in Each Sub-Interval
100-110
Pa
Pat
.940
.847
Sub-Interval Range (cm)
110-120
120-130
130-140
,955
.875
.970
.761
.993
.793
140-150
.997
.852
.824
115
1.8,....---------------------------,
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
OL-.---L------'------'------'-----..L...
100
120
---L..----J
140
Height in cm
Figure 4.3.13:
Predicted lines from linked cross-sectional design.
116
As one other way of evaluating the effect that the spacing of test points can have,
several different sets of points were selected for one example with design b, the
complete, balanced data. The results are reported in Table 4.3.6.
TABLE 4.3.6
Expected Tracking at Selected Points
From Complete Balanced Data Example
Selected Points
100,112.5,125,137.5,150cm
100,110,120,130,140,150
100,105,110,115,120,...,145,150
106,109,112,115,118,...,145,148
105,117.5,130,142.5,155
100,125,150
Sub-intervals:
100,105,110,115,120
130,135,140,145,150
No. of
Points
(t
Tt
5
.866
.798
6
11
15
5
3
.873
.888
.888
.928
.836
.824
.869
.877
.892
.617
5
5
.829
.995
.782
.993
Comparing to the first set of selected points, which are the same as the
observed measurement occasions, it can be seen that as the number of points increases
and th'e spacing between points decreases that the estimates for the unadjusted index,
(t, increase. The adjustment factor is particularly affected by the spacing between
points.
As the distance between points increases, the adjustment factor becomes a
much larger percentage of the estimate.
This pattern held in all examples run,
though just one is reported to illustrate.
Evaluating sub-intervals also had a
considerable effect. For the first 20 centimeters,
20 cm,
(130__ 150=.995.
has a considerable effect:
(100--120
=.829, while for
the latter
Even a shift of 5 centimeters higher from the observed points
(105,117.5. .. 155 =
.928.
117
If interest lies in evaluating individuals, for example identifying individuals
who do not track as well as expected, inconsistently-timed data may pose some severe
problems.
Figures 4.3.14--4.3.16 show plots of the individual indices, Pai' PaW and
T i' against the mean, minimum, maximum heights, and the range of heights for the
individual from one example of the inconsistently-timed data.
The values of the
adjusted observed indices, Pai' tend to cluster at the extremely high end, so the
patterns are not as clear as for the expected individual indices, T i' but in general the
low values for individual tracking (both Pai and T i) at the observed measurement
occasions are associated with lower mean height, lower minimum height, greater
maximum height, and a greater range of heights.
For the individual indices, after
adjusting to the test points, Pati' the pattern is not as clear, but again, there appears
to be an association between a low value for tracking, and lower individual mean and
minimum height.
Thus it would appear that evaluation of individuals for poor tracking may not
be possible with inconsistently-timed data, even after adjusting to a selected set of
points. The effects of having clumps of data, or of only partially covering the interval
of interest appear to overwhelm anything else that may be going on.
This association of individual indices with both the range of time spanned, as
well as the mean and minimum time of observation also causes problems for the idea
of using a weighted average of the individual indices to compute the overall indices.
Any weighting scheme would become fairly complex to handle the combination of
factors associated with expected increases or decreases in the tracking estimate.
addition, the indices
T
i
In
are already weighted averages of the pairwise correlations (see
section 2.2.5), with the variances of the measurement occasions as the weights.
Computing weighted functions of weighted functions becomes too complicated to
interpret.
•
118
Pai
1.0· +
* ***** ****** *
** * * ** *
*******
*
*
*
I
I
I
I
0.5 +
I
* *
I
I
I
0.0 +
I
*
I
I
I
-0.5 +
-+-----+-----+-----+110
120
130
140
1.0 + ********** *
I ** **
I ***
I **
I
0.5 +
1*
* **
**** ****
*** **
* * * *** *
* *
I
I
I
I
0.5 +
*
*
I
I
I
I
0.0 +
I
*
I
I
I
-0.5 +
-+-----+-----+-----+120
130
140
150
MAXIMUM Height in em
*
I
I
I
0.0 +
I
I *
I
I
-0.5 +
-+--------+--------+100
120
140
MINIMUM Height in em
MEAN Height in em
1.0 +
*
1.0 +
***********
I
*** **
I
***** *
I
*
I
0.5 +
I
*
*
I
I
I
0.0 +
I
I
*
I
I
-0.5 +
-+-----+-----+-----+0
20
40
60
RANGE in Heights
FIGURE 4.3.14:
Plots of Pai' adjusted individual tracking estimates vs. mean,
minimum, maximum and range in heights, for inconsistently-timed data (design a).
119
Ta
1.0
0.8
0.6
I
I
+
*
*
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I
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0.8 + **
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+
--+-----+-----+-----+110
130
120
+
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-+---------+---------+
0.8
0.6
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--+-----+-----+-----+130
140
150
MAXIMUM Height in em
*
**** *
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****
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+
I
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120
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120
MINIMUM Height in em
MEAN Height in em
1.0
•
*
100
140
* *
*
0.6
*
**
*
+
--+-----+-----+-----+0
20
40
60
RANGE in Heights
FIGURE 4.3.15:
Plots of T i' individual adjusted expected tracking, vs. mean,
minimum, maximum and range in heights for inconsistently-timed data (design a).
120
Pati
1.0 +
I
I
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I
* ***** ****
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**** * **** *
*
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0.5 +
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-+-----+-----+-----+-
1_10
130
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-+--------+--------+100
*. **** ****
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-0.5 +
-+-----+-----+-----+-
120
140
0.0 +
0.0 +
I
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120
MINIMUM Height in em
MEAN Height in em
I
I
I
I
*
*
0.0 +
0.0 +
1.0 +
*
*
130
140
150
MAXIMIM Height in em
-+-----+-----+-----+0
20
40
60
RANGE in Heights
FIGURE 4.3.16: Plots of Pati' adjusted individual tracking estimates at
selected heights, vs. mean, minimum, maximum and range in heights for
inconsistently-timed data (design a).
121
4.4 Discussion
Many of the issues that plague the analysis of tracking for inconsistently-timed
data are issues that have not been fully explored for complete balanced data. One of
the most basic is the clarification of the steps used in modeling the data before
estimation of tracking.
Model selection can have a large effect on the resulting
tracking indices, as demonstrated in section 4.1. A poor model smooths the data too
much, assigning more of the variability to
(T2,
the within-individual error, and making
the individual curves appear more similar, with resulting higher estimates for
tracking. This is as true for complete balanced data as it is for inconsistently-timed
data.
The use of a model at all implies some assumptions of commonality across
individuals.
This causes smoothing, which, in effect, assumes in some tracking.
Alternatives include methods that do not require modeling the data, such as ranking
methods, or modeling within individuals only. It is important to note, however, that
for inconsistently-timed data, some form of modeling must be used or individuals
cannot be compared or ranked at any particular time.
Given the use of the mixed effects model with
attendant
normality
assumptions, what then are the effects of assuming normality on the resulting
tracking estimation?
The assumption of a normal distribution across the slopes smooths the more
extreme slopes so that they are more like the others.
towards tracking.
In other words, smooths
An alternative could be the use of a t-distribution rather than a
normal, which allows more extremes. Another possibility would be to scrap the use of
the mixed model and use individual least squares (LS) regressions followed by some
evaluation of the distribution of the covariance of the parameter estimates across
individuals.
Yet another idea is the use of the mixed model to estimate population
122
~,
percentile lines (i.e, estimate the population mean,
and covariance, 1), from the
mixed model) and then get the individual parameters from individual LS regressions,
and compare these curves to the nearest contour.
In this case, the actual percentile
would be under- or over- estimated at the extremes of the distribution, but the focus
here has been less on the exact percentile, as upon whether or not it is maintained
across time.
It is interesting to note, however, that the mixed model assumption of
normality for the slopes may actually be useful for inconsistently-timed data.
The
extreme slopes, in all real data examples so far examined, have come from the
individuals with the smallest range of heights.
The smoothing resulting from the
assumption of normality may be reasonable in this case.
Even after deciding to use the mixed model with normality, model selection
can still have a large effect on the subsequent tracking estimation.
Over-fitting the
degree of a polynomial results in problems with convergence, but reducing the degree
of the polynomial causes smoothing, which results in higher correlations between
estimated points.
For a fixed degree polynomial, as the number of measurement
occasions in an interval increases, tracking estimates will also increase, since it will not
be possible to fit the curve through all observed points. Thus, the correlation between
predicted values cannot be zero.
Some firm criteria for model selection must play an important role in any
tracking analysis that requires modeling the data.
It may be worth questioning
whether it is apppropriate to use the same data for model selection as for evaluation
of tracking. One way to avoid this issue is to assume that the correct model for this
type of data is well established in the literature, and to use this model for tracking
analysis.
123
The comparison of the different designs for data collection points up several
problems in the use of these indices for inconsistently-timed data and for incomplete
data.
The tracking indices are functions of the covariance of the measurement
occasions.
As these become very close, the correlation between predicted values at
these points becomes high, and so does tracking.
Inconsistently-timed data, where
individuals have many clumps of data closely related in· time will always give
extremely high estimates for the indices based upon observed measurement occasions,
e.g., for p, Pa, (, and
T.
At the other extreme, where individuals have only a few very
widely spaced observations, the correlations from this model tend to be much lower,
resulting in low values for the tracking indices.
This problem would become more
apparent as the number of observations varied widely among individuals. It was seen
in the incomplete data examples (design d) that the individuals with fewer, and
therefore usually more widely spaced observations had lower mean indices.
The other factor that has a
l~rge
effect on the tracking indices is the interval
spanned by the data. In the examples considered above, most of the crossing occured
in the first half of the interval, so that tracking estimates from this range tend to be
low, with higher tracking toward the second half of the range.
The tracking estimates based upon a selected set of test points were much
more consistent across the different designs.
This is of course to be expected, since
the indices are functions of the covariance structure of the selected measurement
occasions, ~hQ~f, rather than of the covariance of the random effects.
The
variability in the indices based upon the selected points reflects variability in the
estimates of Q from the different designs.
For a given process in a particular
population Q is not fixed, but depends of the measurement occasions represented in
the sample.
What is fixed, are the parameters of the non-stationary process.
The
mixed effects model can adequately fit the data, even when inconsistently-timed.
.
124
However the dependence of J) on the measurement occasions in the sample means
that the estimates of the covariance at the selected points, :tJtJ):tJ~, will depend upon
the design. The indices at the selected points, Pti and Pati were much more variable
for the inconsistently-timed data than for the complete and mistimed data, results of
data clumps, and restricted range for some individuals, as discussed above.
The similarity in the results for the mistimed data and the complete balanced
data, however, are encouraging, even while noting that the degree of mistiming in the
examples was small.
Inconsistently-timed data that looks more like mistimed data
should work fairly well for this type of tracking analysis.
In other words, if
individuals have data that span the whole range of interest, and have no clumps of
data closely related in time, then the results of the tracking estimates based upon a
selected set of points should be reasonable.
One possible approach to tracking analysis with inconsistently-timed data or
complete plus incomplete data would be to use all of the data to estimate the
population mean,
t, covariance of the random effects,
J), and individual deviations,
9i' and then evaluate tracking with these estimates using only those individuals whose
data spans the full range of interest.
Another possibility for inconsistently-timed
data, would be to first define a set of times of interest, and select data from
individuals that approximate those points, and the spacing between these points -that is, select data to make it look like mistiming.
Then it may be possible to
evaluate individual levels of tracking as well as overall attribute tracking.
Some final recommendations for an approach to analyzing tracking for
inconsistently-timed data with the mixed model are as follows:
125
de~ree
1.
Begin with a well defined approach to model selection for determining the
of the polynomial used for estimating tracking.
2.
Have a pre-specified set (or sets) of 'times' of interest for evaluating tracking,
since tracking is a function not only of the range spanned by the data, but of the
specific measurement occasions.
3.
Focus on the indices that are based upon the set of selected points. These are
more easily interpretted since they refer to tracking at the known set of pointll.
4.
Evaluate expected tracking (T t ) for sub-intervals. These can indicate if there are
regions within the range of interest where most of the change in relative
deviation occurs.
5.
If there is interest in evaluating individuals, do this only for data where
individuals span the full range of interest, and do not have observations closely
related in time.
This is best left to complete balanced data, or to other
definitions of tracking.
.
Chapter 5
AN EXAMPLE
In the previous chapters a method was developed for analyzing tracking for
inconsistently-timed data in the mixed model, where tracking was defined as the
maintenance of relative deviation from the population mean over time.
In this
chapter an example is given, applying the method to study the tracking of pulmonary
function with respect to height in children.
The first section describes the data and its source. The second section
describes the selection criteria for inclusion in the tracking analysis, followed by a
section describing the model selection and fitting.
The fourth section gives the
tracking results, and the final section provides a summary.
5.1 Description of the Data
The data used in this example were taken a study on respiratory physiology
being conducted at the Frank Porter Graham (FPG) Child Development Center in
Chapel Hill, NC.
The selection of subjects and measurement of pulmonary function
has been previously described in Strope and Helms (1984) and Fairclough and Helms
(1984).
Briefly, the children in the study were voluntarily enrolled in a day care
program that included longitudinal studies of development, respiratory illness, and
physiology. Six to twelve children were enrolled each year as soon as possible after 6
weeks of age. From the time of enrollment complete clinical and microbiological
documentation of respriatory illness was obtained on the children.
127
Beginning as early as two-and-a-half years of age the children were trained to
perform forced respiratory maneuvers. When they could reliably reproduce their best
maneuver within 10% on five successive days they were included in the respiratory
physiology study.
Children were then evaluated at three month intervals. However
illness, vacations, moving away from the area, and other factors unrelated to
pulmonary function testing led to irregularities, both in the in frequency of testing
and the period of follow-up.
Children were considered well at the time of testing if
they had been free of lower respiratory illness in the preceding 6 weeks, and free of
upper respiratory illness in the preceding 4 weeks.
Description of pulmonary function testing is given in Strope and Helms (1984).
Two pulmonary functions measures are evaluated here for tracking. These are
1.
FVC - Forced Vital Capacity: the volume in liters of air expired after full
inspiration, with expiration performed as rapidly and completely as possible.
2.
FEF 25-75 - Forced Expiratory Flow in liters per second during the middle half of
the FVC.
Previous studies of the growth of lungs through childhood have shown that
pulmonary function grows most closely with height, and that age and/or weight add
little, if anything to a model that includes height (see literature review in Fairclough
and Helms, 1984). In particular, Fairclough and Helms, and Strope and Helms (1984)
using an early portion of this data found a linear relationship with respect to height to
provide the best fit for all pulmonary functions for the range of height under
consideration.
Interest here focuses upon whether or not pulmonary function tracks
with respect to height.
It was this problem that led to an interest in developing an
analysis of tracking for inconsistently-timed data.
128
5.2 Data Selection
In Chapter 4 several problems with analyzing tracking in inconsistent data
were discussed.
Data with times of observation that are highly inconsistent, with
many individuals having data that only partially overlaps the region of interest, wide
variation in the number of observations per individual, and many clumps of
observations closely related in 'time' will lead to large variability in the tracking
estimates, as well as extremely high estimates for the individuals with very short
intervals or many clumps of data.
data considered in this example.
All of these problems are characteristics of the
Therefore certain selection criteria were applied to
the data to make it more amenable for studying tracking. The downward selection of
data in this example is somewhat extreme.
However it was done in part because of
limitations on computer equipment. "The repetetive use of an iterative algorithm for
jackknifing estimation, particularly, can take several days on a personal computer
when the number of parameters estimated, as well as the number of observations per
person is large. This was one of the overriding reasons for restricting the number of
observations per person so"severely for this example.
For the purposes of studying tracking, interest was restricted to pulmonary
function testing done on children from age five to under age thirteen. This was done
to avoid the adolescent growth spurt. Using data collected from 1971 through 1988,
there were 93 children with at least 2 measurements from age 5 to 13. However it is
standard to evaluate pulmonary function separately by race and gender (see literature
review in Fairclough and Helms, 1984; Hibbert et al., 1990). Since there were only 10
white boys and 7 white girls, the analysis Was restricted to the African-American
children, leaving 32 girls and 44 boys.
129
There were a total of 705 exceptable evaluations available on these 76
children, with the number of evaluations per child ranging from 2 to 39. Figures 5.2.1
and 5.2.2 show the distribution across height of the number of studies for each child
for girls and boys, respectively. Reference lines at heights of 100 and 140 centimeters
were included. From these figures it can be seen that the observations are often very
closely related in height, and that for many of the children there are repeated
measurements at the same heights, particularly at the time they first enter the study.
There is also some inconsistency in the interval of heights covered. For example, in
Figure 5.2.1, girl number 53 has observations from approximately 105-118 cm, while
girl number 52 has data ranging from approximately 122-175 cm.
Neither of these problems, the clumping of data over heights, and the lack of
overlap of interval of interest bode well for a tracking analysis.
Based upon these
figures, it was decided to restrict the height range of interest to 110-140 cm for the
boys, and 115-145 cm for the girls, the region of greatest overlap of the data.
The
following was then done to select among the measurements for each individual:
1. Anyone with a height range of less than 15 cm was deleted.
2.
For the girls, any observations at heights of less than 105 cm or greater than
155 cm were deleted.
For the boys, any observations at heights of less than 95 cm or greater than
145 cm were deleted.
3. Among the remaining observations for each individual, starting from the first
observation, the next observation at least 3.5 cm greater was included, etc.,
so that the minimum height distance between observations was 3.5 cm.
4. Any child with only 1 remaining observation was deleted.
5.
For the few children with more than 6 remaining observations, any
observations outside the exact range of interest were deleted. (This last was
done largely to keep the computer time down for the purposes of this
example.)
•
130
+
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144
128 +
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117 +
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110 +
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101 +
98 +
93 +
92 +
89 +
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81 +
77 +
76 +
75 +
71 +
70 +
68 +
64 +
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43 +
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39 +
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29 +
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--+-------------+-------------+-------------+-------100
120
140
160
HEIGHT in CM
Legend: A
= lobs,
Figure 5.2.1:
ages 5-13.
Distribution of observations across height for African-American girls,
B
=2
obs, etc.
131
C
H
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D
,
S
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B
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192
162
152
151
148
143
138
137
136
135
134
129
126
125
122
121
120
119
115
114
112
111
109
107
105
102
100
97
96
91
88
87
86
84
80
79
78
72
60
57
56
37
36
30
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
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----+----------+----------+----------+----------+100
120
140
160
180
HEIGHT in CM
Legend: A = lobs, B
Figure 5.2.2:
ages 5-13.
=2
obs, etc
Distribution of observations across height for African-American boys,
.
132
This left 26 African-American girls with a total of 115 observations, and 30
African-American boys with a total of 102 observations. The number of observations
per child is given in Table 5.2.1. The boys average 3.4 observations per child, with
mean and median spacing of 9.9 and 8.6
observations.
centimeters, respectively, between
The girls average 4.4 observations per child, with mean and median
spacing of 8.8 and 7.6 centimeters.
Selection of the reference or 'test' points for estimating tracking across the
interval of interest was based upon the number of observations per child, and the
spacing between observations.
As a compromise between accomodating the mean
number of observations per child and the average spacing between observations, four
heights at a spacing of 10 centimeters were selected for the boys, and five heights at
7.5 centimeter spacing for the girls. That is, the test points for the boys are 110, 120,
130 and 140 em, and for the girls, 115, 122.5, 130, 137.5, and 145 em.
Table 5.2.1
Number oT Observations Per Child in the Analysis Dataset
aQ ATrican-American Boys 22 ATrican-American Girls
No. oT
~
No. oT
Boys
No. oT
~
No. oT
Girls
2
7
3
11
3
4
4
9
4
5
5
7
5
6
11
2
133
5.3 Model Selection and Fitting
As mentioned earlier, both Fairclough and Helms (1984), and Strope and
•
Helms (1984), using an early portion of this data, found that the pulmonary functions
were best described by a linear relationship with height in centimeters, over
approximately the range of heights considered here.
..
In addition, models with
quadratic fixed and random effects were considered, as well as models relating the
natural logarithm of FVC and FEF 25-75 to height. This last model has been used in
other studies (Dockery, et 0/., 1983; Voter et 0/., 1988) to describe the relationship of
pulmonary function and height, over greater ranges of height.
Separate models were fit for the African-American boys and girls, with the
intercept defined at the left endpoint of the height range of interest, i.e., 110 cm for
the boys and 115 cm for the girls.
Maximization of the log-likelihood function was achieved using the EM
algorithim, and parameter estImates with asymptotic standard errors are reported in
Tables 5.3.1 -- 5.3.4. The tracking estimates reported in these tables are the expected
tracking at the selected heights.
The models with quadratic fixed and random effects did not converge in a
reasonable number of iterations for FVC for either the boys or girls, nor did the
model for FEF 25-75 for the girls converge in a reasonable number of iterations. The
only quadratic model that converged was for FEF 25-75 for the boys, and is the only
one reported, in Table 5.3.1.
section 4.1 quite closely.
This appeared to match the example of overfitting in
There was very little change in the estimates of the fixed
effects and within-individual error with the addition of the quadratic terms, and the
additional covariance parameters were all very small.
In addition, the tracking
estimates remained unperturbed, even with the quadratic term included. So the linear
relationship was preferred to the quadratic.
.
134
The tracking estimates were quite unchanged for FEF 25-75 for both boys and
girls, whether the natural or log scale was used.
For FVC, the tracking estimates
were not quite as invariant, with lower estimates from the log scale for the girls, and
the reverse for the boys. In either case, the tracking estimates were quite high. The
final model· used for estimating tracking for both FVC and FEF 25-75 for the boys as
well as girls was a linear relationship with height with the intercept at the left end of
the height range of interest.
135
Table 5.3.1
Results of Model Fitting for FEF25-75 and Height
African-American Boys
Parameter(s.e.)
t
FEF 25- 75
and Ht
~int(llOcm) 1.55
(.08)
.0112
~slope
(.0033)
~quad
I)
°int(llOcm) .117
.0000284
°slope
.00171
°is
O~uad
°i,q
°S,q
In(FEF 25_75 )
FEF 25- 75
and Ht, Ht 2 and HT
1.56
(.08)
.00908
(.0083)
.0000874
(.00030)
.399
(.06)
.00587
(.0021)
.117
.0000456
.00204
4.56xlO- 8
-1.7xl0- 5
-6.17xl0- 7
.0649
.00000716
.000496
(7'2
.1096
.1067
.0421
-2log likelihood
116.52
116.39
25.59
Tracking
Ct
Tt
.9986
.9982
.9986
.9982
.9963
.9954
Table 5.3.2
Results of Model Fitting for FEF25-75 and Height
African-American Girls
Parameter
t
~int(1l5cm)
~slope
I)
0int(1l5cm)
Oslope
OJ,s
-2log likelihood
FEF 25- 75
and Ht
In(FEF25-75)
and HT
1.59
(.08)
.0259
(.0041)
.406
(.07)
.0124
(.0024)
.114
.000248
.00520
.0715
.0000231
.00119
.1114
.0757
143.48
82.01
.9990
.9988
.9980
.9976
Tracking
Ct
Tt
.
136
Table 5.3.3
Results of Model Fitting for FVC and Height
African-American Boys
FVC and
Height
In(FVC)
and HT
.957
(.027)
.0338
(.0018)
-.021
(.026)
.0243
(.00098)
.0131
.0000575
.000143
.0142
.0000084
-.000028
(1'2
.0107
.00653
-2log likelihood
-88.65
-149.72
.9292
.9028
.9916
.9832
Parameter
t
<Pjnt(llOcm)
<Pslope
1)
Ojnt(llOcm)
°Slope
°is
,
Tracking
't
Tt
Table 5.3.4
Results of Model Fitting for FVC and Height
African-American Girls
FVC and
Height
In(FVC)
and HT
1.08
(.026)
.0361
(.0016)
.081
(.023)
.0237
(.0010)
.00551
.000029
.000232
.0080
.0000095
-.000071
(1'2
.0219
.0104
-2log likelihood
-67.38
-149.78
.9753
.9678
.8955
.8738
Parameter
t
<Pint(llScm)
<PSiope
1)
°int(llScm)
°slope
OJ,s
Tracking
(t
Tt
137
5.4 Tracking Estimation
Once the final models were selected, tracking was estimated from each model.
The parameters estimated were as follows:
1. p
the mean of the individual tracking indices, based upon expected values
at observed measurement occasions.
2. Pa
the mean of the individual tracking indices adjusted for zero correlation,
based upon expected values at the observed measurement occasions.
3.
(
the mean of the individual expected tracking indices after adjusting for
0'2, based upon observed measurement occasions.
4.
T
the mean of the individual expected tracking indices after adjusting for
0'2, adjusted for zero correlation at the observed measurement occasions.
5. Pt
the mean of the individual indices, based upon expected values at a set
of test heights.
6. Pat the mean of the individual indices after adjusting for zero correlation,
based upon expected values at the test heights
7.
(t
the expected tracking after adjusting for
0'2
at the test heights
adjusted for
the expected tracking after adjusting for
correlation at the test heights.
zero
The indices described above are all based upon the tracking of an individual's
expected values. In addition to these, the tracking of the observed values, or tracking
without prior adjustment for within-individual error were also estimated. These are:
9.
p*
the mean of the individual tracking indices, based upon observed values.
10.
p~
the mean of the individual tracking indices adjusted for zero correlation,
based upon the observed valuess.
11.
(*
the mean of the individual expected tracking without adjustment for
based upon observed measurement occasions.
0'2,
12.
T*
the mean of the individual expected tracking without adjusting for
adjusted for zero correlation at the observed measurement occasions.
0'2,
13.
(t
the expected tracking without adjusting for
14.
T*
the expected tracking without adjusting for
correlation at the test heights.
t
0'2,
at the test heights
0'2,
adjusted for zero
•
138
Each index was estimated with the full data and then estimated by
jackknifing, as described in section 3.3. Standard errors and 95% confidence intervals
are reported for the jackknifed estimates.
Tables 5.3.5--5.3.8.
Results for all indices are reported in
139
Table 5.3.5
Tracking Estimates for FEF 25-75 with respect to Height
African-American Boys
Full Data
Estimate
Jackknifed
Estimate
Standrrd
Error
95% Crnfidence
Limits
1.00
1.00
.999
.998
1.00
1.00
.998
.997
.0000116
.0000164
.000624
.00104
.99996, 1
.99994, 1
.997, .999
.995, .999
1.00
1.00
.998
.998
1.00
1.00
.997
.997
.0000113
.0000155
.000675
.000919
.99996, 1
.99995, 1
.996, .999
.995, .998
T*
.732
.608
.730
.592
.747
.640
.742
.616
.0663
.102
.0578
.0877
.633,
.462,
.642,
.464,
'i
T*
.712
.613
.717
.623
.0650
.0866
.607, .826
.475, .768
Parameter
P
Pa
,
T
Pt
Pat
't
Tt
p*
p~
C
t
.857
.808
.838
.761
Table 5.3.6
Tracking Estimates for FEF 25-75 with respect to Height
African-American Girls
Full Data
Estimate
Jackknifed
Estimate
Standard
Errort
95% Confidence
Limitst
1.00
1.00
.999
.998
1.00
1.00
1.00
1.00
.0000432
.000065
.000729
.00112
.99993, 1
.99990, 1
.998,1
.997,1
1.00
1.00
.999
.999
1.00
1.00
1.00
1.00
.0000232
.0000316
.000529
.000734
.99996, 1
.99995, 1
.999, 1
.998, 1
T*
.789
.702
.802
.725
.797
.723
.816
.753
.0645
.0934
.0537
.0734
.671,
.539,
.711,
.609,
'i
T*
.803
.743
.813
.762
.0538
.0694
.708, .919
.626, .898
Parameter
P
Pa
,
T
Pt
Pat
't
Tt
p*
p~
'*
t
tNote: For Jackknifed Estimates
.924
.906
.921
.897
140
Table 5.3.7
Tracking Estimates for FVC with respect to Height
African-American Boys
Full Data
Estimate
Jackknifed
Estimate
Standard
Errort
95% Confidence
Limitst
.920
.834
.907
.832
.903
.825
.904
.832
.0648
.116
.0693
.107
.796, 1
.626, 1
.787,1
.651,1
Pt
Pat
.966
.947
.923
.882
.977
.963
.931
.891
.0463
.0692
.0739
.109
.897,1
.844,1
.805,1
.706, 1
p.
r·
.670
.403
.749
.589
.664
.402
.758
.604
.0728
.134
.0687
.0931
.542,
.174,
.641,
.445,
C;
r·t
.763
.658
.778
.675
.0836
.107
.634, .917
.487, .859
Parameter
p
Pa
C
r
Ct
rt
Pa
C
.788
.629
.873
.759
Table 5.3.8
Tracking Estimates for FVC with respect to Height
African-American Girls
Full Data
Estimate
Jackknifed
Estimate
Standard
Errort
95% Confidence
Limitst
P
Pa
.994
.992
.965
.946
.992
.987
.930
.894
.0110
.0159
.0399
.0581
.970,1
.956,1
.852, 1
.781,1
Pt
Pat
.996
.995
.974
.964
.994
.991
.947
.926
.00782
.0106
.0310
.0420
.978,1
.970,1
.886, 1
.843, 1
p.
r·
.588
.445
.573
.422
.608
.472
.579
.430
.0945
.124
.0812
.107
.422,
.229,
.420,
.220,
C;
r·t
.570
.449
.578
.458
.0879
.110
.406, .750
.242, .674
Parameter
C
r
Ct
rt
Pa
C
tNote: For Jackknifed Estimates
.793
.714
.738
.641
141
5.5 Results
The discussion of tracking results will focus upon the indices based on the set
of test points, in particular T t , although the results for all the indices were included in
the tables. The comparison of designs in section 4.3 indicated that the indices based
upon a selected set of measurement occasions are the most easily interpretted, since
they refer to tracking at known heights and height intervals.
Certainly it is clear
from the estimates reported below, that the same conclusions would be drawn
regardless of the particular index evaluated:
both FVC and FEF 25-75 track
extremely well with respect to height, from 110 to 140 cm for African-American boys,
and from 115 to 145 cm in height for African-American girls. In particular FEF 25-75'
demonstrates near perfect tracking for both boys and girls. For the African-American
boys, the jackknifed estimates for the expected tracking after adjustment for withinindividual error at heights of 110, 120, 130 and 140 centimeters are Tt=.997 for
FEF 2 5-75' with 95% confidence interval, .995<Tt <.998; and for FVC, T t =.891, with
95% confidence interval .706<Tt <1.
For the African-American girls the jackknifed
estimates at heights of 115, 122.5, 130, 137.5, and 145 cm are Tt=1.00 for FEF 25-75'
with 95% confidence interval .998<Tt <1; and for FVC, Tt=.926, with 95% confidence
interval .843<Tt <1.
The tracking estimates for FVC are similar to the results of Dockery et al.
(1983) which found tracking estimates for log(FVC) with respect to height of .90 for
girls age 6-11, and .93 for boys age 6-11.
This study defined tracking in the same
manner, as maintenance of relative deviation from the population mean over time (or
height), but estimated population percentiles from 44,664 observations on 13,299
children, and then estimated tracking from the 3,395 children with 5 annual
observations.
'.
142
The results are also in agreement with those of Hibbert et al. (1990) who
estimated tracking, using McMahan's
age in years.
T,
for the log of lung functions with respect to
For 8-13 year olds, their estimates were for FVC:
.885 for girls and
.936 for boys; and for FEF 25-75: .845 for girls, and .924 for boys. Since lung function
has been shown to be slightly more highly correlated with height than with age, the
higher estimates found in this study are not unreasonable.
Estimates of tracking without adjusting for within-individual error were also
made.
For the African-American boys, the jackknifed estimates for the expected
tracking without adjustment for within-individual error, at heights of 110, 120, 130
and 140 centimeters are 1't=.616 for FEF 25_75 , with 95% confidence interval,
.464<Tt<.761j and for FVC, 1';=.604, with 95% confidence interval ,445<Tt<.759.
For the African-American girls the jackknifed estimates at heights of 115, 122.5, 130,
137.5, and 145 cm are 1't=.753 for FEF 25_75 , with 95% confidence interval
.609<Tt<.897jand for FVC, 1'i=.430, with 95% confidence interval .220<Ti<.641..
Tracking estimates without adjustment for within-individual error for both
FVC and FEF 25_75 are considerably lower than when considering tracking of
expectations.
This is to be expected when the magnitude of the within-individual
variation is as large as the variation among individuals, i.e., the variation in locations
(intercepts) of the curves.
·2
•
For FEF 25- 75 for the boys, u =.110 and 0int=.117, and
fo~ the girls, 0'2=.111 and 0int=.114, while for FVC for the boys, 0'2=.011 and
•
2 ·
0int=.013, and for the girls 0' =.022, and 0int=.0055.
The exact values that constitute good tracking vs. poor or non-existent
tracking are not clearly established. Hibbert et al. (1989) considered a value greater
than .5 to indicate tracking. If estimates greater than .75 constitute good tracking,
then clearly the expectations of FVC and FEF 25_75 both track very well, while
including u 2 in the indices leads to conclusions of poor tracking or no tracking.
143
For the boys, the estimates that include 0- 2 are greater than .5, but the 95%
confidence intervals reach below .5.
including
(T2,
For the girls, FEF 25-75 tracks even when
.
While FVC fails to demonstrate tracking over the range of heights
considered.
Figures 5.4.1--5.4.8 are included, so that these conclusions can be compared
with a visual impression of tracking.
The individual predicted lines are plotted for
each pulmonary function to give a visual impression of tracking of expectations,
followed by a plot of the line segments connecting observations, for an impression of
the tracking of the observed values.
estimated indices.
These appear to correspond well with the
For FEF 25-75 for both boys and girls, the individual predicted
lines do not cross (figures 5.4.1 and 5.4.3), while the connected line segments cross
within a limited range of FEF 2 5-75 (figures 5.4.2 and 5.4.4).
Similarly, a small
amount of crossing of the individual regression lines for FVC -- more for the boys
than the girls, as seen in figures 5.4.5 and 5.4.7. Much more crossing is seen for the
line segments connecting observations, particularly for the girls, where a few
individuals cross just about everyone else.
5.6 Discussion
The tracking estimates in this example appear to be in reasonable agreement
with both the visual impression of tracking and the tracking estimates for pulmonary
function found elsewhere in the literature.
While the selection of data was somewhat extreme, the example still served
the purpose of demonstrating the methodology and interpretation of results.
Computer facilities permitting, it is probably more reasonable to select and fit the
model with all of the data, and then estimate tracking, avoiding clumps of data, for
individuals with data spanning the range of interest.
•
~
4
Forced Expiratory Flow between 25 and 75% Vital Capacity
African-American Boys
I
,
3.5
3
2.5
FEF25 _ 75
(I/s)
2
-d
~
1.5
1
0.5
o'
9'5
'
Figure 5.4.1:
105
'
1'15
I
125
I
135
'
Height in em
Predicted lines for FEF25-75 vs height for 30 African-American boys.
145
'
~
t
4
Forced Expiratory Flow between 25 and 75% Vital Capacity
African-American Boys
I
I
3.5
3
2.5
FEF 25-75
2
(I/s)
1.5
1
----------
0.5
o'
9'5
'
105
I
1'15
I
125
I
135
I
145
I
Height in em
Figure 5.4.2:
~
Line segments connecting observations of FEF25-75 vs height for 30 African-American boys.
.
.....
~
C11
Forced Expiratory Flow between 25 and 75% Vital Capacity
African-American Girls
4
3.5
3
2.5
FEF25 - 75
(I/s)
2
.-:
----
1.5
1
=
0.5
0
100
120
140
160
Height in em
......
Figure 5.4.3:
Predicted lines for FEF25-75 vs height for 26 African-American girls.
~
0)
Lt
Forced Expiratory Flow between 25 and 75% Vital Capacity
African-American Girls
I
I
3.5
3
2.5
FEF 25 - 75
2
(1/5)
1.5
1
0.5
o
I
100
I
120
I
Height
In
140
I
160
I
em
.....
Figure 5.4.4:
..
Line segments connecting observations of FEF25-75 vs height for 26 African-American girls.
w
~
-!
2.4
Farced Vital Capacity
African-American Boys
I
I
2.3
2.2
2.1
2
1.9
1.8
1.1
FVC
in liters
1.6
1.5
1.4
1.3
1.2
1. 1
1
0.9
0.8
0.7
0.6
0.5'
1
95
~I
105
1
1
1 15
1
125
1
135
1
1~5
I
Height in em
......
Figure 5.4.5:
Predicted lines for FVC vs height for 30 African-American boys.
II:>.
00
2 .4
2.3
2.2
2. '1
2
1.9
1.8
1.7
1.6
FVC
1.5
in liters 1.4
1.3
1.2
1. 1
Forced Vital Capacity
African-American Boys
I
I
1
0.9
0.8
0.7
0.6
0.5!
9'5
1
105
1
1
1 15
I
125
I
135
I
145
I
Height in em
Figure 5.4.6:
•
Line segments connecting observations for FVC vs height for 30 African-American boys.
....
of>.
ec
FVC
in liters
2.8
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1. 1
1
0.9
0.8
0.7
Forced Vital Capacity
African-American Girls
100
140
120
Height
In
160
cm
......
Figure 5.4.7:
Individual predicted lines for FVC vs height for 26 African-American girls.
CJl
o
Forced Vital Capacity
African-American Girls
3
2.8
2.6
2.4
2.2
2
FVC
In
1.8
liters
1.6
1.4
1.2
1
0.8
0.6
100
140
120
160
Height in em
Figure 5.4.8:
•
Line segments connecting observations of FVC vs height for 26 African-American girls.
•
......
......
c.n
Chapter 6
SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH
This work has been aimed at the development and evaluation of an analysis of
tracking based
upon
the ideas of McMahan (1981), which
can accomodate
inconsistently-timed longitudinal data using the mixed effects model. Inconsistentlytimed data are data where the measurement occasions vary from subject to subject.
This may arise from mistiming of observations, missing observations, deliberately
incomplete designs, or the use of a time scale such as growth in height that does not
lend itself easily to measurement at the same heights or height intervals.
Following McMahan, tracking was defined as the maintenance of a constant
relative deviation from the population mean over time.
This definition effectively
represents tracking very simply, as well as lending itself to easy interpretation as a
function of the variance components of the mixed effects model.
The mixed effects
model with linear covariance structure was selected as the vehicle for the tracking
analysis, since it can easily accomodate inconsistently-timed data without imposing a
strict correlation structure on the observations across time.
Several tracking indices were defined within this model, based upon the actual
observation times, as well as a selected set of measurement occasions. Indices were
defined that evaluated the tracking of predicted values after adjusting for withinindividual error, as well as indices that evaluate the tracking of the observed values.
Estimators were defined for each index, along with standard errors computed by
jackknifing.
.
153
Generated data were used to explore some of the the effects of a poorly fitted
model.
In particular, underfitting a polynomial model leads to over-smoothing and
overly high estimates for tracking of predicted values.
The use of the mixed effects
model with the asssumption of a normal distribution across the individual parameter
deviations may also lead to excessive smoothing.
Some possiblilities for future
reasearch include exploration of such alternatives as use of a t-distribution in place of
the normal, which would allow more extremes in the intercept, slope and higher order
deviations, or use of individual least- squares regressions with evaluation of the
covariance of the parameters across individuals for tracking.
Other alternatives
include use of a generalized linear model, assuming only that the distribution across
individuals is exponential in family, or use of some type of smoothing technique to
estimate the population central tendency and percentiles, followed by comparison of
predicted curves from individual least squares regressions to these.
Generated data were also used to explore the effects of inconsistent spacing of
observations on tracking estimation. Designs with inconsistent-timing, mistiming, and
complete plus incomplete data were compared to an example with complete balanced
data.
Data that is highly inconsistent, with individuals having little overlap in the
time range of their observations, or many clumps of observations closely related in
time will not function well for a tracking analysis.
Inconsistently-timed data where
individuals have data that span the whole time range of interest can give reasonable
estimates of tracking for an attribute, particularly for the indices based upon a prespecified set of measurement occasions of interest.
However the correlation between
the individual indices and the mean, minimum and maximum time of observation
indicate that the individual indices are not useful for identifying indivdiuals who do
not track well, when the data are inconsistent.
154
Beyond the task of evaluating whether or not a particular attribute, such as
lung funtion or blood pressure tracks, a major goal of many tracking analyses is the
identification of individuals who do not track well, or who exhibit different patterns of
change over time that may put them at risk of developing disease.
Another
possibility for future work in tracking analysis is the further development of the
methods of Lauer et al. (1986) and Foster, et al. (1989), described in section 1.2.6, for
inconsistently-timed data.
Other areas to explore for inconsistent longitudinal data include the use of
other computing algorithms that may speed up the process, especially for jackknife
estimation. These could include improved Scoring algorithms, a REML algorithm, or
one that produces estimates for unstructured covariance rather than linear covariance.
The extension of other types of tracking estimation for inconsistently-timed data, such
as the method of Ware and Wu (1981), emphasizing the ability to predict future
observations would also be of interest ..
The definitions of tracking and methods available for tracking analysis for
complete, consistent data are many, and some coordination of these and description of
their applicability would be a useful contribution to the literature. Beyond this, some
.directions for future research in tracking analysis for both complete balanced data, as
well as inconsistently-timed data include:
1. Tracking with respect to a vector -- for example, tracking of pulmonary
function with respect to height and age, or blood pressure with respect to age
and weight.
2. Incorporation of covariates into the model -- for example, for pulmonary
function given race and gender, with respect to height, are different means by
group adequate, or are separate estimates of variability also necessary.
3. Tracking of groups -- e.g., do the race-gender groups track relative to an
overall mean.
155
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..
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