A COMPARISON OF ERROR PROBABILITIES FOR TWO
STANDARD ANALYSES OF VARIANCE IN UNBALANCED
TWO-WAY DESIGNS
by
Robert E. Johnson
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo
August 1983
Seri~s
No. 1447
A CO~1PARISON OF ERROR PROBABILITIES FOR TWO
STANDARD ANALYSES OF VARIANCE IN UNBALANCED
TWO-WAY DESIGNS
by
Robert E. Johnson
A Dissertation submitted to the faculty of
The University of North Carolina at Chapel
Hill in partial fulfillment~f the requirements for the degree of Doctor of Philosophy
in the Department ofBi~£tatistics.
Chapel Hill
1983
Apr:ed
£.
by:
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ROBERT EARL JOHNSON. A Comparison of Error Probabilities
for Two Standard Analyses of Variance in Unbalanced Two-Way
Designs (Under the direction of DAVID G~ HERR.)
The test of no difference in main effects in two-way
factorial designs usually involves a two step procedure:
first test for the presence of interaction and, if no significant interaction is detected, proceed with a main
effects test. It would be ideal if the interaction test
were independent of any main effects test and if the
presence of interaction had no effect on the tests for main
effects.
In this work joint errors for the two step procedure
are studied for each of three tests: (1) an idealized (but
not. realizable) independent test, (2) STP analysis which
tests equality of simple column (row) averages of cell
means, and (3) EAD analysis which tests columns (rows)
adjusted for tows (columns).
These analyses are first studied in light of their
geometric properties and then, with the aid of existing
distribution theory and new theory developed here, the
erraprobabilities are compared. EAD is fourtd to be the
more powerful test for detecting column (row) differences
in an additive m6del, but a biaSed test in the presence of
interaction.
ii
ACKNOWLEDGEMENTS
David G. Herr has served not only as the director of
this dissertation but also as my instructor, colleague,
and friend. I wish to express my deepest appreciation
for the interest he has shown over the past nine years in
my professional development.
I wish to thank Dana Quade for serving as chairman and
co-advisor on the dissertation committee. His friendship
has been , from the time we worked together on the SENIC
project until the present, a sustaining force for me.
A special thanks goes to my family, to whom this work
is dedicated. I wish to thank my wife Janice for her love
and support and for sharing the job of typing the manuscript
and to my two year old son Joseph for being such a joy.
iii
•
CONTENTS
ACKNOWLEDGEMENTS
1.
INTRODUCTION.
2.
LITERATURE REVIEW
2.1
Nonorthogonal ANOVA . . . • . . • . . . • • . ". 11
2.2
Central Multivariate Gamma-Type Distributions. 21"
2.3
Noncentral Multivariate Gamma-Type
Distributions • • • .
24
2.4
Bivariate T Distributions •• ,
26
2.5
Characteristic Function of the Multivariate
Chi-squared Distribution. . • • • . •
28
Computational Procedures for Evaluating
Multivariate Gamma-Type Distributions
29
2.6
•
2.7
Applications: Simul~aneous Hypoth~sis Testing. 32
2.8
Applications: Conditional Type I Error
Probabilities •
. • . . 32
3.
THE GEOMETRY OF ANOVA AND THE STP AND EAD
ANALYSES. . . • . .
••••
. • . . . 34
4.
DISTRIBUTION THEORY
5.
4.1
X 2 Distribution.
50
4.2
T Distribution
58
4.3
Computational Methods.
·
.
65
A COMPARISON OF THE STP AND EAD ANALYSES TO THE
INDEPENDENT ANALYSIS
. ·
. . ." . . .
5.1
Introduction
5.2
Case 1 : HI True and H
True
C1
Case 2: HI True and H
False.
C1
Case 3 : HI False and HC1 True.
5.3
•
1
5.4
5.5
5.6
Case 4: HI False and HC1 False
Summary.
•
...·
..
iv
70
77
81
89
. 95
"
.106
6.
SUGGESTIONS FOR FURTHER RESEARCH • . • .
. . . .113
REFERENCES
APPENDIX A.
PROOF OF ORTHOGONALITY CONDITIONS ON n
APPENDIX B.
PROOF OF EQUATION 4.23
APPENDIX C.
SAS PROGRAMS
•
•
•
v
Chapter 1
Introduction
One of the most widely used statistical procedures
today is the claBs~cal analysiB ~f v~riance (ANOVA).
Procedures for calculating sums of squares as test statistics
a~e
well known and there is wide agreement on how to
interpret the results when the design is balanced. For a
balanced axb two-way fixed effects design, the usual
hypotheses of interest are:
HI
.
no interaction effects
H : no difference in main row effect.s
R
H : no difference in main column effects.
C
If the true mean of the (i,j)th cell is u .. (i=1, ••• ,a;
1J
j=1~ ••• ,b), then these hypDtheses may be parametrically
stated as:
HI: Yij=O, i=1, •.• ,a-1; j=1, ... ,b-1
H : (Xi=O, i=1, ••• ,a-1
R1
H : (3j=O, j=1, ... ,b-1
C1
where the ith row effect is
effect is B . = u . - U
J
i s y..
1J =
u·1·
JJ..
1J
=
•J
-
=
u.u •• ,
1·
the jth column
, and the (i,j)th interaction effect
••
u·l ·-.uJ. - U •. for:
Iu
.. /b ;
. 1J
J
(x.
1
2
jJ
•
•J
jJ ••
= LjJ: ./a
i
=
LjJ.
. /a
~
and
~J
~.
= \'jJ
./b
~J • J
Suppose that H
is the hypothesis of primary interest
C1
and that the row factor is included in order to remove any
syst~matic
variability due to differences in the row
effects. The usual analysis procedure begins with a test of
HI. If the data fail to provide significant evidence against
HI' then test H • Accepting H
further implies that there
C1
01
are no differences in column means over any given row.
Rejecting H
implies that differences in main column
C1
effects exist and that these differences will reflect the
differences in column means over any given row. If HI is
rejected then the above interpretations of H being true or
C
false are not satisfactory since differences in column
means need not be consistent over rows in the presence
of interaction •
. The possible errors incurred with the implementation
of the above procedure.aresummarized in Table 1.1. The
errors are indexed by double indices (since two hypotheses
are tested) with the first index indicating the type e~ror
made testing HI
error, and
'2'
('0'
means no error,
'1' means a type I
means a type II error) and the second index
indicating the type error made testingH
cl (,.,
means that
H
is not tested).
C1
The analysis that one might presume is being performed
with the prescribed procedure is one where the
for the tests have independent central
r
r statistics
distributions
Table 1 .1
Possible Errors in Testing Procedure
True State of • HI and HC1 !iI False
!iI True
True . H
False
True H
False H
C1
C1
C1
Analysis
Results
+
Accept
Accept
HI
HC1
Reject
H
no error
type (0,2) type(2,0) type(2~2)
1--
type (0,1) no error
type(2,1) type(2,0 )
C1
Reject
type C1
,. )
no error
HI
under their respective null hypotheses. Henceforth we
shall refer to this as the independerit analysis. Throughout
this work we will assume that the marginal tests of HI and
H
have type I error probabilites of 0.10 and
C1
0.05
respectively.
Consider for example a 3x3 design (based on an example
in Kleinbaum and Kupper, 1978) for studying the relationship of patient perception of pregnancy and physicianpatient communication to patient satisfaction with medical
care. The contrived data shown in Table 1.2 are values of
a measure of patient satisfaction with medical care. The
row categorization divides the data into three levels of
patient worry and the column categorization divides the
data into three levels of a measure of affective communication.
These data were generated by using a standard random
4
Table 1.2
ANOVA Example: The Data
Affective Communication
Patient
Worry
Negative
Neutral
Positive
H'12 h
6.67 5.50 4.77
4.29
Me d'1um
4.49 4.17 3.86
5.18 4.66 2.52
5.43
3.90 6.27 3.27
4.98 3.68 3.56
5.50
4.61
3.08 5.67 4.34
6.62
5.56 5.13 4.47
4.36 3.98 3.67
3.74
L ow
4.10 7.51 5.98
5.40
4.97 4.41 4.21
3.90 3.20 3.73
4.81
8.07 6.24 6.74
4.08
Estimated Cell Means
j
:
1
5.31
4.33
3
5.75
2
4.45
4.61
4.18
4.41
3
4.93
4.42
6.28
5.21
,:
4.90
4.45
5.40
1
i
1\
]J
2
1\
]J ,
•J
l'
5.13
number generator ahd adding the true cell means shown in
Table 1.3. Thenonsystemic error
varianc~
was set to 1.
For the true cell means in Table 1.3, it can be seen
that HI is false and HC1 is truB. The possible errors in
this case are type(2,0) and type(2,1). For the example, the
marginal probability of not rejecting HI is 0.4426. If the
design were balanced with five observations per cell, and
\
5
Table 1.3
ANOVA Example: TrlieCell Means
lJ i
/
lJ.
j
5.3
4.4
5.3
5
4.4
6.2
4.4
5
5. 3
4.4
5.3
5
5
5
5
.
•
.
thus the same number of observations over all, this probability would reduce to 0.068. Since the marginal probability of rejecting HC1 is 0.05, then the probability of a
type(2,1) error for the independent analysis is
(0.4426)(0.05)=0.0221 or if the balanced design is used,
(0.068)(0.05)=.0034.
The independent analysis is one which is in general
not realizable with the prescribediprocedure.Even with a
balanced design, the Tstatisticsare not independent if
the same mean squared error is used for each. We could of
course use the error sums of squares to make two or more
independent estimates of the error variance by ,say,
diViding it over groups of
cells~
however this would result
in a loss of power. T statistics which are dependent only
by their common denominator we will call singly dependent.
For the purpose of comparison with the other analyses, we
vlill make reference to the independent analysis even though
one may not exist.
With unbalanced designs many pairs of tests will also
have dependent sums of squares as numerators of the
6
r
statistics.
r
statistics with this dependence in conjunc-
tion with a common mean squared error will be called doubly
dependent.
The existing
comp~ter
packages that offer AN OVA proce-
dures for unbalanced designs provide several analyses.
Some of these
analysesadvert:i,se "orthogonality", that is,
that the sums of squares add up to the total sums of
squares. The hope in uBingorthogonal sums of squares is
that the confounding of effects may in some sense be
unraveled. Another "plus" for the orthogonal analysis user
is that the
r
statistic~
are at most singly
d~pendent.
Some of the analyses f6r unbalanced designs have the
property that, unlike the independent analysis, the null
distribution of the
r
statistic for the test of H
is
C1
noncentral. The noncentrality parameter will be. unknown
unless certain combinations of the cell means are known,
which is rarely the case.
We are thus faced with a choice among analyses that
are less ideal than the independent one. If the designjs
balanced, all of the alternative analyses offered by most
computer packages are the same. If the design is unbalanced, each possible analysis differs from the independent
analysis in some
way~ ~he
problem is to be able to choose
the app·ropriate analysis for the situation at hand. To do
this one needs to know wha~ effect the differences with the
independent analysis will have on the error probabilities.
In Chapter 2, unbalanced ANOVA is reviewed in general along
7
with the literature on some of the various analyses.
Two of the common anal~ses for unbalanced designs are,
as termed by Herr and Gaebelian (1978), the standard
parametric analysis (STP) which is Overall and Spiegel's
Method 1 (1969) and the "each adj usted for the other·"
(so named because the sum of squares for each of the two
factors are the "extra" sum of squares explained by the
model when the factor is added to the model containing the
other factorY which is Overall and Spiegel's Method 2. In
SAS(Statistical Analysis System, SAS Institute 1979), for
example, one Dan perform theSTP analysis by adding the
option SS3 to the MODEL statement and the EAD analysis by
adding the option SS2. In general the STP analysis has the
property that the
r
statistics for HI and H
are doubly
C1
dependent where they are singly dependent for the EAD
analysis. The null distribution forH
C1
for STP but may be noncentral for EAD.
will be central r
The results of the ANOVA example for STP and EAD are
shown in Table 1.4. Per the prescribed procedure, we would
not reject HI using either analysis. Using STP, HC1 is not
rejected, which constitutesatype(2,O) error. That is, we
would conclude that there are no differences in column
means. While this is true for the average column means,
such is not the case for a given row due to interaction.
Using EAD, HC1 is not rejected, which eonstitutes a
type(2,1) error. In this case we would conclude that there
are differences in column means which are uniform over rows.
8
Table 1.4
AN OVA Example: Results of the
STP and EAD Analyses
STP
SOURCE
--
DF
SS
F
PROB>F
---
Affective Communication
2
4.51
1 .88
0.168
Patient Worry
2
3.26
1.35
1.10
0.271
Interaction
Error
4 5.27
36 43.28
SOURCE
DF
0.373
EAD
SS
F
PROB>F
---
Affective Communication
2
9.78
4.07
0.026
Patient Worry
2 10.00
4.16
1.10
0.024
Interaction
4 5.27
36 43.28
Error.
0.373
It is the uniformity conclusion about the column means that
is in error.
The purpose of this work is to compare the STP and EAD
analyses to the independent analysis in ord~r to arrive at
an assessment of both analyses
for various designs and
patterns of cell means. This will be done by means ofa
comparison of error probabilities. In order to calculate
these probabilities, WB viII draw on current distribution
theory and make further developments where needed.
To facilitate a discussion of the distributions needed,
we
introduce the following notation. Let F(~1'W2's) and
~(~1'~2,d) denote bivariatB
noncentrality parameter
~or
r
distributions where ~1 is the
the marginal interaction test,
_.
9
Table 1.5
Distributions Corresponding to
STP and EAD Analysis Errors
Error
type(O,1)
type(O,2)
type (2, 1)
type(2,2)
1
EAD Analysis
F(O,O,s)
1
STP Analysis
F(O,O,d)
1
F (0 ,w ,d) 2
2
F(w ,O;d) 2
1
F(W1,w 2 ,d)
F(O,w 2 ,s)
F (W l' W2 ' s)
F(w ,w ,s)
2
1
2
2
2
2
The density has been derived, and some computational
methods have been developed.
2
The density has not been derived in the literature (see
Chapter 4 for some developments).
W2 is the noncentrality parameter for the marginal column
main effects test,
IS'
denotes single dependence, and 'd'
denotes double dependence. Thedistributibns needed to
compute the various probabilites are shown in Table 1.5.
In Chapter 2 the current: distribution theory is revieltled. We begin with the
generalgamma~type
distributions and
move to the bivariate chi-squaredandT distributions.
Since some of the theory is based on inversions of the
characteristic function, we review the characteristic
function of the bivariate chi-squared distribution. Finally
vIe
consider current computational techniques.
In Chapter 3 we review the geometry of the EAD, STP,
and independent analyses. A discussion or the distribution
theory specific to this work along with further develop-
10
ments and computational considerations is presented in
Chapter 4.
In Chapter 5 we present the comparison of the error
probabilities. Under thaas8umption of no interaction, we
consider the effects of single dependence and nonorthogonality on the type(0,1) and type(0,2) errors. When interaction is present we will compare the type(2,1) and
type(2,2) errors.
Throughout we will illustrate the
results with variations on theANOVA example given in this
chapter.
Suggestions for further research are offered in Chapter
6.
Chapter 2
Literature Review
2.1
Nonor~hogonal
ANDVA
A classical paper dealing with nonorthogonal designs
was presented by Yates
(1934). He suggested that under the
assumption of·no interaction, ordinary least squares methods
may be used. When this assumption is not made, the recommended procedure is one he termed "the method of weighted
squares of means. n For the over-parameterized model
Yl'J'k
= w + a l, + SJ' + y,.
+ e.lJ·k·•
lJ
la, = Is, = Iy·, = ty·· =
i
lj J
. i lJ
j
lJ
with the usual restrictions
0, Searle
.
(1971) showed that the
hypothesis tested by the row· statistic of this ·method is the.
equality of theai's.·Thefunctional form of this analysis
proved cumbersome for practitioners of the pre-computer era,
so Yates further suggested an "unweighted means analysis"
for which the computations are simplified. Searle showed
,
that this analysis also tests the equality of the a· s. The
1
corresponding mean square for rows divided by the mean
squared
~rror
however does not in general have an
r
distri~
bution as do the variance ratios in the weighted means
analysis.
(In Chapter 3 we give general conditions on the
cell sizes for these statistics to have exact
butions J
r
distri-
12
Eisenhart (1947) presented the "necessary" assumptions
for testing hypotheses for a fixed effects design. These
assumptions are summarized as:
1. Randomness. The cell means are fixed and the observations
are values of random variables distributed about the
means.
2. Additivity. This is equivalent to assuming no interaction.
3. Equal variances and zero correlations. The variancecovariance matrix of the vector of observations is
0
2
>0 times the identity matrix.
4. Normality. The vector of observations is an observed
random vector whose elements have a jointly normal
distribution.
Scheffe (1959) discussed the impact of the violation of one
or more of these assumptions. One common assumption that is
not made·above is that of equal cell sizes. This leads to
the question t
~What
is the impact upon the ANOVA when the
cell sizes are not all equal?"
Searle (1971) studied this problem in terms of various
decompositions of the total sums of squares via the regression approach. Let R()J
j:i,
A y) denote the total corrected sum of
squares one obtains when fitting the unrestricted model
= lJ + "a. + S. + y .. +.e. --ok- by least squares. Omission
lJ
1
J
lJ
lJ
of on~ or more of the terms ~tatatY indicate~ fitting the
Y. -k
model without including these terms. For example t R(jJ,a)
refers to the
totalco~racted
sum of squares for the model
13
+ a. + e" k ~ The method proposed for testing
1J
l
1J
hypotheses about the parameters of ~hefull model involves·
Yo Ok = lJ
a technique that is used to test the significance of the
additional variation explained by an independent variable
ina regression model in the presence of other regrBssors.
Searle showed that the sum of squares R(lJ,a)-R(lJ) =H(a/lJ)
tests the hypothesis:
HR2 : (1 /n.) In .. lJ· .
1·
. 1J 1J
J
equal for all i,
and R (lJ,(3 ) -R (lJ) = R((3llJ) tests
HC2 : (1/n .) In. ·lJ ..
"J i 1J 1J
equal for a-II j •
These hypotheses may also be stated as:
HR2 : a i + (1/n. ) In . . ((3. + Y .. )
1"
1J
. 1J J
equal for all i, and
J
HC2 : (3 . + (1/n .) En .. (a i + Y •• ) equal for all jo
oJ . 1J
1J
J
1
The sum_ of squares R(lJ, a, (3)-R(lJ, (3) = R(a/lJ,S) and
R(lJ,a,S)-R(lJ,a) =R(SllJ,a) tasts
H :
R3
Hc3 :
°.1
= 0
for all i. and
= 0
for all j ,
J
respecti vely, l<There:
0.
1
E •
I(n.
.
1
=
,1 .
0
n.2 ./n
1J
.)0'..
"J
1
I
i'~i
Z(n . . l1 .... /n .)a.,+
j
1J 1 J "J 1
I. (n1J
..
- n{j/n"J' )Y iJ' - L l(ni.J·n .... /n . )y ....
i ' ~ij
. 1 J
"J
1J
A similar definition is given for E. ° These parameters
J
J
can also be defined as:
01' = l{n .. lJ ..
. 1 -J 1J
J
E.
J
=
H n 1..J ]J.1J•
i
I
·1"
(n .. n.,./n .)lJ.,.}, and
1J 1 J " J. 1 J
I (n .. n .. , / n.1" ) lJ 1....
}•
j , ·lJ 1 J
J
Each of these hypotheses can be tested under Eisenhart's
14
4 with exact methods. Burdick .e.t
assumptions 1, 3, and
a.e.
(1974) describe these exact methods from a geometric point
of view. They show that the sum of squares for testing any
estimable linear hypothesis of J1
ij
'S can be expressed as
the squared length of the perpendicular projection of the
data vector onto some vectur space corresponding to violations of the hypothesis. The spaces for testing H
R1 and
H
C1 are shown to be in general nonorthogonal as are the
spaces for testing H
R2 and HC2 • The spaces cor~esponding to
H
R3 and HC3 are always orthogonal to the spaces corresponding to H
C2 and HR2 , respectively, but are in general not
orthogonal to each uther. Conditions fbr orthogonality
of
the first pair of spaces are given as functions of the cell
R1 and HC1 ' the conditions are
sizes. For testing H
lin . ..
lJ
= r. + s. where r. and s. are real numbers. For
l
J
l
J
R2 and HC2 ' the conditions are n ij
testing H
p. and
=
Piqj where
~.
are positive real numbers. The latter condition
J
is oft termed the condition of proportionality.· Two other
l
.
analyses are studied by Burdick
.e.t
a.e.
which involve the
paired hypotheses HR2 , H
C3 and HC2 ' HR3 . The spaces
corresponding to these spaces will be, as stated above,
orthogonal.
_ The practitioner is faced with choosing among these
various methods when dealing with unbalanced designs, or so
it seems. The question "how to choose?"
sparked a series of
interesting articles in the psychological research methods
literature. Overall and Spiegel
(1969) described three
15
methods for testing HR and He • Method 1 is equivalent to
the weighted squares of means analysis and the STPanalysia
Method 2 is equivalent to testing the sums of squares
•
R(afu,S) and R(Slu,a) which is the EAD analysis. Method 3
is testing the sums of squares R(alv) and R(S/u,a) or
R(slu) and R(a/u,S). They claim tha~ ~ ••• there is no way to
compare and to say (for example) that Method 1 is the proper
generalization (of the equal Bell sizes case) while Methods
2 and 3 are not."Their recommendations are:
Method 1: " ... should be used if conceptualization of the
problem is in general Ilnear regression terms and
termS.~
not in experimental design
Furthermore,
" ••• care should be exercised not to interpret
significant 'main effects' in the same way that
one might interpret a significant main effect
from a balanced factorial design."
Method 2: should be used
"~ •• whenever
the problem is
conceived as a nulticlassification factorial
design and where conventional analysis of variance
might have been employed except for unequal cell
frequencies ••.• "
Method 3: should be used " ••• if a logical apriori ordering
exists among hypotheses to be
t~sted.u
Rawlings (1972) claimed that Method 1 and 2 do not provide
tests of the simple hypothesis (H
R1
or H ) in the presence
C1
of interaction. He further claimed that Method 1 does not
result in "exact statistics" since certain orthogonality
conditions are not met.
(It needs td be pointed out here
16
that the sums of squares for Method 1 are the squared
lengths of the perpendicular projections of the data vector
onto the spaces described by Burdick .e.iaf...
test HR1 and H '
C1
(1974 ) that
Method 1 nat only tests the simple hypoth-
eses, but it is an exact method.) Rawlings recommended an
alternative to Method 1, Method 1', which tests first for
interaction and then if nonsignificant, tests for main
effects with the interaction sum of squares included with
the error sum of squares.
Overall and Spiegel (1973) felt that Rawlings! new
method was their Method 2. They also pointed out some difficulties with pooling interaction with error.
Ra~lings
(1973)
returned
with stating that (in spite of his earlier
.
.
claims) when there is no interaction, Methods 1, 2, and l '
all test the simple
hypo~heses
- so use the one with the
greatest power. Rawlings felt that Method l ' will have
power at least as large as the other two methods.
Appelbaum and Cramer (1974) treat the problem in terms
of mutiple regression as a
co~parison
of competing linear
models. They claimed that this resolves the problem in an
"obvious manner." These models "are logically independent
of the observed numbers of observations per cell •.. ",
however !' ••• the analyses will be affected by the cell
fre~
quencies." They advocated Method 2 stating that it always
gives the "correct estimate!'. Their algorithm involves three
steps:
1.
Test for interaction. If significance is found then stop.
•
17
~tep
Otherwise, proceed to
2.
2.
Test the sums of squares R(al~,B) and R(BI~,a).If both
are significant, accept the model Y =
is significant, accept the model Y =
~+a+B+e.
~+a+e
If one
or Y =
~+B+e,
respectively. Otherwise, proceed to step 3.
3.
Test R(a Ip) and R(B Ip). If neither is significant,. then
accept the model Y
the model Y =
=~+e.
~+a+e
If one is significant, accept
or Y =
~+B+e,
respectively, but
interpret "cautiously". Otherwise no conclusion is made.
This algorithm actually constitutes a model selection
process, however they imply that accepting the model
Y = w+B+e over Y = w+a+B+e, for example, says that H is
R
true. O'Brien (1976) amended Appelbaum and Cramer's algorithm by adding to step 2 the following: when only one of
the tests performed is significant, also test R(alw) .or
R(Bfw) respectively. If this test is also significant, then
accept the corresponding model. If the additional test is
not significant, "adopt the model Y= w+a+B+e, but cautiously interpret its meaning."
·The final article in this series was presented by Herr
and Gaebelein (1978). They termed the method of Appelbaum
and Cramer as a "regression-oriented view", wheras the
method of Burdick .e.t al.
(1974), for example, is an "effect-
oriented view". The testing ofH
and H should include
C. .
the desired interpretation. Five analyses are presented:
R
2.
"STandard Parametric" (STP) analysis tests HR1 and HC1 .
"WeighTed Means" (WTM) analysis tests HR2 and HC2 •
3.
"Each ADjusted" (EAD) analysis testsH
1.
R3
and H
.
C3
18
4.
"Hierarchical: A (row) first, then B (column)" (HAB)
analysis tests H
and H
•
R2
C3
5.
"Hierarchical: B (column) first, then A (row)" (HBA)
analysis tests H
and H
•
C2
R3
The sums of squares for the above analyses are obtained via
the methods described by Burdick ~i at.
(1974).
Gaebelein pointed out that in SAS (SAS Institute,
Herr and
1979) the
sums of squares for STP are the Type III SS; for WTM two
runs are required alternating rows and columns being listed
first in the MODEL statement and the Type I SS ror the one
listed first is used; for EAD the sums of squares are the
Type II SS; for HAB, the Type I SS with rows listed first;
and for HBA, the Type I S$ with columns listed first.
They
surmised that the practitioner's dilemma is not to choose
one of the five analyses, but to choose which of the parametric hypotheses he wishes to test.
Once a hypothesis is
chosen, the appropriate analysis is determined.
It follows
that if one wishes to test H
and H ' which would have
C1
R1
been tested if the design were balanced, then one should
use the STPanalysis (which is Yates' weighted squares of
means analysis and Overall and Spiegel's Method 1 as stated
by Hinkelmann,
Hocking
~i
1975).
at. (1980) investigated the STP, WTM, and
EAD analyses to determine under what conditions each test
the simple hypotheses (H
R1
and H ).
C1
For example, they gave
model restrictions that provide sufricient conditions for
the analyses to test H
R1
•
These are:
19
STP: all cells must be filled;
WTM: HC1 and HI are true; and
EAD: HI is true.
Burdick et at.
(1974) gave conditions on the cell sizes
that are sufficient for the analyses to test HR1 and HC1
.
These are:·
STP: n .. > 0;
lJ
HR1~-
WTM: to test
n ij = p.1 ,
to test HC1 -- n ij
=
to test both -- n ..
lJ
= p;
and
EAD: same as for WTM.
Under the restriction of no interaction, Hocking et at.
showed that STP and EAD test the same hypotheses, thus
a preference should be based on power.
Buidick and Herr
(1980) showed that EAD has power at least as large as STP.
The proof is in recognizing that the leg of a right triangle
is shorter than the hypotenuse.
The leg and the hypotenuse
correspond to the noncentrality parameters of the T statistics associated '>lith STP and EAD respectively.
Figure 6, Chapter
(See
J.)
. Two other papers which cite parametric versions of the
hypotheses and appeared about the time of Burdick's and
Appelbaum and Cramer's were Carlson~nd Timm (1974) and
Kutner (1974).
Both recommended testing HR1 and HCl for
general situations.
Carlson and Timm along with Bryce (1975)
equated the sums of squares associated with the hypotheses
to Searle's
R(·) notation.
Gianola (1975) revisited a
20
He concluded that "since, in the model with interaction,
differences between rows or between columns cannot be
estimated in the absence of interaction effects, it does
not seem reasonable to compute sums of squares for main
effects without interaction effects in their expectations."
Kutner (1975) resolved Gianola's confusion (or did he~) by
poin ting out tha.t if the usual
La.
i
1
=
LS,
j J
=
LY ..
i lJ
=
LY ..
jlJ
==
II
estima bili ty" as sumptions,
0, are made then R(allJ,S,Y) and
c
R(S!lJ,a,Y)'arenonzero and do provide the appropriate sums
of squares.
Unfortunately NeIder (1975) added to the con-
fusion by stating that fitting y .. subject to constraints is
l.J
undesirable. His reasons were: (1) if interaction is negligible, loss of power occurs, and (2) if interaction is
present (not negligib~e?), the sums of squares correspond
to uninteresting hypotheses.
Obviously NeIder did not
realize what hypotheses STP testB.
The papers by Burdick
(1978), and Hocking QL al.
QL at. (1974), Herr and Gaebelein
(1980) among others resolved the
problem of how to proceed with hypotheses testing in unbalanced two-way designs.
Two recent textbooks bear out the
fact that the message has hot been heard by
all.
Keppel
(1982) treats the unweightedmeans analysis much as Yates
presented it in his 1934 paper.
Kleinbaum and Kupper (1978)
21
mention Yates' unweighted means analysis and present four
additional analyses under the heading of "regression approach".
The four methods turn out to be RAB, RBA, EAD,
and 8TP (when the estimability assumptions are made).
is confusing is that under the
h~ading
What
of the null hypoth-
esis tested, the same parametric representation appears for
each method.
It seems thay are saying that the same hypoth-
eses are tested under each analysis.
2.2
Central Multivariate Gamma-Type Distributions
A gamma-type multivariate distribution can be constructed by considering random vectors of the form
X.1
=1
(X. l'1
• ~ ., X. )' for i =·1,
m
••• , n where the n vectors
are independently and. identically distributed as multivariate normal, each having a common variance - covariance
matrix V where the diagonal elements of V are all ones.
n
The statistics 8. = L
J i=1
(X .. - -X.. )2 , j
J
lJ
=
1, ••• , m, are
distributed as chi-squared random variables with n - 1
degrees of freedom.
81 , 82 ,
.•• , 8
m
The joint distribution of
is a multivariate chi-squared distribution
(Krishnaiah .e.t.af..., 1961).
The joint distribution of
~, ... , 18 m ·is a generalized Rayleigh distribution
(Miller, 1964).
These distributions are described by
Johnson and Kotz (1972) under the general heading of
multivariate gamma-type distributions.
p
When m = 2 and
= corr (Xi l ' Xi2 ), i = 1, ••• , n ,the joint distribution
function of 8 , 8 2 , (the standard bivariate central
1
22
chi-squared distribution) is given by:
(2.1)
Prob(S1 ~ s1' S2 ~ s2) =
co
I
c.Prob(X 2 1+2' ~.
j=O J
n-J
(1-p2)-1s1)prob(Xn2_1+2J'
~
(1- p 2)-1 s2 )
1tlhere c. = r«n_1)!2+j)(1_ p 2)(n-1)/2 p 2j/ j !
(Bose 1935,
J
Finney 1938). A general standard bivariate gamma distribution as defined by Johnson and Kotz is obtained by replacing
(n-1)in 2.1 with v where v>O. Jensen (1970b) derived a
further generalized form by allowing different variancecovariance matrices for the X. random vectors. Kibble (1941)
1
derived the joint density of S1~S2 where corr(S1,S2) = p2
and S. (j =1 ,2) has a standard central gamma distribution
J
.
with density
(2.2)
functio~:
PS (s.)
j
J
=
0.-1
-s
s.J
e j/r(0.)
J
J
for 0 j >O, Sj>O. The joint density of S1 and S2 is:
x
where 0=0 >0; 0
j
L~-1(s)
"
~
p < 1; s1' s2 >0; and
[r(e)r(e+k)/kl]~
Jo(-n
b
[~]sb/r(e+b)
is the Laguerre polynomial. Sarmanov (1968) labeled 2.3 a
nymmetrical (bivariate) gamma distribution sinco 01
-- ()2 .
D'jackenko (1961, 1962a, 1962b) calculated tables for
Kibble's result for p=0.9 and 0=2,3,4,5. Sarmanov (1970a,
23
1970b) constructed an asymmetrical bivariate gamma distribution with density
where for some A> 0 and A< 1,
= A/8 18
2
• A generalized
1
symmetrical gamma distribution is ohtained by replacing
p2,p4,
•••
in 2.3 with a sequence
or
nonnegative numbers
00
• • • such that
I
C
k=1 k
c6nverges~
. )~k.sds
8-1( )
s 8-1 exp (-s
= 0
00
fa
Since
for
k~1,
Sarmanov (1968) showed that PS1,S2(s1,s2) >
if c ,c 2 , · · · are
1
the
then
a
if and only
central moments of a random variable
Y for whichP(02. Y< 1)=1
• Under these conditions,
PS1 ,S2(s1,s2) is a density function.
Jensen (1970a) derived a bivariate
.~~
n
~~
for S1 = ~Xi1 and S2 =
l
chi~squared
density
n
where each X.. isa standard
IX~2
. l
lJ
l
normal random variable, corr(X i1 ,X i2 ) = Pi' and
corr(X. o,X.,.,) = 0 for i 1= i ' (i,i'=1,···,n; j,j'=1,2) .
lJ
l J
Note that (X i1 ,X i2 ) (i=1,···,n) are canonical pairs with
canonical correlationsP1,P2,···,Pn •
derived via the probability integral
~he
joint density,
transformation~is:
24
00
I
(2.5)
k=O
k
k
Gk(o) I
I b (k)\jI(s1 ;n/2+p)
p=O q=O pq
.
x \jI(s2;n/2+q)
~,
where
u
= (p2 p2 ••• p2).
l' 2'
, n '
= I·· · · I
a · a . ···a .
. j1+ ••• + j n=k . 1 J1 2 J2
nJ n
a ..
1J
b
pq
r(j+~) {r(j+1)r(~)}-1
= p~j
1
(k) =
(-1)p+q(~J (~J;
and
Note that \jI(s;v) is the gamma density with parameters v
and
2.3
~
•
Noncentral Multivariate Gamma Distributions
Johnson and Kotz (1972) described what they called
biased Rayleigh distributi6ns fot which the marginal distributions of 8 1 , ••• ,8 are noncentral X2 distributions with
m
n-1 degrees of freedom and noncentrality parameters
n
.L (t,: i 1- ~ • 1 )2,
1=1
n
...,
.I
1=1
(t,:im-~.m)2
n
res p.ectivel y .where'E(X 1· ;) =. t,: .. and ~.J' = It,:··/n.
J
1J
i=l 1J
Blumertson and Miller (1963) derived a joint density assuming
that V = (v pq ) is of the Jacobi form, that is, v pq =0 if
I p-q I> 1 • Miller and 8ackrowi tz (1967 ) derived a form of the
density of the ratio of a noncentral chi-squared random
variable with d degrees of freedom to a central chi random
variable with d+1 degrees of freedom. Miller et al.
(1958)
25
derived a jointly noncentral chL distribution where Xi1'
Xi2 (i=1 ,···,n) are canonical variates with correlation p,
and Var(X .. ) = 0 2 (i=1,··· ,n;j=1,2) • The form of the
lJ
density is:
(2 • 6)· PR 1 ,R 2 (r 1 ,r 2) = {2 nr 1 1'2 r (n ) /0 2 (n - 2 ) }{ 2/ p w 202} (n - 2 ) /2
2
x exp{-(11(r +r )+Aw)}
1
x
I
k=(n.. 2)/2
x
where R. =
J
A =
1-
S~
J
2k(nl~=~-21Ik(2P11r1r2)
.I (wr )I (wr )
2
k
1 k
(j=1,2); w = A/0 2 {t+p);
n
L E;,~.
i=1 lJ
.
.
(j=1,2); 11 = {20 2 (1_ p 2)}
-1
; and I
k
(·) is
the modified Bessel function (Watson,f962). The above
density function assumes n>2 and l;il=l;i2(i=1, ••• ,n).
Miller.e.t al. further derived the density relaxing the
equal expectations assumptions and allowed for unequal
variances (but still required equal canonical correlations).
Johnson (1978) derived the jointly noncentral chisquared distribution of Y1= xtt andY 2 = X12' where
Var (X 11 )=Var (X 12 ) = 0 2 and p= Carr (X 11 ,X . 2 ), which has density
1
where
.
,
KO = {87T0 2 ~-1
"'p}
K1 = pl;11
-
.
K1 = -{2a2 (1_p)}-1
1;12 , and K3 = pl;12 -l;11 •
26
(2.8)
f
31
=I81J I J{ z 2: I z 2/ =1"8';) IF z1 ' Z2 (z 1 ' z 2 ) dz 1 dz 2
I {z 1 : I z 1 I
where
Iz I
=
,3 (s1 ,s2)
2
denotes the norm of z, and ISI denotes integration
over the. set S.
Krishnam (1976) also derived a jointly noncentral
chi-squared density which is a transformed version of 2.6.
2.4
r
Bivariate
A bivariate
r
Dis~ributions
distribution may be constructed by
considering the random variables SO' 3 , 3 , each having a
2
1
chi-squared distribution with t, r, and s degrees of freedom respectively. 3
is assumed to have a central distri-
0
o~
bution and is independent
F
1
=
t3 /r3 and F
2
1
0
=
both 3
1
and 3 • The statistics
2
~S2/s30 each have
r
distributions
with (r,t) and (s,t) degrees of freedom respectively. It is
evident that F
1
andF
2
may not be independent since they
share a common denominator, 3 • When 3 and 3 are indepen1
2
0
dent, F
are
1
and F
2
dependent,
are "singly dependent". When 3
F 1 andF
2
1
and 3
2
are "doubly dependent". A form
of the density for singly dependent bivariate central!
27
distributions with r=s was derived by Finney (1938).
Krishnaiah and Armitage (1965) tabulated Finney's distribution for the upper
10th~ 5th~
2.5th, and 1st
perc~ntiles
with t=1,---,19 and r=s=5,···,45. Kimball (1951) d.erived
an inequality for the singly dependent doubly central T
distribution which states that the dependence between F 1
and F 2 increases the jDint probability that F and F 2 are
1
less than some upper bounds over the case when F and F
1
2
are independent. That is:
Olkin (1971) showed that when r=s and £1,=f 2 =f,'
Jensen (1969b) showed Kimball's ineqtiality true for doubly
dependent, doubly central T distributions. Jensen further
derived a form of the density given by:
r
00
(2.9)
PF
F(f 1 ,f 2 ) =
l' 2
k
k
G (6) I ..Idmn{k)
k=O k
m=O neO
f(r+2m-2)!2 f(s+2n-2)!2
12
{1+Cr/ t )£ 1+ ( s ! t )f 2} (r +s +2m +2n +t ) J2
d
mn
(k) = (-1 )m+n(~J (~J r(r+2m)!2
r{(r+s+2m+2n+t)!Z)
and Gk (6) is as defined in 2.5.
S
x
where
(s+2n)!2
.x
28
2.5
Characteristic Function of the Multivariate
Chi-squared Distribution
Many of the density functions presented in the
previous sections were derived by means of inverting the
characteristic function of the distribution. Kibble (1941),
for example, derived the joint characteristic function of
X~!2
and X~/2, which is written here as:
''.There X1 and X2 are standard normal random variables and
Corr(X 1 ,X ) =P. Jensen (1969c) showed that the character2
istic function for the m-dimensional chi-squared distribution is:
which is defined for real T where:
M = I - 2iTC;
T is the mxm diagonal matrix with real
{ t 1 ' • • • , t m}
elements
;
I is the mxm identity matrix;
C is the mxm correlation matrix Of the vector of
m chi-squared variables;
v is the common degrees 'of freedom of the m
variables; and
lJ'
= (lJ ,···,]lm) where ].lj (j=1,···,m) is the common
1
expectation of v independent, unit variance, normal
variates whose sum of squares is the jth marginal
chi-squared variable.
Lancaster (1958,1963} gave much of the theorical basis
29
needed to invert these characteristic functions in
t~rms
of orthogonal· systems of polynomials • Gurland (1955)
derived the inverse Fourier exponential transform of the
function:
which is given by:
(2.11)
(21T)-1 J : exp(-itx)w(t;g,h)dt
=
h! r (g)L~g-1 ) (x)x g - 1 exp( -x) {r(g)r (g+h)}-1
where
L~g-1)(x) is the Laguerre polynomial of order
h.
This transform was used by J 6nSEm (1970a) in the derivation
of 2.5 and 2.9 •
2.6 Computational Procedures for Evaluating Multivariate
Gamma-Type Distributions
The problem considered here is
th~
evaluation of
P(3 1::.s 1 ;3 2::'s 2 ) or P(F 1 <f ;F 2::.f ) where 3 and 3 2 are
1
1
2
distributed as Chi-squared random variables with rand s
(r::.s, say) degrees of freedom and F1
F2
= t3 1!rB o
and
is a central chi-squared random
0
variable with t degrees of freedom and is independent of
=
t3 Z/s3 0 where 3
3 1 and 3 2 • The second probability can be derived, given
knowledge of the first, as]
Let X = (X.'; X2)' be an (r+ s) -dimensional random vector
with X.l distributed as normal with mean vector
dispersion matrix
0 2
10'
where
Ll
l
..
p~
l
and
is the identity matrix
30
with the same dimension as Xi' Furthermore, let Cov(X
a 2 C where C
= (Dio)
,x
1 2
is rxs, D is an rxr diagonal matrix
)=
with diagonal elements P1,···,Pr' and 0 is an rx(s-r)
matrix of zeros. Suppose that S. = X~X./a2 (i=1,2). Then
111
(2.13)
p(S1~i1;S2~s2) ~ fixeS <s .S <s }fPX(x)dx,
. - 1- l ' 2- 2
where PX(·) is the density function of X.
One method of evaluating 2.13, following Miller 2t ale
(1958) and Johnson (1978), is to make a change of variables
to polar coordinates and solve the multiple integral
explicitly or numerically. In lieu of the dehsities derived
by Jensen (19?Oa) and Kibble (1941), it appears that
quadrature ,procedures using gamma type weights are the most
promising numerical methods. One such procedure is described by Ralston (1965). The quadrature function
n
Q(s1,s2) =
l
H.PX(a.) is used to approximate the probabili=1 1
1_
ity in 2.13, where Hi and a i depend only on s1' s2' and n.
The als are vectors of zeros of Laguerre polynomials and
the HIS are known functions of these zeros. The als-and HIS
have been tabulated for various n by Shao .et ale
(1964).
Krishnaiah (1965), Amos (1972), and Gunst (1973b)
considered the evaluation of bivariate doubly central '1
probabilities via the bivariate '1 density for specific
cases where p. = p (i=1,···.r). Gunst and Webster (1973a)
1
suggested using the the equicorrelation density to approxr
1
imate the general case by substituting (lP~/r)§ for each
i=1 1
p~. Based of Monte Carlo estimates, they showed that these
1
31
approximations overestimate the true probabilities when
not all p.l · (i~1,···,r)
are
equal.
.
.
...
Dutt and Soms (1976) used the results of Gurland (1948)
to derive an integral transformation approximation
fo~
calculating probabilities of the central multivariate chisquared distribution. The method involves calculating the
real parts of probability integral transformations using
characteristic functions of all possible marginal distributions and substituting them into Gurland'i density expansion. This procedure appears to be praQtical only for .the
cases when .Pi~ p (i~1,···,r)~ s1 ~ s2' and r
= s.
Tan and Wong (1978) presented a promising approximation for the general noncentral case. Let X1 and X2 be
any nonnegative continuous random variables with density
F(x 1 ,x 2 ) and the following properties:
(i) E(X~ X~»oo for any u,v~O;
( i i) E(X.)
1
~
m.1 > 0 (i =1 ,2); and
(iii) Var(X.1 »0
(i~1
,2).
Define a function gk , k (xl' x 2 ) such that
1 2
(2.14)
J~J~x~X~gk1,k2(X1'X2)dX1dX2= E(X~X~)
for any u
0,1, ••• ,k 1 and v ~ 0~1, ••• ,k2' The function g
is used as an approximation of F(x ,x ) where k 1 and k 2 are
1 2
chosen by the user. Tan and tvong reported that their g used
~
as an approximation to a doubly noncentral gamma distribution is "good"for "!lloderate ll canohical correlations
and noncentrality parameters.
McAllister
~t ale
(1981a) presented a procedure for
32
computing the
central
r.
probabil~tiesof
Jensen's bivariate doubly
Their approximation is based on the first few
terms of the infinite sum:
00
(2.15)
I
k k
G (6) I Ib (k)I 2 (r/2tm,s/2tn,rf 1 /t,sf 2 /t,t/2)
k=O k
m n mn
..
where I 2 (o) is the integ~al derived by Amos and Bulgren
(1972).
2.7
Applications: Simulataneous Hypothesis Testing
Many authors have noted that simultaneous tests of
several variance ratios all sharing a common denominator
will be dependent. Singly dependent r tests have been
studied by Kimball (1951), Ghosh (1955),·Ramachandran (1956),
and Hurlburt and Spiegel (1976). Doubly dependent
r
tests
with common canonical correlations have been studied by
Krishnaiah (1964), Jensen (1969a), and McAllister et ale
(1981b). Jensen (1970a) and McAllister et ale
(1981b) also
allowed for unequal canonical correlations. These authors
us~d
the distribution theory in the previous sections as
a basis for their discussions.
2.8
Applications: Conditional Type I·Error Probilities
Pertinent to this work are the previous attempts at
quantifying probabilities relating to the simultaneous
testing of any two of the three hypotheses HR , HC' and HI.
Hurlburt and Spiegel (1976) investigQted the probability
of rejecting HR conditioned on rejecting HC for singly
dependen~r tests. They noted that when the numerator
33
degrees of freedom are small relative to the denominator
degrees o·f freedom, the variance ratios are approximately
independent. Their examples also suggest that when all the
degrees of freedom are nearly equal, the conditional
type I error probability is significantly larger than the
unconditional probability.
McAllister et at.
(1981b) focused their attention on
the EAD analysis and reported probabilities obtained from
using evaluation techniques (McAllister et at .. , 1981a) on
Jensen's bivariate
r
distribution. Their investigation
allowed for various combinations of canonical correlations. Because they were dealing with the EAD analysis·,
they made the necessary assumption that there is no
interaction (otherwise Jensen's formula would not apply).
They advocated consideration of experimentwise error rates
when performing multiple Ftests of main effects much as
one would do when "assessing differences among levels of
a main effect.
1I
None of the papers reviewed dealt with the STP analysis
directly, the EAD analysis in the presence of interaction,
or the relative power of these analyses.
CHAPTER 3
The Geometry of ANOVA and the STP and EAD
An~lyses
The ANOVA of fixed effects two-way factorial designs
is bften studied and applied in terms of its relationship
to multiple linear regression. To many users the mathematics of AN OVA is a collection of least squares equations
and
r
ratios with little to help those seeking a basis for
understanding. The foundations of linear regression are
embedded in the geometry of finite dimensional Euclidean
vector spaces. Within this Tealm an intuitively appealing
development of the theory of linear models is possible.
For more on this and the history of geometry in linear
models,
se~
Herr 1980.
In this chapter we will discuss some of the geometric
aspects of ANOVA and define ideas and terminology needed
for subsequent chapters.
Consider an aXb, fixed-effects, no-empty-cells design
with a total of n observations arranged as an nx1 vector
in
~xicographic
order, first by rows and then by columns.
This data vector, Y, is contained in n-dimensional Euclidean spac e
mn .
ren •. The
mean ve ctor, .jl
=
E (Y ), will al solie in
Proposing a linear model for Y with respect to the axb
design corresponds to identifying a subspace, say (1, where
35
~
is assumed to reside. The method of least squares for
fitting the linear model is the process of finding the
vector in ~ which is closest to Y. This vector is the_
perpendicular projection of Y. onto
~,
and is denoted by
p(Y:~)~ see Figure 1. The space~ is called the "estimation
space" since estimates of
}.I
suggested by the model are in
~.
Let X be a full ranknxs matrix and let the span of the
columns of X be denoted by [X]. If we assume that ~ = XS
for some unknown but fixed vector of parameters
S
in
then [X] is an example of a subspace ~.Consequently
'fIf,
mS
is
called the "parameter ~pace." The vector Y is thus modeled
by ~
=
E(Y) plus some random error vector, e, and written
as Y
=
XS + e. If S is chosen to be the vector of expected
cell means in a two-way design, then X is the incidence
p(Y:~)
Figure 1: The perpendicular projection of Y onto
the vector space
~.
36
matrix (with one (1) in each row corresponding to the
appropriate element of
a and
zeros elsewhere) and s = abo
In what
follows we shall only be concerned with the case
in which
a
is the abx1 vector of cell means.
The parameter spacem
ab
can be written as the direct
sum of orthogonal subspaces J, AIJ, BjJ, and AB which are
defined below.
J: the subspace spanned by the abx1 vector of ones,
(we will also use J for the analogous subspace
ab
ofm n and depend on the context to indicate which
1
J is intended);
A: the subspace corresponding tothe row factor and
equal to the span of all vectors that could be the
true parameter vector if there were no column and
no interaction effects;
AIJ: the orthogonal complement of J in A+J, or just A
since J isa subspace of A;
B: the subspace c6rresponding to the column factor and
equal to the span of all vectors that could be the
true parameter vector if there were no row and no
interaction effects;
B.IJ: the orthogonal complement of J in B;
AB: the
~rthogonal complement of A+B inm ab and equal
to the span of all vectors that could be the difference between the true parameter vector and a
parameter vector corresponding to a no interaction
model.
37
Using the orthogonal decomposition of the parameter space
mflb= J lD AIJ lD "BIJ lD AB, wherelD denotes the direct sum
of subspaces, we may write the mean vector as
.
. ab·
B = BJ+BAIJ+BBIJ+BAB where for any subspace Wofm ,
B W = p(B:VV). An additive model is one such thatB AB = 0 or
BEA+B. The existence ofB AB ~ 0 thu~ indicates a nonaddi~
tive or interactive model.
To make the connection with. the usual over-parameterized parametric representation of the linear model,
E(Yijk) = ll+ai+Sj+(aS)ij' note that E(Yijk) is the (i,j)th
cell mean which is a. coordinate
of (3 and
..
ll,
a.,
..
1 ·S.,
J . (as) 1J
BAB
with the constraints l(aB) .. ::: l(aB)i' :: lao = IS'= o.
are the corresponding coordinate~ of (3J' BAtJ~ BBIJ'
i
1J
JJ
i
1
jJ
Three hypotheses of interest arel
HI: no interaction;
HR : no difference in main row effects} and
He: no difference in main column e£fects.
Although different interpretations are possible, these may
be expressed parametrically as:
HI: (as)
.. = 0 for i=1,· .. ,a and j=l,"·,b;
. 1J
HR : a i =
Ct
i +1 for i=1,···,a-1; and
He:. Bj = Sj +1 for
j =1 , • • • , b-1 •
Geometrically we express these hypotheses as:
HI: BAB = 0;
HI{: BAEJ or BAIJ = 0; and
He: BB EJ or BBIJ = O.
The geometric representation leads us to a natural
38
method for testing the hypotheses. Suppose U is some sub, ab
'
space ofm
and we hypothesize li : BU = O. ~his means that
U
the squared length of aU' .
an estimate of 13, say
a,
II BU W ,
is zero under HU ' Using
large values of S~~
=
IIp(s:U)W
may be regarded as evidence against H ' To quantify this
U
evidence however we must know the null distribution of S*.
Assume
that [U]
=
S~
N(B,Z) and let U be a full rank matrix such
1
U. Then A- S*
~ X~dim
U) for some
posi~ive
( , )';"1 ,
constant A if and only if A-1 UU
U
U Z is idempotent
(Rao,1968). It is easily shown that this is equivalent to
requiring U'ZU= AU'U. Consider also S~ =
some V in
mab •
IIp(s:V)W for
Let V be a full rank matrix such that
[v] = V. Then S* and S~ are independent if and only if
=
U'ZV
0 (see Chapter
4,
Corollary
4.4).
If U and V are
orthogonal subpaces, then the requirement is
(3.1)
u'zv = AU'V, which is 0 since U'V = O.
Let Y ~ N.(XI3,ifI) and
a=
(x 'X)-1 X'Y, the best linear
unbiased estimate of the cell means vector
Z
B.
Then
= a 2 (X'X)-1. LetTl' = a 2 (n -1 n -1 ••• n -1'.' n -1), the
11
12
1b
ab
diagonal of Z. Table 3.1 shows for various vector spaces
U and V orthogonality conditions on n for 3.1 to hold
(see Appendix A for a proof).
From Table 3.1 we see that if one requires that the
S* statistics for HR1 and H
be independent chi-squared
C1
random variables, then n .J. (A/J ffi BIJ ffiAB) or ne J for.
a>2 and b>2. If one only requires independence of all three
tests, then the same conditions prevail. Finally if one
39
Orthogonality Conditions on T)=diag(a 2 (X'X)-1)
Table 3.1.
such that U'(a 2 (X'X)-1)V=;>..U'V. t
u
V
AIJ
AI J
for a> 2 ; n.l. AI J t t
for a=2; no restrictions
;>. -1 S~~'VX (a-1) ~~
BIJ
BIJ
for b> 2 ; T).l.; BI J
forb=2; no restrictions
A
AB
AB
for
for
for
for
AIJ
·B/J
Comments
Orthogonality Conditions
-1
a> 2, b> 2 ;T).L~abIJ
a>2,b=2;T).L AIJ
a =2 , b> 2 ; T).L B J
a=2,b=2; no restrictions
S~~'VX
. ..( b-1.. )
-1 S~~ X
'V (a -1 ) (b-1 )
A
A-1 s~~ , A-1 S ~
T)..L AB
independent
AB
AIJ
AB
BIJ
for a> 2; T).L (BIJ
for a=2;T)J. BIJ
(9
for b>2;T)J. (AIJ
for b=2; .J.AIJ
(9
AB)
A-1s~~,A-1s~
AB)
independent
A-1S~~,A-1S~
independent.
t
A=a2IIn:~/ab, the_mean eigenvalue of a 2 (X'X)-1
tt
T)
ij lJ -
"is orthogonal to" AIJ
~~ . S ~~ =
"p(a:
~H~ s~ =
II P ( a:V) W
U)
11 2
simply requires that the test of HI bea chi-squared test,
then T) .L ma b I J or T)£J for a> 2 and b> 2. If T) £J, then
n .. = n/ab, that is the design is balanced. Thus departure
lJ
from equal cell sizes may lead to undesirable properties
of the S* statistics.
Some further notes on S* are:
i.
A is unobservable unless
a2~s
known. When a2 is
unknown the residual sum of squares for regression
40
lip (Y: mnl [X] ) W divided by n-dim ( [X]) will provide an
unbiased independent estimate of a 2 , say 0 2, if
Hence
F~~
=
s~qab/(dim(U)02?~ni~)}
'V
lJ
T (dim(U)
Xa.
l.l =
,n-dim( [X]))
when aU = 0 and a 2 U'(X'X)-1 U = (a2IIn:~/ab)U'U.
ij lJ
ii.
The statistic F* above was originally proposed by
Yates (1934) and the resulting analysis.is usually referred
to as the unweighted meanB analysis. The c6nditions listed
in Table 3.1 are thB conditions for which
test is
Yat~sl
an exact F test.
iii. As will be shown next, A- 1 S* ( or F*for unknown a 2
is the likelihood
)
ratio statistic when the conditions of
Table 3.1 hold.
The model Y
=
may be rewritten as
Y = Xa
+ Xa + e where U (B V = mab and U .1. V• Thus
U
V
l.l
Xa+ e
= XaU + Xa V as shown in Figure
V,
in
where for any W inm ab ,
=
2. If aU
W=
0, then
A
=
lies
({Xw:ws W}), as shown
. in Figure 3. It is readily seen that p(U: mnlv)
p(V:V) ~Ounder Hu: aU
l.l
O. When Y
=
N(1.l,a 2 r) and
l.l
-
l.l
= Xa, the statistic for the likelihood ratio test of HU
'V
is given by:
(3. 2 )
for known a 2 where
G = P(Y:[X]).
Since
ns
[X1, then 3.2
may be written as:
A
IIP(Y:[x]
Iv)W la 2
•
The idea here is to view large values of 3.3 as evidence
41
U
/'
(3
u"
lJ
Figure 2:
v"
v
8V
XS V
S as Slm of components in U and V and lJas sum of
"
components in U
and V" whete for any W inm a b ,
W= ({Xw:we:W}).
u
P (D:
"
mn Iv)
~--------1t---PV
Figure 3.
" under HU:SU=O and the
lJ as an element of V
perpendicular projection of lJ" onto the orthog" inffi n •
ganal complement of V
against HU (see Figure 3).
It is easily shown that since U
for any tv in
~ab, ~
" = U where
V, [X] I V
= ({X(X'X)-1 w: w e:W}). For W in mab the
.L.
two spaces Wand Ware arrived at via twa different transformations from the parameter space to the estimation space.
Each transformation, X and X{X'X)-1, may be thought of as
42
a bridge from
mab
to
[xl.
The maximum likelihood estimate of
"
the perpendicular projection p(Y:W).
XS
wis
y
~ N(~,02I)
and
~
If
= XB, then the likelihood ratios
statistics for HI' HR1 , and HC1 are:
'"
2
HI: 0 S I = "p(Y :AB) W ;
R1 :
0
:
0
H
. H C1
'"
2
SR 1 - "P (Y : A I J )W
2
Se 1
and
'"
= 1/ P ( Y : B IJ.) W
Furthermore:
SI
~ X(a-1)(b-1);
SR1 ~ X(a_1); and
x( X' X) - 1 W( W' ( X' X) -1 W) -1 W( X' X) - 1 X' (0 2 I ) / 0 2
Not e that II P (Y:
w) 11 2
/0 2 =
is i de mpot en t .
a' W( W' ( X' X) - 1W) - 1W' ~ / 02 Whie h
is equal to A-1~'W(W'W)-1W'a if W'(X'X)-1 W = AW'W. Thus
if the conditions of Table 3.1 hold, the statistics SI'
SR1' and SC1 are the S*
st~tistics
discussed earlier.
When considering the three hypotheses HI' HR, and HC'
it is common practice to first test HI and if insufficient
evidence is found against it, assume the additive model is
correct and test HR and/orHC.When the additive model is
correct, r3 = r3 J + 8 AIJ + r3 B]J. Tha parameter space is thus
reduced to AIJ ffi B ana the estimation space is reduced to
".
"
(1 a = AIJ + B. The likelihood ratio statistic for testing
A
A
A
HR1 : r3 AIJ = 0 is IIp(Y:(1a!B)W. Since (1a = A + B, then
(1 a I~ = ArB. The likelihood ratio statistics under the
additive model are:
43
'"
a 2 S I = IIp(Y:AB)W;
"'A
a 2 S R2 = IIp(Y:AIB)W; and
1\
a 2 SC2
1\"
= IIp(Y:BIA)W
when Y rv N(1J, a 2 I) and 1J = XI3 • Furthermore:
SI
rv X(a-1)(b-1);
SR2 rv X(a_1); and
SC2 rv X (b-1 ) •
The subscript 2 on SR2 and SC2 is there because in general
these sums of squares are not the same aa the earlier sums
of squares derived without
the additivity "assumption.
Therefore the t.wo sets of test statistics provide two
different analyses. The first (SI' SR1' Sel) is the STP
analysis mentioned
in the previous chapters and the second
(SI' SR2' SC2) is the EAD analysis. We now explore the geometric properties of these analyses.
The properties of STP and EAD are summarized in
Table 3.2. To demonstrate these properties first consider
vector spaces inm n which are the results of mappings
fromm ab either by X (the best-fit-bridge) or X(X'X)-1
(the best-estimate-bridge) (Burdick, e.t al., 1974). Neither
transformation preserves orthogonality under general conditions, but spaces orthogonal inm ab mapped to mn via
different bridges will still remain orthogonal. That is if
U
..1..
V, then U
.L
V. 1"11 th this in mind we readily see that
certain subspaces are always Qrthogonal (see Table 3.3)
while others may be orthogonal under certain conditions on
the cell sizes.
44
Table 3.2:
Geometric Properties of the
STP and EAD Analyses
x2
Hypothesis
Noncentrality Parameter
~
~
IIp(lJ:AB)W = IIp(XSAB:AB)W
STP: . HI
= 0 if and only if HI is true
~
~
IIp(lJ:AIJ)W= IIp(Xt3 A IJ: A !J)W
= 0 if and only if H
R1
is true
~
II P (lJ: B I J ) If = II P eX S BI J : BI J ) W
HC1
= 0 if and only if H
C1 is true
AA
EAD:
H
AA
IIp(lJ:AIB)W= IIp(Xt3AIJ+Xt3AB:AIB)W·
R1
= IIP(Xt3
AB
: AII3)Wif H
R1
is true
= 0 if. H 1 . and HI are true
R
A
.A"
_
II P (lJ :B I A) W=
HC1
_
A
A
II P ( Xt3B I J +Xt3 AB: B I A) W
= II P (XS
AB
= 0 if H
:BIA) Wif
H
C1 is true
C1 and HI are true
Additive Model Properties (HI true):
~
~
A
IIp(lJ:A/J) W~ /Ip(lJ: A IB)J!2
~
II P (lJ :B I J)
W~
.-
A
A
II P (lJ :B I A)
W
45
Table 3.3:
Orthogonal Subspaces of Ex]
1\
J
B AB
!].
0* 0
0
0
0
0
AIJ
-
-
0
BIJ
0
0
AB
0
0
1\
0
0
0
0
1\
BIA
~~
0
1\
AlB
1\
1\
1\
J ·A
0
0
0
0 denotes orthogonality
The STP analysis has the property that the effect being
tested for will not he affected. hythe presence or absence
of the other effects. For example if there is interaction
(B
AB
~
0) then it will show up in AB
o~
the estimation
1\
space. We see from Table 3.3 however that AB is orthogonal
-
~
to both AIJ and BIJ which is where we would test for row
and column effects. Thus the presence of interaction will
-
~
not affect the component of
~
in AtJ or BtJ. The result of
this property is that under the null hypotJ:mis HR1 or HC1 '
the X 2. distribution will be central regardless of the
validity of the other hypotheseS.
The EAD analysis does not share the above property
under general conditions. For example, the presence of
A
,..
A
A
interaction may show up in AlB and/or BIA because these
1\
spaces are not orthogonal to AB. The X 2 distribution under
46
the null hypothesis for HR1 and/or HC1 may be noncentral
with unknown noncentrality parameter, To illustrate this
problem we consider two cases concerning HR1 ,
Case 1: HR1 true implies that xa A/ J = 0, Now under HR1
"
""
u = xaA/J + Xa B + Xa AB = xa B + Xa AB , Since xa B E B ~AIB,
the noncentrality parameter f·or SR2 is "p(u:AI~) W
A
A
A
A
=
A
//p(XaAB:AIB)Wwhich is zero when xa AB = 0 or AB ..L.AjB
(see Figure 4), Thus one may be more likely to reject HR1
when HR1 is true using SR2 than SR1 in the presence of
interaction,
" "
AlB
~f
'"
p( U : A I J) i s at
the origin
'"
A/J
Figu~e
4:
Component
~~
" "
U in AlB in the presence of
interactionwhenH
R1
is true,
47
A
A
A
Suppose P(lJ: A/ B)= O. Then
Case 2:
A
A
A
A
A
p(lJ:AIB) = P(X 13
:A/B) + P(XI3 AB :AIB) = 0 or
A1J
P(XI3AIJ:A/B) = -P(XI3 AB :AIB).Thissuggeststhat in the
A
A
A
"
presence of interaction, XI3AIJmay not be the zero vector,
Le., HR1 may be false (see Figure 5). Thus one may be less
likely to reject HR1 when HR1 is false using SR2 than SR1
in the presence of interaction.
In the above.discussion a key phrase was "in the presence of interaction". How do the STY and EAD analyses compare when the additive model is the correct model? Suppose
A
A
"
A
A
R1 is true. Then lJ = XI3 B andP{lJ:AIB) = 0 since B .J..AIB.
The noncentrality parameters for SR1and SR2 are both zero.
H
A
A
A
Y(XI3 AB :AI B)
A
AlB
A
* p(lJ:il~)
is at
the origin
AIJ
A
A
Figure 5: Component of lJ in ArB
~eing
zero in the presence
of interaction when H
is false.
R1
48
Suppose HR1 is false. Since pEA + B and A + B = AlB ffi B,
" " +P(p:B).
"
then p = p(p:AIB)
The noncentrality parameter
A
~
for SR1 is thus IIp(lJ: AIJ)
E B ~ A/J.
"
"
P(lJ:B)
~
W =llp(p(lJ:AIB) :AIJ) W since
It is easily shown that
A
~
A
A
A
~
A
/jp(lJ:AIJ) W~ IIp(lJ: A/B )W since p(lJ:AIJ) and P(lJ:A/B)
. represent respectively the leg and the hypotenuse ofa
right triangle (see Figure 6). Thus under the additive
model the probability of rejecting HR1 when HR1 is false
may be greater using SR2 then SR1' That is, the EAD analysis
provides at least as powerful a test as the STP analysis
when the additive model is correct.
A desirable property of any analysis of the main effects
is that the sums
o~
squares be independent. This means that
the spaces on which Y is projected to obtain the sums of
squares must be orthogonal. For the EAD anlysis independence
" "
AJB
" "
,P(lJ:A/B)
.
!
.
!
I
.c;,....--~-'"'iJl--__---+
-
AIJ
P (p:A jJ)
Figure 6:
-
" "
Components of lJ in AlB and AIJ
(cor~esponding
to the degree of noncentrality in SR2 and SR1)
when the model is additive.
e.
49
A
-'"
1\"
of SR2 and SC2 requires AlB ~BIA. The well known ~ondition
for orthogonali ty is that the cell sizes are such that
n ij = ris j for positive constants r i (i=i,···,a) and
s. (j=1,···,b). This is the condition of proportionality.
J
An important note to make here is that proportionality does'
AI\
1\
1\/\
not mean that AlB or BIAwill be orthogonal toAB. So SR2
and SC2 may still be affected by the presence of interaction.
....
....
For the STP analysis the condition for AJJ ~ BIJ is
given in Table 3.1 as the condition for U'(X'X)-1 V/ a2 =
~U'V = 0 when
U = AIJ and V= BIJ. The requirement is that
the vector of reciprocal cell sizes, fl, be orthogonal to
AB. This may also be written as
-1
n~.
1J
~r~+
.
1
s.J for real
constants r. (i=1,···,a) and s. (j=i,···,b) (Burdick, e.t al,
J
1
....
1974). Note here that this condition does not leave AIJ or
....
....
BIJ orthogonal to AB (see Table J.1 f6rconditions on n
to achieve this orthogonality). Thus SRi and/or SCi may not
be independent of ~, but recall that ATJ andBTJ are always
orthogonal to AB, the location of any interaction that
might be present.
In this chapter we have investigated the geometri6
aspects of the ANOVA and in particular the STP and EAD
analyses. In Chapter 5 we will quantify some of the differences between STP and EAD using the properties mentioned
here and the distribution theory developed in Chapter 4.
Chapter 4
Distribution Theory
In this chapter we discuss the distribution theory
relevant to testing multiple hypotheses in the fixedeffects, all-eells-filled ANOVA setting. We begin with the
chi~squared
(X 2 ) distribution and the t distribution. We
conclude with a discussion of computational methods for
evaluating the cumulative density functions (cdf's) of the
mentioned distributions.
4.1 X2 Distribution
Definition 4.1 (Incomplete Gamma Function)
Letr(.) denote the gamma function defined by
r(a) = f~ta-1exp(-t)dt; a>O.
The function r(·,·) defined by
r(a,x)
==
{r(a)}-1J~ t a - 1 exp(_t)dt; a>O, x~O
is called the incomplete gamma function.
Clearly r(·,·) is acdf, thus
r(a,O) =
a
and limr(a,x) = 1x+oo
Theorem 4.1. (Noncentral Incomplete Gamma Function)
The series
G(x;v,w)
0
=1
I
forx<O,
.ex p (-W/ 2 )(w/ 2 )kr (v/ 2 + k,x/2)/k! for
k=O
.
is a cdf in x. for v>O and\jJ ~O.
x~O
51
Proof
The theorem is a direct result of the properties.of
the incomplete gamm~ function (Bearle, 1971~~.49).
One family 'of distributions represented by the noncentral ihcomplete gamma function is the family of X 2
distributions.
Definition 4.2 (X 2 Distribution)
A random variable X with cdf G(x;v,l/J),(v=1 ,2,···
is said to have a X2 distribution,
~enoted
by X2
;tiJ~O),
,I,'
V,'ll
The parameter v is the degrees of freedom and l/J is
the noncentrality parameter. When
~=O,
the distribu-
2
tionof X, denoted by Xv'
is called a centralX 2 and
itscdf is given by:
G(x;v)=
10
forxCO
r(v!2,x!2) for x,:.O
For the remainder of this chapte;cwe.assum$ that the
nx1 vector Y is distributed as a multivariate normal with
vector 8 and positive definite, dispersion matrix
me~n
v
(that is, YrvN(a,V) ).
The following theorems are useful for discussing proper-
ties of the X2 distribution.
Theorem 4.2
y'AY rv X 2
where v=rank(A) and l/J=8'A8 if and only if
v,l/J
AV is idempotent.
Proof
See tearle (1971, p.57).
52
Theorem 4.3
E(y'AY)
= tr(AV)+ 8'A8.
Proof
Searl~ (1971~
p. 55). The function tr(A) denotes the
trace of A.
Theorem 4.4
Var(y'AY)
=
2tr(AVAV) + 48'AVA8.
Proof
Searle (1971, p.57).
Corollary 4.4
Cov{y'Ay,y'BY)
=
2tr(AVBV) +
48'AVB~.
Proof
First note that 2Cov(y'Ay,y'BY)
= Var(y'(A+B)Y) Var(y'AY) -
Var(Y'BY~
From Theorem 4.4 we have that:
Var(y'(A+B)Y)
=
2tr«(A+B)V(A+B)V) + 48'(A+B)V(A+8)e
=
2tr(AVAV) + 48'AVA8 + 2tr(BVBV) +
48'BVB8 + 2(2tr(AVBV) + 29'AVBe +
28'BVA8)
=
Var(y'AY) + Var(y'BY) + 2(2tr(AVBV) +
48' AVB8).
Hence Cov(y'Ay,y'BY)
= 2tr(AVBV) + 48'AVB8.
For the remainder of this
chap~er
let U and W be nxn
symmetric matrices with the following properties:
i. Ux = p(x:[U]) and Wx = P(x:[W]), that is U and Ware
projection matrices and therefore
idBmpotent~
ii. rank(U) = p and rank(W) =q where, without loss of
53
general'i ty,
iii. tr(UW)
=
p~q;
f A.
. 1
1=,
1
and
where {A.}
1
P
are the p largest eigen-
1
values of UW (the other q-p eigenvalues will all be
zero}. Note that as a property of idempotent matrices,
A> 0 (i =1 , • • ., p ) •
Also for the remainder of this chapter, let S1 = y'UY/d 2
=
S2
Y'WY/0 2
Theorem
L
,
and V
=
,
0 2 1 n·
4.5
2
2
S1 '\; XP,w1
and
- - S2 '\; Xq,W
2
where W1 =
II P (E Y: [W}) W/ 0 2
= e'ue/0 2 and
=
1/
P (EY :[UJ ) /-/2/ 02
e' we /0 2 ;
ii. Corr(S1,S2)
and
iii. S1 and S2 are independent if and only if U and Ware
orthogonal.
Proof
i. Since U and Ware idempotent, the result follows from
Theorem 4.2.
ii. By Corollary 4.4,
Cov(S1,8 )
2
U(021 n }W(021 n )/0 4 +
= 2tr(
- 2tr(UW) + 4a'UWa/0 4
= 2(IA. + 2a'UWa/a4 ) .
1
1
By Theorem 4.4,
Var(8 )
1
= 2trCU(021 n )U(02I n })/a 4
+4a'U(02I )Ue/0 4
n
54
=
2(p + 2W ).
1
= 2(q
Similarly Var(S2)
+ 2W 2 ). Hence·
1
Co~r(S1,S2) = Cov(S"S2){Var(S1 )Var(S2)}-§
=
{f1
A.
+
28' UWe/0-2}{ (p
+ 2W 1 ) (q + 2W 2 )} -! .
1
iii. Searle (1971, p.59).
The characteristic function (cf) is defined by
¢(t)
=
f_:exp{itx)dF(x) where F(x) is a cdf. This is·
useful in deriving distributions by means of the inverse
Fourier exponential transform f -00
ooexp(-itx)¢(t)d~.
The cf
ofaX~,w random variable is given ~y Oberhettinger (1973):
(4.1)
¢X 2
(t) = (1_2it)-V/2 exp(itW/(1-2it)).
v,:ljJ
A bivariate cf is defined by
where F(x ,x 2 ) is a bivariate cdf. The bivarite cf of S1
1
and S2 is given in Theorem 4.6.
Theorem 4.6 (Characteristic Function of S1 and S2)
(4.2)
¢S1,S2(t ,t )
1 2
=
(1-2it )-P/2(1-2it )-q/2
2
1
p
x
l-
II ( 1- CA . ) j =1
2
X
J
exp(c 1
f L1'J
j=1
+
c2
f L2'J
j=1
c rff.L1,L2')'
j=1 J J J
~~ .(
)-!
where Lkj = 8 k . 1- CA.
J
J
+
55
8 1j is the jth element of 8 1
~~
~f
= u'e
1
~~
~~
= w'e
1
8 2j is the jth element of 8 2
U and W are nxp and nxq full column-rank matrices
1
1
such that U U l ' = U, U 'U 1 = I p ' W W1 ' = W, W1 'W 1 = I ,
1
1
1
q
and U1 'W 1 = C = (diag(!X1, ••• ,/Ap)iO) ; .
A.
J
=
0 for j>p;
c k = it k /(1-2it k ); and
c = 4c1c2 •
Proof
[::JY/O
Let Z •
*
.~~
.0 that Z ~ NIB ,V ) where
:r
q
By the definition of ¢
.8 1 ,8 2 '
( 4 •3 )
<P 8 1 ' 8
2
(t 1 ' t 2) = E ( e xp (it 1 8 1 + i t 2 8 2 ) )
.." here T =
[t
= E(exp(iZ'TZ))
1I p
o
4.3 can be written as:
Generalizing equation 2.10,
56
where M = [M 11
M
21
1
f.( 1-2i. t 1 ) I p
-2it 1 C
l-2it 2 C'
(1-2it 2 )I
q
J
The partitioned representation of M allows its
determinant to be written as:
IMI =
-1
IM1111M22 - M21M11M121 •
Thus,
IMI =
(4.5)
(1-2it 1 )PI(1-2it 2 )I q - 4it1it2C'C/(1-2it1)1
.
p
= (1-2it 1 )P(1-2it 2 )q IT
j =1
(1-d~")
J
Now M- 1 can be written as:
M-1 = {( 1- 2i t ) ( 1- 2i t 2 )} -1 x
1
1-2it }(I - cCC·)-1
2
p
( 2it (I - cC'C)-1 C'
2
q
Thus,
i8
~~,
-1
M T8
~~
= {(1-2it 1 )(1-2it 2 )}-
1
x
{it1(1-2it2)8~~'(I _cCC,)-18~~
p
*,
,
it 2 (1-2it 1 ) 8 2 (Iq-cC C
~~,
2it 1 it 2 8 1 (Ip-cCC
,)-1.
)-1
*
.8 2
~~
C8 2
+
+
+
2it1it28;~(I -cC'C)-1C'8~) ,
~~
where 8
=
~~
le~l
.
82
L
~~
8 '(I -cCC
1
p
,)-1
.
8
q
.
By making the substitutions:
.
~~
1
. ii. 8;'(I -CC'C}-1 8;
q
!2.
= 2 (1-cA.
j =1
S\.
= L
)-1 ~~ 2
8 1J"
J
(1-c~")
j =1.
J
-1
~~ 2
8 2j
~.
, and
57
iii.
=
J2.
~~
-1
~~
(1-c>...) 1T.8 1 ·8 2 ·
j ;::1
J J " J . J."
1..
·
into 4.6 and 1 e tt lng
e"k*.' we h ave
LkJ. -- (1-c,.)-i
A
J
r
f
='C 1
L1Z • + c 2
L2Z •
J
. j=1·
j=1 J
J
+c
f IrL1·L2··
J J J
j=1
Upon substituting 4.5 and 4.7 into 4.4, the result is
obtained.
We now consider the bivariate cdf of 3
(4.8)
1
and 3 2 •
f31,32(x1'X2) = f_:f_:exp(-it 1 x 1 -it 2 x 2 )
Let
x
¢3 ,3 (t 1 ~t2)dt1dt2
1 2
be the probability density function of 3 1 and 8 2 • Jensen
(1970a) solved 4.8 for the case where ~1= ~2 = 0 (both 3 1
and 3 2 are central X Z variables here). This in turn leaves
Lk j = 0 (k =1 , 2;
00
as
L a.kc
k=O J
k
f= 1 , • • • ,"p).
1.
J en s en first exp an ded (1- c >.. j ) - 2
,where
a jk = >..~r(k+i){r(k+1)r(i)}';'1
(j=1,···,p; k=O,1,2,.·.).
Collecting terms in c k we have that
p .
1.
00
k
IT (1_C>...)-2 = kL='oGk(A)c
j =1
J
where A = (>... , ••• ,>.. )' and Gk(A) = L
La
a ···a ..
1
p
k +••• +k =k 1k 1 2k 2 pk p
Thus
1p
I
G (A)(2c )k(1-2it )-p/2
1
1
k=O k
(2c )k(1-2it )-q/2 •
2
2
x
58
Applying the inverse Fourier exponential transform given
in equation 2.11 to 4.9,4.8 has the form given in equation
2.5 wi th 0
=
A.
Removing the restriction of centrality and using
Jensen's method, the general cfof 8
(4.10)
¢8 ,8 (t ,t ) =
1
2
1
2
1
and 8 2 is:
L {G k (A)(2c 1 )k(1-2it 1 )-P/2
k=O
~ 2
exp ( c1tT1j
) (
2ci
)
k(
1-21t 2
0
)
-
q/ 2
eXP(c2rT2o)exp(cXIX:T1oT2')} •
1
J
1
J
J
J.
.
In 4.9 each term of the summation can be factored into
two products, each one involving only one of t 1 and t 2 •
This facilitates the use of 2.11. Unfortunately this
algebraic nicety is not possible with the general case of
4.10, at least iri its present form. A problem, which to
our knowledge remains unsolved, is to find the appropriate
inverse Fourier exponential transform to invert 4.10.
4.2
r
Distribution
Definition 4.3 (Incomplete Beta Function)
Let B(·,·) denote the beta function defined by
B(a,b)
=
rlta-1(1_t)b-1dt; a>O and b>O. The furiction
o
I(a,b~x) = B(a,b)-1 r x t a - 1 (1_t)b-1 dt , for 0~x~1,a>0
o
and b>O, is called the incomplete beta function.
Itis evident thatI(a,b,O)
=
0 and I(a,b,1)
Theorem 4.7
The seriesH(x;v,t,¢)
=
0
for x<O
=
1.
59
00
I
=
*
•
exp(-¢/2)(¢/2)JI(v/2+j,t/2,x )!j!
j=O
for x>O
where x
~~
= vx/(vx+t), is a cdf in x for v>O, t>O and
Proof
The thBorem is a direct result of the propBrties of
the, incomplete beta functi'on (Searle, 1971, p.51).
One family of distributions represented by, the in'complete beta function is the family of 1 distributions.
Definition 4.4 (1 Distribution)
A random variable X with cdf H(x;v,t,¢), (v=1,2,···;
t=1,2,·.·;¢2:,.0), is said to have an 1 distribution,
denoted by '
v, t
,J.
,0/
,
with, (v, t) degrees of freedom and
'
non centrality parameter ¢. When ¢=O, X is said to have
a central 1 distribution, dBnoted by
1~,tj
and its
. cdf is given by:
. H(x; v, t) = [0
for x<
°*
I(v/2,t/2,x ) for
x~O
The usefulness of the 1 distribution is perhaps best
seen in its relationship to the X 2 distribution.
Theorem 4.8
IfX 1 andX 2 are independent random variables such
that X1 ~ X~~¢ and X2 ~ Xl' then F
= tx,/VX 2
~ ' v ,t,¢.
Proof
Bickel and Doksum (1977, p.16).
Let So
~ X~
and be independent of both S1 and 8 2 as
60
previously defined. Let F1 = t8 1 /p8 0 and F 2 = t8 2 /q8 o for
t>2. We then have the following theorem about F and F •
2
1
Theorem 4.9 (Correlation Coefficient of F 1 and F 2 ;
(4.11)
Corr(F 1 ,F 2 ) = Corrs(F1,F2){1+w1w2Corr(81,82)}
.1.
where Corr s (F 1 ,F 2 ) = {(1+w )(1+w )}-2 is the correlation coeffi~ient of F 1 and F 2 when 8 1 and 8 2 are
1
2
independent, that is , when F 1 and FZ are singly
.1.
dependent; and w1 = {(t-2)(p+2~1)}2/(p+~1)'
.1.
w2 = {(t-2)~+2~2)}2/(q+~2) •
·Proof
is independent of 8 1 and 8 2 ,
Cov(8 1 /8 0 ,8 2 /8 0 ) = E(~8iE(8;2)-E(81)E(82)E2(801)
= COV(8 1 ,8 2 )E(S02)-E(8 1 )E(S2)x
Var (8 1 ) •
0
Now E(S;k) = f~x-kdG(x,t) = f(t/2-k){2 kf(t/2)}-1
for t>2k. Therefore E(8;1)={2(t/2-1 )}-1=(t_2)-1 for
t>2; E(S;2)={4(t/2-1 )(t/2_2)}-1={ (t-2)(t-4)},.1 for
-1·
·2
t> 4; an d Va r (SO) = 2/ { (t - 2 ) ( t - 4 )} .
Us in g the abo v e
and E(S1) = P+~1' Var(S1) = 2(p+2~1)' E(8 2 ) = q+~2'
Var(S2) = 2(q+2~2)' 4.12 may be w~itten as:
(4.13)
.
. .
..1
Cov(8 1 /S 0 ,8 2 /S 0 ) = Cov(S1,S2){(t-2)(t-4)}
+
.
2(p+~1) (q+~2){ (t_2)2(t_4)}-1
.'
1
= 2(P+~1)(q+~2)1{(P+2~1)(q+2~2)}2(t-2)
]
--------.~.
Corr(,,8 )+1 •
(t-2)2(t-4)
(P+~1)(q+~2)
2 .
61
The variance of F1 is Var(F 1 ) = t Zvar(S1!SO)!p2 and
2
(4.14) Var(S1!SO)= Var(S1 )"E(S02) + E (S1 )Var(S01)
= 2(p+2\IJ1){ (t-2) (t-4)} -1
+
2(p+\lJ1 )2{ (t_2)2(t_4)}-1
=
2(Pt w;)2
.!(t_2)(Pt2~1)
(t-2) (t-4) _
t 1] .•
(P+\lJ1).
Similarly,
(4.15)
Var(SiSo)
= 2(qtw~)2
!(t-2)(qt2~2)
t 1]••
(t-2) (t-4) -(q+\lJ2)
2
(t !pq)rrov(S1!SO,SZ!Sn)
x
2
2
{(t !p2)Vir(S1!SO}(t !q2)var(S2!SO)}-k ,
1=
w
using 4.13, 4.14, 4.15, Btidletting
2
2
(t-2)(p+2\IJ1)!(P+\lJ1) and w = (t.;.2) (q+2\IJ2)!(q+\lJ2)
2
we obtain 4.11 •
The form of 4.11 illustrates how the common denominator
of SO' which affectsCorr s (F 1 ,F 2 ),· and the dependence of
S1 and S2' which affects Corr(S1,S2)' together affect the
dependence of F 1 and F 2 • It can be .shown that
lim Corr(F1 ,F 2 ) = Corr(S1,S2) •
t+oo
We now considei the bivariate cdf of F
1
and F 2 • First
note that the cdf can be written as:
The cdf can be derived, at least up to an integral form,
62
given the bivariate cdf of S1 and S2' A general expression
for 4.17 has not been found in closed form except for some
special cases.
Theorem 4.10
When r, s, and t are positive,
even integers and
S1 and S2 are independent, 4.17 can be written as:
H(f 1 ;r,t,w 1 ) + H(f 2 ;s,t,w 2 ) - 1 +
k k'
(4.18)
exp(-(w 1+w 2 )/2)
A f, 72r (t / 2 )
k=O
.
n
n2
r1 r
I
00
k~=O
W1W2
kl
k!k !2 k +
'
• I
x
Z JzJ
1 2
r(t/2+j+j') ,
j =0 j' =0 j! j I !
where A = rf 1 /t + sf 2 /t + 1; z1 = rf 1 /tA; z2 = sf 2 /tA;
n 1 = r/2+k-1; and n 2 = s/2+k l -1 •
Proof
From elementary probability theory, if
~1
and
two events, then P(~1~2) = P(~1) + P(~2) -1 +
where
I1
and
I2
~2
denote
p(I1I 2 ),
are the respective complementary
events. Thus in order to establish 4.18, it will
suffice to show that the last term of 4.18 is:
( 4. 19 )
P (F1> f 1 ; F2> f 2) •.
Since S1 and S2 are independent, 4.19 can be written as:
For even degrees of freedom v, Abramowitz and Stegun
(1964, equation 26.4.5) show that:
G(z;v) = 1 - exp(-z/2)
v/2-1
I
j=O
.
(z/2)J/j!
63
Thu~
applying Theorem 4.1,
1 - G(zp/J,v) =
(4.21)
k
v / 2 +k -1 .
exp(-$/2){(¢/2) }kl}exp(-z/2)·
I (z/2)J/ j l
k=O
j=O
co'.
I
Let n 1 = r/2+k-1 and n 2 = s/2+kf~1 and n6te that
g(x)=dG(x;t)/dx = xt/2-1exp(_x/2)/£2t/2r(t/2)} • Then
using 4.21, 4.20 becomes:
co
where
K.
l
(x) =
I
k=O
a. (x,k) and
l
a. (x,k) = exp(- ¢./2){(¢./2)k/kl}exp(-x/2)
l
l
l
r
i (x/2)j/jl.
'0
J=
Making a change of variablew = Ax, where
A = rf 1 /t + sf 2 /t + 1, and interchanging the integral
and summations (see Appendix B for justification of
this step), we obtain:
co
(4.23)
I
co
I
k=O k'=O
f:a 1 (x,k)a 2 (x,k')g(x)dx
.
Each term of 4.23 will bB the product of a factor not
involvin~
w times a factor of the form:
A-(t/2+j+j') f:w(t/2+j+j '-1) /(2(t/2+j+jf) )exp(-w/2)cLw =
r(t/2+j+j' )/(A (t/2+j+j' ))J:dG(w;t+2j+2j') =
r(t/2+j+j'}/(A(t/2+j+j')).
Letting z1 = rf /tA and
1
z2 = sf 2 /tA, we obtain the last term of 4.18.
From Jensen's (1970a) bivariate X2 , we have an expression for 4.17 when 3 1 and 3 2 are central X2 random. variablEs.
64
Theorem 4.11
When W1 = W2 = 0, the bivariate cdf of F1 and F2 is
given by:
00
k
k
I GkO.) I
I b (k) x
k=Om=O n=O mn
P s {T r +2m ,t ~ rf 1 /(r+2m);T s +2n ,t ~ sf 2 /(s+2n)}
1
where G (>..) is as defined for 4.9; bmn·
(k)=(_1)m+n(k).[k
· · m n, ;
k
and P {T
s
v1 '
t < x 1 ;T
t
v2 '
~
x 2 } denotes the joint cdf of
two singly dependent T variates with distributions
T
and T
v1 , t
v 2 ,t •
Proof
Consider the cdf of F 1 and F2 in the form of 4.17.
Using 2.5,
(4.25)
J
P(S1 ~ rf1x/t;S~ ~ sf 2 x/t) =
rf 1 x/t sf 2 x/t ~
k
k
I G ( >.. ) I I b (k ) G' (t 1 ; r +2 m)
k=O k
m=O n=O mn
J
o
0
G' (t 2 ;s+2n)dt 2dt 1
k
06
x
=
k
I G (>..) I
Ib (k)G(rf 1 x/t;r+2m)G(sf 2 x/t;s+2n)
k=O k
m=O n=O mn
(the justification for the last step is similar to
that made for 4.23, see Appendix B). Therefore the
cdf is:
k
k
I Gk (>..) I
I b (k)
k=Om=O n=O mn
00
(4.26)
x
65
From 4.17, we recognize the iritegral in 4.26,as
Ps(Trt2m,t ~ rf1/(rt2m);Tst2n,t ~ sf 2 /(st2n», th~s
4.24 is established.
4.3 Computational Methods
The computation of probabilities from univariate X2
and T cdf's is facilitated by the availability of functions
in SAS which were contributed by Hardison, Quade, and
Larigston (1980). These functions,
and a noncentrality
gi~en
degrees of
paramete~, canp~ovide
fr~edom
the probability
given the argument of the cdf, or provide the argument
given the probability.
To calculate probabilities of singly dependent T's,
one may use 4.18, for the, cas e of even degrees of freedom,
by truncating the infinite series. For the case r=2 and
s=4" a limit of 50 for k gave us results accurate to four
decimal
plac~s
for even values of t. The programming was
done in SASe A listing is included in Appendix C.1.
To calculate probabilities of doubly dependent, but
central T's, we employed 4.18 together with 4.24., The
method used fo'r calculating values of G(A) is similar to
one suggested by McAllister .et al.
(1981 a). This method
also involves the truncation of an infinite series. For
r=2 and s=4, a limit of 20 for kyielded results accurate
to three or four decimal places £or even values of t. A
listing of the SAS program is in Appendix C.2.
Finally we considered a method for evaluating the cdf
66
of doubly dependent, doubly noncentral T's. Because this
distribution has not been found in closed form, we must use
an approximate method. None of
the methods reviewed in
Chapter 2 was deemed optimal for various reasons:
i. (Gaussian quadrature based on orthogonal Laguerre
polynomials.) This method proved accurate for the·
univariate case, but not for the bivariate, doubly
central case. We thus assumed the same would be true
for the doubly noncentral case.
r
ii.
(Substitution of IA./r
for each A. (i=1,···,r).)
1 1
As
1
mentioned in Chapter 2, this method overestimates the
probabilities.
iii. (Dutt and SomIs integral transformation approximation.)
This procedure is rather cumbersome and proved
only for the cases·when A.1 = A
f 1
=
(i=1,···~),
accur~e
r=s, and
f 2•
iv. (Tan and Wong's moment approximation.)
This method is
extremely cumbersome since it involves higher order
joint moments. Furthermore Tan and Wong (1977) reported that this approximation is not good for "extreme"
values of Ai (values close to 1) or "large" noncentrality parameters.
We will use a simulation method that will take advantage of the normality assumptions. Consider an axb fixedeffects design with no empty cells. We can calculate any
of the sums of squares in terms of the cell means and cell
denote the sample mean for the (i, j ) th
sizes. Let Y..
. lJ
67
cell. Under normality, Y.. 'VN(j.l .• ,a 2 /n .. ). Using a stan1J
1 J · · 1J
d~rd normal random number g~nerator (such as the function
NORMAL in SAS), we can generate a random sample of cell
means and thus a random sample of sums of squares. An
independent sample of error sums of squares can be generated by using the inverse X 2 function with uniform (on the
interval (0,1»
~
Suppose
that is, ~
=
random numbers as the arguments.
is the intersection of the events
~1
and
~2'
~1~2 • With knowledge of P(~1) and P(~2) we
can approximate p(~) in the following manner. Let n
number of occurrences of
~
in N
simulations~
1
be the"
Let n 2 be the
number of occurrences of ~1~2 • Then the st~tistic "
1\
P~
(4.27)
=
w1n/N + w2(P(~1) - n2IN)",
=
where w + w
1 and
1
2
" following properties.
wk ~
0 (k=1,2), will have the
Theorem 4.12
1\
i. E(P~) = p(~) ;
1\
i1. Var(P~)
=
{P(~1) (1-P(~1) )w2+(w1+(2P(~1 )-1 )w2)P(~)
P 2 (~) )/N}
;
1\
iii. Var(P~) is minimal when w2
=
P(~21~1)
=
P(~)/P(~1).
Proof
i. E(n 1 /N)
p(~)
=
= p(~)
and E(n 2 /N)
= P(~1!2).
P(~1)- P(~1t2) and w1 + w2
expectation of 4.27 is p(~).
ii. Var(n /N)
1
Var(n /N)
2
= P(~){1-P(~)}/N,
=
P(~1~2){1-P(~1~2)}/N
=
Since
1, then the
68
=
{P(1;1 )-P(I;)}{1-P(1;1 )+P(I;)}/N, and
Cov(n 1 /N,n 2 /N)
= E(n 1n 2 /N 2 )
=
E(n 1 /N)E(n /N)
2
(N-1)P(I;)P(1;1"[2)/N - P(I;)P(1;1~2)
=
~P(I;){P(1;1)
-
- P(I;)}/N •
Thus,
1\
Var(PI;)
= w1Var(n 1 /N)+w 2Var{P(1;1)-n 2 /N} +
=
2Cov(w 1n 1 /N,w 2 {P(1;1 )-n 2 /N})
{P(1;1) (1-P(1;1) )w +(w 1 +w 2 ) (w 1 +(2P(1;1 )-1)wi P (t;;)
2
+ (w 1 +W 2 )P (1;)}/N •
2
Noting that w1 + w2 = 1, the result is obtained.
iii. The derivative of Var(PI;) ~ith respect to w2 is
1\
( 4. 28 )
{ 2P ( I; 1 ) ( 1- P ( I; 1 ) )w2+ «( 2P ( I; 1) -1 ) -1 ) P ( I; ) }/ N
=
-e
2{1-P{1;1)}{P(1;1)w 2 -P(I;)}/N •
Now 4.28 is zero when w2 = P(I;)/P(1;1). Since the
second derivative of Var(PI;) is 2{1-P(1;1 )}P(1;1 )jN
1\
~
0,
1\
w2
= P(I;)/P(1;1 ) yields the minimum value of
Va~(Pi;).
In the context of the problem at hand, i;1 is the event
that (F 1 ~ f 1 ) and 1;2 is the event that (F 2 ~ f 2 ). ~(I;) is
the probability that HC (no difference in main column
effects) and HI (no interaction) ate both accepted, where
.f 1 and f 2 are the respective critical values.
Since the optimal weight, w2 ' is d~pendent upon p(i;),
we must approximate w2 iteratively by first letting
Pi; = n 1 !N, calculate w2 ' use w2 in 4.27 to obtain a new Pi;'
1\
1\
and repeat the process until essentially no change in w2 .
. occurs. A 95%. confidence interval for p(l;) may be found ~y
69
A
the
A
~
use of {P~ - P(~)}/{Var(p~)}2
theorem. Due to limitatioris of
and the central limit
prog~am effici~ncy,
10,000
simulations were used. A listing of the SAS program is in
Appendix C.3.
Chapter 5
A Comparison of the STP and EAD
Analyses to the Independent Analysis
5.1 Introduction
In this chaptBrwe compare the STP and EAD analyses to
the independent analysis, and thus to each other, by way
of the
hypothesis-tes~ing
error probabilities as defined
in Chapter 1. We make the following assumptions for all
three analyses:
1. HC1 is the hypothesis of primary interest,
2. the analysis starts with a test of HI and concludes
with a test of HC1 only if HI is not rejected,
3. the marginal type I error probabilities are set to 0.10
for the test of HI and 0.05 for the test of HC1 (alternatively we may set the type I error probability for the
test of HC1 conditioned on not rejecting HI to 0.Q5),
and
.5. the design is
~3x3
ECART (equal column and row totals)
design with no empty dells.
The comparison will be divided into fqur cases: HI and
HC1 true, HI true and HC1 false, HI false and HC1 true, and
both HI and H
false. For·each case the discussion will
C1
begin with an example derived from the example in Chapter 1.
71
A two-parameter family of ECART designs as shown in
Table 5.1 will be used throughout this chapter. ECART
designs are useful in that they allow for various patterns
of cell sizes, and thus degrees of nonorthogonality, by
simple adjustment of the parameters.
Table 5.1
Relatlve Cell Sizes* for a Two-Barameter
Family of ECART Designs
< q < ~,
-t
*
Ipi
1tptq
1-2q
1-p+q
1-2q
1+4q
1-2q
1-ptq
1-2q
1+p+Q
< 1 +q
cell sizes divided by the average cell size
Various values of p and q were considered. Those used
f~r
this study were chosen in order to display the effects
of nonorthogonality in what may be viewed as extreme cases
according to an eyeball inspection. Indeed values of p and
q near zero have minimal effect, but the purpose
here is to
,
investigate the effects of nonorthogoriality and the extreme
cases merely serve to make the effects identifiable. Three
designs were chosen to represent those considered and are
given in Table
5.2.
Values of the average cell size, say k, were chosen to
be 3, 5, and 15 yielding error degree? of freedom 18, 36,
and 126 respectively. Here individual cell sizes may be
72
Table 5.2
Relative Cell Sizes for
. Three ECART Designs
Design 1 (D1)
Design 2 (D2)
Design 3 (D3)
1
1
1
p=O
1
1
1
q=O
1
1
1
.8
1.4
.8
1.4
.2
1.4
.8
1.4
.8
1.3
1.4
.3
1.4
.2
1.4
.3
1.4
1.3
p=O
q=-.2
p=.5
q=-.2
non-integral.
The independent analysis, as mentioned in Chapter 1,
will be assumed to have the properties below:
1. The noncentralityparameter of the F statistic for
testing HI is kl~(S:AB)JF and the noncentrality parameter for testing HC1 is k/lP(S:BIJ) W .
2. The F statistics for testing HI and HC1 are independent.
The first property forces the noncentrality parameters to
be that which would be realized for both the STP and EAD
analyses if the cell sizes were all replaced by k.The idea
is to make the noncentrality parameter (and therefore the
tests) not depend on the degree of unbalancedness or upon
the presence
of other effects. This is accomplished since
the vector spaces AB and BIJ are orthogonal and in the
1'3
par~meter
space which is the same regardless of the cell
sizes.
The noncentrality parameters for the'independent
analysis differ from those of theSTP and EAD analyses due
to the
uribalanc~dness
action tests may be
of the design. As a result the
d~ssimilar
inter~
when interaction is present
due to the different noncentrality parameters. When interaction is present we will consider an additional analysis
which is the independent one with the interaction noncentrality parameter replaced by the interaction noncentrality
paramBterof the EAD and STP analyses. This is the
noncen~
trality parameter we would attain by reducing k, say to k*.
,e
In effect
k-k~~
indicates what reduction in the average cell
size would be possible in order to achieve the same power
for the test of HI with a balanced design with known
variance.
We now define
patterns of true cell means that we can
" manipulate to yield varying magnitudes of effects. Four
patterns were chosen as shown in'Table 5.3.
Table
5.3
Patterns of True Cell Means
Pattern 1 (P1):
HI true and HC1 false
Pa ttern 2 (P2):
HI true and HC1 ' false
c
0
-c
0
c
0
c
0
-c
0
c
0
c
0
-c
0
c
0
74
Table 5.3 continued
Pattern 3 (p 3) :
HI false, HC1 true
d
-2d
d
-2d
4d
d
-2d
Pattern 4 (p 4) :
HI false, HC1 false
d
c-2d
d
-2d
-2d
c+4d
-2d
d
d
c-"2d
d!
The non centrality parameters in terms of the ECART
design parameters are given for each of the test statistics
in Table 5.4.
Table 5.4
Analy.
Type
ECART Design Noncentrality Parameters
H Geometric a
NC P
Indep. HI
P1
kllp (a: AB )11 2
Cell Means Pattern
P2 & P':3 & Pl,
36d 2k
a
Indep. HC1 kj I P ( a:B I J )11 2
6c 2k
2c 2k
~
1+39- 69 2 - 8 q 3 36d 2k
1+q
a
STP &
EAD
HI
11 P (lJ : AB )11 2
STP
HC1
11 P (lJ : B I J )11 2
EAD
HC1
11 P ( lJ : B I A)11
~
A
18
c 2k
n1+ 2n 2
18 c 2 k
n1
A
2
r
2(9-4p 2)c 2 k 2 (1 - 4q 2) c +
~
.
.
£9L1 2k
1+?n
n1= ( 1+p+q ) -1
. + ( 1-2q )-1
. + (. 1-p+q )-1
a The noncentrality parameter,~, is as defined in Chapter 4
Pearson-Hartley charts (1951): <l>=/~7ma2, m=degrees of freabm.
75
Values of c were chosen so that6c 2 for P1 and 2c 2 for
P2 are equal to 1/3, 2/3, 1, and 4/3. Two values of d were
chosen so that the power of the interaction test using the
independent analysis would be approximately 0.66 and 0.99
. while being 0 ..33 and 0 .. 75 using the STP and EAD analyses
(k=15). This corr~sponds to 36d 2 being 0.5 and 1.5.
As stated before, the discussion of each case will
begin with a numerical example. The following items will be
displayed:
1. Pattern of true cell means.
2. Data obtained by adding computer generated random .errors
with standard normal distributions to the true cell
i. the marginal p-value which is the univariate
~robabil-
ity that F>observed-F,
ii. the conditional p-value which is the conditional
probability that F>observed-F given HI is not rejected
at the 0.10 significance level (for both the marginal
and conditional p-values we assume HI and H
are both
C1
true), and
iii. the values of the noncentrality parameters.
76
5. The joint error probabilities for the independent, STP,
and EAD analyses that apply given the true cell means.
For each example we use an ECART design with p=O,
q=-.2, and k=5.
77
5. 2 Case 1: HI true
and He1 true
Example 1:
True Cell Means
lJ. j
lJ
*".
,J
(lJ,*)*,
l
J
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5.24 3.54
2.;37 3.50
4.71
4.67
4.45
4.84
4.58
4.96
4.28
1 4 • 20
4.61
4.58
4.53
4.61
4.52
4.53
..
5.77 4.97
3.64 4.29
3.61 5.69
4.29 4.10
4.94 4.08
4.44
4.84
5.28 5.18
3.60 5.54
5.15 3.99
3.30
4.22 3.05
6.53 6.04
2.88 3.51
4.82 5.34
4.04 3.86
5.54
5.72 3.23
4.15 3.69
Estimates *
4.41
Data
4.53 3.78
5.45 4.90
4.93 3.98
5.42
ANOVA
Source
DF
SS
F
Margipal,
p-value
Conditional
p-value
NCP
EAD: Column
2
0.0452
0.02
0.9758
0.9750
0
STP: Column
2
0.1370
0.07
0.9286
0.9254
0
Interaction
4
1.8395
0.50
0.7372
Error
36 33.2360
Joint Error
Probabilities·Error
Independent
type (0,1.)
~~
0
Note that each
Y~,<5.
lJ
0.0450
EAD
STP
0.0423
0.0375
This is due to random chance only.
78
The analysis indicates insufficient evidence to conclude there is
intera~tion.
Since neither the STP nor the
EAD sum of squares for columns indicate a differBnce in
column
effects~
we would conclude that H
should not be
C1
rejected. If we had rejected H ' thBn a type(0,1) error
C1
would have occurred. The probability of a type(0,1) er~o~
is largest with the independent analysis and smallest
with the STP analysis. Both the STP and EAD analyses may
be considered conservative in this context with STP the
most conservative.
Table 5.5 contains the type(0,1) error probabilities
and conditional probabilities of a type I error rejecting
H
given HI not reject~d. By 'k=ool we mean las k+oo'.
C1
Alternatively we may view the corresponding probabilities
as those that would be realized if perfect information
about the error variance were available. In this case we
would perform chi-squared tests instead of
r
tests. These
probabilities were thus calculated via a bivariate chisquared distribution.
desig~
The probabilities under EAD are invariant to the
This follows from the single dependence of the F statistics
for testing HI and H
and from HI and H
being true
C1
C1
leaving the EAD noncentrality parameters zero. When k is
equal to 3, the error probability is
~mallest.
With k=15,
the limiting value of 0.0450 is almost reached. This might
be expected in light of the correlation
coeff~cient
two F statistics. The correlation coefficients for
of the
79
Type(0,1) Unconditional and Conditional a
Probabilities of Rejecting HC1 Given HI
Not Rejected
Table 5.5
Design
k
D1
3
5
15
00
D2
3
5
15
00
-
D3
3
5
15
00
Analysis Type
Independent
EAD
Uncon •. Condo Uncon. Condo
STP
Uncon. Condo
.0450
.0450
.0450
.0450
.05
.05
.05
.05
.0396
.0423
.0443
.0450
.0440
.0470
.0492
.05
.0396
.0423
.0443
.0450
.0440
.0470
.0492
.05
.0450
.0450
.0450
.0450
.05
.05
.05
.05
.0396
.0423
.0443
.0450
.0440
.0470
.0492
.05
.0350
.0375
.0394
.0401
.0389
.0417
.0438
.0446
.0450
.0450
.0450
.0450
.05
.05
.05
.05
.0396
.0423
.0443
.0450
.0440
:0470
.• 0492
.05
.0314
.0338
.0356
.0364
.0349
.0376
.0396
.0404
aCQnditional probability
=
(Unconditional probability)';'0.9
k = 3,5,15 are 0.149, 0.076, and 0~022 respectively.
The F statistics of the STP analysis are doubly dependent and thus nonorthog?nality affects the error. probabilities. Design 2 makes the STP a more conservative analysis
than either the independent or EAD analyses. Design 3
leaves STP most conservative compared to the other designs.
The correlation coefficients of the F statistics in STP
for Design 1 are the same as for EAD with Design 1; but for
Design 2 the correlations are 0.156, 0.080, 0.023 and for
80
Design 3 the correlations are 0.162, 0.083, and 0.024.
For each design, theSTP correlation coefficient is
(1+f)*(EAD correlation coeffici~nt) where f is proportional
to the correlation between the interaction andSTP column.
sums of squares (see Theorem 4.9, Chapter 4). The effects
of this correlation can be seen in the reduction of the
error probabilities from Design 1. In fact for each k the
ratio of the reduction iromDesign 1 to Design 2 and from
Design 1 to Design,3
of the
~s
1 :1.8 which is similar to the ratio
sums-of-square~~orrelations.
A problem with
a.
conservative test ('conservative' in
that the assumed a-level is an overstatement) is that its
. power is less than what could be achieved if the rejection
~egion
were increased to bring the true a-level up to the
assumed one. For the subsequent cases, we will also display
the error probabilities when the critical values of the
STP and EAD F statistics for testing H
are adjusted so
C1
as to make the conditional type I errors equal to 0.05.
81
5.3 Case 2: HI true
and HC1 false
Example 2:
True Cell Means
5.3
4.7
5
lJ
lJ
•j
v '.
"X"J
(lJ.*)*.
J
l
5.3
5
4.7
5.3
5
4.7
5.3
5.3
5
5
5
5
4.7
4.7
5
Data
6.48 5.29 4'.43 5.11
5.75 4.65 '4.79 7.19
4.744.50
4.23
3.71 5.61
4.38 3.64 r
3.82 4.92
6.45 7.19
6.13 5.62
5.73
5.10
6.16 3.74
6.27 2.62
5.48 4~ 19 '
4.72
5.63 5.86
4.04 6.60
4.39 3.43
4.40 4.99
5.77 4.49
5.32
5.77 5.18
4.16 3.02
Estimates
5.54 5.00 4.34
,e
5.69
5.10
4.74
I 5.53
4.68
4.53
5.59
5.61
5.05
4.93
4.86
4.94
4.54 + Y j
4.58 + Y*j'
5• 05 . + ( Yi* ) * j
-
ANOVA
Source
DF
j
SS
!
Marginal
p-value
Conditional
p-value
NCP
EAD: Column
2
8.1655 3.72
0.0340
0.0317
2.70
STP: Column
2
7.8654 3.58
0.0381
0.0311
2.52
Interaction
4
0.3959 0.09
0.9850
Error
Joint Error
Probabilities
0
36 39.5249
Error
ty'pe(0,2)
Independent
0.6552
EAD
0.6623
STP
0.6837
±.0016
83
Table 5.6
Column Noncentrality Parameters
for Values of c Versus k
Design 1
Cell Means Patterns: P1, P2
Analyses: Independent, EAD, STP
c:(Pl t P2)
k
(17/6,/6/6) (1/3,1"3"13) (./6/6,12/2)
1
2
3
3
1 .67
5
3.33
5
15
10
15
5
(12/3,./6/3)
Design 2
Cell Means Pattern: P1
Analysis: EAD"
c
..J11
15
/7./6
1
1 .67
5
3
5
15
0.93
1.56
4.67
1 .87
3.11
9.33
k
3
5
./6/6
3
2
3.33
10
15
/7./3
4
6."67
20
2.80
4.67
14
3.73
6.23
18.7
/7./2
2.52
4.20
12.6
ib/3
3.36
5.60
16.8
5
Analysis: "STP
Cell Means Pattern: P2
Analysis: EAD
c
k
3
5
15
ib/6
0.84
1 .40
4.20
1"3"/3
1.68
2.80
8.40
4
6.67
20
85
Figure 5.1
Type(O,2) Error Probabilities
for k=3,5, and 15
.
.
-------------------~-------------'-------------------- ------Legend: Vertical- type(O,2) error probability
Horizontal- column noncentrality
Solid line- Independent analysis
top., middle ,bottom linek=3, 5, and 15 respectively
Dashed line-EAD or STP analysis.
. 90
.80
.80
.70
.70
.60
.60
.50
.50
.40
.40
.30
.30
~,
~k=15
-*I.::-:...::
J ..
'2.0
.10
o '--_ _
..,--._~-:-
1/3
2/3
3/3
~:::
.20
.10
o 1--~-:....~_ _-:-_ _~_-=,>-"k_=.:...:.15
1/3
4/3
2/3
&2 (2c2)
3/3
6c
a. EAD and STP: Dl.P1 (D1.P2)
. EAD:· D2.P1
2
4/:
(2c"-)
b. Adju.ted EAD and adjusted
STP: D1.P1 (D1.P2)
Adjusted EAD: D2.P1
.90
.80
'~'
--,---,
\
~\
.70
'-
'
,
:\
.
'- . .
.• 60
.20
-
-
--
.60
.50
........
-
.
--.. k=5
.40
\
\
.30
\
\
i
\
\,
\
.20
1
.10'1
o
~k=3
.70
-'" k=3
"~,._"-
\,
t
.
'''''-
\,
.40
.30
.
. -""'. .
,-.
.50
.80
-... .,;-~----
~k=15
1'--_ _..,--._ _~-~-----i,':':;"O"""'
1/3
c. STP: D2.P1
2/3
6c 2
3/3
",._--
.10
,--
4/3
0
1/3
2/3
6c
~k=15
2
d. Adjusted STP: D2.P1
3/3
4/3
86
.90
Figure 5.1 continued
.80
.70
.60
.50
,,
.40
.
• 20
,
\ "
.30
\~"'"
J
~~->-__.
.10
'---=-::::
OL--
~
1/3
2c
_ __ +
k_
= 1_5
3/3
2/3
4/3
2
e. EAD: D2.P2 D3.P2
.90 __
.90
-
.80
.70
.60
.40
.30
.20
.10
g. STP: D2.P2
.90
-"'-- ... -
.70
-------
.80
---. k=3
.70
.60
.50
.40
.30
2/3
2c
2
h. Adjusted STP: D2.P2
3/3
---.-- ...
- -. k=3
k=5
.50
.80
.... ---
4/3
i. EAD: D3.P1
87
.90
Figure 5.1 c~ntinued
.80
.70
.60
.50
.40
.30
.20
.10
0
--.
•90
1/3
2/3
3/3
6c 2
j • AQjusted EAD: D3.P1
4/3
.90
--- -'- '"
.80
.70
'\- -:::-,,-
.80
,,
,70
..........
,
\
\',
:::.• 1
~---
".
-,. k=3
\>., ~k"
: ~ -:-_\_\_~-:-'~_-_'_'~. _IL._ _
__
1/3
2/3
3/3
_k_=_15
. ---'"
....
4/3
6(:2
k. STP: D3.P1
• 90
1. Adjusted STP: D3.P1
k .
~~_._-so ...
-...._-------....
- ...... \ ',~---
.
---
.70
,50
.40
j
............
\~""',.,'~
\
.30
'
\
\,
.20 i
e
I
1
i
oL
1/3
ro. STP: D3.P2
--.
k"'5
.80
k=3
.70
.60
·.50
.40
-.
.30
,
~----,
2/3
2
2c
t
'
""-
\,\
I
i
.10
-----. k=3
\ ...
-._"~~
' ,
• 60 1
.90
3/3
4/3
k=15
.20
.10
0
1/3
n. AQjusteQ STP: D3.P2
3/3
4/3
88
As shown in each of the graphs the type(O,2) error
probabilities are greater for EAD and STP analyses than
for the independent analysis. The effect of single dependence of the F statistics can be seen in Figure 5.1(a)
where the design is balanced. The EAD and STP error
probabilities are slightly less when theH C1 rejection
regions are adjusted. As would be expected the error probabilities decrease with increases in k.
Designs 2 and 3 have two properties not shared by
Design 1 which
affec~
the error probabilities. The first,
which affects both EAD and STP, is that the nonorthogonality reduces ,the noncentrality parameter for HC1 • This in
turn increases thechancea of a type(O,2) error. The second,
which affects STP only is the dependence between the inter!
action and column sums of squares induced by nonorthogonality. The second property is most readily seen when oellmeans-pattern 2 is used. This illustrates the important
point that for a fixed pattern of cell sizes, different
patt'erns of true cell means with HI true may cause more
extreme error probabilities. When the rejection regions are
adjusted, the probabilities are
x~duced,
but not dramatical-
ly. It also appears that for a fixed cell means pattern,
Design 3 gives more extreme error probabilities than Design
2.
89
5. 4 Case 3:. HI false and HC1 .true
Example 3:
Da·ta
True Cell Means
5.3
4.4
5.3
4.4
6.2
4.4
5.3
4~4
5.3
W. i
.5
5
5
lJv.
""J
(lJi*)*j
4.88 4.52 4.88
4.71 4.86 4.71
.-
6.67 5.50
4.77 4.29
4.33
5.75
4.45
4.61
4.18
4.93
4.42
6.28
4.90
4.81
4.70
4.45
4.39
4.97
5.40
5.16
4.70
4.10 7.51
5.98 5.40
4.97 4.41
4.21 3.90
3.20 3.73
4.81
3.08 5.67
4.34 6.62
8.07 6.24
6.74 4.08
5.56 5.13
4.474.36
3.98 3.67
3.74
i
I
I
I
I
3.90 6.274.61
3.27·4.98
3.68 .3.56
5.50
Estimates
5.31
4.49 4.17
3.865.18
4.66 2.52
5.43
ANOVA
Source
DF
SS
Margihal
p-value
F
Conditional
p-value
NCr
EAD: Column
2
9.7777
4.07
0.0256
0.0237
3.024
STY: Column
2
4.5097
1.88
0.t679
0.1197
o
Interaction
·4
5.. 2710
1.10
0.3733
Error
Jain tError
Pro babili ti as
6.048
36 43.2810
Error
Independen t
type (2,1)
0 .. 0034
Adjusted: 0.. 0221
_EA_D_",,""-
STP
0.11850.0175
±.0009
90
The data do not provide sufficient evidence to conclude
that interactioh is present, thus we would test H • The
C1
EAD ~nalysis suggests rejecting HC1 in favor of, coupled
with the assumption of no interaction, accepting that a·
difference in column main effects exists uniformly over
rows. The STP analysis on the other hand suggests that
insufficient evidence exists to reject HC1 •
Since HI was not rejected, a type II error occurred.
The interaction noncentrality parameter under the independent analysis is 16.2 which yields a power of 0.932. However due to nonorthogonality, the interaction noncentrality
parameter decreases to 6.048 for EAD and STP yielding a
power of 0.557. Obviously nonorthogonality can severely
handicap the EAD and STP analyses' ability to detect interaction.
An average cell size of 1.87 instead of 5 would have
made the interaction noncentrality parameter of the independent analysis equal to that
tively
~
of EAD and STP. Alterna-
value of d (for cell means pattern P3 which is
used in the example) equal to 0.183 as opposed to 0.3 in
the example would reduce the noncentrality parameter to
6.048.
Using the EAD analysis we would commit a type(2,1)
error. The EAD column noncentrality is due entirely to
~
th~
"
presence of interaction and the nonorthogonality of BIJ
"
(the space used to test HC1 ) and AB (the space where the
interaction resides, see Table 3.2 and Figure 4 in
91
Chapter 3).
With the STP analysis,H cl is not rejected. Note that
the STP columnnoncentralityparameter is zero. This is due
..
to the orthogonality of BIJ Cthespace used to test HC1 )
a,'nd AB.
1\
'In Figure 5.2 we show the type(2,1) error probabilities
for the example designs and cell means patterns. With the
independent analysis the noncentralities
for interaction
are 1.5, 2.5 and 7.5 for k= 3, 5, and 15 when d';' /2/12 and
4.5, 7.5, and 22.5 fork= 3, 5, and 15 when d= 1"6/12. These
are also tJ:1e EAD and STP interaction . n oncentrali ties for
Design 1, however for Designs 2 and 3 the noncentralities
are approximately 37% of those for the independent analysis.
The column noncentrality parameters for EAD are zero with
Design 1 and 50% of the EAD interaction noncentrality
parameters for Designs 2 and 3.
For the balanced design (D1) we see that the type(2,1)
error probabilities are less for EAD and STP than for the
independent analysis. Adjusting the column test
criti~al
region so to equate the type(0,1) error probabilities
increa~es,
although not dramatically, the type(2,1) error
probabili ties.
For Designs 2 and 3, the EAD error probabilities take
on a strikingly different pattern •. The probe-bili ties
increase with increasing d and with increasing k. As stated
earlier,' the EAD analysis "thinks" it sees a column effect
which is in reality due only to intera.ction. Increases in
92
Figure 5.2 Type(2,1) Error Probabilities for
_______________
_
f~!'_!~_2!_2!_!:!:~_12
Legend: Vertical- type(2,1) error probability
Horizontal- inte~action non centrality
Solit line- Independent analysis
top, middle, bottom linek=3~5~ and 15 respectively
Dashed line-EAD orSTP analysis
.05]
. 04 1
'03']:
•
02 1
• 010-1·'-..
360
".
.
_
,"", ..,~_.
-,
"-
2
k =.J
~' .. k=5
~_'_-_
__L__'~
'--""""...,. k=15
1/2
3/2
1/2
:,---:~~~::=,-",'-.
36d
3/2
2
b. Adjusted EAD and adjusted
STP: 01
a. BAD andSTP: 01
.1 i-
, k=15
• k=15
1 r, ...
.1 ,·
-'
.12'
,,
/
.12
,
.11i
18
• 08
;,- k=5
/
1
/
/
:..1
,.'
.06i
I/
• 05;
I
/
.. '
,,
.08
/
. O/!
,
/
/
I
j.
.10
,"
k=5
.09
/
/
,,
.,
I
.11
I
1
09
. 1
•
,
,,
.07
/ .
.06
.
.
,.-t k=3
/
/
~.
.05
/'
..04!.. "x- /~ . .' . '
• D3~.····
~
.
-.
.cJ·
""',~ ""
.1
.
.0:1
..• ~
3/2
1/2
36<3
c. lOAD: 02 03
2
.04
.03
.02
.01
O~--------'----
1/2
360 2
3/2
·d. Adjusted EAO: 02 03
(Adjusted Independent)
93
Figure..
5 2 .continued
.05
.04
"'"-----_--
.03
.02/
• 01 1
.k=3
~:::,
oll---:-::;--~3j;/2~.
1/2
360
e. STP: D2
f.
2
Ad ' t e d STP: D2
)
. JUSt·e ·d. · Independent
(Adjull
!
;
. D5 1
.04
.04
03
•
1.1
.02;
.01
j
o Il·---'-:--:-::--'---;3/;'/2>
1/2
. 360
g, STP: D3
2
Ad' ustedSTP: D3
)
h .(4dju.s
· J t e d Independent
94
sample size enhances EAD's ability to detect this phantom
column effect. Adjusting the column test's critical region
only serves to increase the likelihood of error (most
noticeable for k=3).
The STP error pr6babilities also increase for Designs
2 and 3 over Design 1, but still retain similar patterns,
that is, the probabilities decrease with increases in d.
It is interesting to note that Design 2 yields larger
probabilities than Design 3. First recall that the non·
coincidenDe of AB (the space where interaction is tested)
,.,
and AB will decrease the power of STP's (and EAD's)
interaction test. This alone will cause an increase in the
type(2,1) error probabilities. However the more nonorthog-
-
onal BI J is to AB , the more the failure to .rej ect HI will
tend to cause failure to reject H
(see again Table 5.5)."
C1
This in turn tends to bring down the type(2,1) error
pro babili ties. The n"onorthogonality of BI J and AB is
greater with Design 3 than with Design 2. Again it should
be noted that a large decrElase in the type(2,1) error
probability compared to the independent analysis may not be
desirable since the power to detect differences in column
effects would alsob&reduDed.
95
5.5 Case 4:' HI false and HC1 false
Example 4:
True Cell Means
5.1 5.3 5.1
4.8
5.9
4.8
5.1
5."3
5.1
4.47
1. 5 •11
5.47
I5.78
4.71 5.16
4.71 5.50
4.97, 5.28
5.15
5.01
4.97
"
3.61 4.48
5,.954.64
4.52<4.67
5.04
4.12 5.82
4.23 3.89
3.71 5.71
3.80
4.31
5.76 3.885.234.54 ,6.10 5.84
5.46,5.35 t.45 5.79 5.19 6.05
6.99 4.88
I
6.38
'----l'---£.
-,..-;---JI
Estimates
4.32 5.70 5.19
4.31
6.40 4.74
5.95 3.68
4~12
"
5
5.5
5
4.96 5.34 4.96
4.97 5.28 5.04
4.70
3.29 4.85
3.75 5~38
Data
6.155.75
5.08 7.19
7.26 4.35
,
ANOVA
Source
DF
SS
F
Marginal
p-value
Conditional
p-value
NCP
'EAD: Column
2
2.0145
1 .10
0.3435
0.3371
0.756
STP: Column
2
1.6586
0.91
o. 4130
0 •3945
1 •400
Interaction
,4
4.0616' 1 .11
0.3669
Error
0.672
36 32.9344
•
Joint Error
Probabilities'
Error
Independent
type(2~2)
0.5695
Adjusted: 0.7489
EAD
0.7663
STP
0.7461
±.0037
'
96
The data do not p!'ovidesufficient evidence to reject
Hr. Both the EAD and STP tests
~f
HC1 suggest not rejecting
HC1 ' thus we would conclude that there are no differences
in columns means over the rows·.
Since HI was not rejected, a type II error occurred.
Since H
was not rejected, another type II error occurred.
C1
Thus for both analyses, we have committed a type(2,2) erro~
Figure 5.3 is a plot of the true 'column means by rows.
Although the differenoes in column means are not all the
same over rows, the differences are in the same direction.
6.0
5.8
5.6
5.4
'''*
5.2
~~
column 2
columns 1 & 3
5.0
4.'8
o
Lt-----+-----+-1
2
row
3
Figure 5.3 . True Column Means of Example 4· By Rows'
"
It is open to interpretation whether the interaction that
is present is of any practical significance. Surely one
would not want to conclude that
~here
·in column means, thus the second
~ype
are no differences
II error is an error
97
still to be protected
against~
In Figure 5.4 we show the typef2,2) error probabilities.
The interaction noncentrality parameters are the same as in
Figure 5.2 and the·STP column honcentrality parameters are
the same as in Figure 5 .1. TheEAD coulmn notlcentrality
parameters are given in Table 5.7 for Designs 2 and 3 with
cell means pattern 4. Note that we use both d=-12112 and
d=I2/12~
While both yield the same results for the inde-
pendent and STP analyses. they yield different results for
EAD.
Table 5.7
EAD Column NoncentralityParameters for
Designs 2 and 3 with Cell Means Pattern 4
k
d
-/2/12
n/1 2
..
1""6/6
1)"/3
12/2
1b/3
2.09
3 •.33
4.48
5.58
3.48
5. 55
9~30
15
10.45
16. 66·
22. 40
27. 90
1b/6
./513
.r1/2
VEl3
o•15
0.25
1 .00
1 .85
2.85
0.75
3.00
5. 55
8.55
0
0.25
0.75
1.40
0
0.75
2.25
4.20
c
1b/1 2 VE/6
./5/3
,f"Z/2
1""6/3
....L. -L
0.60
1 ~ 11
1.71
0
0.15
0.45
0.84
7.47
98
Figure 5.4
Type{2,2) Error Probabilities for
____________ ~~2!_2!_~~~_12_~B~_2§~:~!I..~!_2l~
_
Legend: Vertical- type(2,2) errorprobaility
Horizontal- column noncentrality
Solid line- Independent analysis
top, middle, bottom linek=3, 5, and 15 respectively
Dashed line-EAD or STP analysis
.80
.80.
.70
.60
.50
,,
,,
,
.., ,
.40
,,
,,
'. k=5
.30
,,
,,
,
,,
,,
,
". k=5
.20
,,
, ,.
.10
.~
b. Adjusted EAD and adjusted STP: D1
99
Figure
5.4 continued
1
36d 2 =3/2
I
J
.• 70
.60
.5°1
.30
.20
:]
o
l::=:::~::::=::~~~"F"'~~'
1/3
2/3
2c
c. EAD and STP: D1
2
3/3
kp=15
4/3
= =1; :/~3= =2; :j~3:= =:; '3 /~3;='~- ;4j.i~ =;.l:1
.'1::.
0
2c
2
d • . Adjusted EAD and adjusted STP: D1
~
1--"
<rq
C
fi
CD
36d 2 =1/2
d=-n/12
I1
• 80
r------ --.. . -
d=n/12
-._-
.....
..... '
..... ,
,,
.70
.....,
•60
", ,
-....
,,
'
\J1
I
.80!
36d 2 =1/2"
_
36d 2 =3/2
d=,IO/12
Q
.JI1-
•
o
.... _
~
/
k=3
.
c+
/
/
.70
- ... k=3
1
/
;
k=5
.l:-
.60
.. -------
~
C
CD
-e_
'"
/
1--"
'"
p..
-. k=5
\
.50
\
.50
.50
.40
•.40
.30
.30
\
\
\
\
.40
\
\
\
,30
""
\
"-
\
\
\
,,
I
'ok=5
\
.20
"-
/
-.,
I
.20
k=15
.20
/
I
/
.10
-. k=15
I
I
.10
.10
1
-_~_=-k=1
2/3
2c
e. EAD: D2 D3
2
3/3
4/3
"~
01
1/3
2/3
2c 2
3/3
f. EAD: D2 D3
4/3
1
ot
'.
1/3
2/3
2c
2
3/3
•
4/3
g. EAD: D2 D3
~
o
o
e
it
e
e
--
e
I-:I:j
f-l.
(JQ
l:::
I-i
CD
I
36d'=1/2
d=-1'2/12
I
• 80
•
.80 L
.....
- ........
--
"""'"'
.60
.60
k=5
o
..... -------
-- ... _-
~~
.70
.60
"-,,-
-.,... k=3
l:1
c+
f-l•
l:1
t ..
,,
I" .........
l:::
_-._~
.......
...-
t'
.50
-tQ
I
.... k=3
\J1
36d'=3/2
d=Ib/12
.801
.....
• 70
.70
:
36d 2 =1/2
d=I'2/12
I
CD
p..
... k=5
J
.50
•
.40
.40
.30
.• 30
S
1
'-.. k=3
.40
\
\
.30
\
.... --
\
\
.20
......:
..,
.20
\
'tI
k=15
\
~
.1.0
"-
oI
"-
"-
"-
"-
.
. --.,.. - -.-
1/3
2/3
2c
3/3
-
01
. =-'.
.• -
2
h. Adjusted EAD: D2 D3
(Adjusted Independent)
1/3
.
2c
i. Adjusted EAD: D2 D3
(Adjusted Independent)
~
2/3
3/3
~
4/3
2
j. Adjusted EAD: D2 D3
(Adjusted Independent)
-'
o
-'
":cj
f-J.
Gtl
~
fi
CD
.--
.80t
I
'_
.70
-0-
,
.p-
.801
Q
I
~'-'_~,
-----.,
-I'":' - - - -- - - - -...._-
-
'0
k=3
• 70
,
',. k=3
.60
.''1-
'<'"
.60 i
o
\
",,
,,
,
~
\
.30
\
\
\
\
'-"
"
"-
• 20
"
"-
''0
k=15
1
~
CD
P-o
"
I
.30
\
0"
\
1/3
.40
,,
\
":1
"-."-"'~
~'-'j
2/3
3/3
4/3
• 20
10
.
t
1. Adjusted STP: n2
(Adjusted Independent'
---._-<~
,
"',
t
' . k=15
I
I
ot ·
1/3
2c~
k. STP: D2
:3
' . k=5
.40
,
'-3
- -0-
---
.50
\
f-J•
-, -
k=5
.50
:3
c+
------
_
I
.40
\J'1
36dO=3/2
36d'=1/2
.80
_
--_0,_
~
i
I
36d'=1/2
2/3
2c
ill.
3/3
.
4/3
2
STP: D2
--"
o
l\)
e
--
e
e
--
e
f-:I:j
1-'.
Q""q
~
Ii
CD
\J1
36d 2 ;:::3/=
~'-.-:.- .....
.80.... ...... _-
8'"
!
---.-
...
.70
----- ...............
• 60 1
~
-·-"'---.k=3
.........
I
.50 i
_-...,
.....
.60J,'
.......
. . . ............. k=5
I
.5
0j
... ,
. , ... ,
,
-~ k=3
'"
t
,20
...................
I
\.\:~
.
t
2/3
3/3
2c:?
n. Acjusted STP: D2
(Adj~sted
Independent)
'
\
\
•"\
\
\
.30
"\
'-,
,,
20
,
1
I
• 10 .f
•
.10
4/3
p.,
\
'.
"
.,
' . ·k=15
I
i
1/3
~
CD
\
........k=15
oI = : : : : : =
::3
,
\
".\
.40
.~
,,
1-'•
.... k=5
\
'" k=15
..........
..........
... ,k=2
............
\
"' .....
........
• 1 O~
I
1\'
t
\, ''"
1\ "
..........................
::3
c+
...
.60 }\'"
.50 i
"-
,
(;)
o
"',
'.. k=5
'\
.f'-
.
... ...
I
._~--'-- ........
.20
.70,
.
.30
.30
'\.. . .~·...... . -.. --.. l
",
"40
•.40
.80J~--'
'
.70 t."""
~
36e'=3/;'
t- ...
"-..,
,
"'~..
I
'.
...............
........
36d 2 =1/2
01
1/3
2/3
_ 2
Lc
o. ST?: D3
3/3
--4/3
I
OJ
1,I";.
2"I:;
... ;.'
3'~
1./
Lc·
p. Adjusted STF: D3
(Adj~st~d Independen,t~
...J.
(:)
\..>J
104
Figure 5.4 continued
.801
.80."
i
t------.-
--e_
.70·
r-----
- -.-. k=3
;
.601
!,
t------..
I ......
50
.
1 ~
-'•• k=5
~
- ........
• 20
..... ,
.10
otl==~==========o==s=~-­
1/;
2/;
3/;
l.i3
2c~
q. ST.P:
~(:'
.
,
1
o
1-1_-----::"-r;c-_-----::-;7'-_-_-----_=-=====~-=:1n
1
r.
2/ ';
2~;
Adju~ted 0~",
~,
3,':;
(Adj~sted-i~~e~;ndent)
1.1:,
105
~Vith
the balanced design, the type(2,2) error prob-
abilities fortheEAD and STP analyses
aregreate~
than the
independent analysis probabilities •. This difference
decreases with increases in k, but it appears to remain
about the same over values of the col1.lmn noncentrality
parameter. Increases in the intera.ction noncentrality
parameter increase the differe.ncesbetween the EAD ... STP and
the independent analyses' error probabilities, at least
for small k. Adjusting the critical region for the column
test reduces the differences, but not dramatically.
EAD performs quite differently with an unbalanced
design depending upon the magnitude and pattern of the
interaction. With d= -12/12 theEAD type(2,2) error probabilities are less than those of the independent analysis
for sufficiently large va.lues· of the column noncentrality
parameter. However when the direction is changed so that
d= 12""/12 (but the interaction noncentrality parameter is
the same since d 2 is used in its formula.tion), the type(2,2)
error probabilities are dramatically larger than the
independent analysis error probabilities. These large
differences still prevail when d is increased to 16/12~
Adjusting the column test critical region and adjusting the
interaction noncentrality parameter for the independent
analysis does very little to change the differences,
although they appear slightly less dramatic.
A key point about the EAD analysis that is illustrated
by the graphs is that the error curves may hot be monoton-
106
ically decreasing in the column noncentrality parameter.
In fact when·d= 1b'/12, the EAD type(2,2) error is most
probable for 2c 2 equal to 1/3. This was predictable since
the column noncentrality parameter for EAD is zero for this
case (see Table 5.7}. This results in the EAD test of HC1
being a biased test in the presence of interaction.
The STP type(2,2) error probabilities are cOnsistently
larger than the error probabilities for the independent
analysis, and dramatically so. Although the differences are
reduced when adjusted column test critical regions and
adjusted interaction" noncentrality parameters are used,
the differences are still substantial. The efrects of
Designs 2 and 3 show very small differences.
Whether the EAD error probability exceeds or falls
below the STP probability depends on the noncentrality
parameters and the pattern of interaction. Generally EAD
performs better than STPwhen d= -12/12, but not as well
as STP, except for sma"11 column noncentrali ty, for d= /2'/12
or 1b'/12.
5.6 Summary
In the analysis of two-way, fixed effects, a1l-ce11sfilled designs where no difference in main column effects
(H C1 ) is the key hypothesis of interest, the procedure
begins with a test for interaction (HI)' or an assumption
of no interaction, and concludes with a test of HC1 only
if HI is not rejected (see Figure 5.5). The possible errors
107
r-Rej.ect
I HI
~t
Fa.il.J
to
It.j.e. .••••
HI·· .
~-<-.-.,--
/ Do not
\, test HC1
Reject
HC1
Conclude a
difference in
column effects
exists uniformly
over rows
Figure 5.5
Fa.il
to
Reject
HC1
Conclude no.
difference in
column effects
exists
Procedure for testing Er and HC1
108
CS)
.,
type
I
:.FIC
.,.
erro!'
type
II
error
no
error
no
error
type
+
(0,2)
error
a. HI true and HC1 true
b~
HI true and HC1 false
no
~
error
no
error
type
(2,1).,.
error
Figure 5.6
II
erro!'
type
type
error
+(2,2)
error
(2,0)
c. HI false and HC1 true
+-type
false
Possible Errors in the Testing Procedure
109
in this two step procedure depends on the truth of HI and
HC1 (see Table 1.1 and Figure 5.6).
Two analyses have been considered, EAD (each-adjustedfor-the-other) and STP (standard-parametric). A practioner
must make a choice between at least thBsetwo analyses •
.
For example, if one has the SAS output from the general
linear models program, the choice is between the type 2
sums of squares (EAD) and the type 3 sums of sqi.HtreS (STP).
How should one decide?
The answer to the posed question may be that one uses
the
presenceo~
absence
o~interactien
to decide. If
interaction is present, then no choice needs to be made
since the main effects test will not be done. If interaction is not present, then the choice 'is EAD, since EAD is
more pOi-lerful than STP. Even when both analyses are adj usted so that their respective conditional alpha levels are
the same, EAD is still more powerful than STP. This difference in power, or reduction in the type(O,2) error probability, is demonstrated in Figure 5.1. For example, in
Figure 5.1(c) with k=5 and 6c 2 =4/3 (Design 2 and cell means
pattern 1), the STP type(0,2) error probability is 0.40
compared to 0.38 for EAD (F·igu:re 5.1 (a) ). When cell means
pattern 2 is used instead (Design 2, k=5, and 2c 2 =4/3),
the STP probability is 0.58 (Figure 5.1(g)) compared to
0.44 (Figure 5.1 (e)) forEAD.
The fallacy of this "presence of interaction" strategy
is that it is unusual to know whether or not interaction is
110
present. Usually one only has evidence provided by an
interaction test. It has been stated that there is almost
always some interac~iod. If this be the case, then one
should make the decision about which analysis to use based
on how they compare in the presence of interaction.
In the presence of interaction, EAD can greatly exceed
its assumed a-level. The EAD analysis, with an unbalanced·
design, "thinks" it sees a column effect even when there is
none. This phantom effect is due only to interaction.
Increases in sample size only enhance EAD1sability to
detect the phantom effect. This is demonstrated in Figure
5.5. For example, in Figure 5.2(c) with k=5 and 36d 2 =3/2
. (Design 2 or Design 3), the EAD type (2,1) error pro ba bili ty
is 0.098. For Design 1 (balanced design) the probability
is 0.013 (Figure5.2(a).
The STP type(2,1) error probabilities are also affected
by the presence of interaction, but the probabilities
generally decrease for increases in the interaction none entrality a.nd increases in the average cell size (for k=3,
there may be some increase in the type(2,1) error probabilities for small
interaction noncentrality). For example,
the same one noted for EAP, in Figure 5.2(e) with k=5 and
36d 2 =3/2 (Design 2), the STP type(2,1) error probability
is 0.027 compared to 0.013 for Design 1 (Figure 5.2(a».
The power of the EAD analysis is unpredictable in the
presence of interaction. In general, the EAD column test
is biased in that the power curve may not be minimal under
•
111
the null hypothesis. The magnitude of the EAD type(2,2)
error probability depends heavily ort the magnitude and
pattern of interaction. For example. when k=5, 36d 2 =1/2,
and 2c 2 =1/3, the EAD type(2,2) error probability for
d=-/2/12 is 0.55 (Figure "5.4(h)) and for d=/2/12 the probability is 0.775 (Figure 5.4(i}). Note that 0.775 is larger
than the value obtained for 2c 2 =0 with d=/2/2.
The STP column test is unbiased whether interaction is
. present or not. The STP type(2,2) error probability is
affected by interaction, but not by the pattern of interaction. Whether STP has a smaller or larger type(2,2) error
probability than EAD does depend however on the pattern and
magnitude of interaction. The probabilities are
consi~tent~
larger for STPwhen d=-n/12 (compare Figure 5.4(h) with
Figure 5.4(k)) and smaller for STP whend=n/12, with the
following exception (compare Figure 5.4(i) with Figure
5.4(k)): EAD has a smaller type(2,2) error probability for
values of 2c 2 close to zero, that is, EAD has better power
for
det~cting
minute differences in column effects.
The analysis procedure shown in Figure 5.5 could be
modified for the EAD analysis in the presence of interaction by replacing HC1 with HI: no interaction and no
difference in main column effects. Failure to reject HI
would lead one to the same conclusion of no difference in
main column effects uniformly over rows. Rejection of HI
would lead one to conclude that either:
i. no difference in the main column effects exists, but
112
differences in cell means may exist within a row, or
ii. differences in the main column effects exist, but the
differences may not be uniform over rows.
The EAD test of HI is in general still biased.
In summary, since the choice of analysis should be
made with the awareness of possible interaction, STP should
be used unless one knows that interaction is not present.
The reasons for this choice are:
i. EAD is a biased test of H '
C1
ii. the EAD conditional alpha level may exceed the assumed
level , but the STP conditional alpha level cannot
exceed the assumed level, and
iii. the power of'EAD is unpredictable.
If indeed interaction is
no~ present~
then some power is
lost by choosing STP over EAD with the magnitude of the
loss depending on the degree of nonorthogonality of the
STP .column and interaction sums of squares. The "clo ser"
the design is to being balanced, the smaller the loss in
power.
Chapter 6
Suggestions for Further Researnh
Since interaction is inevitable· in most experimental
multifactor designs, one must be prepared to deal with it
when testing for differences inrnain effects and upon
interpretation of the
analys~s resuats~
sented in Chapters 3 and 4
po~nt
The material pre-
out that the STP analysis
can be used aa a safeguard against interaction creeping
into main effects. This does not provide an escape from
interaction however when an interpretation of detected
differences in main effects.is presented. So what patterns
of interaction, perhaps such that the power of a test for
interaction is small, would be deemed
practically insig-
nificant? Should it be such that the differences in cell
means are all in the same direction over rows or columns?"
A study of patterns of interaction that lead to these kinds
of differences would be useful with respect to how the
various analyses preform.
Even with a small amount of interaction, EAD could be .
" lies ~n a direction
sever1y affected. SupposethatP(JJ:AB)
" "
which is highly nonorthogonal to BlA.
The mean sum of
squares for interaction may be small, but the sum of
squares for columns underEAD may be larger than expected
114
under H
due to the nonorthogonality. The same degree of
e1
A
interaction could be represented however byP(~:AB) lying
in a direction
tha~is
orthogonal to BjA, if such a direc-
tion exists, thus leaving the column sum of squares
unaffected. The practitioner w0uld have no way of knowing
which is the case since such knowledge would require knowledge of the interaction. A study of the 'worst case'
interaction problem needs to be done so that for a partic~
ular unbalanced design, a potential EAD user may be forewarned of the degree of
poss~blainfluence
that undetected
interaction could play on his main effects tests.
It has been oft suggested to pool, when the additive
model is indicated, the insignificant interaction sum of
squares into error (e.g. Rawlings 1972). How do the error
probabilities for STP and EAD compare when this new error
term is used? The column sum of squares for EAD will still
A
A
be independent of the error sum of squares since B I A
.J.
AB,
but for STP they may be dependent. In either case the new
error sum of squares will be noncentral if some interaction
is present.
Suppose the purpose of the factorial design and resulting analysis is to provide an estimate of
a,
the vector of
cell means. An obvious unbiased estimate is the vector of
sample cell means. This is what one obtains when the full
model including interacti0n terms is used. When might it be
more desirable to use the additive model to estimate 8?
.Certainly the estimate would be more precise when there is
11 5
in fact no interaction, but for what patterns of interaction would this continue to be the
~ase?
How does nonor-
thogonalityof the design and overall sample size enter
in? A measure of precision that may be used is
where
a
A
•
18
an estimate of
~
• Since
a ±s
Elle-a W
the vector of cell
means, this error measure ii simply the sum of the mean
squared errors of the individual mean estimators. The focus
should be on the component of
a
contained in AB. It is
conceivable that this component must lie in some convex
subset of AB in order for the additive model to provide
the more precise estimate.
The last suggestion refers to the development of a
package of information that could be made available to the
user of an ANOVAcomputer program that deals with unbalanced designs. Such possible items of interest might be:
i. The corr.elation coefficients of the various sums of
squares under their respective null hypotheses;
ii. The observed conditional alpha level of any particular
statistic
given
hypothese~ is
that interaction (and poss·ibly other
not rejected ata
user-specified
level);
iii. The possible effect of interaction upon the EAD sum
of squares (perhaps in terms of the maximum percentage
of the interaction noncentrality parameter); and
iv. A more complete analysis of interaction broken up into
logically selected directions so that one may determine where the interaction lies.
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Appendix A
Proof of Orthogonality
Cond~tions
on
n
Definition
For any pxr matrix U, let u'ir
he the vector carrel
sponding to the ith row of Ufor i=1,···,p and let
u",
;, J
be. the vector corresponding· to the J'th column of
U for j=1,·-;·,r. Then
U
=
Definition
r
Let
III u ~~ r
. 1
J=
denote the
partition~ngof
U into its
r
individual columns. We say thatU =
IIJui~r.· Note that
j =1
U
= Ill(
III u.. )"
j =1 i=1 lJ
and U" =
IIUfnu.,}".
i=1 J;::1 lJ
Definition (Kronecker product)
Let U be pxr and V be qxs. The Kronecker product of
U and V is
U€9V =
u 11 V u
V
V
1Z •• • u 1r
u Z1 V u 2Z V • • • u V
2r·
•
••
•
•
• • ••
•
upZV a.a. U pr V
[~~1 V
..
Some of the properties of the Kronecker product are listed
below (Muirhead, 1982).
i. (aU)€9(BV) = aB(U€9V) for any scalars a, B.
ii. If U and V are both pxr and W is qXs, then
(U+V)€9W = U€9W+ V€9W
iii. (U€9V)€9W = U€9(V€9W)
iv. (U€9V)'= U'€9V' •
r
r
v. U€9V =
III (u~~ .€9V)
j =1
=
J
s
DI
BI
j=1 k=1
(u*.€9 v*k) for U pxr and
J
V qxs.
Definition (J-product)
Let U be pxr and V be pxs. The J-product of U and V is
U~V = ( i=1
III (u.1 ~~~v.1 ~~))'
Some of the important properties of the J-product ara given
I
in Theorem A. 1 •
Theorem A.1
Let U be pxr and V be pxs.
i. If r=1 or s=1, then
ii.
(aU)~(BV)
=
aB(U~V)
U~V
=
V~U
•
iii.
r
III
iv.
j =1
(u ~~ . ~v
J
)
v. If U2 is pxr and V2 is qxs, then
(U+U2)~V =(U~V) + (U2~V) and U~(V+V2) =
(U~V)+(U~V2).
vi. Let C be qxc and D be qXd. If c=1 or s=1, then
(U~C) ~
(V€90) = (U@V)
~
(C@D) •
vii.· [( U@C) @ (V€90) J = [( U@V) €9 (C@O) l
Proof
i. Suppose r=1. Then uev = (
= (
=
rll(u·19v.~~»'
J.. J.
'1·
J=
·rn (v.J. *9u'.1J. ) )' .
i=l
V@U
Supposes=l. Then veu = U@V
ii. (aU)E)((3V) = (
iii. UE)l
III(au.~~~(3v.*»)"
i=1
J.
= 1 @U from i. UE)!
p
P
=
J.
P
=
aBl,~1(ui*~vi4»'
J.=1
= a(3(UE)V).
(rll (u. *~1 ) )' =
i=lJ.
(
fllui~~)'
i=1
= u.
iv.
r
UE)V=
III
j =1
(u~~ ~@V)
J
•
v. (U+U 2 )E)V = ( fl/(u'~~+U2'*)~v,*)'=
i=l J.
J.
J.
(J,,(u,~~®v.*)+
i=l J.
J.
(u 2' *®v , *))' = (III (u , *~v , ~~) v+
J.
J.
'··1 J..
J.
J.=
(i~~(U2i*&Vi*»)~=
The remainer of the proof
vi. Suppose c=l.
r
( III
j=l
vls slmilar.
Then(U~C)@(V~D)=
(u~~,~C~q»
J
o~
(U@V) + (U 2 iV) •
E)
s
d
(III Ill(v*k~d*l»
k=l 1=1
(i=1IlIrl!
(u .. @c. f1@v· k @d. '1))'
i' =1
::LJ
1
1
1
(III
Ell (u.lJ;@v.1 k@C'1 , 1 @d.1, 1)
)'
i=1 i' =1
·
(( i=1
III u lJ
.. @v. k)@
1
( III u .. @V . k) '@
i=1 lJ
1
(
(
=
Ell
c., 1 @d. t 1))·' =
i'=1 1
1.
D
I c. , 1 @d. , 1)' =
i'=1 1
1
The proof is similar for s=1.
r
vii. From vi,
(U@C) E> (V@O) =
211
c
211
(u~~.~k~~k)E>(V@O)
=
J
j=1 k=1
A different partitioning of the same vectors is:
c
Df(c*kE)O) = (UE>V) @ (CE>D).
k=1
[(U@C) @ (V@O)]
= [(U@V) @ (CE>O)]
•
Thus
Definition
= (III
For U pxr, vee(U)
rnu .. ) ' . Some of the
j=1 i=1
1J
properties of the vee operator are listed below
(Muirhead, 1982) •
i. vee{U'WV)
ii.tr(U'V)
=
=
(U@V)'vec(W}.
vee(U)'vee(V).
iii. vee(aU+BV) = a vee(U) +6 vee(VL
Theorem A.2
If W= (w .. ) is diagonal and n
= (
1J
rllw .. )' , then
i=1
11
vee(U'WV) = (UElV)'n •
Proof
r
Consider vee(W)'(U@V) =
s
III III (
k=1 1=1
vec(W)'(u~~k@V~~l))'
ff
w•.u·kv·
j =1 i=1 1JJ 1 l
f·. w•• u 'k v '1·
i=1
11 1
. 1
=
=
( Blwi·)(Dlu·k@v· l )'·
'. 1 1
'·1 1
1
1.=
1=
•
n'
Hence vee(U'WV)
=
(UQV)
(UElV)'n •
=
Definition
Let U and V be subspaces ofmP-and let W be a subspace
offfi. q • Let:
U @ W = [{ U@W: UEU and WEW} ] and
U El V = [{uElv: UEU and VEV}].
-'
Theorem A.3
If U is pxr, V is pxs, and W is qxt such that
rU]
= U, [V] = V, and -[w] = W, then
U @ W = [U@W] and U
@ V = [U@V) •
Proof
The proof follows directly from properties i and v of
the Kronecker product and ii and v of the J-product.
Theorem A.4
Let [C~) = mal [Ial for a~2. Then:
i.
ii.
[C EJC
a a
J
= ffi.a if a>2 and
= [1 a ] if a=2.
[C EJC ]
a
a
Let
E.
Proof
be the vector in
J
ma with a1 in the jth posia
tion and O's elsewhere. Then let Ca = III (E 1 -E.).
j =2
_J
C
a
~C
a
=
•
= fe;.
if j=k
So if a> 2 then
jJ
(0
[C @C
a
J
a
if j#k
a
LII
(E +E.))]
j=2 1 J
= ma
and if a=2,
Theorem A.5
Let U be pxr, V be pxs, and W be pxp.
U'wv = kU'V for k=tr(W)!p if and only if
vee(W)
.L.
[UgvJI [vee(I)].
Proof
First note that vee (U' (W-kI) V) = (UgV) 'vee (W-kI) •
Now P(vee(W-kI): [UgV]-) =
P(vee(W):[U@V]I[vee(I)}) - kP(vee(I):[UgV]
I [vee(I)])
+ P(vee(W): [vee(I)]) - kP(vec(I): [vee(I)]).
Since P(vee(I): [U@vj
I [vee(I)])
-;: 0,
P(vec(Ih [vee(I)]) = vee (I), and
P ( vee ( W) : [vee ( I )]) = (vee ( I ) , vee (W ) ! p) • vee ( I )
= (tr(W)!p)·vec{I) ;
then
P(vee(W-kI): [UgV]) = P(vec(W): [U@V]
I [vee(I)] ).
Hence U'WV = kU'V ++ U'(W-kI)V =0 ++
(UgV) 'vee(W-kI)= 0 ++ vec(W)
.L
[UgV]
I [vee(I)].
Corollary A.5
Suppose W = (w .. ) is diagonal and n= ( filw .. )' • Then
1J
i=1 11
U'WV = kU'V where k=~r(W)/p if and only if
n
.
.L.
[UElvj I J.
Proof
The proof follows
dire~tly
from the theorem upon
noting that (U~V) 'vee (W} = (UElV)'n •
Let A, B, AB, andJ be vector sUbspacesof fiab such
that A = ma 8 [l bJ, B = [l ] 8 ffib, AB = [C 8C J, and
a
a b
J = [lab]. Note that AIJ = [C a 81 bJ andBIJ = [la8CbJ.
n
tions below on
are met for the various [UJ and [VJ
spaces, then U'WV: kU'V where k = tr(W)/p •
.M-
ill
i. AIJ
AIJ
Restrictions on n for U'WV = kU'V
for a>2, n ..a.. AIJ
ii. BIJ
BIJ
for b>2, n ..L. BIJ
AB
for a>2 and b>2, n ..L.ffiab lJ
iii. AB
for a>2 and b=2, n ..L.AIJ
for a=2 and b>2, n ..L.BIJ
iv. AIJ
BIJ
n ..L. AB
v.AIJ
AB
for a>2, n
..L.
(BIJ ffi AB)
for a=2, n ..L.BIJ
vi. BIJ
for b>2, n
AB
.J.
(AIJ ffi AB)
for b=2, n .J..AIJ
Proof
For each case it will sUffice to show that
[U~VJ
equal to the mentioned space orthogonal to n.
i.
[U~)V] = [(Cag1b)~(Cag1b)J
=
[(Ca~Ca)~(lb~lb)J =
mag [ l
[
1 =A
b
if a>2
[IaH [I b ] = J if a=2
IJ is
Hence [UElV] IJ
=
[A I J
i f a>
2
null space if a=2
ii. [UElV]
=
[(la@Cb)!)(la@C )]
b
=
[(laElla)@(Cb~Cbn :::
b
[[email protected] " B i f b>2
'.
[ [1 a @[1 b J = J if b=2
J
Hence [UElV] IJ = [BIJ. if b>2
null apace ifb=2
iii.
[UElV] = [(Ca@Cb)El(Ca@C )] =
b
[(CaElCa)@(Cb!)C b )] =
ffia@ ffi b = mab if a>2 and b>2
ma @[l b ] =
A i f a>2and b=2
[1 ] @m b = B ifa=2and b> 2
a
[1 a] @[1 b] = J if 'a=2 and b=2
Hence. [UElV] IJ =
mab/J if a>2 and b>2
A IJ
if a>2 and b=2
BIJ
if a=2 and b>2
null space if a=2 and b=2
iv.
[UElV] = [(Ca~Hb)El(la.@Cb)J =
[(Ca~na)@(lbElCb)] -
[Ca@C b]
=
AB
Hence [UElVJ IJ = AB( since AB
v.
[U~)VJ. -
[(C @l )El(C @C )] =
a b
a b
[(C ElC )@(l @C )] =
a a
b B
.J.,
J).
ma8[CbJ
=
I [laJ 8[C bJ
Hence {UE>V]
vi.
[UE>V]
= BIJ if a=2
IJ = f.B1J
lBIJ
•
ffi AB if a>2
if a=2
= [(la 8C b)E>(C a 8C b )] =
[(laE>Ca)8CCb~Cb)] =
f [Ca] 8 mb = [ Ca 18 [lb]ffi [CaJ 8 [C b]
I
I
1[C a J8[lb]
if b>2
=AIJ
ifb=2
.
Hence [UE>V]~ =IA IJ ffi AB i f b>2
.AIJ
iY b=2
.
= AIJ ffi AB
Appendix B
Proof of Equation 4.23
We show here that:
00
00
oo
•
fooK (X)K 2 (X)dG(x;t) = I
I. f a (x,k)a2 (x,k l )g(x)dx
1
~
k=D k'=O I 1
1vhere
K. (x) = lim K.(x,n);
1.
n-+ oo 1
K.
1
(x,n) =
ex. (x,k)
1
=
n
I
k=O
a. (x,k);
1
.. k. . .
n.
.1
exp(-~./2){$./2) exp(-x/2)
1
1
I
j=O
.
(x/2)J/ j lkl,
n.1
=1
v./2
+ k - l £or i=1,2;
·
g(x) = G'(x;t); and
v.1
,~.,
1
t, Gare as given in Chapter 4.
The following thElorems will be called upon (Rudin, 1976).
Theorem B.1
Suppose K is compact, and:
i. lf n } is a sequence of continuous functions on K,
ii. lf n } converges pointwise to a continuous function
f on K,
iii. tn(x) ~ f n +1 (x) for all xeK, n=1,2,3,.·· •
Then f + f uniformly on K.
n
Theor'em B. 2
If f is continuous on [a,b] then f is Riemann
integrable on [a,b].
Theorem B.3 (A weak form of Legesque's dominated convergence theorem.)
Suppose g and f n (n=1~2,3,···) are defined on (0,00),
are Riemann-integrable on fa,bJ whenever O<a<b<oo,
If n I<g, f n + f uniformly
on every compact subset. of
.
(0,00) and foog(x)dx<oo; then lim foof (x)dx = foof(x)dx.
o
n+oo 0 n
0
Let fn(x) = K1 {x,n)K 2 (x)g(x}. We will state without
proof that f.n (x) (n=1,Z,3,···) and g(x) are continuous on
(0,00). Then f n + f = K1 (X)K 2 (X)g(x) for every XE(O,oo) and
f is continuous on (0,00). Now f n +1 (x)-f n (x) = Q1(x,n+1) ~ 0
for every XE(O,oo). Hence for every compact set K on (0,00)
we have by Theorem B.1 that f
Since f
n
n
f uniformly on K.
+
and g are continuous on (0,00), Theorem B.2
gives us that they are Riemann integrable on (0,00).
The summands Q1 (x,k) > 0, thus K1 (x,n) > O. Since
K1 (x) = 1-G(x;v 1 'W1) < 1, then lim K1 (x,n) = K1 (x) < 1
n+oo
(where G(XN 1 ,W1) defined in Chapter 4 is a cumu,lative
probability distribution function). Hence If n I < g.
Furthermore, f:g(x)dx = G(x;t)< 1.
We may now use Theorem B.3 to state:
= limf6K1 (x,n)K 2 (x)g(x)dx
n+oo
n
r
= limf:
Q1(x,k)K 2 (X)g(x)dx
n+oo k=O
00
= If: a 1 (X,k}K 2 (X)g(x)dx
k=O
Redefine f n now as fn(x) = K2 (x,n)a 1 (x,k)g.(x). By
making analogous arguments as above, we may apply Theorem
B.3 again to obtain
00
I
k=O
f~~1(x,k)K2(x)g(x)dx =
n
00
I
limf I a (x,k)a 2 (x,k')
k=O n+oo k'=Q 1
.
g(x)dx
00
00
I J:a~(x,k)~2(x,k')
k=O k,=ot
.
= I
g(x)dx
Appendix C
SAS·Programs
C•1
* IHI!> PROGRAM CAlCUlA1rS PROBABILITIES OF SINGLY
* ~DS!>IBlY NONC~NlkAl BIVAkIAT~ F DISTRIBUTION!>
Of-TION!> l!>"!'OI
"'HUC MATRIXI
u=J. C51.5cl I H=i.151.5111 ll=J.
C~l.ll,l
DfP~NUENT
Hul
L2=J. 151.1 I I
••••••• ~ ••• ~ •••••*.o •••••••••••••••••••••••••••• o.o.o •••• ~.o.o••• ;
*
•
*
•
K=~l
1=IK-llll*91
NC~1=2.701
"'Cf-2=5.401
AVEHA&l CELL !>IZEI
DENOMINATOk DEGREES OF ~HEEDOMI
COLUMN NONCENTHAllTY PAkAME1EHI
INTERACTION NONCENTkALJlY PARAMt HRI
• • • • • • 0 • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ~.o •• o
Fl=FINVI.95.2.T.811
f2=fINVI.90.~.T.011
l.I=fPR08Cfl.2. TtNCPlll
P=fPROBlf2.4.T.NCP211
N=h/21
Cl"211/T*fll
CZ=411/T·f21
A"Cl'CZ'll
Cl=Clfo/AI
CZ=CZII/AI
G(1.11 =11
DO Jl=l TO 501
GIJ1.1.11=cl·CN·J1-11111l,J1·GIJ1.111
ENOl
DO J1,,0 TO 501
00 JZ=l TO 511
~IJ1.1.JZ.11=C2·IN.J1.JZ-11111l/J2·GIJ1.I.JZII
ENOl
ENOl
H·l1.11 "1.N*C21
00 1'\1=1 TO 501
HIKJ.l.ll=HIK1,ll·GIK1.1.11·(,IK1.1.ZII
ENOl
00 IIZ=l TO 501
&0=01
DO K1=O TO 50 I
GO=GOtGIK1·1.KZ-ZII
HIK1·1.KZ·11=HlIl1·1.1<21-GOI
ENOl
ENOl
Llll.l=lI
lZIl.lall
00 1<1=1 TO 501
llIKl.l.l=llIK1.I·NCP1w/I<111/21
LZ (KIo 1. I "-lZ IKl"·NCPZ#K1ltl2l
ENOl
SUM=OI
DO K1=1 TO !Ill
00 K2=! TO 511
SUM=5UM.L1IK1.I·LZIK2.I·HCK1.KZII
ENOl
.
ENOl
SUM"l-Q":OP I-INCPI.HCP21 .nllll IAuNI *SUMI
SUMaP-SUM! . .
.
ALL"NCP11INCPiltCfLLIISUM!
NoTE NCP1 NCP2. CELL-SIZl
PROBI
PRINT ALL!
•• ;
C.2
.. THI!> P~Ur,PAM CALCULATf!> JOI"'T fo'HOBAtHLIlIf.S Ot DOUBLY OEPE"'n[NT.1
.. UUUULY CfNTRAL F DISTRIBUTIONS.'
UPlIONS LS=1lO1
I'HUC MATRIAI
• • • 000
000 • •·0
l=ll:ll
El=l-/JI
Ec=l#/l51
0 •••• 0
-
00011' 00 • • • ;
.. DENOMINATOR DEGREES Of fREEDOM;
.. FIRST SQUARED CANUNICAL CORRELATIONI
.. SErONO SQUARED CANONICAL CORRELATIONI
•••• o*oo •• o~ •• o •• o.o.o.o ••••••• O • • • • • • • • • • • • • • • O • • • OO • • • • • 0
fl=fINVI.9S.2.1.011
DO I =I TO 221
f.!lhll=I1
00 J=Z TO ~21
811.JI=0'
ENO'
ENOl
DO
",EO'ON:
fZ=FINVI.yt.4.T.OII
1=1 TO 211
DO ..1=1 TO 2ll
B I 1+1 ,..1.11 =8 II.J)+BI I • ..1+1/'
If BI1.hJ+lhITHEIII 60 TO KEEPON'
ENOl
END'
PRO~=O'
"'=T#/2'
cl=211/T*fll
C2=411/T*1"2'
A=CI +c2+ 1J
Cl=CII1/A'
C2=C211/AI
Gllo1) .. 11
DO ..11=1 TO 21'
G 1..11+10 11=( 1* IN • ..11-11 II/JJ *r-I",I.111
ENOl
00 ..11=0 TO 211
00..12=1 TO 221
6IJl+l.J2+11=C2*IN+Jl·J2-1111/J2*6IJI.l.J21'
fNo'
ENOl
HII,II=11
DO 111=1 TO. Zlf
Hllll·l.11="11I1,11+61111.1.111
ENOl
DO 1(2=1 TO 221
60=01
DO 111=0 TO 2Jl
60=60·61111+1,K2.JI'
Hllll·J,1(2.1)=HIKl+l.1121+601
ENOl
ENOl
DO 1(1=1 Tu 221
DO K;:=1 TO 2JI
HIKI.~21=HIKI.K21./A*·NI
ENOl
ENOl
uo 1\=0 TO 211
If K = 0 THEN 001
f AcTOR:J I
GO To OONU
£.1"0 I
If El = 0 AND E2
0 THEN DOl
f AcTOR=1I1
GO TO DONt.;
ENOl
MULT « Ie I =11
MUL T 12, I =•51
00 1=2 TO KI
MUL T« 1.1.1 =MlILT 11.) -11-1.51111/1'
ENOl
•••• ;
IF El·EZ "II THEN 001
F ACTOR:oMI./L 1 (K'" I 01 El'E21 ··K I
&0 TO DONEI
END;
FACTOR=O;
DO 1=0 TO K;
FACTOR=FACTOk.MULTII'I.I·MULTIK-I.l.I·El··I·EZ·"IK-111
END;
..
DONE: INTER=O;
DO ""=0 TO K;
DO N=O TO K;
INTER=I~TER.(1-121.MOOIM'N.211.HIM'1.N'Z)
·AIK·l.l'4·ll·~IK.ltN'lIl
[111M
END;
PROA=PROH • FACTOR.· INTER;
END;
PFlOH= I-FPWOB IFl.2. T .01 -PROIH
Al,L=TIIElIIE.21IPiolQBI
NOTE DENOM-DF·
EI
PRINT ALL!
PROb;
E2
C.3
• THI~ PROGRAM ESITMATES PRObABILITES • VIA A SIMULATION PROClDURl,1
• OF I:IIVAWlAlE F OIS'rHIBUTlONS AS~OC1ATEIJ WPM THtSTP ANALYSI~
I
•••••• o •••••••••••••••••••••••••••• ~ ••• o •••••••••••••••• o~~ •••• o ••• o.1
AOJUSTEO=H
• I=YES. II=NO - ADJUST THE ALPHALEIIEL ? ;
PATTEkN=ll
K= 3 ;
p .. o;
Q=,-21
C=OI
0"0;
0
0
0
•
•
0
CELL MEANS PATTERN I
AVEkAGE CELL SllE;
ECART DESIGN PARAMETER;
ECART DESIGN PARAMETER;
COLUMN EFFECT PMlAMETERI
INTERACT IONPARAIolEtEIH .
•••••• 0 •••• 0 ••••••••••••••••••••••••••••••••••••••••••••••••••••••••• ;
IolACR(l BOB
,
• INSERT AN ASTERISK IN ORDER TO SUPPRE~S
CHANGING OklENTATION OF PROB STATEMENT;
U=5.0115-IZ·DII15.DI15~(Z.DII15.~·DI15-IZ·DII15+DI15-(2*011/5+01
IF PATTERN=I THEN 112=CII01I-C/lCII.01I-CIICI/IIII-C;
ELSE U2=011C11011011C11011011C1101
U=U.U21
D=I·(J+PIII-12*OIIII·Q-PIII-IZ*OIIII·4·0/1l-IZ"OIII
1 +O-F-/Il-IZ·ullll'(J+PI
D=K~DI
1.1=0_1./; STpI=OI
~TP2=0;
DF=IK-llll"Y;
OtlS=J.19.11;
C= 1 1 II -10 II 0 -I I
1:1= J,13.1IIt1C;
Atl=CliC;
X=INvIDIAGIOIII
"·INfOkMATJO~
B=X·S"INVIB'·X*BI"B,·al
A8=X"AB"INV(A~'''X''ABI''AB'''XI
NCPl=U'·~·UI
NCPZ=U'·Ail"UI
MA1RIAI
• COLUMN PROJECTION ~ATHIAI
" INTERACTION PROJECTION MA1HIA;
o COLUIolN NONCENTkALI1Y PARAIolETEkl
* INTERACTION NO~CENlkALJ1Y PAkAMETEkl
IF NCPl< 0 THEN NCPl=OI
IF NCPZ < 0 THEN NCPZ=OI
IF AOJUSTEO=O THEN 001
Fl= fINVC.95.2.bf.Ot;
F2= FINVI.90.4.DFtOII
END;
o
• COLUIolN CRITICAL VALUEI
INTERACTION CRITICAL VALUE;
•
•
If ADJUSTED,,1 THEN nOI
If P"O ANO 0-0 THEN cRnV. 3.381>58& 3.18931 3.0507i:!81
If P=O AND O"-.i THEN CRITV"3.i"Z&2c 3.0&15&c i.9359301
If P=.5 THEN CIHTVI03.2i9280 3.0"9Si9l.92S01131
If K=3 THEN KK=H If 1'0_5 THEN KK"i' If 1'0=15 THEN KII:=31
fl=CRpVl.KIO I
fZ=fINVI.90 .... Df.0I'
ENOl
U=I-FPRoalfl.2.0f.NCPlI1
P=FPI<OBlfZ.4.DF''''Cpcl'
/)0 1=1 TO 100001
00 J=I TO 91
08SIJ.I"NORM,LIClI·$QRTIOIJ.11 -UCJ.II
ENOl
• THE SIIo1ULATlONS'
,I)' ~ Ollf/f)' I
MSE"t I NVIUN lfORM CO}
MSC= IOtlS··1:l.08S1 "/21
If MSC
< n.MSE. THEN GO TOSKJpI1I
MSI .. COtl~, .Aa.Oas 111/.'"
IF 10151< fc.",SE. THEN STPl=STPl.·11
ELSE 5TPc"STp2_11
SKIPP: ENOl
1111".51 w2=.51
PROtl=QlfIZI
PROH2=01
DO "'HILEIABSIPROB-PRORZI ) .OOOOSII
pRDElc=PkOBI
PROB.. 1Il1.S1PU/I0000 _ Wi.10-S1Pilf/lOuOOII
1Il2=PROHlf/UI
"'hi-life I
BOA
ENOl
CONF_ERR= SURl I Iw1.1 2*0- Ill) ·w21·PROB -PROBu c •
O· (1-01.Wc••ZIIJ/SURT (100001 9 1.91>1
pROB=P-PROBI
NOTE NCPl
NCPl
Of
PROal
ALL=NCPIIINtPZIIOFIIPROBl ALL=ALL'I
PRINT ALL!
PRINT CONF_EI<RI
•