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Solutions to the multivariate G-sample Behrens-Fisher problem based upon generalizations of the Brown-Forsythe F* amd Wilcox Hm tests

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Title:
Solutions to the multivariate G-sample Behrens-Fisher problem based upon generalizations of the Brown-Forsythe F* amd Wilcox Hm tests
Creator:
Coombs, William Thomas, 1954-
Publication Date:
Language:
English
Physical Description:
v, 148 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Degrees of freedom ( jstor )
Error rates ( jstor )
Mathematical dependent variables ( jstor )
Mathematical robustness ( jstor )
Matrices ( jstor )
Population distributions ( jstor )
Population mean ( jstor )
Population size ( jstor )
Statistical discrepancies ( jstor )
Statistics ( jstor )
Education -- Research -- Statistical methods ( lcsh )
Multivariate analysis ( lcsh )
Statistical hypothesis testing ( lcsh )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1992.
Bibliography:
Includes bibliographical references (leaves 141-147).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by William Thomas Coombs.

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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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SOLUTIONS TO THE MULTIVARIATE G-SAMPLE
BEHRENS-FISHER PROBLEM BASED UPON GENERALIZATIONS
OF THE BROWN-FORSYTHE F* AND WILCOX H TESTS












By


WILLIAM


THOMAS


COOMBS


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY


OF FLORIDA


1992












ACKNOWLEDGEMENTS


would


individuals


like


that


express


have


sincerest


assisted


appreciation


in completing


to the


study


First


, I would


like


to thank


. James


. Algina


, chairperson


doctoral


committee,


suggesting


topic


dissertation,


theoretical

providing


guiding


barriers,

editorial


professional


through


debugging


suggestions


personal


growth


difficult


computer


and


through


applied


errors,

fostering


encouragement


support,


and


friendship.


Second,


am indebted


and


grateful


the


other


members


committee


, Dr


. Linda


Crocker


. David


Miller


, and


. Ronald


. Randl


patiently


reading


the


manuscript,


offering


constructive


suggestions,


providing editorial assi


stance,


and giving continuous support.


Third,


must


thank


John


Newell


who


fifth


unoffi


cial


member


committee


still


attended


committee


meetings,


the


read


progress


manus


of the


script,


project.


and


Finally,


vigilantly


would


inquired


like


as to


express


heartfelt


thanks


wife


Laura


son


Tommy


Space


limitations


prevent


from


enumerating


many


personal


sacrifi


ces


both


large


and


small


, required


wife


so that


was


able


to accomplish


task


Although


shall


never


a


-


,,


..






committee and


family,


let me begin


simply


and


sincerely--thank


you.













TABLE


OF CONTENTS


ACKNOWLEDGEMENTS


ABSTRACT


CHAPTERS


INTRODUCTION


The Problem
Purpose of the
Significance of


. .
tudy
the


Stu
Study


* a
a a a a a a


REVIEW


OF LITERATURE


The Independent
Alternatives to
ANOVA F Test .
Alternatives to
Hotelling's T2
Alternatives to
MANOVA Criteria
Alternatives to


Samples
the Ind

the ANO
'est
the Hot

the MAN


t Test
dependent

VA F Tes

selling's

OVA Crit


Sampl

t .


a 2
mn2


t Tes


Test. .

I a a a a


METHODOLOGY


Development of Test
Brown-Forsythe
Scale of
with
Equality
of


Statistics .
Generalizations
the measures <
in group variabi
of expectation <
between and


of between
lity
of the mea!
within


and

sures
group


Wilcox
Invariance
Brown-
Wilcox
Design .
Simulation
Summary


disp
General
propertyy
orsythe
General


version
ization
of the
Genera
ization


st Statisti
nations .


cs a a
. .


Procedure







RESULTS


AND


DISCUSSION


. 67


Brown-Forsythe


General


zations


. 74


Johansen


Test


cox


General


zation


CONCLUS IONS


S ~ ~ S 4 4 S S 4 0 4 5 S 96


General


Observations


S S S 4 4 4 S 4 4 496


Suggestions


to Future


Rese


archers


APPENDIX


ESTIMATED


TYPE


ERROR


RATES


. 100


REFERENCES


. 141


BIOGRAPHICAL


SKETCH


. f148













Abstract


of the


of Dissertation


University


Requirements


SOLUTIONS


BEHRENS-FISHER


Presented


of Florida


Degree


TO THE
PROBLEM


the


Partial I
of Doctor


MULTIVARIATE


BASED


UPON


Graduate


School


Fulfillment of
of Philosophy


G-SAMPLE


GENERALIZATIONS


OF THE


BROWN


-FORSYTHE


AND


WILCOX


TESTS


William


Thomas


Coombs


August


1992


Chairperson:


Major


James


Department:


J. Algina
Foundations


of Education


The


Brown


-Fors


ythe


and


Wilcox H_
-In


tests


are


generalized


form


multivariate


alternatives


MANOVA


use


situations


where


dispersion


matrices


are


het


eroscedastic.


Four


generalizations


the


Brown


-Forsythe


test


are


included.


Type


error


rates


for the Johansen


test


and


the five


new


general


nations


were


estimated


using


simulated


data


variety


of conditions


The


design


experiment


was


a 2


factorial


The


factors


were


type


distribution,


number


of dependent


variables


number


groups,


ratio


total


sample


size


number


dependent


variables


form


the


sample


size


ratio,


degree


of the sample


size


ratio


, (g)


degree


of heteroscedasticity,


relationship


sarmni e


s1. A


disnersion


matrices.


Only


conditions


, (e)







which dispersion matrices


were heterogeneous were


included.


controlling


Type


error


rates,


the


four


generalizations


Brown-Forsythe


test


greatly


outperform both


Johansen


test


and


generalization


the Wilcox H
-ll


test.












CHAPTER


INTRODUCTION


Comparing


two


population


means


using


data


from


independent

statistical


sample


hypothesis


one of

testing<


the most


fundamental


One solution


to thi


problems


problem,


the


independent


samples


test


, is


based


the


assumption


that


the


samples


are


drawn


from


populations


with


equal


variances.


According


Yao


(1965),


Behrens


(1929


was


first


solve


testing


without


making


assumption of


equal


population


variances


Fisher


(1935


, 1939)


showed


that


Behrens


solution


could


derived


from


Fisher'


theory


of stati


stical


inference


called


fiducial


probability


Others (Aspin,

solutions to th


The


1948;


two


independent


Welch,


-sample


193


Behrens


samples


test


8, 1947

-Fisher


has


) have

problem


been


proposed


as well


generalized


analysis


variance


(ANOVA)


test,


test


equality


population


means.


This


procedure


assumes


homoscedasticity,


that


22 ..
-- *


. Several


authors


have


proposed


procedures


to test


without


assuming


equal


population


variances.


Welch


(1951)


extended


1938


work


arrived


an approximate


degrees


of freedom


(APDF)


solution.


Brown


and


Forsythe


(1974),


James


= ~CL2









(1951) ,


and


Wilcox


(1988,


1989)


have


proposed


other


solutions


to the


-sample


Behrens


-Fisher


problem.


Hotelling


test


(1931)


to a test of the e

procedure makes


generalized


quality

the a


the


of two


independent


population mean


assumption


equal


samples


vectors.


population


dispersion


(variance-covariance)


matrices


, that


1 =- 2".


Several

without


authors

assuming


have

equal


proposed pr

population


ocedures


dispersion


test


matri


1=t- Lz


ces


James


(1954)


solution.


generalized


Anderson


1951


(1958)


work


, Bennet


and


arrived


(1951)


, Ito


series


(1969)


van


der


Merwe


(1986)


Scheffe


(1943),


and


Yao


(1965)


have


proposed


additional


solutions


the


multivariate


two-


sample


Behrens


-Fisher


problem.


Bartlett


(1939)


, Hotelling


(1951)


, Lawley


(1938),


Pillai


(1955)


Roy


(1945),


Wilks


(1932


have


proposed


multivariate general


zations


of the ANOVA F test,


creating the


four


basic


multivariate


analysis


variance


(MANOVA)


procedures


testing


These


procedures


make


the


assumption


of equal


population


dispersion


matrices


James


procedures


test


(1954)


the


and


Johansen


equality


mean


(1980)

vectors


proposed

without


making


assumption


of homoscedasticity,


that


1 =2:2


.
. ZG


James


extended James


(1951)


univariate


procedures


produce


first


-order


second


-order


series


solutions.









The


Problem


To date,


neither the


Brown


-Forsythe


(1974)


nor the


Wilcox


(1989)


procedure


has


been


extended


the


multivariate


setting.


test


G Brown


Forsythe


(1974)


proposed


the


statistic


- X


- 12
N


where


denotes


the


number


of observations


in the


group,


the


mean


the


group,


the


grand


mean,


the


variance


of the


group,


N the


total


number


of observations,


and


G the


number


groups


. The


statistic


approximately


distributed


as F


with


and


f degrees


freedom


where


ni
N


The


degrees


freedom,


, were


determined


using


procedure


due


to Satterthwaite


(1941).


To test


. = .


0o : 1


Wilcox


(1989


proposed


statistic


-y


where


i cxi


Clz


n)


N











n, ( n


-X
+ 1) 2


+ 1)


and


i

i=1


t w.
1


In the


equation

sample.


denotes


The


statistic


last


approximately


observation


distributed


as chi


-square


with


degrees


freedom.


Purpose


the


Study


The


purpose


thi


study


extend


the


univariate


procedures


proposed


Brown


and


Forsythe


(1974)


and


Wilcox


(1989)


test


and


compare


Type


error


rates


the


proposed


multivariate


generalizations


the


error


rates


Johansen


(1980)


test


under


varying


stributions


, numbers


dependent


(criterion)


variabi


numbers


groups


, forms


of the


sample


size


ratio


, degrees


the


sample


size


ratio,


ratios


total


sample


size


to number


dependent


variable


, degrees


heteroscedasticity,


relationships


of sample


size


to dispersion


matrices.


Sicqnificance


of the


Study


The


application


of multivariate


analysis


of variance









the


future


data


analysis


(Bray


& Maxwell,


1985,


p.7)


Stevens


suggested


three


reasons


why


multivariate


analysis


prominent:


1.
subject


the
ways


inves
the


sensitive
variables


Any


more


worthwhile


than


tigator 2
subjects


one


treatment


way,


hence


is to determine


will


measurement


affected
techniques


will
the


affect
problem


which
and


the
for


specific


then


find


those


Through


the


use


multiple


criterion


measures


des


we can


obtain


cription


a more


phenomenon


complete
under i]


and


detailed


investigation


Treatments


while


the


cost


can be


expensive


obtaining


data


to implement,
on several


dependent
maximizes


variable


information


relatively


gain


. (1986


small


. 2)


Hotelling


sensitive


violations


homoscedasticity


, particularly


when


sample


S1zes


are


unequal


(Algina


Oshima,


1990;


Algina


Oshima


, &


Tang,


1991;


Hakstia]

& Clay,

(1954)


n, Roed,

1963;

first-


& Lind


Ito

and


1979;


Schull,


second-ordei


Hollow

1964).

r, and


& Dunn,


Yao


1967;


(1965),


Johansen


(1980)


Hopkins


James


tests


are


alternatives


Hotelling


that


have


underlying


assumption


of homoscedasticity


In controlling


Type


error


rates


under


heteroscedasticity


, Yao


test


superior


James


Algina,


first-order

Oshima, and


test

Tang


(Algin

(1991)


Tang,


studied


1988;


Type


, 1965)


error


rates


the


four


procedures


when


applied


data


sampled


from


multivariate


distributions


composed


independent


- & .


I I r


,,


r I


k *









can be


seriously nonrobust with extremely skewed distributions


such


as the


exponential


and


lognormal,


are


fairly


robust


with moderately


skewed


distributions


such


as the


beta(5


They


also


appear


robust


with


non


-normal


symmetric


distributions


such


the


uniform,


, and


Laplace.


The


performance


Yao


test,


James


second


-order


test


, and


Johansen


test


was


slightly


superior


the


performance


James


first


-order


test


(Algina,


Oshima,


Tang


, 1991)


MANOVA


criteria


are


relatively


robust


non-normality


(Olson,


1974,


1976)


but


are


sensitive


violations


homoscedasticity


(Korin,


1972


Olson,


1974,


1979;


Pillai


Sudj ana


1975;


Stevens


, 1979)


The


Pillai


-Bartlett


trace


criterion


most


robust


of the


four


basic MANOVA criteria


for protection against non-normality and heteroscedasticity of


dispersion

to MANOVA


matri


ces


criteria


(Olson,


that


are


197r

not


4, 1976,

based or


1979)


Ithe


Alternatives


homoscedasticity


assumption


include James


s first-


and


second-order


tests


, and


Johansen


test.


When


sample


sizes


are


unequal,


dispersion matrices


are


unequal


, and


data


are


sampled


from


multivariate normal


distributions


Johansen


s test and James


second


-order


test


outperform


the


Pillai


-Bartlett


trace


criterion


and


James


first


-order


test


(Tang


, 1989)


the


Wilcox


univariate


test


case,


require


Brown-Forsythe


equality


* test


population









test.


Thi


suggests


that


generalizations


the


Brown-


Forsythe


procedure


and


the


Wilcox


procedure


might


have


advantages over the commonly used MANOVA


procedure


in cases


heteroscedasticity.


Brown


and


Forsythe


(1974)


used Monte


Carlo


techniques


examine


the


ANOVA


test,


Brown


-Forsythe


test,


Welch


APDF


test,


and


James


first-order


procedure.


The


critical


value


proposed


Welch


a better


approximation


small


sample


than


that


proposed by


James.


Under


normality


and


inequality


variances


both


Welch


s test


and


the


test


tend


to have


actual


Type


error


rates


near nominal


error rates


wide


variety


conditions


However,


there


are


conditions


which


each


fail


control


terms


power,


the


choi


between


Welch


(the


specialization


Johansen


s test


and,


the


case


of two


groups


of Yao


s test)


and


depends


upon


magnitude


the


means


and


their


standard


errors


The


Welch


test


preferred


the


test


extreme


means


coincide


with


small


variances


When


the


extreme


means


coincide


with


large


variances


power


of the


test


greater


than


that


the


Welch


test.


limited


simulation


Clinch


and


Keselman


(198


indicated


that


under


conditions


heteroscedasticity,


Brown


-Forsythe


test


ess


sensitive


to non-normality than


Welch


s test.


In fact,


Clinch


Keselman


concluded


the


user


- .-- .. --I--- -- t- -


LI ~1.


MI _-L


--


_


*


-L r__.









normal


data,


in some


conditions


test


has


better


control


over


r than


does


James


s second


-order


test


, Welch


s test,


Wilcox


test.


other


conditions


test


substantially worse control.


Oshima and Algina


concluded


that


James


second


-order


test


should


used


with


symmetric


distributions


Wilcox


test


should


used


with


moderately asymmetric distributions.


With markedly asymmetric


distributions


none


the


tests


had


good


control


Extensive


simulations


(Wilcox,


1988)


indicated


that


under


normality


the


Wilcox


H procedure


always


gave


the


experimenter


more


control


over


Type


error


rates


than


the


or Welch


test


and


has


error


rates


similar


James


second


-order


method,


regard


ess


degree


hetero


scedasticity


Wilcox


(1989)


proposed H ,


an improvement


to the


Wilcox


(1988)


H method;


improved


test


is much


easier to


use


than James


second-order method.


Wilcox


(1990)


indicated


that


the


test


more


robust


non


-normality


than


the


Welch


test.


Because


the


Johansen


(1980)


procedure


extension


of the


Welch


test


, the


results


reported by


Clinch


Keselman


and by


Wilcox suggest general


zations of the Brown


-Forsythe procedure


the


Wilcox


procedure


might


have


advantages


over


Johansen


procedure


in some


cases


of heteroscedasticity


and/or


skewne


SS.


Thus


the


construction


and


comparison


new


procedures


which


may


competitive


even


superior


under













CHAPTER


REVIEW


OF LITERATURE


Independent


Samples


t Test


The

the


independent


equality


samples


two


used


population


to test


means


the


when


hypothesis

independent


random


samples


are


drawn


from


two


populations


which


are


normally distributed and have equal


population variances.


The


test


statistic


1 1)
+,n


where


R1 R


has


at


distribution


with


n'+n2


degrees


of freedom.


The


degree


robustness


the


independent


samples


test


to violations


the


assumption


of homoscedasticity


been well


documented


(Boneau,


1960;


Glass,


Peckham


, & Sanders,


1972


Holloway


Dunn,


1967 ;


Hsu,


1938;


Scheffe,


1959).


cases


where


there


are


unequal


population


variances,


the


relationship


between


the


actual


Type


error


rate


nominal


Type


error


rate


influenced


sample


ccii 'Tnp Yun el a aa1ff1 nT


amr41


Wkan


e ~mnl n


Inin \


E '1 rt n~


P ra









large,


7 and


a are


near


one


another.


In fact,


Scheffe


(1959,


p.339)


has


shown


equal


-sized


samples


is asymptotically


standard


normal,


even


though


two


populations


are


non-


normal


have


unequal


variances.


However,


Ramsey


(1980)


found


there


are


boundary


conditions


where


longer


robust


to violations


of homoscedasti


city


even with equal


-sized


samples


selected


from


normal


populations.


Results


from


numerous


studies


(Boneau,


1960;


Hsu


1938;


Pratt


, 1964;


Scheffe


1959)


have


shown


that


when


the


sample


zes


are


unequal


and


the


larger


sample


selected


from


the


population


with

test


larger variance

is conservative


(known


as the


(that


7<


positive


condition),


Conversely,


when


larger


variance


sample


selected


(known


the


from


negative


population


condition),


with


the


smaller


test


liberal


(that


, r > a)


Alternatives


the


Independent


Samples


Test


According


to Yao


(1965)


Behrens


(1929


was


the


first


propose


a solution


the


problem


testing


the


equality


population


means


without


assuming


equal


population


variances


Fisher


problem


problem.


has


Fisher


come


to be known


(1935,1939)


noted


as the


that


Behrens-


Behrens


solution


could


be derived


using


Fisher


s concept


of fiducial


distributions.








A number


of other


tests


have


been


developed


test


the


hypothesis


1 = "2


in situations


which


Welch


(1947)


reported


several


tests


in which


the


test


statistic


+


The


critical


value


different


the


various


tests.


There


are


two


types


of critical


values:


approximate


degrees


freedom


(APDF),


and


series.


The


APDF


critical


value


(Welch,


1938)


fractile


Student


s t


distribution


with


2 2
a, (2]


( L


n1 -i


degrees

obtained


freedom.

replacing


practice,


parameters


the


estimator


statistic


that


replaces


1,2)


the


literature


the


test


using


estimator


referred


as the


Welch


test.


Welch


(1947)


expressed


the


series


critical


value


function


and


, and


developed


seri


critical


value


in powers


- 1)


The


first


three


terms


series


critical


value


are


shown


Table


The


zero-


order


term


simply


fractile


the


standard


normal


stribution


using


the


zero-order


term


critical










Table


Critical


Value


Terms


Welch's


(1947)


Zero-


F First-


. and


Second-Order Series Solutions


Power of
(n1 1) 1 Term


Zero


2
Si)2


z [ 2
4


2

1=1


z [-


Sc
i=1


-1)


i in


3+5z2+Z4 i=l


2
Si
n.
.1


15+32z


+9z4 2=1









whereas


the


second-order


critical


value


the


sum


three


terms.


the


sample


sizes


decline


, there


is a greater


need


for


the


more


complicated


critical


values.


James


(1951)


and James


(1954)


generalized


Welch series


solutions


to the


G-sample


case


and


multivariate


cases


respectively


Consequently,


tests


using


the


series


solution


are


referred


as James


s first-order and second-order tests.


The


zero-order


test


often


referred


the


asymptotic


test.


Aspin


(1948)


reported


the


third-


and


fourth-order


terms


, and


investigated


, for equal


-sized samples


variation


in the


first-


through


fourth


-order


critical


values.


Wilcox


(1989)


proposed


a modification


the


asymptotic


test.


The


Wilcox


statistic


2
s1
2+


where


224-1


n, (n+l1


.i (n1i -1


asymptotically


stributed


standard


normal


distribution.


Here


(i=l


are


biased


estimators


population


means


which


result


improved


empirical


Type


error


rates


(Wilcox


, 1989).


The


literature


suggests


following


conclusions


two-sample


case


regarding


the control c


Type


error


rates


2Xii









series


tests,


Brown


-Forsythe


test,


and


Wilcox


test:


performance


Welch


test


and


Brown


-Forsythe


test


superior


to the


test;


the


Wilcox


test


and


James


second-


order


test


are


superior


Welch


APDF


test;


and


most


applications


in education


and


the


social


sciences


where


data


are


sampled


from


normal


distributions


under


heterosceda


sticity,


Welch


APDF


test


is adequate.


Scheffe


(1970)


examined


different


tests


including


the


Welch


APDF


test


from


standpoint


Neyman-Pearson


school


thought.


Scheffe


concluded


Welch


test


, which


requires


only


the


easily


accessible


t-table,


sati


factory


practical


solution


to the


Behrens


-Fisher problem.


Wang


(1971)


examined


Behrens


-Fisher test


Welch APDF test


and


Welch-


Aspin


series


test


(Aspin,


1948;


Welch


, 1947)


Wang


found


Welch APDF


test


to be superior to


Behrens


-Fisher test


when


combining


over


the


experimental


conditions


considered.


Wang


found


T-aO


was


smaller


the


Welch-Aspin


series


test


than


the


Welch


APDF


test.


Wang


noted,


however


, that


Welch


-Aspin


series


critical


values


were


limited


select


sample


sizes


and


nominal


Type


error


rates.


Wang


concluded


, in


practice,


one


can


just


use


the


usual


t-table


carry


out


the


Welch


APDF


test


without


much


loss


accuracy


However,


the


Welch


APDF


test


becomes


conservative


with


very


long-tailed


symmetric


stributions


(Yuen,


1974)


Wilcox


* a


__








Wilcox


test


tended


to outperform


the


Welch


test.


Moreover,


over


conditions,


the


range


r was


.032


, .065)


a=.05, indicating


the


Wilcox


test


may


have


appropriate


Type


error


rates


under


heteroscedasticity


and


non-normality


summary,


the


independent


samples


test


is generally


acceptable


in terms


of controlling Type


error rates


provided


there


are


sufficiently


large


equal


--S1Z


sample


even


when


the


assumption


of homoscedasticity


violated.


For unequal


sized


samples


, however,


alternative


that


does


assume


equal


population


variances


such


the


Wilcox


test


James


second


-order


series


test


preferable.


ANOVA


F Test


The


ANOVA


used


test


the


hypothesis


equality


of G population means when


independent random samples


are


drawn


from


populations


which


are


normally


distributed


have


equal


population


variances.


The


test


statisti


i x1i.


N-G)


has


an F


distribution


with


G-1 and


N-G


degrees


of freedom.


Numerous


studi


have


shown


that


the


ANOVA


test


is not


robus


violations


assumption


homoscedastic


(Clinch


Keselman


, 1982;


Brown


Forsythe,


1974;


Kohr









test


with


one


exception.


Whereas


the


independent


samples


generally


robust


when


large


sample


zes


are


equal


, the


ANOVA

rates


may


even


not

with


maintain

equal-:


adequate


sized


control


samples


Type


the


degre


error

e of


heteros


Serlin,


cedasticity


1986)


conservative


larg

the

the


e


(Rogan


positive

negative


Keselman


condition

condition


1977 ;


the

the


Tomarken


test


test


liberal

1974; H


(Box


1954;


:orsnell,


1953


Clinch


Rogan


Keselman

& Keselm


1982


an,


197;


Brown

2; Wi]


l[


& Forsythe

cox, 1988)


Alternatives


ANOVA


F Test


number


tests


have


been


deve


loped


test


hypothesis


*. = S


in situations


which


(for


at least


one


pair


of i


and


Welch


(1951)


generalized


the Welch


(1938)


APDF


solution


proposed


statisti


w (x1


G 1
2 f (1
l=1 1


where


G
i-1


W
w










G

-=1
2=1


wix


and


=i2


=l ,...,G


The


statistic


approximately


distributed


with


and


G3-
G2-l i=1


degrees


of freedom.


James


(1951)


generalized


the


Welch


(1947)


series


solutions


, proposing


the


test
G

i=l


statistic


where


S -,


1
- i
ni


-t
1 =1


w.x
w


and


t a S a S aI S a Sa


1
i


- .


1(I(I ~


r


r, lr


1









freedom.


sample


sizes


are


not


sufficiently


large,


however,


distribution


test


statistic


may


not


accurately


approximated


a chi


-square


distribution


with


degrees


which


of freedom.


a function


James

of the


(1951)

sample


derived a series

variances such


expression

that


S2h


- a


James


found


approximations


to 2h(


of orders


Sand


1 = n
i -


the


first-order


test,


James


found


order


1 the
1


critical


value


2
- XG-I


2(G2


W
- )
Ff


null


hypothesis


hypothesis


> 2h(


rejected


James


favor


the


provided


alternative


ond-order


solution


which


approximates


order


James


noted


that


second


-order


test


very


computationally


intensive.


Brown


Forsythe


(1974)


proposed


test


statistic


i c(x.
1' I


-X


- n
N


statistic


approximately


distributed


with


and


a sec


P [C










nI2
N


degrees


freedom.


the


case


two


groups,


both


Brown


-Forsythe


test


and


Welch


(1951)


APDF


test


are


equivalent


the


Welch


(1938)


APDF


test


Wilcox


(1989)


proposed


the


states


- )


-
i11


where


G

2=1


ni (n,+l1)


n, (n,+1)


i= G


G
2=1


w
W


The


statistic


approximately


stributed


-square


with


G-1 degrees


freedom.


The


literature


suggests


the


following


conclusions


about









Brown-Forsythe,


performance


Wilcox


each


and


ese


Wilcox


tests


alternatives


ANOVA


superior to


Welch


test


outperforms


the James


first-


order


test;


generally


Welch


competitive


with


and


one


Brown


another,


-Forsythe


however


tests


, the


are


Welch


test


is preferred


with


data


sampled


from


normal


stributions


while


the


Brown


-Forsythe


test


is preferred


with


data


sampled


from


skewed


distributions


and


the


Wilcox


James


second


-order


test


outperform


these


other


alternatives


ANOVA


under


the


greatest


variety


conditions.


Brown


Forsythe


(1974)


used Monte


Carlo


techniques


examine


ANOVA


Brown


procedures


-Forsythe


when


equal


Welch

and


APDF


James


unequal


zero


samples


-order

s were


selected


from


normal


populations;


was


or 10;


ratio


largest


the


smallest


sample


size


was


the


ratio


the


largest


smallest


standard


deviation


was


total


sample


size


ranged


between


small


sample


sizes


critical


value


proposed


Welch


a better


approximation


true


critical


value


than


that


propose


d by


James.


Both


Welch


APDF


test


and


Brown


-Forsythe


test


have


r near


under


the


inequality


variances.


Kohr


Games


(1974)


examined


ANOVA


test


, Box


test,


and


Welch


APDF


test


when


equal


unequal


-t a aa


, or


; (d)


1


,,,,1


,,,,1


t,,,


r rr ur








1.5,


or 2


the


ratio


the


largest


the


smallest


standard


deviation


was


4/10,


or J13;


and


total


sample


size


ranged


between


and


The


best


control


Type


error


rates


was


demonstrated


the


Welch


APDF


test.


Kohr


and


Games


concluded


the


Welch


test


may


used


with


confidence


with


the


unequal


-sized


samples


and


heteroscedastic


conditions


examined


their


study


Kohr


Games


concluded


the


Welch


test


was


slightly


liberal


under


heteroscedastic


compared


conditions;


inflated


however


error


this


rates


bias


the


was


test


trivial


and


test


under


comparable


conditions.


Levy


(1978)


examined


Welch


test


when


data


were


sampled


from


either


the


uniform,


-square


, or exponential


stributions


and


found


that


under


heteroscedasticity


, the


Welch


test


can


liberal


Dijkstra


and


Werter


(1981)


compared


James


second


order,


Welch


APDF


and


Brown


-Forsythe


tests


when


equal


unequal


-S1Z


samples


were


selected


from


normal


populations;


was


ratio


largest


smallest


sample


was


total


sample


size


ranged


between


12 and


and


ratio


of the


largest


to the


smallest


standard


deviation


was


or 3


Dijkstra and


Werter concluded


the James


second


-order test gave


better


control


Type


error


rates


than


either


the


Brown-


Forsythe


or Welch


APDF


test


Clinch


(-J-I itt C.


(198


studied


the


ANOVA


. Welch


, or


J7,


U i IV


r.









when


equal


unequal


-sized


sample


were


selected


from


normal


stributions,


chi-square


distributions


with


degrees


freedom,


or t


distributions


with


five


degrees


freedom;


was


ratio


largest


smallest


sample


size


was


or 3


total


sample


size


was


144 ;


variances


were


either


homoscedastic


heteroscedastic


assumption


The


violations


ANOVA


Type


test


error


was


most


rates


affected


Welch


test


were


above


, especially


negative


case.


test


provided


the


best


Type


error


control


that


generally


only


became


nonrobust


with


extreme


heteroscedasticity


Although


both


Brown


-Forsythe


test


and


Welch


test


were


liberal


with


skewed


distributions,


the


tendency


was


stronger


the


Welch


test.


Tomarken


and


Serlin


(1986)


examined


tests


including


the


ANOVA


test,


Brown-Forsythe


test


, and


Welch


APDF


test


when


equal


and


unequal


-sized


samples


were


selected


from


normal


populations;


was


the


ratio


largest


the


smallest


sample


size


was


(c1)


total


sample


size


ranged


between


36 and


and


ratio


of the


largest


smallest


standard


deviation


was


Tomarken


though


Serlin


generally


found


acceptable,


that


Brown


was


least


-Forsythe


slightly


test,


liberal


whether


sample


sizes


were


equal


directly


inversely


S S -


, 6


, or


* *-









Wilcox,


Charlin,


and Thompson


(1986)


examined Monte Carlo


results


on the


robustness


the


ANOVA


Brown-Forsythe


and


the


Welch


APDF


test


when


equal


and


unequal


-sized


samples were


selected


from normal


populations;


G was


or 6;


ratio


of the


largest


to the


smallest


sample


was


, 3


, 3.3


total


sample


size


ranged


between


smallest


and


standard


and


deviation


the


was


ratio


or 4.


the


Wilcox


largest


, Charlin,


Thompson


gave


practical


situations


where


both


the


Welch


and F*


tests


may


not


provide


adequate


control


over


Type


error


rates.


Welch


unequal


For


test


equal


should


-sized


variances


but


be avoided


samples


and


unequal


favor


possibly


samples


the


unequal


test


, the


but


variances


the


Welch


test


was


preferred


the


test.


Wilcox


(1988)


proposed


competitor


Brown-


Forsythe


Welch


APDF


, and


James


second-order


test.


Simulated equal


and unequal


-sized samples


were


selected where


distributions


were


either


normal


, light-


tailed


symmetric,


heavy-tailed


symmetric,


medium-tailed


asymmetric,


exponential


-like;


was


, or 10;


the


ratio


of the


largest


smallest


sample


size


was


, or


total


the


ratio


sample


size


largest


ranged


the


between


smallest


and


100;


standard


deviation


was


, 4,


, or 9


These


simulations


indicated


that


under


..









than


did


the


test


or Welch


APDF


test.


Wilcox


showed


that,


under


have


normality


r much


, James'


closer


second


than


-order


the


test


Welch


Wilcox'


Brown


test


-Forsythe


tests


The


Wilcox


gave


conservative


results


provided


(i=l


. ,G)


Wilcox'


results


indicate


H procedure


Type


error


rate


that


similar


to James'


second-


order method


, regard


ess


of the


degree


of heteroscedasticity


Although


computationally


more


tedious,


Wilcox


recommended


James'


second


-order


procedure


general


use.


Wilcox


(1989)


proposed


, an


improvement


Wilcox'


(1988)


method,


designed


to be


more


comparable


power


James'


second


-order


test


Wilcox


compared


James'


second-


order


test


with


when


data


were


sampled


from


normal


populations


was


or 6


ratio


of the largest


small


sample


size


was


, or


total


sample


size


ranged


between


121;


and


ratio


largest


Wilcox'


to the


results


smallest


indicate


standard


that


deviation


when


was


applied


or 6


normal


heteroscedasti


data,


has


T near


a and


slightly


ess


power


than


James'


second


-order


test.


The


main


advantage


improved


Wilcox


procedure


that


much


easier


use


than


James'


second


order


, and


easily


extended


higher


way


designs.


Oshima


and


Algina


press)


studied


Type


error


rates


-A- *-----1 -a--- .-


.._LL


L~ 1


r ....


F L









These


conditions


were


obtained


crossing


the


31 conditions


defined


sample


sizes


and


standard


deviations


Wilcox


(1988)


study


with


five


distributions--normal,


uniform,


beta(1.5,8


, and


exponential.


The


James


second


-order


test


and


Wilcox


test


were


both


affected


non-normality


When


samples


were


selected


from symmetric


non


-normal


distributions


both James'


second-order test and


Wilcox'


test maintained


r near


When


the


tests


were


applied


to data


sampled


from


asymmetric


distributions,


T-a


increased.


Further,


degree


of asymmetry


increase


ed, I


v-a


tended


increase.


The


Brown


-Forsythe


test


outperformed


the


Wilcox


test


James'


second-order


test


under


some


conditions


, however,


reverse


held


under


other


conditions.


Oshima


Algina


concluded


the


Wilcox H
-m


test


and James'


second


-order test


were


preferable


Brown-Forsythe


test,


James'


second


-order


test


was


recommended


data


sampled


from


symmetric


distribution,


Wilcox'


test


was


recommended


data


sampled


from


moderately


skewed


distribution.


summary,


when


data


are


sampled


from


normal


distribution


have better


Wilcox


control


of Type


test


and


error


James


rates


second-order


, particularly


test


as the


degree


heteroscedasticity


gets


large.


All


these


alternatives


the


ANOVA


are


affected


skewed


data


t(5)








Hotellina


s T2


Test


Hotelling


(1931)


test


equality


population


mean


vectors


when


independent


random


samples


are


selected


from


populations


which


are


distributed


multivariate


normal


and


have


equal


dispersion


matrices.


The


test


stati


stic


given


nn2


n, + n2


-x^2


I s-i


X2C


where


-1),2


Hotelling


demonstrated


transformation


ng +n2


-p-i


n, +n2


has


an F


distribution


with


nl+n2


degrees


of freedom.


The


sensitivity


Hotelling


violations


assumption


of homoscedasticity


well


documented


been


investigated


empirically


both


(Algina


analytically


Oshima


, 1990;


(Ito


Schull,


Hakstian,


1964)


Roed,


Lind,


1979;


Holloway


Dunn,


1967 ;


Hopkins


Clay,


1963)


Schull


(1964)


inves


tigated


the


large


sample


properties


presence


of unequal


dispersion


matrices


Schull


showed


that


in the


case


two


very


large


equal


sized


samples


well


behaved


even


when


dispersion


gC


n,+n,


of T2


r T2









inequality


dispersion


matrices


provided


the


samples


are


very


large.


However,


the


two


samples


are


of unequal


size,


quite


a large


effect


occurs


on the


level


of significance


from


even


moderate


variations.


Schull


indicated


that,


asymptotically,


with


fixed


n,/ (n1+n2)


and


equal


eigenvalues


of E2 -S


a when


eigenvalues


are


greater


than


one


T >


when


eigenvalues


are


ess


than


one.


Hopkins


Clay


(1963)


examined


stributions


Hotelling'


with


sample


sizes


, 10


, and


selected


from


either


bivariate


normal


populations


with


zero


means,


dispersion


matri


ces


the


form


aI -


where


a,/01


was


circular


bivariate


symmetrical


leptokurtic


populations


with


zero


means


, equal


variances,


was


. Hopkins


and


Clay


reported


robust


violations


of homoscedasticity


when


n1=n2


but


that


robustn


ess


does


extend


to disparate


sample


zes.


Hopkins


Clay


reported


that


upper


tail


frequencies


distribution


Hotelling'


are


substantially


affected


moderate


degrees


symmetrical


leptokurtosis.


Holloway


and


Dunn


(1967)


examined


the


robustness


Hotelling'


violations


homoscedasticity


assumption


when


equal


and


unequal


-sized


samples


were


selected


from


multivariate


normal


distributions;


was


, 1


,,


__


*


L









eigenvalues


s2;-'I


were


Holloway


and


Dunn


found


equal


-sized


samples


help


keeping


r close


Further


Holloway


and


Dunn


found


that


large


equal


-sized


samples


control


Type


error


rates


depends


number


dependent


variable


example


, when


i = 50


(i=l1


the


and

and


eigenvalues


but


r markedly


of S2Z,


departs


= 10,

from


T is near


a when


for

or p


= 10


Holloway


and


Dunn


found


that


generally


number


dependent


variable


increases,


sample


size


decreases


, T Increases


Hakstian,


Roed


and


Lind


(1979)


obtained


empirical


sampling


stributions


of Hotelling'


when


equal


unequal


-sized


samples


were


selected


from


multivariate


normal


populations;


was


or 10;


(n1+n2)


was


or 10;


was


or 5


dispersion


matrices


were


form


where


was


d2I,


diag( 1


S. d2, d2


= 1


I...,


, or


Hakstian


Roed,


Lind


found


that


equal


-sized


sample


procedure


is generally


robust.


With


unequal


-sized


samples


was


shown


become


increasingly


ess


robust


disper


sion


heteroscedasticity


number


independent


variable


Increase.


Consequentially


, Hakstian,


Roed,


Lind


argued


against


use


negative


condition


cautious


use


in the


p05


itive


condition.


n1/ n2


r








number


of dependent


variables


was


or 20;


and


the


majority


conditions


= d2Z1


3.0) .


Algina


Oshima


found


that


even


with


a small


sample


size


ratio


example,


procedure


with


can


and


be seriously


.25S1,


sample


nonrobust


size


For


ratio


small


Algina


1.1:1


and


can


Oshima


produce

also


unacceptable


confirmed


Type


earlier


error


findings


rates.

that


Hotelling'


test


became


ess


robust


the


number


dependent


variable


and


degree


heteroscedasti


city


increased.


summary


, Hotelling'


test


robust


violations


assumption


homos


cedasti


city


even


when


there


are


equal


-sized


samples


, especially


the


ratio


total


sample


size


to number


of dependent


variable


small.


When


the


larger


sample


selected


from


the


population


with


larger


ected


dispersion


from


matrix


population


When


with


larger


smaller


sample


dispersion


matrix


, r > a.


These


tendenci


increase


with


the


inequality


the


size


the


two


samples


the


degree


heteroscedasticity,


and


the


number


of dependent


variables


Therefore


the


independent


behavior

samples


of Hot


test


selling'

under


test


similar


violations


assumption


homoscedasticity.


Hence,


desirable


examine


robust


alternatives


that


require


basic


~~ FI~IIIVC: Aa


4-I-n Ua~n ln 114-.InrraA


CkA


nrhnn~lrrrh









Alternatives


the


Hotellincr'


Test


number


tests


have


been


develop


test


hypothesis


situation


which


it?


Alternatives


to the


Hotelling


procedure


that


do not


assume


equality


James'


test


the


(1954)


two


population


first-


Johansen'


dispersion


second-order


(1980)


test.


matrices


tests


Differing


Yao'


only


include


(1965)


their


critical


values


four


tests


use


the


test


statistic


-x2


+2
t4 ,J


-'C2


where


I are


respectively


the


sample


mean


vector


sample


dispersion


matrix


sample


The


literature


suggests


the


following


conclusions


about


control


of Type


error rates


under heteroscedastic conditions


Hotelling'


test


, James'


first-


and


second-order tests


Yao'


test,


and


Johansen'


test


Yao'


test


, James'


second


-order test


and Johansen'


test are


superior to James'


first-order


test;


ese


alternatives


Hotelling'


are


sensitive


data


sampled


from


skewed


populations.


Yao


(1965)


conducted


a Monte


Carlo


study


compare


Type


error


rates


between


the


James


first


-order


test


test


when


equal


unequal


-sized


samples


were


selected,


was


, (c)


ratio


total


sample


size


to number









were


unequal.


Although


both


procedures


have


r near


a under


heteroscedasticity,


Yao'


test


was


superior


to James'


test.


Algina


and


Tang


(1988)


examined


performance


Hotelling'


James'


first


-order


test,


and


Yao'


test


when


was


of the


or 10;


largest


N:p


smallest


, 10


was


sample


or 20;


was


ratio


, 1.25


and


dispersion


matri


ces


were


form


and


where


was


diag{3,1,1


..., 1)


, diag{3,


. a


...,1)


diag{1/3,3,3


S.. .,3)


or


diag{ 1/3,1/3,


,3,3,S


.,3}


Algina


and


Tang


confirmed


the


superiority


of Yao'


test.


Yao'


test


produced


appropriate


Type


error


rates


when


, and


For


appropriate


error


rates


occurred


when


applied


both


specific


cases


where


one


dispersion


matrix


was


multiple


the


second


d2ES)


and


more


complex


cases


of heteroscedasticity


When


N:p


and


, Algina


Tang


found


Yao'


test


to be


liberal


Algina,


Oshima,


and


Tang


(1991)


studied


Type


error


rates


James'


first-


and


second


-order


Yao'


Johansen'


tests


various


conditions


defined


the


degree


heteroscedastic


non-normality


(uniform,


Laplace,


beta(5


exponential


, and


lognormal


distributions)


The


study


indicated


ese


four


alternatives


to Hotelling'


, 4,


, or


1: n2


t(5)


115),












positive


kurtosis.


Although


four


procedures


were


serious


nonrobust


with


exponential


lognormal


distributions,


they


were


fairly


robust


with


remaining


distributions.


The


performance of Yao


s test,


James


s second-


order


test,


Johansen


s test


was


slightly


superior


the


performance


of James


s first


-order


test


Algina,


Oshima


, and


Tang


indicate


that


test


also


sensitive


to skewn


ess.


summary


Yao


test


, James


second-order


test,


Johansen


test


work


reasonably


well


under


normality.


Although


of these


alternatives


to Hotelling


s T2 test


have


elevated


Type


error


rates


with


skewed


data,


Johansen


s test


practical


advantages


general


zing


to G


being


relatively


easy


to compute.


MANOVA


Criteria


The


four basi


multivariate


analyst


of variance


(MANOVA)


criteria


are


used


test


the


equality


of G


population


mean


vectors


when


independent


random


samples


are


selected


from


populations which are distributed multivariate normal


and have


equal


dispersion


matri


ces


Define


-z


-X) (X


-E Ii)


-x









The ba


sic


MANOVA criteria are


functions


of the eigenvalues


Define


to


the


eigenvalue


(i=1,.


where


- min(p,G-


Those


criteria


are


Roy


(1945)


largest


root


criterion


+x71


Hotelling


-Lawley


trace


criterion


(Hotelling,


1951;


Lawley


, 1938)


trace


1w-i


-z


. Pillai


-Bartlett


trace


criterion


(Pillai,


1955;


Bartlett,


1939)


trace [H


H+E)


and


Wilks


(1932)


likelihood


ratio


criterion


H+E


1


Both


analytic


(Pillai


Sudj ana,


1975)


and


empirical


(Korin


1972


Olson,


1974)


investigations


have


been


conducted


the


robustness


MANOVA


criteria


with


respect


violations


examined


homoscedasticity.


violations


Pillai


and


homoscedasticity


Sudjana


the


four


(1975)


basic


MANOVA


criteria.


Although


the


generalizability


the


study


- a -


IS)


ft f


I .


1 I..


m









heteroscedasticity,


results


were


consistent--modest


departures from a


for minor degrees of heteroscedasticity and


more


pronounced


departures


with


greater


heteroscedasticity


Korin


(1972)


studied


Roy's


largest


root


criterion


the


Hotelling-Lawley


likelihood


ratio


trace


criterion


criterion


when


equal-


and

and


Wilks'


unequal -


sized samples were selected from normal populations;


p was


or 4;


G was


or 6;


the


ratio of total


sample


size


to number of dependent variables was 8.25,


, 15.


, 18 or


dispersion matrices were of the form I or D,


where


was


2d2I


1.5


10) .


For


small


samples,


even


when


the


sample


sizes


were


equal


dispersion


heteroscedasticity produced Type I error rates greater than a.


Korin


reported


the error


rates


R were greater than those


for U


and L.


Olson


(1974)


conducted


Monte


Carlo


study


comparative robustness of six multivariate tests including the


four basic MANOVA criteria


when


equal-sized


samples were selected


; (b)


p was


or 10; (c


G was


was


dispersion


matrices


were


form


where


represented either a low or high degree of contamination.


the low degree of contamination,


= d2I,


whereas for the high


degree


of contamination,


= diag(pd2-p+l, 1,1,..., 1)


= 2,


, 10









should


avoided,


while


may


recommended


the


most


robust


of the


MANOVA


tests.


In terms


of the


magnitude


of the


departure


of r from


tendency


order


increased


the


was


typically


degree


hetero


> V.


scedasticity


increased.

increased


The

with


departure


from

the


increase


number


dependent


variable


, however,


the


impact


was


well


defined.


Additionally


, for


, and


7 decreased as


sample


increased


except


when


When


, 7 increased


four


basi


MANOVA


procedures


, although


the


increase


was


least


for


Stevens


(1979)


contested


Olson


(1976)


claim


that


superior


to L


and


general


use


multivariate


analysis


variance


because


greater


robustness


against


unequal


dispersion


matri


ces.


Stevens


believed


son


conclusions


were


tainted


using


an example


which


had


extreme


subgroup


variance


differences,


which


occur


very


infrequently


practice.


Stevens


conceded


Vwas


the


clear


choice


diffuse


structures,


however,


for concentrated noncentrality


structures


with


dispersion


heteroscedasticity,


actual


Type


error


rates


, U,


and


are


very


similar


Olson


(1979)


refuted


Stevens


(1979)


objections


practical


grounds.


experimenter,


faced


with


real


data


unknown


noncentrality


and


trying


follow


Stevens


recommendation


use









Alternatives


MANOVA


Criteria


number


tests


have


been


developed


test


hypothesis


1 = P2


* *= .G


in a situation


in which


(for


at least


one


pair


James


(1954)


generalized


James


(1951)


seri


solutions


and


proposed


the


stati


stic


-
1=1


where


G

i=1


"j
-Ej=


- If.-1


S G


iwii
i=1


James


(1954)


zero-


, first-


, and


second-order


critical


values


parall


those


developed


James


(1951).


Johansen


(1980)


generalized


Welch


(1951


test


proposed


using


the


James


(1954)


test


statistic


divided


- p(G-1)


+ 2A


G-l) +


1 f cj











2=1


trace


-w1w.)


+ trace


2 1W


The


critical


value


Johansen


test


fractile


distribution


with


p(G-


and


p(G-


1) [p(G


degrees


of freedom.


The


literature


suggests


the


following


conclusions


about


control


of Type


error


rates


when


sampling


from


multivariate


normal


populations


under


heteroscedast ic


conditions


four


basi


MANOVA


criteria


James'


first-


second-order


tests,


and


Johansen'


test


the


Pillai


-Bartlett


trace


criterion


most


criteria;


with


robust


unequal


the


-sized


four


samples


basic


, Johansen'


MANOVA


test


James


s second


-order


test


outperform


the


Pillai


-Bartlett


trace


criterion


and


James'


first


-order


(1969)


analytically


examined


Type


error


rates


James'


zero


-order


test


showed


showed


T-a
I


increased


the


variation


the


sample


sizes


degree


heteroscedasticity


and


number


dependent


variables


increased,


whereas


r-a


decreased


the


total


sample


size


increase


Tang


James'


(1989)


first-


studied


and


Pillai


second-order


-Bartlett


tests,


trace


criterion


Johansen'


test


when


equal


unequal


-Siz


ed samples


were


ected


from


multivariate


normal


populations;


was


or 6;


was


-rw1


-1)+


3/ (3A








number


dependent


variabi


was


dispersion


matri


ces


were


either


form


or D


, where


was


, diag{(l


,d2,d2)


or diag{ 1/d2


,dd2,d2}


for p=3


or D was d'I


diag(l,1,1,d2


,d2)


or diag( 1/d2


,1/d2


,1/d2


or 3).


Results


study


indicate


when


sample


zes


are


unequal


dispersion


matri


ces


are


unequal,


Johan


sen'


test


and


James


s second-order


test


perform


better


than the


Pillai


-Bartlett


trace


criterion and James


first


-order


test


Whil


both


Johansen'


test


and


James'


second


-order


test


tended


have


Type


error


rates


reasonably


near


Johansen'


test


was


slightly


liberal


where


eas


James'


second-


order test


was slightly


conservative.


Additionally,


ratio


total


sample


size


to number


of dependent


variable


has


strong


impact


performance


tests


Generally,


as N:p


increases


the


test


becomes


more


robust.


summary


, the Pillai


-Bartlett


criterion


appears


most


robust


four


asic


MANOVA


criteria


violations


assumption


of dispersion


homoscedasticity


In controlling


type


error


rates


the Johansen


test


and James


second-order


test


are


more


effective


than


either


the Pillai


Bartlett


trace


criterion


or James


first-order


test


Finally,


Johansen


test


computationally


practical


intensive


than


advantage


James


of being


second-order


ess


test.













CHAPTER


METHODOLOGY


In this


chapter


, the


development


of the


test


stati


stics,


design


and


the


simulation


procedure


are


described.


test


states


extend


the


work


of Brown


and


Forsythe


(1974)


Wilcox


(1989)


The


design


based


upon


review


relevant


literature


and


upon


the


cons


ideration


that


experimental


conditions


used


the


simulation


should


similar


those


found


educational


research.


Development


of Test


Statisti


Brown


-Forsvthe


Generalizations


test


*** = L


Lo :1


G Brown
u


and


Forsythe


(1974)


proposed


the


statistic


pmX


- x


Ni
N


The


statistic


approximately


distributed


with


f degrees


of freedom


, where


n.
- 'i)
IN
N


u n








Suppose


. XG


are


-dimensional


sample


mean


vectors


and


I SG


are


p-dimensional


dispersion


matri


ces


independent


random


samples


S1zes


respectively,


1,.-',niG,


from


multivariate


normal


stribution


,Zg)


To extend


the Brown


-Forsythe


statisti


the


multivariate


setting


, replace


means


corresponding


mean


vectors


and


replace


variances


their


corresponding


dispersion


matri


ces.


Define


-E


-K)


and


-z
1=1


The


(i=1,


S. .,G)


are


stributed


independently


Wishart


,S1)


and M


said


to have


a sum


of Wi


shares


stribution,


denoted


and


van


as M


der


~ SW(n,


Merwe


(1986)


- n1


have


generalized


- n,/N) ZG)


Satterthwaite


(1946)


results


and


approximated


the


sum


Wisharts


distribution


~ Wp(f


Applying


and


van


der


Merwe


results


to M


the


quantity


the


approximate


degrees


freedom


of M and


is given


trace


Ci i


+ trace


{ tra


In',


+ trace


is']


S. ,Np(G


ni
N


N,(p,


rC1)


WP(ni


/N) C1


ei C f









ni
N


The problem is


to construct


test statistic


and determine


critical


values


The


approach


used


this


study


construct


test


statistic


analogous


those


developed


Lawley-Hotelling


Pillai-Bartlett


(V)1


and


Wilks


Define


-r


-X)


and


- r


Then


the


test


stati


Hotelling-Lawley


trace


criterion,


the


Pillai-Bartlett


trace


criterion,


the


Wilks


likelihood


ratio


criterion


are,


respectively


trace


trace


flE-i


E) -1]


+ '1


Approximate


trans format ions


can


used


with


each


these


test


statisti


CS.


Define


the


following


variable


es:


= number


(the


independent


degrees


variables


of freedom


.....1 A. -- 1..- -- --- -- -- aa. a -


G


(V)


a(a+


,,,1


rHLU~I


1I-









=-- min(p,h)


(the degrees of


freedom


for the


multivariate analog to sums of

within groups)


squares


- h


.5(e


For the Hotelling-Lawley


criterion,


transformations


developed


Hughes


Saw


(197


McKeon


(1974)


respectively


are given by


2 (sn+1)
(2m+s+l


s(2m+s+l),2(sn+1)


and


F (2)
U


2n
a-2


- F
Sph,a


where


= 4


ph +


and


+ h)
- 1)


2n + p)


2n +


For


the


Pillai-Bartlett


criterion


(1985,


p.12)


transformation


is given by


2n+s+l
2m+s+l


F
- ) smn+s*l),s(2n+s*l)


For


Wilks


criterion,


(1952,


p.262)


transformation


is given by


rt 2q


F (1)
U


N


F









where


p2h2


= 1


, otherwise


and


= e


_ P


Scale


the


measures


between


within


qrouD


variability.


Consider


the


univariate


(p=1)


case


denominator


the


Brown-Forsythe


statisti


-z


- z


= G [


1


- i
N


G
_n .S


G 2
--' N


--2
= Gs


Here


is the


arithmetic


average


G sample


variances


the


their


respective


average

'e sample


the


sizes.


G sample


Because


variances

both are a


weighted


approaches


+h2


t
Y








freedom


for the


sum


squares


between


groups.


Because


numerator


the


between


group


sums


squares,


Brown-Forsythe statistic


is in the metric of the ratio of two


mean


squares


Now the


MANOVA


criteria


are


the metric


ratio of two sum of


squares.


Consider the common MANOVA


criteria


univariate


setting.


For


Hotelling-Lawley,


Pillai-Bartlett,

SSBG/(SSBG+SSWG),


and


Wilks


and L


respectively,


= SSWG/


(SSBG+SSWG)


SSBG/SSWG,


In each case the


test


statistics


are


functions


the


sum


squares


rather


than mean squares.


Hence,


in order to use criteria analogous


to U,


E must be


replaced by


(f/h)M.


i=1,...


eigenvalue


characteristic


equation


-r (f/h)M|I=0.


One


statistic


consider


would


analogous


to Roy


largest


root


criterion


(1945)


where


four


basic


MANOVA


criteria


, Roy's


largest


root


criterion


most


affected


heteroscedasticity


(Olson,


1974


, 1976,


1979;


Stevens,


1979).


Consequentially,


will


omitted.


Lawley-Hotelling trace (Hotelling,


1951;


Lawley,1938)


is based


upon


the


same


characteristic


equation


Roy'


largest


root


criterion


(194


this


case,


the analogous statistic U*


trace(H[(f/h)M]


provides one of the


test statistics


interest.


i=1,...


denote


eigenvalue


characteristic


equation


8e [H+(f/h)M]


-11=0.


Then


(1.









(Bartlett,1939;


Pillai,


1955)


= trace(H[H+(f/h)M]


= s ei


provides


another


test


statistic


interest.


Similarly,


F .1


the


eigenvalue


of the


characteristic


equation


(f/h) M


(H+(f/h)M)


, then


analogous


Wilks


(193


criterion


defined


(f/h)M


H+(f/h)M


conduct


hypothesis


testing,


approximate


tran


sformations


were


used


with


each


ese


analogous


test


statistics,


replacing


N-G,


the


degrees


of freedom


, by


the


approximate


degrees


freedom


Thus,


the


variables


are


defined


follows:


= number


of independent


variables


(the


degrees


freedom


the


multivariate


analog


sums


squares


between


group


= min(p,h)


trace


ciS


+ trace [


ciS2


{trace


[ci s,


+ trace


where


= 1


- i
N


-E
2=1


i-1,


S=G


G


iSi3 2)









For the modified Hotelling-Lawley


criterion,


the Hughes


and Saw


(197


and McKeon


(1974)


transformations respectively


are now given by


2(sn'+l)
s(2m+s+l


s(2m+s+l) 2 (sn'+1)


fU'


where


= 4


ph +


and


-' -


2n' + h)


2(n*


- 1) (2n'


+ p)
+ 1)


For the modified Pillai-Bartlett criterion the SAS (1985,


p.12)


transformation


now given by


2n'+s+l
2m+s+l


Fs(2m+s+l) ,s(2n'+s+l)


For the modified Wilks criterion,


the Rao


(1952,


p.262)


transformation


is now given by


t r*t


- 2q


where


p2h


Fph r't


p2 +h


p2 + h2


a'-2


Fphr d


- V'









=3-


Eaualitv


expectation


the


measures


of between


and


within


crouo


dispersion.


The


Brown


-Forsythe


statistic


was


constructed

expectations


that,


the


under


the


numerator


null


denominator


hypothesis


are


equal


show


the


proposed


multivariate


general


zation


Brown-


Forsythe


statistic


possesses


the


analogous


property


(that


E(H)=E(M),


assuming


true)


following


results


are


useful:


E(x1


=~1'


- IL


=11


E(x x'


= var


+ pp'


- ~ii)


- Var"


1


var


i=1


2i
n


Using


results


, E(M)


given


E(M5d


= E[


7n


Ir


P +











-E
2=1


- 12
n


Similarly,


using


results


, E (H)


given


- i)Ij


-IL),


-
.1=1


+
1=1


E(X


-Il,'


-z
2=1


var


VarZ


X-i


- -I
[xx1


- x F


- lx L


~- I


-I


+ IL I"


+ L I'] }


0
-z
2=1


G
1
l2i=1


n'El


x x I
X X


/ 2


IL j.L/


- n


ii x' 2 +


Ip p' + n


I 1'


- n


JAILr'


-Sn1
i~rI


1 =1
. i=1


nii


-2n [


var


+ iA CI


tA IL'


- -


Gn l. ]


= E[


- E=
i=1


- EC


Ir Cr/


i cx~i


- x~ ex,


it Ici
n
i











- 2


E
1 =
n=1


nE
ni
n


+ u Iu" ]


+ 2


.1' IL'


- Ei
Sil
1=1


i=1 n.
i- a.


-21
2=1


niS
n


-2n ppl'


+2n ppI


-SEI1
2=1





1=1


-n,
F^E.


n


Hence,


E(H)


= E (M).


Thus the modified Brown


-Forsythe general


nations parallel


basi


MANOVA


criteria


terms


the


measure


of between


group


dispersion,


the


measure


of within


group


dispers


metric


between


and


within


group


dispersion


, and


equality


the


expectation


the


measures


between


within

Wilcox


group dispersion.

Generalization


test


Wilcox


(1989)


proposed


using


test


statistic


-E
2=1


where


approximately


distributed


-square


with


degrees


freedom.


extend


thi


the


multivariate


. ._1


A .* I -.


jZ)


"


1 I I


1


L1~











- i)


where


- ni,1


+1) i12i


i=1


+ 1)


-1 I~~
1=1


The


statistic


approximately


stributed


-square


with


p(G-


degrees


freedom.


Invariance


ProDertv


Test


Stati


stics


Samples


experiment


were


selected


from


either


contaminated


population


or an uncontaminated


population.


subset

matrix


of populations


as their


labeled


common


uncontaminated


dispersion


had


matrix.


the


The


identity


subset


populations


categorized


as contaminated


had


a common


diagonal


matrix


generality


That


beyond


ese


the


matrix


limited


forms


form


entail


loss


heteroscedasti


city


investigate


due


well


known


theorem


Anderson


and


invariance


characteristic


test


statistics.


I..- .., .1t- .w!


-%/iK


---


n" 1


L.


I


F









positive


definite,


there


exists


pxp


nonsingular


matrix


such


that


TZ.T


TZ.T


, where


pxp


identity


matrix


and


pxp


diagonal


matrix


(Anderson,


1958)


Hence


, when


the


design


includes


two


population


subsets


with


common


dispersion


matri


ces


within


given


subset,


including


only


diagonal


matri


ces


each


simulated


experiment


additional


limitation


on generalizability.


Second


, the test


stati


are


invariant


with respect


transformations


where


a pxp


nonsingular


transformation.


Brown-Forsvthe


General


zations


=Tx1


denote


the


sample


mean


vector


and


well


sample


known


dispersion


that


- = Tx
I


. and
I


TS.T
1


sample


calculated


using


*and


well


known


that


THT


Now


-
N


n.
N


2'S1


=TM T


For


the


modified


Hotelling


-Lawley


trace


criterion


trace{


M i


matrix


It is


* and


* be


i=1


n.
N


=2":










trace {


T H T


TMT ] -1


trace


Similarly,


the


modified


Pillai-Bartlett


trace


criterion


H'[H*


trace{T H T'[T H 2


)TM ] -1}


trace {H[H+ tM] -1}
h


For


modified


Wilks


likelihood


ratio


criterion,


. + f
h


T M T'


TH2"'


f
+-
12


TM?1 '


STjIZI


if
-MI
h


f
+ M
h


Wilcox


Generalization


trace {


hM
h"


f
+ -M
h


J











- 12i


rs;1-


[ T'] -lnis-1


t~w~i
2 =1


G
(T') [r
2=1


ni9;1]-2.


1=1


- Tii


G
2=1


'-n S
i


{Z w1}T11
1i-


r21~


2=1


n'Si.i,


2 =1


G=1
i-1


i=1


Using


results


1-4,


is


shown


to be


invariant


follows


- T) 'W*


i(TX,


- Ti)


G

-T
i=1


- ) )


TI) -W1T-[T(fi


- 2)]


-r
. 11


- f) 'wi


-X)


ml-.~~ ~ ~ ~ ~ -% -~ a C a Sr ew ~ -r,.4 e TV -*t -


4-1,a


4-, 1-4-


.1=1


= T(


- Ti


alr


i=1


mk AHA CAHA


|


ann










loss


generality


solely


using


diagonal


matrices


simulate


experiments


which


there


are


only


sets


dispersion matrices.


It should be noted,


however


, when there


are more


than two sets


of differing dispersion matrices


matrices


cannot


always


simultaneously


diagonalized


transformation matrix T.


Design


Eight


factors


were


considered


study.


These


are


described


following paragraphs.


Distribution


tvDe


(DT).


Two


types


distributions--


normal


exponential--were


included


study.


Pearson


and Please


(1975)


suggested that studies of robustness should


focus


distributions


magnitudes


less


with


than


skewness


0.6,


kurtosis


respectively.


having


However,


there


evidence


suggest


these


boundaries


are


unnecessarily


restrictive.


example,


Kendall


and


Stuart


(1963,


p.57)


reported


the


time


marriage


over


300,000


Australians.


The skewness and kurtosis were


2.0 and


respectively.


distributional


Micceri


(1989)


characteristics


investigated

achievement


psychometric measures.


Of these 440 data sets,


15.2%


had both


tails


with


weights


about


Gaussian,


49.1%


least


one extremely heavy tail,


and 18


.0% had both tail weights less


*h~rn


a
fl~iicci an -


U


1-n ri',


found


28.4%


-IUI aUY aL 'a)( S.. a- -F -


~h P


YLIU








being


extremely


asymmetric.


Of the


distributions


considered,


11.4%


were


classified


within


category


having


skewness


extreme


The


Micceri


study


underscores


the


common


occurrence of


distributions that are non-normal


Further


Micceri


study


suggests


the


Pearson


and Please criterion may


too


restrictive.


For


the


normal


stribution


the


coefficients


of skewness


and


kurtosis


(p4/M22


are


respectively


0.00 and


0.00.


For


the


exponential


distribution


the


coefficients


skewness


and


kurtos is


are


respectively


SThe


Micceri


study


provides


evidence


that


proposed


normal


exponential


distributions


are


reasonable


representations


data


that


may


found


educational


research.


Number


of dependent


variables


(Dl.


Data


were


generated


simulate


experiments


which


there


are


dependent


variable


Thi


choi


reasonably


consistent


with


the


range


of variable


commonly


examined


educational


research


(Algina


Oshima,


1990;


Algina


Tang,


1988;


Hakstian,


Roed,


Lind,


1979;


, 1991;


Olson


, 1974;


Tang,


1989)


Number


of DODulations


sampled


Data


were


generated


to simulate


or G=6


experiments


populations


which


there


Dij kstra


sampling


Werter


(1981)


from either


simulated


experiments


simulated


with


equal


experiments


, and


with


eaual


Olson


to 2. 3


(1974)


and


* f


(,/2/23) 1/2


6


.


.









rare


educational


research


(Tang


1989)


Hence,


chosen number of populations sampled


should provide


reasonably


adequate


examination


this


factor


Decree


sample


size


ratio


( NR)


Only


unequal


sample


S1zes


are


used


study


Sample


size


ratios


were


chosen


ratios


range


n1:n2:n3


from


small


used


the


moderately


simulation


large.


when


The


sampling


basic


from


three


different


ratios


from


populations


. :n 6


different


are


used


populations


given


the


are


Table


simulation


given


Similarly,


when


Table


sampling


Fairly


large


ratios


were


used


in Algina


and


Tang


(1988)


study


, with


an extreme


ratio


of 5


In experimental


and


studi


common


to have


sample


-size


ratios


between


(Lin,


1991)


Olson


(1974)


examined


only


case


equal-


sized


samples


Since


error


rates


increase


as the


degree


the


sample


size


ratio


increases


(Algina


Oshima


, 1990),


nominal


error


rates


are


excess


ively


exceeded


using


small


to moderately


large


sample


size


ratios


, then


procedure


presumably


will


have


difficulty


with


extreme


sample


size


ratios


Conversely,


the


procedure


performs


well


under


this


range


sample


ratios


then


should


work


well


equal


sample


size


ratios


question


of extreme


sample


rati


still


open.


Hence


, sample


size


ratios


were


chosen


under


the


constraint


i


4m









Table


Sample


Size


Ratios


~Ln1 Ln2- n


nI : n2 : n3



1 1 1.3

1 1 2

1 1.3 1.3

1 2 2


Table


Sample


Size


Ratios


*: .


nI : n2 : : 4 n5 : n6



1 1 1 1 1.3 1.3

1 1 1 1 2 2

1 1 1.3 1.3 1.3 1.3

1 1 2 2 2 2









sample


size


and


largest


sample


size


populations


sampled.


some


cases


these


basi


ratios


could


be maintained


because


the


restriction


the


ratio


total


sample


size


to number


of dependent


variable


Departure


from


these


basic


ratios


was


minimized.


Form


the


sample


ratio


(NRF


When


there


are


three


groups


either


the sample


size


ratio


form


= n,


< n3


denoted


NRF=


or the


sample


size


ratio


form


-- n3


denoted


NRF=2


When


there


are


six


groups


either


sample


size


ratio


of the


form


- n2


= n3


4 < n5


- n6


denoted


NRF= 1


or the


sample


ratio


form


= n2


denoted


NRF=2.


Ratio


total


sample


size


number


dependent


variables


(N:D


The


ratios


chosen


were


N:p=10


and


N:p=20.


Hakstian


, Roed


and Lind


(1979)


simulated


experiments


with N


equal


With


some


notable


exceptions


(Algina


Tang


, 1988;


, 1991)


current


studi


tend


avoid


smaller


than


. Yao


s test


(which


is generally


more


robust


than


James


s first


-order


test)


should


have


N:p


at least


10 to


robust


(Algina


Tang,


1988)


With


, Lin


(1991)


reasoned


seems


likely


that


will


need


to be at least


robustness


obtained


upper


limit


was


chosen


represent


moderately


large


experiments.


These


*


.l-.-. ,,


r .c -on


I; I J


7


1


7









Decree


of heteroscedasticitv


Each


population


with


dispersion


matrix


equal


a pxp


identity


matrix


will


called


an uncontaminated


population.


Each


population


with


pxp


diagonal


dispersion


matrix


with


at least


one


diagonal


element


not


equal


one


will


called


contaminated


population.


The


forms


the


dispersion


matrices,


which


depend


upon


the


number


of dependent


variables


, are


shown


Table


Two


level


d=J2


were


used


simulate

matrices


the


degree


Olson


of hetero


(1974)


scedasticity


simulated


experiments


the

with


dispersion


d equal


, 3.0,


and


Algina


and


Tang


(1988)


simulated


experiments


(1989)


chose


with


equal


equal


1.5,


and


, and


Algina


Tang


Oshima


(1990)


selected


d equal


to 1.5 and


3.0


For


this


study,


was


used


to simulate


a small


degree


of heteroscedasticity


and


was


selected


represent


larger


degree


heterosceda


sticity


SThese


values


were


selected


to represent


range


heteroscedasticity


more


likely


common


educational


experiments


(Tang,


1989)


RelationshiD


of sample


size


to dispersion


matri


ces


Both

and


positive

dispersion


and


negative


matri


ces


relationships


were


between


investigated.


sample


the


size


positive


relationship


the


larger


samples


correspond


the


negative


relationship


, the


smaller


samples


correspond


to D.


- S U U a- a -- -


d=J


Ilr ii


^


rrrr 1


,


I* r








Table 4


Forms of


Dispersion Matrices


Matrix p=3 p=6



D Diag(l,d2,d2} Diag(l,l, d2 ,d2,d2 ,d)d

I Diag{l,l,l} Diag(1,1,1,1,1,1)








Table


Relationship


of Samile


Size


to Heteroscedasticitv


(G=3)


Sample


Size


1 : n2


Ratios


* n3


Relationship


Positive


Negative


IID


IDD


Table


Relationship


of Sample


Size


to Heteroscedasti


city


(G=6)


Sample


Size


Ratios


Relationship


1 : n2


: n4


: n5


Positive


Negative


IIIIDD


IIIIDD


IIDDDD


IIDDDD


DDDDII

DDDDII


DDIIII

DDIIII









Desicmn


Layout.


sample


sizes


were


determined


once


values


, N:p,


NRF,


and


were


specified.


These


sample


zes


are


summarized


Table


Table


respectively


Each


these


conditions


were


crossed


with


two


distributions


, two


level


heteroscedasticity


, and


relationships


sample


size


dispersion


matri


ces


generate


experimental


conditions


from


which


to draw


conclusions


regarding


the


competitiveness


the


proposed


statistics


establ i


shed


Johansen


procedure.


Simulation


Procedure


The

for each


simulation

condition,


was

with


conducted


replications


separate


per


runs


condition


each


condition,


performance


Johansen


test


(4),


variations


modifi


Hote


lling-Lawley


test


, the


modified


Pillai


-Bartlett


test


modified


Wilks


test


modified


Wilcox


test


were


evaluated


using


generated


data.


For


sample,


nixp


(i=1


S. .,G)


matrix


uncorrelated


pseudo


-random


observations


was


generated


(using


PROC


IML


SAS)


from


target


stribution--normal


exponential


When


target distribution was an


exponential,


the


random


observations


each


variates


were


*


.3 a ~ a a e r a a *1 -. a -A a 4-a ~.3 -


i G


,,1


-YI1L ~ -1


1


LL


-I









Table


Sample


zes


(G=3)


p G N: p N n, n2 n3


Note.
closely


occas


ionally


altered


maintain


ratio


as manageable









Table


Sample


zes


p G N:p N n1 n2 n3 n4 n5 n6


Note.
closely


J is occasionally
as manageable.


altered


maintain


ratio


LG=6)








variates were


identically


distributed


with mean


equal


zero,


variance


equal


one,


and


covariances


among


variates


equal


zero.


Each


nixp


matrix


observations


corresponding


contaminated


population


was


post


multiplied


an appropriate


D to simulate


dispersion


heteroscedasticity


For


each


replication,


the


data


were


analyzed


using


Johansen


s test


the


two variations of


the modified Hotelling-


Lawley


trace


criterion


the


criterion

modified


Wilks


modified Pillai-

likelihood ratio


-Bartlett


criterion


trace

. and


the modified


Wilcox test.


The


proportion


of 2000


replications


that


yielded


significant


results


at a= 0


were


recorded


Summary


Two


distribution


types


[DT=normal


exponential],


level


dependent


variable


(p=3


two


level


populations


sampled


or 6),


two


level


of the


form


of the


sample


size


ratio,


two


levels


of the


degree


of the


sample


size


ratio


, two


level


of ratio


total


sample


to number


dependent


variable


(N:p=10


or 20)


, two


level


degree


heteroscedasticity


(d=J


3.0),


and


two


levels


relationship of


negative


sample size


condition)


to dispersion matri


combine


give


ces


(S=positive


experimental


conditions


The


Johansen


test


('ii


the


two


variations


of the


m-a4 PA^ U a1 1 4r -hT TT^ t m f


TT *\


+Iha mh~; f; 6~


o; 11


taet llT


r T .~ Gt'l d'l)








modified


Wilcox


test


(H )
'-ID


were


applied


each


these


experimental


conditions.


Generalizations


behavior


these


tests will


be based


upon


collective


results


of these


experimental


conditions.











CHAPTER 4


RESULTS AND


DISCUSSION


this


chapter


analyses


a=.05


are


presented.


Results with regard to i


for a=


.01 and for a=.10 are similar.


The


analyses


are


based


data


presented


Appendix.


Distributions


the


six


tests


are


depicted


Figures


labelled


In each


.05 denotes


denotes


.0750


these


.0250


.1249,


SlX


.0749

and


figures,


the


the


interval


forth.


interval


labelled


From


these


figures


rates


it is clear that


in terms of


performance


controlling Type I


Johansen


and


error


modified


*


Wilcox


tests


are


similar;


the


performance


first


modified


Hotelling-Lawley


(U,*) ,


second


modified


Hotelling-Lawley


cn~*


modified


Pillai-Bartlett


modified Wilks


tests are similar;


the performance of


these

the p


two sets of


performancee


tests greatly

'the Johansen


differ


test


from


one another;


superior to


that


the Wilcox generalization;


and


the performance of each of


Brown-Forsythe


generalizations


superior


that


either


the


Johansen


test


or Wilcox


generalization.


Because


the performance of the Johansen and modified Wilcox tests were

so different from that of the Brown-Forsythe generalizations,

separate analyses were conducted for each of these two sets of


-~ a I I


r r


r































.05 .10


.15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00


Figure


Frequency
Johansen


Histocram


Estimate


TyDe


Error


Rates


Test
































m a


.05 .10


.15 .20 .25 .30 .35


.40 .45


.55 .60 .65


.70 .75 .80


.85 .90 .95 1.00


Figure


Freauencv


Modified


stoaram


Wilcox


Estimated


Tvoe


Error


Rates


Test
































.05 .10


.15 .20 .25 .30 .35


.45 .50 .55 .60 .65 .70 .75 .80 .85 .90


.95 1.00


Figure


Freaixan~y


First


Modified


stoararn


Estimated


Hotellinc-Lawlev


TvDe


Error


Rates


Test

































.05 .10


.15 .20 .25 .30 .35


.40 .45 .50


.55 .60 .65 .70


.75 .80 .85


.90 .95 1.00


Figure


Freauencv


Second


Histoaram


Modified


Hot


Estimated


ellinq-Lawley


Type
Test


Error


Rates











250


.05 .10


.15 .20


.25 .30 .35 .40 .45 .50


.55 .60 .65 .70


.80 .85 .90 .95 1.00


Figure


Frequency
Modified


Histocram


Pillai-Bartlett


Estimated


TvDe


Error


Rates


Test
































.05 .10 .15 .20 .25 .30 .35 .40 .45 .50


.55 .60


.65 .70 .75 .80 .85 .90


.95 1.00


Figure


Freauencv


Modified


Histoqram


Wilks


Estimated


Tvne


Error


Rates


Test









was


used


investigate


effect


the


following


factors: Distribution Type (DT), Number of Dependent Variables


(P:'1


Number of


Size Ratio


(NR)


Populations Sampled


Degree of


, Form of the Sample Size Ratio


(NRF)


the Sample


Ratio of


Total


Sample


Size


Number


Dependent


Variables


(N:p),


Degree of Heteroscedasticity


(d) ,


Relationship of Sample Size


to Dispersion Matrices


(S),


and Test


Criteria


(T) .


Brown-Forsvthe Generalizations


Because


there


are


nine


factors,


initial


analyses


were


conducted


determine


which


effects


enter


into


analysis of variance model.


A forward selection approach was


used,


with


main


effects


entered


first,


followed


two-way


interactions,


three-way


interactions,


four-way


interactions.


Because R2 was


for the model


with


four-way interactions,


more complex models were not examined.


models


are


shown


in Table


The


model


with


main


effects


and


two-way


through


four-way


interactions


was


selected.

Variance components were computed for each main effect,


two-way,


three-way,


four-way


interaction.


The


variance


component


i=1,...,255)


for each effect was computed using


the formula 106(MSEF-MSE)/(2x),


where MSEF was the mean square


for that given effect,


MSE was


the mean square error


for the


fnuir--far;tfnr i n'l-'rn rt'-i nn mnrd=1 .


. ant 2? was twhe ninmhtr nf 1 Aval


( G)









Tabl


Man rn i tnri0a


rn1 P2


Main Effects


rrTun -Wa 17


Interaction


hreoo-


Way Interaction, and Four-Way Interaction Models when using
Way Interaction, and Four-Way Interaction Models when using


the


Four


Brown


-Forsvtne


General


zations


Highest-Order Terms R2



Main Effects 0.52

Two-Way Interactions 0.77

Three-Way Interactions 0.89

Four-Way Interactions 0.96








variance components were set to zero.


Using the sum of these


variance


components


plus


MSEx106


measure


total


variance,


proportion


total


variance


in estimated


Type


error


rates


was


computed


for the


effect


,255)


using the


formula


e,/[ (el+... +e55)


106MSE] .


Shown


in Table


10 are effects that


were statistically significant and


accounted


for at


least


1% of


total


variance


in estimated


Type


error rates.


Because N:p


, and GxT are among the largest effects


and--in


contrast to


factors


such as d


DT--do not have to


inferred


from


data


, their


effects


were


examined


calculating


percentiles


each


combination


N:p.


These


percentiles


should


provide


insight


into


functioning of the four tests.


The DTxNRFxSxd interaction was


significant and


second


largest


effect.


Consequently the


effects of the four factors


involved in this


interaction were


examined


constructing


cell


mean


plots


involving


combinations


four


factors.


Other


interaction effects


with


large


variance


components


that


included


these


factors


were


checked


change


findings


significantly.


The DTxG


interaction will be examined because


accounts


for 4.0% of the total


variance in estimated Type I error rates


and


is not explained


in terms of


either the effect of T,


N:p,


and


G or the effect


of DT


, NRF


, S,


and d.


The


factor p


has nei their a larae ma i n effect or larae interactions with any


i-i,









Table


Variance


Comnon F.m t


Fi rst


Mnr i fi sd'


Hotalling-Lawle


Second Modified Hotellincg-Lawley, Modified Pillai-Bartlett.


Modified


Wilks


Tests


Percent


Effect


of Variance


N:p

DTxNRFxSxd

T

DTxNRFxS

NRFxSxd

DTxd

DTxG

NRFxS

G

GxT


DTxGxd

DTxGxNRFxS

Sxd

d

S

NRFxSxdxT









Table


10--continued.


Percent


Effect


of Variance


pxNRFxSxd


DTxGxN


:pxd


GxN:p

NRFxSxT

DTxNRxS

dxT

DTxS


Others









variance,


effect


was


examined by


inspecting


cell


means


and


--C.


Finally,


influence


degree


sample


size


ratios


(NR)


was


minimal


The


main


effect


accounted


error


only


rates.


.1% of the


The


total


three-way


variance


interaction


in estimated


DTxNRxS


Type


was


effect


with


the


largest


variance


component


which


included


and


still


only


accounted


of the


total


variance


estimated


Type


error


rates.


Effect


of T


, and


. Percentil


are


displayed


Tabl


percentil


are


shown


Table


Using


Bradley'


liberal


criterion


.5a) ,


the


following


patterns


emerge


regarding


control


Type


error


rates


the


Brown


-Forsythe


generalizations


first


modified


Hotelling-Lawley


test


*
CM1)


was


adequate


when


N:p


was


however


test


tended


to be


liberal


when


was


the


second modified Hotelling


-Lawley test


CM2)


was


adequate


when


either


was


10 and


was


or when


was


and


was


second


modified


Hotelling


-Lawley


test


tended


to be


cons


ervative


when


N:p


was


10 and


was


whereas


the


test


tended


to be


slightly


liberal


when


N:p


was


and


was


when


the


was


modifi


20 and


ed Pillai


was


-Bartlett


the


test


modified


was


Pillai


adequate

-Bartlett


test


tended


to be conservative


when N:p


was


10 or when N:p was


20 and


was


the


modified


Wilks


test


was


adequate


when









Table


Percentiles of


for the First Modified Hotellinq-Lawlev Test


1(U ) and Second Modified Hotellina-Lawley Test (U ) for
Combinations of Ratio of Total Sample Size to Number of
Dependent Variables (N:p) and Number of Populations Sampled
J-GI


(N:p=10)


(N:p=20)


Test


Percentile


95th

90th

75th

50th

25th

10th

5th

95th

90th

75th

50th

25th

10th

5th


.0795*

.0710

.0555

.0505

.0430

.0375

.0345

.0730

.0625

.0513

.0453

.0385

.0325

.0290


.0770

.0715

.0595

.0500

.0398

.0315

.0295

.0510

.0460

.0388

.0290

.0198*

.0140*

.0135*


.0855*

.0795*

.0610

.0538

.0493

.0460

.0435

.0815*

.0785*

.0590

.0510

.0470

.0430

.0405


.0885*

.0835*

.0708

.0625

.0540

.0490

.0485

.0710

.0650

.0565

.0483

.0388

.0355

.0330









Table


Percentiles


the


Modified


Pillai-Bartlett


and Modified Wilks Test (L ) for Combinations of Ratio of
Total Sample Size to Number of Dependent Variables (N:p) and


Number


of Populations Sampled


(N:p=10)


(N:p=20)


Test


Percentile


95th

90th

75th

50th

25th

10th

5th

95th

90th

75th

50th

25th

10th

5th


.0555

.0495

.0430

.0370

.0318

.0240*

.0200*

.0705

.0635

.0483

.0440

.0388

.0330

.0310


.0365

.0310

.0258

.0210*

.0145*

.0110*

.0070*

.0465

.0425

.0360

.0288

.0215*

.0155*

.0130*


.0695

.0660

.0533

.0480

.0425

.0365

.0345

.0780*

.0745

.0575

.0513

.0455

.0415

.0405


.0510

.0500

.0455

.0380

.0315

.0275

.0235*

.0615

.0580

.0533

.0450

.0375

.0345

.0325


Test







82

modified Wilks test was conservative when N:p was 10 and G was

6.


Effect of


DT, NRF.


shown


Figure


Figure


when data


were sampled


from a


normal


distribution,


regardless


the


form


sample


size


ratio,


mean


increased


degree


heteroscedasticity


increased


positive


condition


whereas


mean


decreased


degree


heteroscedasticity


increased


the


negative


condition.


However


, as shown


in Figures 9


and 10,


when data were sampled


from an


exponential


distribution,


mean


increased


as degree


of heteroscedasticity increased regardless of the relationship


of sample


sizes


and dispersion matrices.


The mean difference


in t between the higher and lower degree of heteroscedasticity


was


greater


positive


condition


when


the


sample


was


selected


first


form


sample


size


ratios


whereas when the sample was selected


in the second form of


the sample size ratio


, the mean difference was greater


in the


negative


condition.


With


data


sampled


from


exponential


distribution


Brown-Forsythe


generalizations


tend


conservative


when


there


was


slight


degree


heteroscedasticity


(that


, (b)


degree


heteroscedasticity


increased


(d=3)


the


first


form of the


sample size


ratio was paired with


the negative condition,


the degree of heteroscedasticity increased and the second


fnrm nf *1


cz mn1 aS


r^31+ i n


- a a -~


.. -* a


Cl ~ ~ ~ ~ ~ u '7 'I. *~ rb.. I aII


.. j~i .


nh~;t~trr


d=J


E 1 '7 d











Mean Type I Error Rate


0.07


0.06


0.05


0.04


0.03


0.02


d = sqrt(


positive
condition


negative
condition


Sample Size to Dispersion Relationship


Figure


Estimated


TvDe


Error


Rates


the


Two


Levels


Degree


of Heteroscedasticity (d = J2 or 3) and Relationship of Sample
Size to Dispersion Matrices (S = positive or negative
condition) When Data Were Sampled as in the First Form of the
-~~~~ ~~ ~~ a-- a-- -* S


Sample


Ratio


from


an Normal


DistriDution










Mean Type I Error Rate
0.07


= sqrt(2)


positive negative
condition condition
Sample Size to Dispersion Relationship


Figure


Estimated Type I Error Rates for the Two Levels of the Degree
of Heteroscedasticity (d = J2 or 3) and Relationship of Sample
Size to Dispersion Matrices (S = positive or negative
condition) When Data Were Sampled as in the Second Form of the
Sample Size Ratio from an Normal Distribution











Mean Type I Error Rate


0.07


0.06


0.05


0.04


0.03


0.02


positive negative


condition


condition


Sample Size to Dispersion Relationship


Figure


Estimated


Mean


TvDe


Error


Rates


the


Two


Levels


Degree of Heteroscedasticity (d = J2 or 3) and Relationship of
Sample Size to Dispersion Matrices (S = positive or negative
condition) When Data Were Sampled as in the First Form of the


Sample


Size


Ratio


from


a Exponential


Distribution


d = sqrt(2)











Mean Type I Error Rate


0.07


0.06


0.05


0.04


0.03


0.02


d=3


d = sqrt(2)
d = sqrt(2)


positive negative


condition


condition


Sample Size to Dispersion Relationship


Figure


Estimated


Mean


TvDe


Error


Rates


Two


Level


Degree of Heteroscedasticity (d = J2 or 3) and Relationship of
Sample Size to Dispersion Matrices (S = positive or negative
condition) When Data are Sampled as in the Second Form of the


Sanpile


Ratio


from


a Exponential


Distribution









distribution,


the


Brown-Forsythe


generalizations


tended


to be


liberal


when


the


first


form


sample


size


ratio


was


paired


with


the


positive


condition,


the


second


form


the


sample


size


ratio


was


paired


with


the


negative


condition.


Effect


of DTxG


interaction.


As shown in Figure 11


mean


the


Brown


-Forsythe


generalizations


was


nearer


a when


was


than


when


was


regard


ess


type


distribution


from which


the


data


were


sampled.


When


data


were


sampled


from


normal


distribution,


the


tests


tended


slightly


conservative.


Mean


was


near


when


data


were


sampled


from


exponential


distribution


and


was


However


when


data


were


sampled


from


exponential


distribution


and


was


Brown


-Forsythe


general


zations


tended


to be conservative


Effect


Shown


Figure


mean


was


near


a for


the


Brown


-Forsythe


general


zations


when


was


When


was


the


tests


tended


to be slightly


conservative.










Mean Type I Error Rate


0.07


0.06


0.05


0.04


0.03


0.02


G=3


Normal Exponential


Distribution


Type


Figure


Estimated


Mean


Distribution


Tvype


TVDe


Error
Number


Rates


Combinations


of Populations


Sampled











Mean Type I Error Rate


0.04


0.03


0.02


6


Number of Dependent Variables


Figure


Estimated


Mean


Tvpe


Error


Rates


Brown


-Forsvthe


Generalizations for the Two Levels of the Number of Dependent
Variables









Johansen Test and Wilcox Generalization


Because


there


are


nine


factors,


initial


analyses


were


conducted


determine


which


effects


enter


into


analysis of variance model.


A forward selection approach was


used,


with


main


effects


entered


first,


followed


two-way


interactions,


three-way


interactions,


four-way interactions.


Because R2 was


.997


for the model with


four-way


interactions


, more complex models were not examined.


The


main


models


effects


are


two-way


shown


through


in Table


four-way


model


interactions


with


was


selected.

Variance components were computed for each main effect


two-way,


three-way


and


four-way


interaction.


variance


component


i=1,


..,255


for each effect was computed using


the formula 104(MSEF-MSE)


, where MSEF was the mean square


for that


given


effect,


MSE was


the mean


square


error


for the


four-factor interaction model


, and


was the number of levels


for the


factors


included


that given


effect.


Negative


variance components were set to zero.


Using the sum of these


variance


components


plus


MSExl04


measure


total


variance,


proportion


total


variance


in estimated


Type


error


rates


was


computed


effect


(i=l,


. *.


using the formula e8/[ (8,+... +255)


+ 104MSE].


Shown in Table


14 are effects that


were statistically significant and (b)









Table


Magnitudes


n.E Pr2


Main Effcrts


Interaction


Tb rPa -


Way Interaction, and Four-Way Interaction Models when usinQ
a a


t~hw


Johan


sen


Test


and


Wi-I


cox


General


action


Highest-Order Terms R2



Main Effects 0.767

Two-Way Interactions 0.963

Three-Way Interactions 0.988

Four-Way Interactions 0.997


~wn -W~ v








Table


Variance Components


for the Johansen and Modified


Wilcox Tests


Percent


Effect


of Variance


36.2


GxN


227


14.8

11.4


GxT


GxNRFxNR

P

pxG


GxNR


NRFxNR


Others









Because


N:p,


, GxN:p,


and


GxT


are


among


the


largest


effects


and--in


contrast


to factors


such


as d and


DT--do


have


to be inferred


from


data,


their


effects


will


be examined


calculating


percentiles


of f


each


combination


of G and


N:p.


These


percentiles


should


provide


insight


into


functioning


these


two


tests.


Effect


of T,.


, and


SPercentiles


are


splayed


.5a)


Table

the l


Using


following


Bradley'


patterns


liberal


emerge


criterion


regarding


control


Type


error


rates


the


Johansen


test


and


cox


general


zation


Johansen


test


was


adequate


only


when N


was


20 and


was


and


Wilcox


general


zation


was


inadequate


over


range


experimental


conditions


considered


the


experiment.


Since


performance


of the Johansen


test


the


Wilcox


general


zation


was


inadequate


further


analyst


was


warranted


either


ese


two


tests.


Summary


clear


that


terms


controlling


Type


error


rates


under


the


heteroscedastic


experimental


conditions


considered


the


four


Brown


-Forsythe


general


zations


are


much


more


effective


than


either


modified


Wilcox


test


Johansen


test


, (b)


Johansen


is more


effective




Full Text

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7DEOH FRQWLQXHG '7 3 1S 15) 15 6 G 9r /r + r P ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 8r 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 9 8r 9r /r + r P ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 V G 8r 9r /r K ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 S 1 S 15) 15 6 G 9 9r /r M + r P ( ( ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 9r /r r ( ; 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G XU 9r /r M K 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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,OO 7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 9 mr 9r /r M K 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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7DEOH f§FRQWLQXHG '7 S 1 S 15) 15 6 G 9 9r /r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 9r /r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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7DEOH f§FRQWLQXHG '7 S 1 S 15) 15 6 G 8r 8r 9r /r + r / L  P 1 1 1 1 1

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7DEOH (VWLPDWHG 7\SH (UURU 5DWHV :KHQ J '7 3 1 S 15) 15 6 G 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH FRQWLQXHG '7 3 1 S 15) 15 V G XLr 8r 9r /r K ( L ( L ( L ( L ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G mr 9r /r K ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 9 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 9r /r K ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 V G 8r 9r /r M ( ( ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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7DEOH FRQWLQXHG '7 3 1 S 15) 15 6 G 9r /r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 V G 9 9 9r /r K 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G XU Xr 9r /r M 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

PAGE 133

7DEOH f§FRQWLQXHG '7 3 1S 15) 15 6 G 8r 9r /r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

PAGE 134

7DEOH f§FRQWLQXHG '7 S 1 S 15) 15 6 G 9 8r 9r /r M K 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

PAGE 135

7DEOH f§FRQWLQXHG '7 S 1 S 15) 15 6 G 8r 8r 9r /r +Pr 1 1 1 1 1

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7DEOH (VWLPDWHG 7\SH (UURU 5DWHV :KHQ D '7 3 1 S 15) 15 6 G 8r 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G 9 9 9r /r +Q ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 6 G XU 8r 9r /r +Pr ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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7DEOH f§FRQWLQXHG '7 3 1 S 15) 15 V G 8r 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (

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FILES



SOLUTIONS TO THE MULTIVARIATE G-SAMPLE
BEHRENS-FISHER PROBLEM BASED UPON GENERALIZATIONS
OF THE BROWN-FORSYTHE F* AND WILCOX H TESTS
By
WILLIAM THOMAS COOMBS
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1992
UNIVERSITY OF FIORWA LIBRARIES

ACKNOWLEDGEMENTS
I would like to express my sincerest appreciation to the
individuals that have assisted me in completing this study.
First, I would like to thank Dr. James J. Algina, chairperson
of my doctoral committee, for (a) suggesting the topic for my
dissertation, (b) guiding me through difficult applied and
theoretical barriers, (c) debugging computer errors, (d)
providing editorial suggestions, and (e) fostering my
professional and personal growth through encouragement,
support, and friendship. Second, I am indebted and grateful
to the other members of my committee, Dr. Linda M. Crocker,
Dr. M. David Miller, and Dr. Ronald H. Randles, for patiently
reading the manuscript, offering constructive suggestions,
providing editorial assistance, and giving continuous support.
Third, I must thank Dr. John M. Newell who as a fifth and
unofficial member of my committee still attended committee
meetings, read the manuscript, and vigilantly inquired as to
the progress of the project. Finally, I would like to express
my heartfelt thanks to my wife Laura and son Tommy. Space
limitations prevent me from enumerating the many personal
sacrifices, both large and small, required of my wife so that
I was able to accomplish this task. Although I shall never be
able to fully repay the debt I have incurred to both my
ii

committee and family, let me begin simply and sincerely—thank
you.

TABLE OF CONTENTS
ACKNOWLEDGEMENTS Ü
ABSTRACT vi
CHAPTERS
1 INTRODUCTION 1
The Problem 3
Purpose of the Study 4
Significance of the Study 4
2 REVIEW OF LITERATURE 9
The Independent Samples t Test 9
Alternatives to the Independent Samples t Test . . 10
ANOVA F Test 15
Alternatives to the ANOVA F Test 16
Hotelling's T2 Test 26
Alternatives to the Hotelling's T2 Test 30
MANOVA Criteria 32
Alternatives to the MANOVA Criteria 36
3 METHODOLOGY 39
Development of Test Statistics 39
Brown-Forsythe Generalizations 39
Scale of the measures of between and
within group variability 43
Equality of expectation of the measures
of between and within group
dispersion 47
Wilcox Generalization 49
Invariance Property of the Test Statistics .... 50
Brown-Forsythe Generalizations 51
Wilcox Generalization 52
Design 54
Simulation Procedure 62
Summary 65
IV

4 RESULTS AND DISCUSSION 67
Brown-Forsythe Generalizations 74
Johansen Test and Wilcox Generalization 90
5 CONCLUSIONS 96
General Observations 96
Suggestions to Future Researchers 97
APPENDIX ESTIMATED TYPE I ERROR RATES 100
REFERENCES 141
BIOGRAPHICAL SKETCH 148
v

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
SOLUTIONS TO THE MULTIVARIATE G-SAMPLE
BEHRENS-FISHER PROBLEM BASED UPON GENERALIZATIONS
OF THE BROWN-FORSYTHE F* AND WILCOX H TESTS
— —m
By
William Thomas Coombs
August, 1992
Chairperson: James J. Algina
Major Department: Foundations of Education
The Brown-Forsythe F* and Wilcox H^ tests are generalized
to form multivariate alternatives to MANOVA for use in
situations where dispersion matrices are heteroscedastic.
Four generalizations of the Brown-Forsythe F* test are
included.
Type I error rates for the Johansen test and the five new
generalizations were estimated using simulated data for a
variety of conditions. The design of the experiment was a 2s
factorial. The factors were (a) type of distribution, (b)
number of dependent variables, (c) number of groups, (d) ratio
of total sample size to number of dependent variables, (e)
form of the sample size ratio, (f) degree of the sample size
ratio, (g) degree of heteroscedasticity, and (h) relationship
of sample size to dispersion matrices. Only conditions in
vi

which dispersion matrices were heterogeneous were included.
In controlling Type I error rates, the four
generalizations of the Brown-Forsythe F* test greatly
outperform both the Johansen test and the generalization of
the Wilcox H test.
Vll

CHAPTER 1
INTRODUCTION
Comparing two population means by using data from
independent samples is one of the most fundamental problems in
statistical hypothesis testing. One solution to this problem,
the independent samples t test, is based on the assumption
that the samples are drawn from populations with equal
variances. According to Yao (1965), Behrens (1929) was the
first to solve testing Ho: /i1 = n2 without making the
assumption of equal population variances. Fisher (1935, 1939)
showed that Behrens solution could be derived from Fisher's
theory of statistical inference called fiducial probability.
Others (Aspin, 1948; Welch, 1938, 1947) have proposed
solutions to the two-sample Behrens-Fisher problem as well.
The independent samples t test has been generalized to
the analysis of variance (ANOVA) F test, a test of the
equality of G population means. This procedure assumes
homoscedasticity, that is, a2 = a 2 = ... = a2. Several
authors have proposed procedures to test Ho: /i1 = \i2 = ... = MG
without assuming equal population variances. Welch (1951)
extended his 1938 work and arrived at an approximate degrees
of freedom (APDF) solution. Brown and Forsythe (1974), James
1

2
(1951), and Wilcox (1988, 1989) have proposed other solutions
to the G-sample Behrens-Fisher problem.
Hotelling (1931) generalized the independent samples t
test to a test of the equality of two population mean vectors.
This procedure makes the assumption of equal population
dispersion (variance-covariance) matrices, that is, Z1 = Z2.
Several authors have proposed procedures to test Hq: = /x2
without assuming equal population dispersion matrices. James
(1954) generalized his 1951 work and arrived at a series
solution. Anderson (1958), Bennet (1951), Ito (1969), Nel &
van der Merwe (1986), Scheffe (1943), and Yao (1965) have
proposed additional solutions to the multivariate two-sample
Behrens-Fisher problem.
Bartlett (1939), Hotelling (1951), Lawley (1938), Pillai
(1955), Roy (1945), and Wilks (1932) have proposed
multivariate generalizations of the ANOVA F test, creating the
four basic multivariate analysis of variance (MANOVA)
procedures for testing HQ: ^x1 = /x2 = ... = These
procedures make the assumption of equal population dispersion
matrices. James (1954) and Johansen (1980) proposed
procedures to test the equality of G mean vectors without
making the assumption of homoscedasticity, that is, S, = Z2 =
... = ZG. James extended James's (1951) univariate procedures
to produce first-order and second-order series solutions.
Johansen generalized the Welch (1951) procedure to form an
APDF solution to this problem.

3
The Problem
To date, neither the Brown-Forsythe (1974) nor the Wilcox
(1989) procedure has been extended to the multivariate
setting. To test Hq: /x, = m2 = ••• = MG Brown and Forsythe
(1974) proposed the statistic
- *..)2
F*
1=1
n.
E (1 ' N)S‘
¿=1
where n¡ denotes the number of observations in the ith group,
Xj the mean for the ith group, x the grand mean, Sj2 the
variance of the ith group, N the total number of observations,
and G the number of groups. The statistic F* is approximately
distributed as F with G-l and f degrees of freedom, where
n
í =
N
d-^)sn2
1=1
n, - l
The degrees of freedom, f, were determined by using a
procedure due to Satterthwaite (1941).
To test Ho : jU, = jU2 = ... = MG Wilcox (1989) proposed the
statistic
G
= £ fc'i (*i - X) 2 .
i =1
wi =
where
-i

4
and
2x\
n1 - 1 -
+ x.
1 ni(ni + 1) ni(ni + 1) 1
WiXi
£*.
2=1
2=1
In the equation for xi , xirii denotes the last observation in
the ith sample. The statistic is approximately distributed
as chi-square with G-l degrees of freedom.
Purpose of the Study
The purpose of this study is to extend the univariate
procedures proposed by Brown and Forsythe (1974) and Wilcox
(1989) to test H0 : /x1 = n2 = ... = MG and to compare Type I
error rates of the proposed multivariate generalizations to
the error rates of Johansen's (1980) test under varying
distributions, numbers of dependent (criterion) variables,
numbers of groups, forms of the sample size ratio, degrees of
the sample size ratio, ratios of total sample size to number
of dependent variables, degrees of heteroscedasticity, and
relationships of sample size to dispersion matrices.
Significance of the Study
The application of multivariate analysis of variance in
education and the behavioral sciences has increased
dramatically, and it appears that it will be used frequently

5
in the future for data analysis (Bray & Maxwell, 1985, p.7).
Stevens suggested three reasons why multivariate analysis is
prominent:
1. Any worthwhile treatment will affect the
subject in more than one way, hence the problem for
the investigator is to determine in which specific
ways the subjects will be affected and then find
sensitive measurement techniques for those
variables.
2. Through the use of multiple criterion
measures we can obtain a more complete and detailed
description of the phenomenon under investigation.
3. Treatments can be expensive to implement,
while the cost of obtaining data on several
dependent variables is relatively small and
maximizes information gain. (1986, p. 2)
Hotelling's T2 is sensitive to violations of
homoscedasticity, particularly when sample sizes are unequal
(Algina & Oshima, 1990? Algina, Oshima, & Tang, 1991;
Hakstian, Roed, & Lind, 1979; Holloway & Dunn, 1967? Hopkins
& Clay, 1963; Ito & Schull, 1964). Yao's (1965), James's
(1954) first- and second-order, and Johansen's (1980) tests
are alternatives to Hotelling's T2 that have no underlying
assumption of homoscedasticity. In controlling Type I error
rates under heteroscedasticity, Yao's test is superior to
James's first-order test (Algina & Tang, 1988; Yao, 1965).
Algina, Oshima, and Tang (1991) studied Type I error rates of
the four procedures when applied to data sampled from
multivariate distributions composed of p independent
univariate distributions. When (a) sample sizes are unequal
and (b) dispersion matrices are unequal, the four procedures

6
can be seriously nonrobust with extremely skewed distributions
such as the exponential and lognormal, but are fairly robust
with moderately skewed distributions such as the beta(5,1.5).
They also appear to be robust with non-normal symmetric
distributions such as the uniform, t, and Laplace. The
performance of Yao's test, James's second-order test, and
Johansen's test was slightly superior to the performance of
James's first-order test (Algina, Oshima, & Tang, 1991).
MANOVA criteria are relatively robust to non-normality
(Olson, 1974, 1976) but are sensitive to violations of
homoscedasticity (Korin, 1972; Olson, 1974, 1979; Pillai &
Sudjana, 1975; Stevens, 1979). The Pillai-Bartlett trace
criterion is the most robust of the four basic MANOVA criteria
for protection against non-normality and heteroscedasticity of
dispersion matrices (Olson, 1974, 1976, 1979). Alternatives
to MANOVA criteria that are not based on the homoscedasticity
assumption include James's first- and second-order tests, and
Johansen's test. When (a) sample sizes are unequal, (b)
dispersion matrices are unequal, and (c) data are sampled from
multivariate normal distributions, Johansen's test and James's
second-order test outperform the Pillai-Bartlett trace
criterion and James's first-order test (Tang, 1989).
In the univariate case, the Brown-Forsythe F* test and
Wilcox test do not require the equality of population
variances for the G groups. Hence, the Brown-Forsythe and
Wilcox tests are more general procedures than the ANOVA F

7
test. This suggests that generalizations of the Brown-
Forsythe procedure and the Wilcox procedure might have
advantages over the commonly used MANOVA procedure in cases of
heteroscedasticity.
Brown and Forsythe (1974) used Monte Carlo techniques to
examine the ANOVA F test, Brown-Forsythe F* test, Welch APDF
test, and James first-order procedure. The critical value
proposed by Welch is a better approximation for small sample
size than that proposed by James. Under (a) normality and (b)
inequality of variances both Welch's test and the F* test tend
to have actual Type I error rates (r) near nominal error rates
(a) in a wide variety of conditions. However, there are
conditions in which each fails to control r. In terms of
power, the choice between Welch (the specialization of
Johansen's test and, in the case of two groups, of Yao's test)
and F* depends upon the magnitude of the means and their
standard errors. The Welch test is preferred to the F* test
if extreme means coincide with small variances. When the
extreme means coincide with large variances, the power of the
F* test is greater than that of the Welch test.
A limited simulation by Clinch and Keselman (1982)
indicated that under conditions of heteroscedasticity, the
Brown-Forsythe test is less sensitive to non-normality than is
Welch's test. In fact, Clinch and Keselman concluded the user
should uniformly adopt the F* test over the Welch test. More
recently Oshima and Algina (in press) reported that, with non-

8
normal data, in some conditions the F* test has better control
over r than does James's second-order test, Welch's test, or
Wilcox's H test. In other conditions the F* test has
—m —
substantially worse control. Oshima and Algina concluded that
James's second-order test should be used with symmetric
distributions and Wilcox's H test should be used with
—m
moderately asymmetric distributions. With markedly asymmetric
distributions none of the tests had good control of r.
Extensive simulations (Wilcox, 1988) indicated that under
normality the Wilcox H procedure always gave the experimenter
more control over Type I error rates than did the F* or Welch
test and has error rates similar to James's second-order
method, regardless of the degree of heteroscedasticity.
Wilcox (1989) proposed H^, an improvement to the Wilcox (1988)
H method; the improved test is much easier to use than James's
second-order method. Wilcox (1990) indicated that the H test
is more robust to non-normality than is the Welch test.
Because the Johansen (1980) procedure is the extension of the
Welch test, the results reported by Clinch and Keselman and by
Wilcox suggest generalizations of the Brown-Forsythe procedure
and the Wilcox procedure might have advantages over the
Johansen procedure in some cases of heteroscedasticity and/or
skewness. Thus, the construction and comparison of new
procedures which may be competitive or even superior under
some conditions than the established standard is merited.

CHAPTER 2
REVIEW OF LITERATURE
Independent Samples t Test
The independent samples t is used to test the hypothesis
of the equality of two population means when independent
random samples are drawn from two populations which are
normally distributed and have equal population variances. The
test statistic
*l"*2
N
n.
Sp ( —+—)
p " ru
where
2 _ (^-1) s± + (n2-1) si
Sp ni+n2~2
has a t distribution with n1+n2-2 deqrees of freedom.
The degree of robustness of the independent samples t
test to violations of the assumption of homoscedasticity has
been well documented (Boneau, 1960; Glass, Peckham, & Sanders,
1972; Holloway & Dunn, 1967; Hsu, 1938; Scheffe, 1959). In
cases where there are unequal population variances, the
relationship between the actual Type I error rate (r) and the
nominal Type I error rate (a) is influenced by the sample
size. When sample sizes are equal (n1=n2) and sufficiently
9

10
large, r and a are near one another. In fact, Scheffe (1959,
p.339) has shown for equal-sized samples t is asymptotically
standard normal, even though the two populations are non¬
normal or have unequal variances. However, Ramsey (1980)
found there are boundary conditions where t is no longer
robust to violations of homoscedasticity even with equal-sized
samples selected from normal populations. Results from
numerous studies (Boneau, 1960; Hsu, 1938; Pratt, 1964;
Scheffe, 1959) have shown that when the sample sizes are
unequal and the larger sample is selected from the population
with larger variance (known as the positive condition), the t
test is conservative (that is, t < a). Conversely, when the
larger sample is selected from the population with smaller
variance (known as the negative condition), the t test is
liberal (that is, r > a).
Alternatives to the Independent Samples t Test
According to Yao (1965), Behrens (1929) was the first to
propose a solution to the problem of testing the equality of
two population means without assuming equal population
variances. This problem has come to be known as the Behrens-
Fisher problem. Fisher (1935,1939) noted that Behrens's
solution could be derived using Fisher's concept of fiducial
distributions.

11
A number of other tests have been developed to test the
hypothesis HQ: /x1 = /¿2 in situations in which a 2 f- a2. Welch
(1947) reported several tests in which the test statistic is
% =
Xi-x2
N
2 2
S, So
n,
no
The critical value is different for the various tests. There
are two types of critical values: (a) approximate degrees of
freedom (APDF), and (b) series.
The APDF critical value (Welch, 1938) is a fractile of
Student's t distribution with
f =
2 2
n\ n2
3>*
n,
no
nx~ 1
^2-1
degrees of freedom. In practice, the estimator of f is
obtained by replacing parameters by statistics, that is, s(2
replaces a2 (i=l,2). In the literature the test using this
estimator for f is referred to as the Welch test.
Welch (1947) expressed the series critical value for tv
as a function of s,2, s22, and a, and developed a series
critical value in powers of (n¡ - l)’1. The first three terms
in the series critical value are shown in Table 1. The zero-
order term is simply a fractile of the standard normal
distribution (z); using the zero-order term as the critical
value is appropriate with large samples. The first-order
critical value is the sum of the zero- and first-order terms,

12
Table 1
Critical Value Terms for Welch's (1947) Zero-, First-, and
Second-Order Series Solutions
Power of
(n, - 1)‘1 Term
Zero
z
One
s?
a i^r)2
ni
z[ 1 + z2 * ^_1 ]*
4 2 2
[ E— ]2
4—' n .
i=l
Two
2 S?
y ( 2 )2
„ r 1+z2 h. ni(ni-l)
2 2 a2
—i2
4—/ n .
2=1 ii2
2 3
r i 1 i2
3+5z2 + z4 fri i^-l
3 2 2
[ E— ]3
H ni
a (-Si)2
T [ n‘ l2
15+32zz+9z4 £í 1 1
32 2 2
tr —i4
h ni
Note. Z denotes a fractile of the normal distribution.

13
whereas the second-order critical value is the sum of all
three terms. As the sample sizes decline, there is a greater
need for the more complicated critical values. James (1951)
and James (1954) generalized the Welch series solutions to the
G-sample case and multivariate cases respectively.
Consequently, tests using the series solution are referred to
as James's first-order and second-order tests. The zero-order
test is often referred to as the asymptotic test. Aspin
(1948) reported the third- and fourth-order terms, and
investigated, for equal-sized samples, variation in the first-
through fourth-order critical values.
Wilcox (1989) proposed a modification to the asymptotic
test. The Wilcox statistic
where
2x,
xi =
nr l -
x,-
ni(ni+1) nJ(i3i+1)
is asymptotically distributed as a standard normal
distribution. Here x¡ (i=l,2) are biased estimators of the
population means which result in improved empirical Type I
error rates (Wilcox, 1989).
The literature suggests the following conclusions in the
two-sample case regarding the control of Type I error rates
under normality and heteroscedasticity for the independent
samples t test, Welch APDF test, James first- and second-order

14
series tests, Brown-Forsythe test, and Wilcox test: (a)
the performance of the Welch test and Brown-Forsythe test is
superior to the t test; (b) the Wilcox test and James second-
order test are superior to the Welch APDF test; and (c) for
most applications in education and the social sciences where
data are sampled from normal distributions under
heteroscedasticity, the Welch APDF test is adequate. Scheffe
(1970) examined six different tests including the Welch APDF
test from the standpoint of the Neyman-Pearson school of
thought. Scheffe concluded the Welch test, which requires
only the easily accessible t-table, is a satisfactory
practical solution to the Behrens-Fisher problem. Wang (1971)
examined the Behrens-Fisher test , Welch APDF test, and Welch-
Aspin series test (Aspin, 1948; Welch, 1947). Wang found the
Welch APDF test to be superior to the Behrens-Fisher test when
combining over all the experimental conditions considered.
Wang found jr-a] was smaller for the Welch-Aspin series test
than for the Welch APDF test. Wang noted, however, that the
Welch-Aspin series critical values were limited to a select
set of sample sizes and nominal Type I error rates. Wang
concluded, in practice, one can just use the usual t-table to
carry out the Welch APDF test without much loss of accuracy.
However, the Welch APDF test becomes conservative with very
long-tailed symmetric distributions (Yuen, 1974). Wilcox
(1990) investigated the effects of non-normality and
heteroscedasticity on the Wilcox and Welch APDF tests. The

15
Wilcox test tended to outperform the Welch test. Moreover,
over all conditions, the range of r was (.032, .065) for
a=.05, indicating the Wilcox test may have appropriate Type I
error rates under heteroscedasticity and non-normality.
In summary, the independent samples t test is generally
acceptable in terms of controlling Type I error rates provided
there are sufficiently large equal-sized samples, even when
the assumption of homoscedasticity is violated. For unequal¬
sized samples, however, an alternative that does not assume
equal population variances such as the Wilcox test or the
James second-order series test is preferable.
ANOVA F Test
The ANOVA F is used to test the hypothesis of the
equality of G population means when independent random samples
are drawn from populations which are normally distributed and
have equal population variances. The test statistic
G
Y ni (xi -x ) 2/ (G-l)
F = —
G
Y (nrl) s\/ (N-G)
i = 1
has an F distribution with G-l and N-G degrees of freedom.
Numerous studies have shown that the ANOVA F test is not
robust to violations of the assumption of homoscedasticity
(Clinch & Keselman, 1982; Brown & Forsythe, 1974; Kohr &
Games, 1974; Rogan & Keselman, 1977; Wilcox, 1988). The
behavior of ANOVA F parallels that of the independent samples

16
t test with one exception. Whereas the independent samples t
is generally robust when large sample sizes are egual, the
ANOVA F may not maintain adequate control of Type I error
rates even with equal-sized samples if the degree of
heteroscedasticity is large (Rogan & Keselman, 1977; Tomarken
& Serlin, 1986). In the positive condition the F test is
conservative and in the negative condition the F test is
liberal (Box, 1954; Clinch & Keselman, 1982; Brown & Forsythe,
1974; Horsnell, 1953; Rogan & Keselman, 1972; Wilcox, 1988).
Alternatives to the ANOVA F Test
A number of tests have been developed to test the
hypothesis Ho:/x1 = = ... = /xG in situations in which a* / a.2
(for at least one pair of i and j). Welch (1951) generalized
the Welch (1938) APDF solution and proposed the statistic
a
£ (Xi-x) V (G-l)
F = —
^2(0-2) A A(1-j^)2
G2-1 fj w
where
w =
7=1
ni i=1
xij
1=1,...,G

17
* = E
7=1
WjXj
w
and
ft = ^-1 i = l, . . • , G
The statistic Fy is approximately distributed as F
and
[
degrees of freedom.
James (1951) generalized the Welch (1947)
solutions, proposing the test statistic
J = £ W^X^X)2
i=l
where
w,-
i= 1, . . . ,G
and
i= 1 G
* = E
7=1
w
with G-l
series
In the asymptotic test the critical value of the statistic J
is a fractile of a chi-square distribution with G-l degrees of

18
freedom. If the sample sizes are not sufficiently large,
however, the distribution of the test statistic may not be
accurately approximated by a chi-square distribution with G-l
degrees of freedom. James (1951) derived a series expression
which is a function of the sample variances such that
G
P [^2 w1(xi-x)2 i: 2h(s¡) ] = a .
i =1
James found approximations to 2h(Sj2) of orders 1/fj and 1/fj2
(fj = nj-1). In the first-order test, James found to order
1/fj the critical value is
2 h{sj)
Xg-l;a t 1 +
3 Xg-1 ; a + G * 1
2(G2-l)
Et-(i--)2] •
^ r w
H f±
The null hypothesis is rejected in favor of the alternative
hypothesis if J > 2h(Sj2). James also provided a second-order
solution which approximates 2h(Sj2) to order 1/fj2. James
noted that this second-order test is very computationally
intensive.
Brown and Forsythe (1974) proposed the test statistic
G
F*
Y, ni (x¿.
2=1
G
E (1 -
i =1
-x.)
The statistic F* is approximately distributed as F with G-l
and

19
f =
1=1
tEu-^isn
n<
a [ (1 - si ] 2
E- "
i=l
72, - 1
degrees of freedom. In the case of two groups, both the
Brown-Forsythe test and Welch (1951) APDF test are equivalent
to the Welch (1938) APDF test.
Wilcox (1989) proposed the statistic
Hm = E Wi{*i " *)2
i=1
where
s?
= [ — ] -1
«i
i=l G
w
7=1
*7 =
72,-1 _
*7
72, (72, +1 ) 72, (72,+1)
i=1,...,G
and
* = E
7=1
w
The statistic is approximately distributed as chi-square
with G—1 degrees of freedom.
The literature suggests the following conclusions about
control of Type I error rates under heteroscedastic conditions
by the ANOVA F, Welch APDF, James first- and second-order,

20
Brown-Forsythe, Wilcox H, and Wilcox tests: (a) the
performance of each of these alternatives to ANOVA F is
superior to F; (b) the Welch test outperforms the James first-
order test; (c) the Welch and Brown-Forsythe tests are
generally competitive with one another, however, the Welch
test is preferred with data sampled from normal distributions
while the Brown-Forsythe test is preferred with data sampled
from skewed distributions; and (d) the Wilcox and James
second-order test outperform all of these other alternatives
to ANOVA F under the greatest variety of conditions. Brown
and Forsythe (1974) used Monte Carlo techniques to examine the
ANOVA F, Brown-Forsythe F*, Welch APDF, and James zero-order
procedures when (a) equal- and unequal-sized samples were
selected from normal populations; (b) G was 4, 6, or 10; (c)
the ratio of the largest to the smallest sample size was 1,
1.9, or 3; (d) the ratio of the largest to the smallest
standard deviation was 1 or 3; and (e) total sample size
ranged between 16 and 200. For small sample sizes the
critical value proposed by Welch is a better approximation to
the true critical value than is that proposed by James. Both
the Welch APDF test and Brown-Forsythe F* test have r near a
under the inequality of variances.
Kohr and Games (197 4) examined the ANOVA F test, Box
test, and Welch APDF test when (a) equal- and unequal-sized
samples were selected from normal populations; (b) G was 4;
(c) the ratio of the largest to the smallest sample size was

21
1, 1.5, or 2.8; (d) the ratio of the largest to the smallest
standard deviation was 1, 2.0, Jl, 710, or 7l3; and (e) total
sample size ranged between 32 and 34. The best control of
Type I error rates was demonstrated by the Welch APDF test.
Kohr and Games concluded the Welch test may be used with
confidence with the unequal-sized samples and the
heteroscedastic conditions examined in their study. Kohr and
Games concluded the Welch test was slightly liberal under
heteroscedastic conditions; however, this bias was trivial
compared to the inflated error rates for the F test and Box
test under comparable conditions. Levy (1978) examined the
Welch test when data were sampled from either the uniform,
chi-square, or exponential distributions and also found that,
under heteroscedasticity, the Welch test can be liberal.
Dijkstra and Werter (1981) compared the James second-
order, Welch APDF, and Brown-Forsythe tests when (a) equal-
and unequal-sized samples were selected from normal
populations; (b) G was 3, 4, or 6; (c) the ratio of the
largest to the smallest sample size was 1, 2, or 2.5; (d)
total sample size ranged between 12 and 90; and (e) the ratio
of the largest to the smallest standard deviation was 1 or 3.
Dijkstra and Werter concluded the James second-order test gave
better control of Type I error rates than either the Brown-
Forsythe F* or Welch APDF test.
Clinch and Keselman (1982) studied the ANOVA F, Welch
APDF, and Brown-Forsythe F* tests using Monte Carlo methods

22
when (a) equal- and unequal-sized samples were selected from
normal distributions, chi-square distributions with two
degrees of freedom, or t distributions with five degrees of
freedom; (b) G was 4; (c) the ratio of the largest to the
smallest sample size was 1 or 3; (d) total sample size was 48
or 144; and (e) variances were either homoscedastic or
heteroscedastic. The ANOVA F test was most affected by
assumption violations. Type I error rates of the Welch test
were above a, especially in the negative case. The F* test
provided the best Type I error control in that it generally
only became nonrobust with extreme heteroscedasticity.
Although both the Brown-Forsythe test and Welch test were
liberal with skewed distributions, the tendency was stronger
for the Welch test.
Tomarken and Serlin (1986) examined six tests including
the ANOVA F test, Brown-Forsythe test, and Welch APDF test
when (a) equal- and unequal-sized samples were selected from
normal populations; (b) G was 3 or 4; (c) the ratio of the
largest to the smallest sample size was 1 or 3; (d) total
sample size ranged between 36 and 80; and (e) the ratio of the
largest to smallest standard deviation was 1, 6, or 12.
Tomarken and Serlin found that the Brown-Forsythe F* test,
though generally acceptable, was at least slightly liberal
whether sample sizes were equal or directly or inversely
paired with variances.

23
Wilcox, Charlin, and Thompson (1986) examined Monte Carlo
results on the robustness of the ANOVA F, Brown-Forsythe F*,
and the Welch APDF test when (a) equal- and unequal-sized
samples were selected from normal populations; (b) G was 2, 4,
or 6; (c) the ratio of the largest to the smallest sample size
was 1, 1.9, 3, 3.3, or 4.2; (d) total sample size ranged
between 22 and 95; and (e) the ratio of the largest to
smallest standard deviation was 1 or 4. Wilcox, Charlin, and
Thompson gave practical situations where both the Welch and F*
tests may not provide adequate control over Type I error
rates. For equal variances but unequal-sized samples, the
Welch test should be avoided in favor of the F* test but for
unequal-sized samples and possibly unequal variances, the
Welch test was preferred to the F* test.
Wilcox (1988) proposed H, a competitor to the Brown-
Forsythe F*, Welch APDF, and James second-order test.
Simulated equal- and unequal-sized samples were selected where
(a) distributions were either normal, light-tailed symmetric,
heavy-tailed symmetric, medium-tailed asymmetric, or
exponential-like; (b) G was 4, 6, or 10; (c) the ratio of the
largest to the smallest sample size was 1, 1.8, 2.5, 3.7, or
5; (d) total sample size ranged between 44 and 100; and (e)
the ratio of the largest to the smallest standard deviation
was 1, 4, 5, 6, or 9. These simulations indicated that under
normality the new procedure always gave the experimenter as
good or better control over the probability of a Type I error

24
than did the F* test or Welch APDF test. Wilcox showed that,
under normality, James's second-order test and Wilcox's test
have r much closer to a than the Welch or Brown-Forsythe
tests. The Wilcox test gave conservative results provided n¡
> 10 (i=l,...,G). Wilcox's results indicate the H procedure
has a Type I error rate that is similar to James's second-
order method, regardless of the degree of heteroscedasticity.
Although computationally more tedious, Wilcox recommended
James's second-order procedure for general use.
Wilcox (1989) proposed H^, an improvement to Wilcox's
(1988) H method, designed to be more comparable in power to
James's second-order test. Wilcox compared James's second-
order test with when (a) data were sampled from normal
populations; (b) G was 4 or 6; (c) the ratio of the largest to
the smallest sample size was 1, 2.5, 2.7, or 5; (d) total
sample size ranged between 44 and 121; and (e) the ratio of
the largest to the smallest standard deviation was 1, 4, or 6.
Wilcox's results indicate that when applied to normal
heteroscedastic data, has r near a and slightly less power
than James's second-order test. The main advantage of the
improved Wilcox procedure is that it is much easier to use
than James's second order test, and it is easily extended to
higher way designs.
Oshima and Algina (in press) studied Type I error rates
for the Brown-Forsythe test, James's second-order test,
Welch's APDF test, and Wilcox's H test for 155 conditions.

25
These conditions were obtained by crossing the 31 conditions
defined by sample sizes and standard deviations in the Wilcox
(1988) study with five distributions—normal, uniform, t(5),
beta(1.5,8.5), and exponential. The James second-order test
and Wilcox test were both affected by non-normality. When
samples were selected from symmetric non-normal distributions
both James's second-order test and Wilcox's H test maintained
—m
r near a. When the tests were applied to data sampled from
asymmetric distributions, ¡ r —o; ¡ increased. Further, as the
degree of asymmetry increased, ¡T-aj tended to increase. The
Brown-Forsythe test outperformed the Wilcox test and
James's second-order test under some conditions, however, the
reverse held under other conditions. Oshima and Algina
concluded (a) the Wilcox H test and James's second-order test
were preferable to the Brown-Forsythe test, (b) James's
second-order test was recommended for data sampled from a
symmetric distribution, and (c) Wilcox's test was
recommended for data sampled from a moderately skewed
distribution.
In summary, when data are sampled from a normal
distribution, the Wilcox H test and James second-order test
have better control of Type I error rates, particularly as the
degree of heteroscedasticity gets large. All of these
alternatives to the ANOVA F are affected by skewed data but
there is some evidence the Brown-Forsythe F* test and Wilcox
H test are less affected.

26
Hotelling's T2 Test
Hotelling's (1931) T2 is a test of the equality of two
population mean vectors when independent random samples are
selected from two populations which are distributed
multivariate normal and have equal dispersion matrices. The
test statistic is given by
T2 = â–  -i-2 - (5q - x2)' S1 (5?1 - x2)
ni+n2
where
g = (nx-l) S1 + (n2-l)S2
ni+n2-2
Hotelling demonstrated the transformation
nx+n2 -p-1 t2
(nx+n2-2)p
has an F distribution with p and n.,+n2-p-l degrees of freedom.
The sensitivity of Hotelling's T2 to violations of the
assumption of homoscedasticity is well documented. This has
been investigated both analytically (Ito & Schull, 1964) and
empirically (Algina & Oshima, 1990; Hakstian, Roed, & Lind,
1979; Holloway & Dunn, 1967; Hopkins & Clay, 1963). Ito and
Schull (1964) investigated the large sample properties of T2
in the presence of unequal dispersion matrices Z1 and S2. Ito
and Schull showed that in the case of two very large equal¬
sized samples, T2 is well behaved even when the dispersion
matrices are not equal and that in the case of two samples of
nearly equal size, the test is not affected by moderate

27
inequality of dispersion matrices provided the samples are
very large. However, if the two samples are of unequal size,
quite a large effect occurs on the level of significance from
even moderate variations. Ito and Schull indicated that,
asymptotically, with fixed ^/(n^n^ > 0.5 and for equal
eigenvalues of Z2S1'1, r < a when the eigenvalues are greater
than one and r > a when the eigenvalues are less than one.
Hopkins and Clay (1963) examined distributions of
Hotelling's T2 with sample sizes of 5, 10, and 20 selected
from either (a) bivariate normal populations with zero means,
dispersion matrices of the form a 21 (i=l,2), where oz/ay was
1, 1.6, or 3.2; or (b) circular bivariate symmetrical
leptokurtic populations with zero means, equal variances, and
62-3 was 3.2 or 6. Hopkins and Clay reported T2 is robust to
violations of homoscedasticity when n1=n2 > 10 but that this
robustness does not extend to disparate sample sizes. Hopkins
and Clay reported that upper tail frequencies of the
distribution of Hotelling's T2 for n¡ > 10 (i=l,2) are not
substantially affected by moderate degrees of symmetrical
leptokurtosis.
Holloway and Dunn (1967) examined the robustness of
Hotelling's T2 to violations of the homoscedasticity
assumption when (a) equal- and unequal-sized samples were
selected from multivariate normal distributions; (b) p was 1,
2, 3, 5, 7, or 10; (c) total sample size ranged between 10 and
200; (d) n1/(n1+n2) was .3, .4, .5, .6, or .7; and (e) the
.4,
•5,

28
eigenvalues of £2Z-|'1 were 3 or 10. Holloway and Dunn found
equal-sized samples help in keeping r close to a. Further,
Holloway and Dunn found that for large equal-sized samples,
control of Type I error rates depends on the number of
dependent variables (p) . For example, when nf = 50 (i=l,2)
and all the eigenvalues of Z^'1 = 10, r is near a for p = 2
and p = 3, but r markedly departs from a when p = 7 or p = 10.
Holloway and Dunn found that generally as the number of
dependent variables increases, or as the sample size
decreases, r increases.
Hakstian, Roed, and Lind (1979) obtained empirical
sampling distributions of Hotelling's T2 when (a) equal- and
unequal-sized samples were selected from multivariate normal
populations; (b) p was 2, 6, or 10; (c) (n1+n2)/2 was 3 or 10;
(d) n1/n2 was 1, 2, or 5; and (e) dispersion matrices were of
the form I and D, where D was I, d2I, or
diag{1,1,...,l,d2,d2,...,d2} (d = 1, 1.2, or 1.5). Hakstian,
Roed, and Lind found that for equal-sized samples, the T2
procedure is generally robust. With unequal-sized samples, T2
was shown to become increasingly less robust as dispersion
heteroscedasticity and the number of independent variables
increase. Consequentially, Hakstian, Roed, and Lind argued
against the use of T2 in the negative condition and for
cautious use in the positive condition.
Algina and Oshima (1990) studied Hotelling's T2 where (a)
p was 2, 6, or 10; (b) the ratio of total sample size to

29
number of dependent variables was 6, 10, or 20; and (c) for
the majority of conditions Z2 = d2Z, (d = 1.5, 2.0, 2.5, or
3.0). Algina and Oshima found that even with a small sample
size ratio, the T2 procedure can be seriously nonrobust. For
example, with p = 2 and Z2 = 2.25E.,, a sample size ratio as
small as 1.1:1 can produce unacceptable Type I error rates.
Algina and Oshima also confirmed earlier findings that
Hotelling's T2 test became less robust as the number of
dependent variables and degree of heteroscedasticity
increased.
In summary, Hotelling's T2 test is not robust to
violations of the assumption of homoscedasticity even when
there are equal-sized samples, especially if the ratio of
total sample size to number of dependent variables is small.
When the larger sample is selected from the population with
the larger dispersion matrix, t < a. When the larger sample
is selected from the population with the smaller dispersion
matrix, t > a. These tendencies increase with the inequality
of the size of the two samples, the degree of
heteroscedasticity, and the number of dependent variables.
Therefore, the behavior of Hotelling's T2 test is similar
to the independent samples t test under violations of the
assumption of homoscedasticity. Hence, it is desirable to
examine robust alternatives that do not require this basic
assumption of the Hotelling's T2 procedure.

Alternatives to the Hotelling's T2 Test
30
A number of tests have been developed to test the
hypothesis Ho:/ii = n2 in a situation in which Z1 f Z2.
Alternatives to the Hotelling T2 procedure that do not assume
equality of the two population dispersion matrices include
James's (1954) first- and second-order tests, Yao's (1965)
test, and Johansen's (1980) test. Differing only in their
critical values, all four tests use the test statistic
^ - ,r= r . 3* ^
Tv = {x1 - x2) [ —
ni n2
(Xi - x2)
where x1 and S- are respectively the sample mean vector and
sample dispersion matrix for the ith sample (i=l,2).
The literature suggests the following conclusions about
control of Type I error rates under heteroscedastic conditions
by Hotelling's T2 test, James's first- and second-order tests,
Yao's test, and Johansen's test: (a) Yao's test, James's
second-order test, and Johansen's test are superior to James's
first-order test; and (b) all of these alternatives to
Hotelling's T2 are sensitive to data sampled from skewed
populations.
Yao (1965) conducted a Monte Carlo study to compare Type
I error rates between the James first-order test and the Yao
test when (a) equal- and unequal-sized samples were selected,
(b) p was 2, (c) the ratio of total sample size to number of
dependent variables was 10 or 13, and (d) dispersion matrices

31
were unequal. Although both procedures have r near a under
heteroscedasticity, Yao's test was superior to James's test.
Algina and Tang (1988) examined the performance of
Hotelling's T2, James's first-order test, and Yao's test when
(a) p was 2, 6, or 10; (b) N:p was 6, 10, or 20; (c) the ratio
of the largest to the smallest sample size was 1, 1.25, 1.5,
2, 3, 4, or 5; and (d) the dispersion matrices were of the
form I and D, where D was d2I (d = 1.5, 2.0, 2.5, or 3.0),
diag{ 3 , 1, 1 , . . . , 1 ) , diag( 3 , 3 , . . . ,3,1,1, ...,1),
diag{1/3,3,3, . . . ,3 }, or diag{1/3,1/3,...,1/3,3,3 , . . . ,3 }.
Algina and Tang confirmed the superiority of Yao's test. For
10 < N:p < 20, Yao's test produced appropriate Type I error
rates when p < 10, n1: n2 < 2:1, and d < 3. For N:p = 20,
appropriate error rates occurred when n1: n2 < 5:1 and d < 3.
This applied for both the specific cases where one dispersion
matrix was a multiple of the second (Z2 = d2Z1) and in more
complex cases of heteroscedasticity. When N:p = 6 and Z2 =
d2Z,, Algina and Tang found Yao's test to be liberal.
Algina, Oshima, and Tang (1991) studied Type I error
rates for James's first- and second-order, Yao's, and
Johansen's tests for various conditions defined by the degree
of heteroscedasticity and non-normality (uniform, Laplace,
t(5), beta(5,1.5), exponential, and lognormal distributions).
The study indicated these four alternatives to Hotelling's T2
may not be robust, when the sampled distributions have
heteroscedastic dispersion matrices, are skewed, and have

32
positive kurtosis. Although the four procedures were
seriously nonrobust with exponential and lognormal
distributions, they were fairly robust with the remaining
distributions. The performance of Yao's test, James's second-
order test, and Johansen's test was slightly superior to the
performance of James's first-order test. Algina, Oshima, and
Tang indicate that Yao's test is also sensitive to skewness.
In summary Yao's test, James's second-order test, and
Johansen's test work reasonably well under normality.
Although all of these alternatives to Hotelling's T2 test have
elevated Type I error rates with skewed data, Johansen's test
has the practical advantages of (a) generalizing to G > 2, and
(b) being relatively easy to compute.
MANOVA Criteria
The four basic multivariate analysis of variance (MANOVA)
criteria are used to test the equality of G population mean
vectors when independent random samples are selected from
populations which are distributed multivariate normal and have
equal dispersion matrices. Define
G
H = Y ni (xi - x) (xi - x)'
1=1
and
E = Y {nt - 1)3, .
i= 1

33
The basic MANOVA criteria are all functions of the eigenvalues
of HE'1. Define t â–  to be the ith eigenvalue of HE'1
(i=l,...,s), where s = min(p,G-l). Those criteria are:
1.Roy's (1945) largest root criterion
2. Hotelling-Lawley trace criterion (Hotelling, 1951;
Lawley, 1938)
U = traceiHE'1) = Y zi ;
i =1
3. Pillai-Bartlett trace criterion (Pillai, 1955;
Bartlett, 1939)
S T â– 
V = trace[H{H+E)-1] = Y 1- ;
+-f 1 +T .
2=1 ‘l
and
4.Wilks's (1932) likelihood ratio criterion
L =
\H+E
n
Ti
2=1 1+T2
Both analytic (Pillai & Sudjana, 1975) and empirical
(Korin, 1972; Olson, 1974) investigations have been conducted
on the robustness of MANOVA criteria with respect to
violations of homoscedasticity. Pillai and Sudjana (1975)
examined violations of homoscedasticity on the four basic
MANOVA criteria. Although the generalizability of the study
was limited by only examining equal-sized samples selected
from two populations with unclear degrees of

34
heteroscedasticity, the results were consistent—modest
departures from a for minor degrees of heteroscedasticity and
more pronounced departures with greater heteroscedasticity.
Korin (1972) studied Roy's largest root criterion (R) ,
the Hotelling-Lawley trace criterion (U), and Wilks's
likelihood ratio criterion (L) when (a) equal- and unequal¬
sized samples were selected from normal populations; (b) p was
2 or 4; (c) G was 3 or 6; (d) the ratio of total sample size
to number of dependent variables was 8.25, 9, 12, 15.5, 18 or
33; and (e) dispersion matrices were of the form I or D, where
D was d2I or 2d2I (d = 1.5 or 10). For small samples, even
when the sample sizes were all equal, dispersion
heteroscedasticity produced Type I error rates greater than a.
Korin reported the error rates for R were greater than those
for U and L.
Olson (1974) conducted a Monte Carlo study on the
comparative robustness of six multivariate tests including the
four basic MANOVA criteria (R, U, L, V) when (a) equal-sized
samples were selected; (b) p was 2, 3, 6, or 10; (c) G was 2,
3, 6, or 10; (d) n¡ was 5, 10, or 50 (i=l,...,G); and (e)
dispersion matrices were of the form I or D, where D
represented either a low or high degree of contamination. For
the low degree of contamination, D = d2I, whereas for the high
degree of contamination, D = diag {pd2-p+l, 1,1, . . . , 1) (d = 2,
3, or 6). Results indicated that for protection against non¬
normality and heteroscedasticity of dispersion matrices, R

35
should be avoided, while V may be recommended as the most
robust of the MANOVA tests. In terms of the magnitude of the
departure of r from a, the order was typically R > U > L > V.
This tendency increased as the degree of heteroscedasticity
increased. The departure of r from a for R, U, and L
increased with an increase in the number of dependent
variables, however, the impact of p on V was not as well
defined. Additionally, for R, U, and L, t decreased as sample
size increased except when G > 6. When G > 6, r increased for
all four basic MANOVA procedures, although the increase was
least for V.
Stevens (1979) contested Olson's (1976) claim that V is
superior to L and U for general use in multivariate analysis
of variance because of greater robustness against unequal
dispersion matrices. Stevens believed Olson's conclusions
were tainted by using an example which had extreme subgroup
variance differences, which occur very infrequently in
practice. Stevens conceded V was the clear choice for diffuse
structures, however, for concentrated noncentrality structures
with dispersion heteroscedasticity, the actual Type I error
rates for V, U, and L are very similar. Olson (1979) refuted
Stevens's (1979) objections on practical grounds. The
experimenter, faced with real data of unknown noncentrality
and trying to follow Stevens's recommendation to use V for
diffuse noncentrality and any of the V, U, or L statistics for
concentrated noncentrality, must always choose V.

36
Alternatives to the MANOVA Criteria
A number of tests have been developed to test the
hypothesis Hq:/i1 = nz =...= /iQ in a situation in which Z; f Z-
(for at least one pair of i and j). James (1954) generalized
James's (1951) series solutions and proposed the statistic
G
J = (Xi-X) 'Wi
2=1
where
= [**]- i = l G
ni
* = iwi
i= 1
xi = ir i=1 G
â– ^2 j=l
and
x = FT1 ¿ Vx,
2=1
The James (1954) zero-, first-, and second-order critical
values parallel those developed by James (1951).
Johansen (1980) generalized the Welch (1951) test and
proposed using the James (1954) test statistic J divided by
6A
c = p(G-l) + 2A -
P(G-1) +2
where

37
A
E
7=1
trace (I-W~1Wi)2 + trace2 (I-W~1Wi)
2(ni-l)
The critical value for the Johansen test is a fractile of an
F distribution with p(G-l) and p(G-l)[p(G-l)+2]/(3A) degrees
of freedom.
The literature suggests the following conclusions about
control of Type I error rates when sampling from multivariate
normal populations under heteroscedastic conditions by the
four basic MANOVA criteria, James's first- and second-order
tests, and Johansen's test: (a) the Pillai-Bartlett trace
criterion is the most robust of the four basic MANOVA
criteria; and (b) with unequal-sized samples, Johansen's test
and James's second-order test outperform the Pillai-Bartlett
trace criterion and James's first-order test.
Ito (1969) analytically examined Type I error rates for
James's zero-order test and showed r > a. Ito showed jr-aj
increased as the variation in the sample sizes, degree of
heteroscedasticity, and number of dependent variables
increased, whereas j r — or J decreased as the total sample size
increased.
Tang (1989) studied the Pillai-Bartlett trace criterion,
James's first- and second-order tests, and Johansen's test
when (a) equal- and unequal-sized samples were selected from
multivariate normal populations; (b) p was 3 or 6; (c) G was
3; (d) the ratio of the largest to the smallest sample size
ratio was 1, 1.3, or 2; (e) the ratio of total sample size to

38
number of dependent variables was 10, 15, or 20; and (f)
dispersion matrices were either of the form I or D, where D
was d2I, diag{ 1, d2, d2 } , or diag{ 1/d2, d2, d2} for p=3 or D was d2I,
diag{ 1,1, l,d2,d2,d2} , or diag {1/d2,1/d2, 1/d2, d2, d2, d2 } for p=6 (d
= 71.5 or 3). Results of the study indicate when (a) sample
sizes are unequal and (b) dispersion matrices are unequal,
Johansen's test and James's second-order test perform better
than the Pillai-Bartlett trace criterion and James first-order
test. While both Johansen's test and James's second-order
test tended to have Type I error rates reasonably near a,
Johansen's test was slightly liberal whereas James's second-
order test was slightly conservative. Additionally, the ratio
of total sample size to number of dependent variables (N:p)
has a strong impact on the performance of the tests.
Generally, as N:p increases, the test becomes more robust.
In summary, the Pillai-Bartlett trace criterion appears
to be the most robust of the four basic MANOVA criteria to
violations of the assumption of dispersion homoscedasticity.
In controlling type I error rates, the Johansen test and James
second-order test are more effective than either the Pillai-
Bartlett trace criterion or James first-order test. Finally,
the Johansen test has the practical advantage of being less
computationally intensive than the James second-order test.

39
CHAPTER 3
METHODOLOGY
In this chapter, the development of the test statistics,
the design, and the simulation procedure are described. The
test statistics extend the work of Brown and Forsythe (1974)
and Wilcox (1989). The design is based upon a review of
relevant literature and upon the consideration that the
experimental conditions used in the simulation should be
similar to those found in educational research.
Development of Test Statistics
Brown-Forsvthe Generalizations
To test Hq: /i1 = )li2 = ... = MG Brown and Forsythe (1974)
proposed the statistic
F*
£ ni (xi. - x..)
2=1
2 = 1
—) s?
N)Sl
The statistic F* is approximately distributed as F with G-l
and f degrees of freedom, where
n.
i E 11 - i2
f =
i= 1
N
1=1
1 (1 - w)s'
ni -1
2 1 2

40
Suppose x1,...,xG are p-dimensional sample mean vectors
and S,, . ..,SG are p-dimensional dispersion matrices of
independent random samples of sizes n1f...,nG, respectively,
from G multivariate normal distributions
, • • • ,Np(^G,ZG) . To extend the Brown-Forsythe statistic
to the multivariate setting, replace means by corresponding
mean vectors and replace variances by their corresponding
dispersion matrices. Define
G
H = - x) (xi - x)1
2 = 1
and
M
= y (i - )3
^ N
2=1
The Sj (i=l,...,G) are distributed independently as Wishart
W (n ,Z;) and M is said to have a sum of Wisharts distribution,
denoted as M - SW(n1, . . . ,nQ; (1 - n.,/N) Z1,...,(1 - nQ/N) ZG) . Nel
and van der Merwe (1986) have generalized Satterthwaite's
(1946) results and approximated the sum of Wisharts
distribution by Z - Wp(f,Z). Applying the Nel and van der
Merwe results to M, the quantity f is the approximate degrees
of freedom of M and is given by
G G
trace2 [ ci^i ] + trace [ ^ ciEi ]2
£ _ 2=1 2=1
g -
Y' [trace2 [c^A + trace [c,E,] 2}
hf n7- - l
where

41
C±
The problem is to construct test statistics and determine
critical values. The approach used in this study is to
construct test statistics analogous to those developed by
Lawley-Hotelling (U), Pillai-Bartlett (V), and Wilks (L).
Define
a
H = Y ni {x1 - x) (xi - x)'
2=1
and
G
E = Y, {ni - 1)S1 â– 
2=1
Then the test statistics for the Hotelling-Lawley trace
criterion, the Pillai-Bartlett trace criterion, and the Wilks
likelihood ratio criterion are, respectively
U = trace [ HE'1 ]
V = trace [H(H + E)_1]
and
Approximate F transformations can be used with each of
these test statistics. Define the following variables:
p = number of independent variables
h = G - 1 (the degrees of freedom for the
multivariate analog to sums of squares between
groups)

42
s = min(p,h)
e = N - G (the degrees of freedom for the
multivariate analog to sums of squares
within groups)
m= . 5 (¡ p — h { - 1)
n = . 5 (e - p - 1)
For the Hotelling-Lawley criterion, the transformations
developed by Hughes and Saw (1972) and McKeon (1974)
respectively are given by
F (i) = 2(sn+l) U _ p
U s(2m+s + l) S s(2m+s+l) ,2 (sn+1)
and
p (2)
r U
2n JL u ~ F
a-2 ph ph-a
where
4 + P,h +. 2
b - 1
and
b = (2n + h) (2n + p)
2 (n - 1) (2n + 1)
For the Pillai-Bartlett criterion the SAS (1985, p.12)
transformation is given by
F = 2n+s+l V _ p
v 2/n + S + l S - V p s (2m+s+l) , s (2n+s+l) •
For the Wilks criterion, the Rao (1952, p.262)
transformation is given by
_i
„ _ 1 - L c rt - 2 q
ph
ph,rt - 2q
1
t

43
where
q = PP— 2
t =
p2h2 - 4
^ p2 + h2 - 5
if p2 + h:
5 > 0
t = 1, otherwise
and
r - e
p + h + 1
Scale of the measures of between and within group
variability. Consider for the univariate (p=l) case the
denominator of the Brown-Forsythe statistic (F*)
n
n<
(1 -
i=1
M - e - E
7=1
AT
- E
7* =1
N
= Gs2 - S2
Here s 2 is the arithmetic average of the G sample variances
and s 2 is the average of the G sample variances weighted by
their respective sample sizes. Because both are approaches to
approximating the average dispersion, M roughly represents a
mean square within group (MSWG) multiplied by the degrees of

44
freedom for the sum of squares between groups. Because the
numerator of F* is the between group sums of squares, the
Brown-Forsythe statistic is in the metric of the ratio of two
mean squares. Now the MANOVA criteria are in the metric of
the ratio of two sum of squares. Consider the common MANOVA
criteria in the univariate setting. For Hotelling-Lawley,
Pillai-Bartlett, and Wilks respectively, U = SSBG/SSWG, V =
SSBG/(SSBG+SSWG) , and L = SSWG/(SSBG+SSWG). In each case the
test statistics are functions of the sum of squares rather
than mean squares. Hence, in order to use criteria analogous
to U, V, and L, E must be replaced by (f/h)M.
Let t *, i=l,...,s be the ith eigenvalue of the
characteristic equation |h -r i* (f/h)M|=0. One statistic to
consider would be analogous to Roy's largest root criterion
(1945) t*/(1 + t i *) where t* > ... > t*. Of the four basic
MANOVA criteria, Roy's largest root criterion is the most
affected by heteroscedasticity (Olson, 1974, 1976, 1979;
Stevens, 1979) . Consequentially, r* will be omitted. The
Lawley-Hotelling trace (Hotelling, 1951; Lawley,1938) is based
upon the same characteristic equation as Roy's largest root
criterion (1945). In this case, the analogous statistic U* =
trace{H[(f/h)M]'1} = Z t* provides one of the test statistics
of interest.
Let ©j, i=l,...,s denote the ith eigenvalue of the
characteristic equation |h - Q. [H+ (f/h) M] '11 =0. Then the
statistic analogous to the Pillai-Bartlett trace

45
(Bartlett, 1939 ; Pillai, 1955) V* = trace{H[H+ (f/h)M] ‘1} = S 0¡
provides another test statistic of interest.
Similarly, if <5jf i=l,...,s is the ith eigenvalue of the
characteristic equation |(f/h)M - 6- (H+(f/h)M) |=0, then the
analogous Wilks (1932) criterion is defined L* =
| (f/h) M | / | H+ (f/h) M | = II 6,..
To conduct hypothesis testing, approximate F
transformations were used with each of these analogous test
statistics, replacing N-G, the degrees of freedom for S = (N-
G)'1E, by f, the approximate degrees of freedom for M. Thus,
the variables are defined as follows:
p = number of independent variables
h = G - 1 (the degrees of freedom for the
multivariate analog to sums of squares between
groups)
s = min(p,h)
G G
trace2 [ ciSi ] + trace [ ^ ci5i ]2
£ _ i=l i-1
G 1
{trace2 [c^J + trace 2}
¿=1 ~ ^
where
i= I,
. =G
N
Eni
7=1
m = • 5 ( ¡ p - h j -1)
n*= . 5 (f - p - 1) .

46
For the modified Hotelling-Lawley criterion, the Hughes
and Saw (1972) and McKeon (1974) transformations respectively
are now given by
„ (!) = 2 ( SJ2 * + 1) U* _ „
U* S ( 2/n+S + l) S s (2in+s+l) , 2 (sJ2*+!)
and
F .(2)
rW
2n’ a* ,
a *-2
where
a* = 4 + Ph_L_2
b* - l
and
_ (2n* + A) (2n’ + p)
2 (n* - 1) (2n* + 1)
For the modified Pillai-Bartlett criterion the SAS (1985,
p.12) transformation is now given by
= 2n * + s + i V _ „
v' 2/77 + S + l s - V' s(2m*s*X) , s (2nm+s+X) â– 
For the modified Wilks criterion, the Rao (1952, p.262)
transformation is now given by
p = 1 ~ L L r't - 2q _ „
L' ,l ph ph’r' t-2Q
L t
where
= p_h r 2
4
t
p2h2 - 4
N p2 + h2 - 5
if p2 + h2 - 5 > 0
otherwise

47
t = 1,
and
r
f - P + h + 1
2
Equality of expectation of the measures of between and
within group dispersion. The Brown-Forsythe statistic was
constructed so that, under the null hypothesis, the
expectations of the numerator and denominator are equal. To
show the proposed multivariate generalization of the Brown-
Forsythe statistic possesses the analogous property (that is,
E(H)=E(M), assuming Hq: =...= ¿tG is true) the following
results are useful:
1. E(x±) = Hi = H
2. E(x) = ii
3. E(x x') = var (x) + |i |i;
4. E[(xi - ji) (x± - jx>7] = variXj) = j Ej
5 . £[ (x - ji) (x - ji)'] = var (x)
Using results 1-5, E(M) is given by
¿ (1 - ^±)E[ 3X ]
E(M)

48
â–  E<*-
7=1
Similarly, using results 1-5, E(H) is given by
G
E(H) = E[ Yni(*i - x) {x± - x)' ]
2—1
G G
= Y, ni E(*l - |i) - |i) ' + Y ni E- Ji) (X - n) ' ,
i=l ¿=1
G G
= Yni var^i> + Y ní var (3?)
i=1 i=1
G
- E{ Y nii Xi*' - *1^' - + JA ] }
1= 1
G
- E{Y ni tx Xi - x \i' - [i x[ -t- JA M.'] } ,
i—1
= E^(
7=1
EM 4E»A 1
2=1
72 2 i=l
-n £"[ x x7 ] + n n + n |i (i7 - n |i |i'
-n E[ x x' ] + ^ |i [i7 + n p, n; - n ji \l' ,
g g _
= Y 2¿ + — E n£i ~ 2n. C var (xl + l1 W 3 + 2 n \i \n' ,
2=1 2=1
- E^<
7* =1
* E
7' = 1
n

49
Ji W ] +2 n |i |i' ,
â–  Eej
7 =1
ni^l
1=1
73
-2 E
7=1
/7
-2/7 nn7 +2/7 up/ ,
â–  EEi
7 =1
E
7=1
n
- E 7=1
Hence, E(H) = E(M).
Thus the modified Brown-Forsythe generalizations parallel
the basic MANOVA criteria in te/rms of the measure of between
group dispersion, the measure of within group dispersion, the
metric of between and within group dispersion, and the
equality of the expectation of the measures of between and
within group dispersion.
Wilcox Generalization
To test Hq: /lx1 = /z2 = ... = MG Wilcox (1989) proposed
using the test statistic
G
Hm = E (*i ’ j?) 2 '
i =1
where is approximately distributed chi-square with G-l
degrees of freedom. To extend this to the multivariate
setting, replace wi with W¡, x{ with x-, x with x , and define

50
- X) 'Wi ~ x)
2=1
where
W;
ni3lX
ni(ni + 1)
n<
1 _
x,-
hi (r¡i + 1)
and
x = [ ¿ Wi 1 -1 ¿ wí*í ■
2=1 2=1
The statistic H^* is approximately distributed as chi-square
with p(G-l) degrees of freedom.
Invariance Property of the Test Statistics
Samples in this experiment were selected from either a
contaminated population or an uncontaminated population. The
subset of populations labeled uncontaminated had the identity
matrix (I) as their common dispersion matrix. The subset of
populations categorized as contaminated had a common diagonal
matrix (D) . That these matrix forms entail no loss of
generality beyond the limited form of heteroscedasticity
investigated is due to (a) a well known theorem by Anderson
(1958) and (b) the invariance characteristic of the test
statistics.
First, denote by and 2T the pxp dispersion matrices for
populations i and j, respectively. Since Z. and Zj are

51
positive definite, there exists a pxp nonsingular matrix T
such that TZjT'= I and TZjT'= D, where I is a pxp identity
matrix and D is a pxp diagonal matrix (Anderson, 1958).
Hence, when the design includes two population subsets with
common dispersion matrices within a given subset, including
only diagonal matrices in each simulated experiment is not an
additional limitation on generalizability.
Second, the test statistics are invariant with respect to
transformations where T is a pxp nonsingular transformation.
Brown-Forsvthe Generalizations
Let Yjj = TXj j. Let , S¡* denote the sample mean vector
and sample dispersion matrix for y^. in the ith sample. It is
well known that y( = Tx; and Sj* = TSjT' . Let H* and M* be
calculated by using y( and S,.*. It is well known that H* =
THT'. Now
M*
rl E (1 - ]*' -
For the modified Hotelling-Lawley trace criterion
U
trace{ H* [ 4 M* J -1 }
h

52
= trace {T H T1 [ -=â–  rAfi*']'1 }
= trace { H[(4)M] 1 }
h
Similarly, for the modified Pillai-Bartlett trace criterion
V* = trace{ H* [H* + (-f)nr] L}
h
= trace[T H 1?[T H I* + (-£) t M T/] 'x}
= trace{H[H+ — M]-1} .
h
For the modified Wilks likelihood ratio criterion,
H* + 4 M'
h
| ^ T M T' I
T H T' + 4 T M T'
h
|r|||M||r,|
|r| I if + —m\\t'\
h
\H
Wilcox Generalization
To show the invariance of H*, the following results are
useful:

53
1. V* = ni[S*i]-1 = [T^^n^^T-1
2. ¿ vfi = (r7) T'1 = (rO [¿ wj r-1
2=1
2=1
2=1
3. y± = Tx± .
4. y = (¿Wií-'íÉ^i) ,
2=1 2=1
= [ (r') {¿ ür^r1](r/)"1r¡i^'i1r-12«i ,
2-1
2=1
= r(J2 Wj) ^(t') -'t'Y, nis¡1sti ,
2=1
2=1
= r(¿ W¿) _1¿ = ric
2=1 2=1
Using results 1-4, is shown to be invariant as follows:
G
H*m = £ (Txi - Tx)'Wi{Txi - Tx) ,
2=1
= E [r^ - x) ] '(r7) '1wiir 1 [T{xi - x) ]
2=1
= E (^i • 5) (5i ~
2=1
Therefore, the test statistics U*, V*, L* and H^* are
invariant to nonsingular transformation. Thus, there is no

54
loss of generality by solely using diagonal matrices to
simulate experiments in which there are only two sets of
dispersion matrices. It should be noted, however, when there
are more than two sets of differing dispersion matrices, the
matrices cannot always be simultaneously diagonalized by a
transformation matrix T.
Design
Eight factors were considered in the study. These are
described in the following paragraphs.
Distribution type (DT) . Two types of distributions—
normal and exponential--were included in the study. Pearson
and Please (1975) suggested that studies of robustness should
focus on distributions with skewness and kurtosis having
magnitudes less than 0.8 and 0.6, respectively. However,
there is evidence to suggest these boundaries are
unnecessarily restrictive. For example, Kendall and Stuart
(1963, p. 57) reported the age at time of marriage for over
300,000 Australians. The skewness and kurtosis were 2.0 and
8.3, respectively. Micceri (1989) investigated the
distributional characteristics of 440 achievement and
psychometric measures. Of these 440 data sets, 15.2% had both
tails with weights at or about Gaussian, 49.1% had at least
one extremely heavy tail, and 18.0% had both tail weights less
than Gaussian. The Micceri study found 28.4% of the
distributions were relatively symmetric, 40.7% were classified
as being moderately asymmetric, and 30.7% were classified as

55
being extremely asymmetric. Of the distributions considered,
11.4% were classified within a category having skewness as
extreme as 2.00. The Micceri study underscores the common
occurrence of distributions that are non-normal. Further, the
Micceri study suggests the Pearson and Please criterion may be
too restrictive.
For the normal distribution the coefficients of skewness
(/i32/M23)1/2 and kurtosis - 3) are respectively 0.00 and
0.00. For the exponential distribution the coefficients of
skewness and kurtosis are respectively 2.00 and 6.00. The
Micceri study provides evidence that the proposed normal and
exponential distributions are reasonable representations of
data that may be found in educational research.
Number of dependent variables (p). Data were generated
to simulate experiments in which there are p=3 or p=6
dependent variables. This choice is reasonably consistent
with the range of variables commonly examined in educational
research (Algina & Oshima, 1990? Algina & Tang, 1988;
Hakstian, Roed, & Lind, 1979; Lin, 1991; Olson, 1974; Tang,
1989) .
Number of populations sampled (G). Data were generated
to simulate experiments in which there is sampling from either
G=3 or G=6 populations. Dijkstra & Werter (1981) simulated
experiments with G egual to 3, 4, and 6. Olson (1974)
simulated experiments with G equal to 2, 3, 6, and 10.
Multivariate experiments with a large number of groups seem to

56
be rare in educational research (Tang, 1989). Hence, the
chosen number of populations sampled should provide reasonably
adequate examination of this factor.
Degree of the sample size ratio (NR). Only unequal
sample sizes are used in the study. Sample size ratios were
chosen to range from small to moderately large. The basic
ratios of n^njinj used in the simulation when sampling from
three different populations are given in Table 2. Similarly,
the ratios of n1: . . . : n6 used in the simulation when sampling
from six different populations are given in Table 3. Fairly
large ratios were used in Algina and Tang's (1988) study, with
an extreme ratio of 5:1. In experimental and field studies,
it is common to have sample-size ratios between 1:1 and 2:1
(Lin, 1991). Olson (1974) examined only the case of equal¬
sized samples.
Since error rates increase as the degree of the sample
size ratio increases (Algina & Oshima, 1990), if nominal error
rates are excessively exceeded using small to moderately large
sample size ratios, then the procedure presumably will have
difficulty with extreme sample size ratios. Conversely, if
the procedure performs well under this range of sample size
ratios, then it should work well for equal sample size ratios
and the question of extreme sample size ratios is still open.
Hence, sample size ratios were chosen under the constraint
n[G]:nt1] is less extreme than 2:1, where nMJ is the smallest

57
Table 2
Sample Size Ratios (n.,: n:: ru)
n
1
n,
n
3
1 1 1.3
112
1 1.3 1.3
12 2
Table 3
Sample Size Ratios fn.,; . . . : n,)
n1 :
n2 :
n3 :
n4 :
n5 :
n6
1
1
1
1
1.3
1.3
1
1
1
1
2
2
1
1
1.3
1.3
1.3
1.3
1
1
2
2
2
2

58
sample size and n[G] is the largest sample size of the G
populations sampled. In some cases these basic ratios could
not be maintained because of the restriction of the ratio of
total sample size to number of dependent variables. Departure
from these basic ratios was minimized.
Form of the sample size ratio ÍNRF) . When there are
three groups, either the sample size ratio is of the form n1
= n2 < n3 and is denoted by NRF=1 or the sample size ratio is
of the form n1 < n2 = n3 and is denoted by NRF=2. When there
are six groups, either the sample size ratio is of the form n,
= n2 = n3 = n4 < n5 = n6 and is denoted by NRF=1 or the sample
size ratio is of the form n1 = n2 < n3 = n4 = n5 = n6 and is
denoted NRF=2.
Ratio of total sample size to number of dependent
variables CN:p). The ratios chosen were N:p=10 and N:p=20.
Hakstian, Roed, and Lind (1979) simulated experiments with N:p
egual to 6 or 20. With some notable exceptions (Algina &
Tang, 1988; Lin, 1991) current studies tend to avoid N:p
smaller than 10. Yao's test (which is generally more robust
than James's first-order test) should have N:p at least 10 to
be robust (Algina & Tang, 1988). With G>2, Lin (1991)
reasoned it seems likely that N:p will need to be at least 10
for robustness to be obtained. An upper limit of 20 was
chosen to represent moderately large experiments. These
selections result in a minimum total sample size of N=30 and
a maximum total sample size of N=120.

59
Degree of heteroscedasticitv (d). Each population with
dispersion matrix equal to a pxp identity matrix (I) will be
called an uncontaminated population. Each population with a
pxp diagonal dispersion matrix (D) with at least one diagonal
element not equal to one will be called a contaminated
population. The forms of the dispersion matrices, which
depend upon the number of dependent variables, are shown in
Table 4. Two levels of d, d=y2 and d=3.0, were used to
simulate the degree of heteroscedasticity of the dispersion
matrices. Olson (1974) simulated experiments with d equal to
2.0, 3.0, and 6.0. Algina and Tang (1988) simulated
experiments with d equal to 1.5, 2.0, 2.5, and 3.0. Tang
(1989) chose d equal to J 1.5 and 3.0. Algina and Oshima
(1990) selected d equal to 1.5 and 3.0. For this study, d=y2
was used to simulate a small degree of heteroscedasticity and
d=3.0 was selected to represent a larger degree of
heteroscedasticity. These values were selected to represent
a range of heteroscedasticity more likely to be common in
educational experiments (Tang, 1989).
Relationship of sample size to dispersion matrices (S).
Both positive and negative relationships between sample size
and dispersion matrices were investigated. In the positive
relationship, the larger samples correspond to D. In the
negative relationship, the smaller samples correspond to D.
These relationships for G=3 and G=6 are summarized in Table 5
and Table 6, respectively.

60
Table 4
Forms of Dispersion Matrices
Matrix p=3
p=6
D
I
Diag {1, d2, d2 } Diag {1,1, d2, d2, d2, d2}
Diag{1,1,1} Diag{l,1,1,1,1,1}

61
Table 5
Relationship of Sample Size to Heteroscedasticitv (G=3)
Sample Size Ratios Relationship
n, : n2 : n3 Positive Negative
1
1
1.3
IID
DDI
1
1
2
IID
DDI
1
1.3
1.3
IDD
DII
1
2
2
IDD
DII
Table 6
Relationship of Sample Size to Heteroscedasticitv (G=6)
n1 :
Sample Size Ratios
n2 : n3 : n4 : n5 :
n6
Relationship
Positive Negative
1
1
1
1
1.3
1.3
IIIIDD
DDDDII
1
1
1
1
2
2
IIIIDD
DDDDII
1
1
1.3
1.3
1.3
1.3
IIDDDD
DDIIII
1
1
2
2
2
2
IIDDDD
DDIIII

62
Design Layout. The sample sizes were determined once
values of p, G, N:p, NRF, and NR were specified. These
sample sizes are summarized for G=3 and G=6 in Table 7 and
Table 8, respectively. Each of these 32 conditions were
crossed with two distributions, two levels of
heteroscedasticity, and two relationships of sample size to
dispersion matrices to generate 256 experimental conditions
from which to draw conclusions regarding the competitiveness
of the proposed statistics to the established Johansen
procedure.
Simulation Procedure
The simulation was conducted as 256 separate runs, one
for each condition, with 2000 replications per condition. For
each condition, the performance of Johansen's test (J) , the
two variations of the modified Hotelling-Lawley test (U^,
U2*) , the modified Pillai-Bartlett test (V*) , the modified
Wilks test (L*) , and the modified Wilcox test (H^*) were
evaluated using the generated data.
For the ith sample, an n(xp (i=l,...,G) matrix of
uncorrelated pseudo-random observations was generated (using
PROC IML in SAS) from the target distribution—normal or
exponential. When the target distribution was an exponential,
the random observations on each of the p variates were
standardized using the population expected value and standard
deviation. Hence, within each uncontaminated population, all

63
Table 7
Sample Sizes (G=3)
p
G
N: p
N
nl
n2
n3
3
3
10
30
9
9
12
7
7
16
8
11
11
6
12
12
20
60
18
18
24
15
15
30
16
22
22
12
24
24
6
3
10
60
18
18
24
15
15
30
16
22
22
12
24
24
20
120
36
36
48
30
30
60
32
44
44
24
48
48
Note. N is occasionally altered to maintain the ratio
closely as manageable.
as

64
Table 8
Sample Sizes (G=6)
p
G
N:p
N
n1
n2
n3
n4
n5
n6
3
6
10
30
5
5
5
5
6
6
4
4
4
4
8
8
4
4
5
5
5
5
4
4
6
6
6
6
20
60
9
9
9
9
12
12
7
7
7
7
16
16
8
8
11
11
11
11
6
6
12
12
12
12
6
6
10
60
9
9
9
9
12
12
7
7
7
7
16
16
8
8
11
11
11
11
7
7
12
12
12
12
20
120
18
18
18
18
24
24
15
15
15
15
30
30
16
16
22
22
22
22
12
12
24
24
24
24
Note. N is occasionally altered to maintain the ratio as
closely as manageable.

65
the p variates were identically distributed with mean equal to
zero, variance equal to one, and all covariances among the p
variates equal to zero.
Each n;xp matrix of observations corresponding to a
contaminated population was post multiplied by an appropriate
D to simulate dispersion heteroscedasticity.
For each replication, the data were analyzed using
Johansen's test, the two variations of the modified Hotelling-
Lawley trace criterion, the modified Pillai-Bartlett trace
criterion, the modified Wilks likelihood ratio criterion, and
the modified Wilcox test. The proportion of 2000 replications
that yielded significant results at a= 0.05 were recorded.
Summary
Two distribution types [DT=normal or exponential], two
levels of dependent variables (p=3 or 6) , two levels of
populations sampled (G=3 or 6), two levels of the form of the
sample size ratio, two levels of the degree of the sample size
ratio, two levels of ratio of total sample size to number of
dependent variables (N:p=10 or 20), two levels of degree of
heteroscedasticity (d=72 or 3.0), and two levels of the
relationship of sample size to dispersion matrices (S=positive
or negative condition) combine to give 256 experimental
conditions. The Johansen test (J), the two variations of the
modified Hotelling-Lawley test (U^, U2*) , the modified Pillai-
Bartlett test (V*) , the modified Wilks test (L*) , and the

66
modified Wilcox test (H^*) were applied to each of these
experimental conditions. Generalizations of the behavior of
these tests will be based upon the collective results of these
256 experimental conditions.

CHAPTER 4
RESULTS AND DISCUSSION
In this chapter analyses of f for a=.05 are presented.
Results with regard to f for a=.01 and for a=.10 are similar.
The analyses are based on data presented in the Appendix.
Distributions of f for the six tests are depicted in
Figures 1 to 6. In each of these six figures, the interval
labelled .05 denotes .0250 < f < .0749, the interval labelled
.10 denotes .0750 < f < .1249, and so forth. From these
figures it is clear that in terms of controlling Type I error
rates (a) the performance of the Johansen (J) and modified
Wilcox (H^*) tests are similar; (b) the performance of the
first modified Hotelling-Lawley (U^) , second modified
Hotelling-Lawley (U2*) , modified Pillai-Bartlett (V*) , and
modified Wilks (L*) tests are similar; (c) the performance of
these two sets of tests greatly differ from one another; (d)
the performance of the Johansen test is superior to that of
the Wilcox generalization; and (e) the performance of each of
the Brown-Forsythe generalizations is superior to that of
either the Johansen test or Wilcox generalization. Because
the performance of the Johansen and modified Wilcox tests were
so different from that of the Brown-Forsythe generalizations,
separate analyses were conducted for each of these two sets of
tests. For each separate set of tests, analysis of variance
67

68
100
.05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00
Figure 1
Frequency Histogram of Estimated Type I Error Rates for the
Johansen Test

69
80
.05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00
Figure 2
Frequency Histogram of Estimated Type I Error Rates for the
Modified Wilcox Test

70
.05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00
Figure 3
Frequency Histogram of Estimated Type I Error Rates for the
First Modified Hotellinq-Lawlev Test

71
200
.05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00
Figure 4
Frequency Histogram of Estimated Type I Error Rates for the
Second Modified Hotellinq-Lawley Test

72
250
.05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00
Figure 5
Frequency Histogram of Estimated Type I Error Rates for the
Modified Pillai-Bartlett Test

73
200
.05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 1.00
Figure 6
Frequency Histogram of Estimated Type I Error Rates for the
Modified Wilks Test

74
was used to investigate the effect on r of the following
factors: Distribution Type (DT) , Number of Dependent Variables
(p), Number of Populations Sampled (G), Degree of the Sample
Size Ratio (NR), Form of the Sample Size Ratio (NRF), Ratio of
Total Sample Size to Number of Dependent Variables (N:p),
Degree of Heteroscedasticity (d), Relationship of Sample Size
to Dispersion Matrices (S), and Test Criteria (T).
Brown-Forsvthe Generalizations
Because there are nine factors, initial analyses were
conducted to determine which effects to enter into the
analysis of variance model. A forward selection approach was
used, with all main effects entered first, followed by all
two-way interactions, all three-way interactions, and all
four-way interactions. Because R2 was .96 for the model with
four-way interactions, more complex models were not examined.
The R2 for all models are shown in Table 9. The model with
main effects and two-way through four-way interactions was
selected.
Variance components were computed for each main effect,
two-way, three-way, and four-way interaction. The variance
component (©j, i=l,...,255) for each effect was computed using
the formula 106(MSEF-MSE)/(2X), where MSEF was the mean square
for that given effect, MSE was the mean square error for the
four-factor interaction model, and 2X was the number of levels
for the factors not included in that given effect. Negative

75
Table 9
Magnitudes of R2 for Main Effects. Two-Way Interaction. Three-
Way Interaction, and Four-Wav Interaction Models when usina
the Four Brown-Forsvthe Generalizations
Highest-Order Terms
R2
Main Effects
0.52
Two-Way Interactions
0.77
Three-Way Interactions
0.89
Four-Way Interactions
0.96

76
variance components were set to zero. Using the sum of these
variance components plus MSExlO6 as a measure of total
variance, the proportion of total variance in estimated Type
I error rates was computed for the ith effect (i=l,...,255)
using the formula Q./ [ (6,+. . . +©255) + lO^SE] . Shown in Table
10 are effects that (a) were statistically significant and (b)
accounted for at least 1% of the total variance in estimated
Type I error rates.
Because N:p, T, G, and GxT are among the largest effects
and—in contrast to factors such as d and DT--do not have to
be inferred from data, their effects were examined by
calculating percentiles of f for each combination of G and
N:p. These percentiles should provide insight into the
functioning of the four tests. The DTxNRFxSxd interaction was
significant and the second largest effect. Conseguently the
effects of the four factors involved in this interaction were
examined by constructing cell mean plots involving all
combinations of the four factors. Other interaction effects
with large variance components that included these factors
were checked and did not change the findings significantly.
The DTxG interaction will be examined because it (a) accounts
for 4.0% of the total variance in estimated Type I error rates
and (b) is not explained in terms of either the effect of T,
N:p, and G or the effect of DT, NRF, S, and d. The factor p
has neither a large main effect or large interactions with any
other factors. However, because it accounted for 1.5% of the

77
Table 10
Variance Components for the First Modified Hotelling-Lavlev.
Second Modified Hotellinq-Lawlev. Modified Pillai-Bartlett.
and Modified Wilks Tests
Effect
0
Percent
of Variance
N: p
81
8.3
DTxNRFxSxd
76
7.8
T
66
6.8
DTxNRFxS
55
5.7
NRFxSxd
46
4.8
DTxd
41
4.2
DTxG
39
4.0
NRFxS
35
3.6
G
33
3.4
GxT
31
3.2
DTxGxd
25
2.6
DTxGxNRFxS
25
2.5
Sxd
23
2.4
d
22
2.3
S
21
2.2
NRFxSxdxT
18
1.8
P
15
1.5

78
Table 10—continued.
Effect
0
Percent
of Variance
pxNRFxSxd
13
1.3
DTxGxN:pxd
13
1.3
GxN: p
13
1.3
NRFxSxT
11
1.2
DTxNRxS
10
1.0
dxT
10
1.0
DTxS
10
1.0
All Others
<10
<1.0

79
variance, its effect was examined by inspecting cell means for
p=3 and p=6. Finally, the influence of the degree of the
sample size ratios (NR) was minimal. The NR main effect
accounted for only . 1% of the total variance in estimated Type
I error rates. The three-way interaction DTxNRxS was the
effect with the largest variance component which included NR
and it still only accounted for 1.0% of the total variance in
estimated Type I error rates.
Effect of T, N:p, and G. Percentiles for and U2’ are
displayed in Table 11; percentiles for V* and L* are shown in
Table 12. Using Bradley's liberal criterion (.5a < i < 1.5a),
the following patterns emerge regarding control of Type I
error rates for the Brown-Forsythe generalizations: (a) the
first modified Hotelling-Lawley test (U,*) was adequate when
N:p was 10; however, the test tended to be liberal when N:p
was 20; (b) the second modified Hotelling-Lawley test (U2*) was
adequate when either N:p was 10 and G was 3 or when N:p was 20
and G was 6; (c) the second modified Hotelling-Lawley test
tended to be conservative when N:p was 10 and G was 6 whereas
the test tended to be slightly liberal when N:p was 20 and G
was 3; (d) the modified Pillai-Bartlett test (V*) was adequate
when N:p was 20 and G was 3; (e) the modified Pillai-Bartlett
test tended to be conservative when N:p was 10 or when N:p was
20 and G was 6; (f) the modified Wilks test was adequate when
N:P was 10 and G was 3 or when N:p was 20; and (g) the

80
Table 11
ÍU,*) and Second
Modified
Hotelling-
-Lawlev Test
(U,’)
for
Combinations of
Ratio of
Total Samóle Size
to
Number
of
Deoendent Variables (N:o)
and Number
of Populations Sampled
161
G
G
(N
:p=10)
(N:
p=2 0)
Test Percentile
3
6
3
6
U,* 95th
.0795*
. 0770
.0855*
.0885*
90th
. 0710
. 0715
.0795*
.0835*
75th
. 0555
. 0595
. 0610
. 0708
50th
. 0505
.0500
. 0538
. 0625
25th
. 0430
. 0398
. 0493
. 0540
10th
. 0375
. 0315
. 0460
. 0490
5th
. 0345
. 0295
. 0435
. 0485
U2* 95th
. 0730
. 0510
. 0815*
. 0710
90th
. 0625
. 0460
. 0785*
. 0650
75th
. 0513
. 0388
. 0590
. 0565
50th
. 0453
. 0290
. 0510
. 0483
25th
. 0385
. 0198*
. 0470
. 0388
10th
. 0325
.0140*
. 0430
.0355
5th
. 0290
. 0135*
. 0405
. 0330
Note. Percentiles denoted by an asterisk fall outside the
interval [.5a,1.5a].

81
Table 12
Percentiles of f for the Modified
Pillai-Bartlett
Test
(V*)
and Modified Wilks Test (L*] for
Combinations of
Ratio
of
Total Samóle Size to Number of Dependent Variables
(N:p)
and
Number of Populations Sampled (G)
G
G
(N:p=10)
(N:p=20)
Test
Percentile
3
6
3
6
V*
95th
. 0555
. 0365
. 0695
. 0510
90th
. 0495
. 0310
. 0660
. 0500
75th
. 0430
. 0258
. 0533
. 0455
50th
. 0370
.0210*
. 0480
. 0380
25th
. 0318
. 0145*
. 0425
. 0315
10th
.0240*
.0110*
. 0365
. 0275
5th
. 0200*
. 0070*
. 0345
.0235*
L *
95th
. 0705
. 0465
. 0780*
. 0615
9 0 th
. 0635
. 0425
. 0745
. 0580
75th
. 0483
. 0360
. 0575
. 0533
50th
. 0440
. 0288
. 0513
. 0450
2 5 th
. 0388
.0215*
. 0455
. 0375
10th
. 0330
.0155*
. 0415
. 0345
5th
. 0310
.0130*
. 0405
. 0325
Note. Percentiles denoted by an asterisk fall outside the
interval [.5a, 1.5a].

82
modified Wilks test was conservative when N:p was 10 and G was
6.
Effect of DT. NRF. S. and d. As shown in Figure 7 and
Figure 8, when data were sampled from a normal distribution,
regardless of the form of the sample size ratio, mean f
increased as degree of heteroscedasticity increased in the
positive condition whereas mean t decreased as degree of
heteroscedasticity increased in the negative condition.
However, as shown in Figures 9 and 10, when data were sampled
from an exponential distribution, mean f increased as degree
of heteroscedasticity increased regardless of the relationship
of sample sizes and dispersion matrices. The mean difference
in i between the higher and lower degree of heteroscedasticity
was greater in the positive condition when the sample was
selected as in the first form of the sample size ratios
whereas when the sample was selected as in the second form of
the sample size ratio, the mean difference was greater in the
negative condition. With data sampled from an exponential
distribution the Brown-Forsythe generalizations tend to be
conservative when (a) there was a slight degree of
heteroscedasticity (that is, d=72), (b) the degree of
heteroscedasticity increased (d=3) and the first form of the
sample size ratio was paired with the negative condition, or
(c) the degree of heteroscedasticity increased and the second
form of the sample size ratio was paired with the positive
condition. With data sampled from an exponential

83
Mean Type I Error Rate
Sample Size to Dispersion Relationship
Figure 7
Estimated Type I Error Rates for the Two Levels of the Degree
of Heteroscedasticitv (d = ,/2 or 31 and Relationship of Sample
Size to Dispersion Matrices (S = positive or negative
condition) When Data Were Sampled as in the First Form of the
Sample Size Ratio from an Normal Distribution

84
Mean Type I Error Rate
Sample Size to Dispersion Relationship
Figure 8
Estimated Type I Error Rates for the Two Levels of the Degree
of Heteroscedasticitv (d = J2 or 3) and Relationship of Sample
Size to Dispersion Matrices (S = positive or negative
condition) When Data Were Sampled as in the Second Form of the
Sample Size Ratio from an Normal Distribution

85
Sample Size to Dispersion Relationship
Figure 9
Estimated Mean Type I Error Rates for the Two Levels of the
Degree of Heteroscedasticity fd = ,/2 or 3) and Relationship of
Sample Size to Dispersion Matrices (S = positive or negative
condition) When Data Were Sampled as in the First Form of the
Sample Size Ratio from a Exponential Distribution

86
Sample Size to Dispersion Relationship
Figure 10
Estimated Mean Type I Error Rates for the Two Levels of the
Degree of Heteroscedasticitv (d = ,/2 or 3] and Relationship of
Sample Size to Dispersion Matrices (S = positive or negative
condition) When Data are Sampled as in the Second Form of the
Sample Size Ratio from a Exponential Distribution

87
distribution, the Brown-Forsythe generalizations tended to be
liberal when (a) the first form of the sample size ratio was
paired with the positive condition, or (b) the second form of
the sample size ratio was paired with the negative condition.
Effect of DTxG interaction. As shown in Figure 11, mean
f for the Brown-Forsythe generalizations was nearer a when G
was 3 than when G was 6, regardless of the type of
distribution from which the data were sampled. When data were
sampled from a normal distribution, the tests tended to be
slightly conservative. Mean f was near a when data were
sampled from an exponential distribution and G was 3.
However, when data were sampled from an exponential
distribution and G was 6, the Brown-Forsythe generalizations
tended to be conservative.
Effect of p. Shown in Figure 12, mean f was near a for
the Brown-Forsythe generalizations when p was 6. When p was
3, the tests tended to be slightly conservative.

88
Figure 11
Estimated Mean Type I Error Rates for Combinations of the
Distribution Type and Number of Populations Sampled

89
Number of Dependent Variables
Figure 12
Estimated Mean Type I Error Rates for the Brown-Forsvthe
Generalizations for the Two Levels of the Number of Dependent
Variables

90
Johansen Test and Wilcox Generalization
Because there are nine factors, initial analyses were
conducted to determine which effects to enter into the
analysis of variance model. A forward selection approach was
used, with all main effects entered first, followed by all
two-way interactions, all three-way interactions, and all
four-way interactions. Because R2 was .997 for the model with
four-way interactions, more complex models were not examined.
The R2 for all models are shown in Table 13. The model with
main effects and two-way through four-way interactions was
selected.
Variance components were computed for each main effect,
two-way, three-way, and four-way interaction. The variance
component (©j, i=l,...,255) for each effect was computed using
the formula 104 (MSEF-MSE)/ (2*), where MSEF was the mean sguare
for that given effect, MSE was the mean square error for the
four-factor interaction model, and 2X was the number of levels
for the factors not included in that given effect. Negative
variance components were set to zero. Using the sum of these
variance components plus MSExlO4 as a measure of total
variance, the pr.oportion of total variance in estimated Type
I error rates was computed for the ith effect (i=l,...,255)
using the formula 9,-/[(©,+ .. .+0255) + 104MSE] . Shown in Table
14 are effects that (a) were statistically significant and (b)
accounted for at least 1% of the total variance in estimated
Type I error rates.

91
Table 13
Magnitudes of R2 for Main Effects. Two-Way Interaction. Three-
Way Interaction, and Four-Wav Interaction Models when usina
the Johansen Test and Wilcox Generalization
Highest-Order Terms
R2
Main Effects
0.767
Two-Way Interactions
0.963
Three-Way Interactions
0.988
Four-Way Interactions
0.997

92
Table 14
Variance Components for the Johansen and Modified Wilcox Tests
Effect
0
Percent
of Variance
G
557
36.2
GxN: p
227
14.8
N:p
174
11.4
GxT
83
5.4
T
73
4.8
GxNRFxNR
34
2.2
P
26
1.7
pxG
26
1.7
GxNR
20
1.3
DT
18
1.2
NR
18
1.1
NRFxNR
17
1.1
All Others
<15
1.0

93
Because N:p, T, G, GxN:p, and GxT are among the largest
effects and—in contrast to factors such as d and DT—do not
have to be inferred from data, their effects will be examined
by calculating percentiles of f for each combination of G and
N:p. These percentiles should provide insight into the
functioning of these two tests.
Effect of T. N:p, and G. Percentiles for J and H* are
1 * — —m
displayed in Table 15. Using Bradley's liberal criterion (.5a
< f < 1.5a), the following patterns emerge regarding control
of Type I error rates for the Johansen test and Wilcox
generalization: (a) the Johansen test (J) was adeguate only
when N:p was 20 and G was 3; and (b) the Wilcox generalization
was inadeguate over the range of experimental conditions
considered in the experiment.
Since the performance of the Johansen test and the Wilcox
generalization was so inadequate, further analysis was not
warranted for either of these two tests.
Summary
It is clear that in terms of controlling Type I error
rates under the heteroscedastic experimental conditions
considered (a) the four Brown-Forsythe generalizations are
much more effective than either the modified Wilcox test or
the Johansen test, (b) the Johansen test is more effective
than the modified Wilcox test, and (c) the modified Wilcox

94
Table 15
Percentiles of
f
for the
Johansen
Test (J)
and Wilcox
Generalization
(O
for Combinations of
Ratio of
Total Sample
Size
to Number
of
Dependent
Variables
(N:p) and Number of
Populations Sampled
(G)
G
G
(N: p=
10)
(N:p
= 2 0)
Test
Percentile
3
6
3
6
J
95th
.1700*
.7535*
. 1085*
.2950*
90th
.1260*
. 6785*
.0915*
.2520*
75th
.1030*
.5590*
. 0733
.1850*
50th
.0765*
.4548*
. 0595
.1568*
25th
. 0648
.3688*
. 0525
.1103*
10th
. 0580
.3245*
. 0480
. 0930*
5th
. 0550
.2780*
. 0460
.0865*
95th
.2260*
.9690*
.1400*
. 6795*
90th
.1985*
.9555*
.1165*
.6170*
75th
.1560*
.8113*
.0905*
.4435*
50th
.1243*
.7240*
.0798*
.3263*
25th
.0900*
.5820*
. 0630
.2285*
10th
.0760*
.5135*
. 0515
.1775*
5th
. 0735
.4575*
. 0490
.1535*
Note. Percentiles denoted by an asterisk fall outside the
interval [.5a,1.5a].

95
test is not sufficiently effective over the set
experimental conditions considered to warrant its use.
of

CHAPTER 5
CONCLUSIONS
General Observations
Two hundred and fifty-six simulated conditions were
investigated in a complete factorial experiment. The results
obtained may be applied to experiments which have experimental
conditions similar to the 256 simulated experiments conducted
in this study. The generalizability of the results is limited
by the range of values for the Distribution Type (DT), Number
of Dependent Variables (p), Number of Populations Sampled (G) ,
Ratio of Total Sample Size to Number of Dependent Variables
(N:p) , Degree of the Sample Size Ratio (NR) , Form of the
Sample Size Ratio (NRF), Degree of Heteroscedasticity (d) , and
Relationship between Sample Size and Dispersion Matrices (S).
With these limitations in mind, the following conclusions may
be set forth:
Conclusion 1. The estimated Type I error rates for the
first modified Hotelling-Lawley test (U^), second modified
Hotelling-Lawley test (U2*) , modified Pillai-Bartlett test
(V*) , and modified Wilks test (L*) were much closer to the
nominal Type I error rate over the variety of conditions
considered in the experiment than either the Johansen test (J)
or the modified Wilcox test (H *) .
96

97
Conclusion 2. The modified Wilks test (L*) is the most
effective of the Brown-Forsythe generalizations at maintaining
Type I error rates. When both (a) N:p is small and (b) G is
large, however, the test becomes conservative. Under
conditions where both (a) N:p is small and (b) G is large, the
first modified Hotelling-Lawley test is most effective at
maintaining acceptable Type I error rates.
Conclusion 3. If one can tolerate a somewhat liberal
test the first modified Hotelling-Lawley test (U^) might be
used, although the shortcoming of this procedure is the test
appears to become more liberal as N:p increases.
Conclusion 4. For all four of the Brown-Forsythe
generalizations f increases as N:p increases, suggesting the
procedures may not work well with very large sample sizes.
Conclusion 5. The Wilcox generalization and Johansen's
test are inadequate in controlling Type I error rates over the
range of experimental conditions considered.
Suggestions to Future Researchers
The generalizability of the study is limited by (a) the
limited number of distributions considered, (b) the limited
forms of dispersion matrices considered, and (c) the limited
variation in the degree of the sample size ratios. First, the
inclusion of the normal distribution and the exponential
distribution gave representation to a symmetric and an
extremely skewed distribution. Further research needs to be

98
conducted to see how these tests perforin and how factors
included in the study are affected when (a) data are sampled
from moderately skewed distributions and (b) data are sampled
from mixed distributions. It is reasonable to assume that the
performance of the tests will fall somewhere between the two
extremes, however, this certainly needs to be confirmed
empirically. Findings in this area might be additionally
strengthened by considering differing levels of kurtosis, as
well. Second, only limited forms of dispersion matrices were
considered. Because of the invariance properties of the test
statistics, the results should be highly generalizable when
data are sampled from populations with heteroscedatic
dispersion matrices limited to two forms. Further research is
needed to examine the influence of greater varieties of
dispersion heteroscedasticity. Although the Brown-Forsythe
generalizations have acceptable Type I error rates under
heteroscedasticity conditions, it remains an open question
whether acceptable error rates will result under homoscedastic
conditions, since only heteroscedastic conditions were
considered in the experiment. Since the Brown-Forsythe
generalizations become slightly conservative as the degree of
heteroscedasticity decreases, this suggests the tests will be
even more conservative under homoscedastic conditions. These
tests need to be examined empirically under homoscedastic
conditions to insure this conservative trend is within
acceptable tolerances. Third, since (a) the degree of the

99
sample size ratios (NR) is typically a factor which
significantly influences Type I error rates and (b) NR was not
a strong influence in this study, this suggests that greater
variation in this factor should be examined to confirm its
influence is indeed minimal when using the Brown-Forsythe
generalizations.
It is clear that the power of the Brown-Forsythe
generalizations needs to be investigated. It would be prudent
to compare the power of the first modified Hotelling-Lawley
test, second modified Hotelling-Lawley test, modified Wilks
test, and modified Pillai-Bartlett test with that of James's
(1954) second-order test, since James's test also tends to be
conservative.
Finally, since the Johansen test is based upon asymptotic
theory, it might be fruitful to examine the test using (a)
larger sample sizes and (b) moderately skewed distributions.
The test may prove to be useful if the boundary conditions
where robustness no longer occurs can be more clearly defined.
Using larger sample sizes may lower Type I error rates
sufficiently to warrant further examination of the power of
the test.

APPENDIX
ESTIMATED TYPE I ERROR RATES
Table 16, Table 17, and Table 18 include estimated Type
I error rates (f) for the First Modified Hotelling-Lawley Test
(U,*) , Second Modified Hotelling-Lawley Test (U2*) , Modified
Pillai-Bartlett Test (V*) , and Modified Wilks Test (L*) at
nominal Type I error rates (a) of .01, .05, and .10,
respectively, for differing levels of Distribution Type (DT),
Number of Dependent Variables (p) , Number of Populations
Sampled (G), Ratio of Total Sample Size to Number of Dependent
Variables (p), Form of the Sample Size Ratio (NRF), Degree of
the Sample Size Ratio (NR), Relationship of Sample Size to
Dispersion Matrices (S) , and Degree of Heteroscedasticity (d)
The levels of the factors are the same as those found in
Chapter 3 (see p.54) with the following modifications: (a) for
DT, "E" denotes the sampled distribution was exponential
whereas "N" denotes the sampled distribution was normal; (b)
for S, "0" denotes the positive condition and "1" denotes the
negative condition; for NR, "1" denotes the degree of the
sample size ratio was 1.3 whereas "2" denotes the degree of
the sample size ratio was 2; (c) for Degree of
Heteroscedasticity d2 was recorded rather then d; and (d)
estimated Type I error rates are of the form 10000 times f.
100

101
Hence, for example, the estimated Type I error rate was
.0075 for the first modified Hotelling-Lawley test when a=.01
when (a) data were sampled from an exponential distribution,
(b) there were 3 dependent variables, (c) data were sampled
from 3 populations, (d) the ratio of total sample size to
number of dependent variables was 10, (e) data were sampled as
in the first form of the sample size ratio, (f) the degree of
the sample size ratio was 1.3, (g) the relationship of sample
size to dispersion matrices was the positive condition, and
(h) the degree of heteroscedasticity was J2.

102
Table 16
Estimated Type I Error Rates When a=.01
DT
P
G
N: p
NRF
NR
S
d2
U1*
V
V*
L*
J
H *
m
E
3
3
10
1
1
0
2
0075
0050
0045
0060
0110
0280
E
3
3
10
1
1
0
9
0325
0285
0105
0255
0270
0490
E
3
3
10
1
1
1
2
0085
0035
0030
0050
0130
0350
E
3
3
10
1
1
1
9
0095
0060
0055
0085
0560
0945
E
3
3
10
1
2
0
2
0060
0045
0030
0025
0105
0315
E
3
3
10
1
2
0
9
0245
0180
0100
0175
0145
0440
E
3
3
10
1
2
1
2
0085
0055
0045
0060
0235
0555
E
3
3
10
1
2
1
9
0110
0075
0080
0085
0810
1650
E
3
3
10
2
1
0
2
0090
0060
0045
0065
0135
0335
E
3
3
10
2
1
0
9
0130
0090
0045
0085
0460
0760
E
3
3
10
2
1
1
2
0070
0045
0025
0035
0145
0350
E
3
3
10
2
1
1
9
0405
0335
0150
0280
0575
0890
E
3
3
10
2
2
0
2
0075
0060
0025
0050
0175
0310
E
3
3
10
2
2
0
9
0120
0085
0025
0075
0350
0590
E
3
3
10
2
2
1
2
0105
0080
0055
0095
0400
0610
E
3
3
10
2
2
1
9
0370
0265
0125
0245
0870
1215
E
3
3
20
1
1
0
2
0145
0115
0090
0115
0165
0210
E
3
3
20
1
1
0
9
0240
0215
0165
0205
0110
0185
E
3
3
20
1
1
1
2
0110
0095
0090
0110
0165
0255
E
3
3
20
1
1
1
9
0125
0115
0085
0105
0350
0505

103
Table 16--continued.
DT
P
G
N:p
NRF
NR
S
d2
0,*
V
V*
L*
J
H *
m
E
3
3
20
1
2
0
2
0080
0065
0055
0075
0115
0155
E
3
3
20
1
2
0
9
0295
0265
0155
0250
0140
0235
E
3
3
20
1
2
1
2
0095
0095
0075
0090
0225
0315
E
3
3
20
1
2
1
9
0140
0120
0105
0135
0555
0740
E
3
3
20
2
1
0
2
0090
0070
0075
0090
0110
0175
E
3
3
20
2
1
0
9
0115
0085
0065
0085
0290
0380
E
3
3
20
2
1
1
2
0125
0090
0080
0105
0200
0235
E
3
3
20
2
1
1
9
0400
0360
0200
0330
0325
0405
E
3
3
20
2
2
0
2
0065
0055
0035
0050
0090
0130
E
3
3
20
2
2
0
9
0140
0105
0085
0095
0195
0250
E
3
3
20
2
2
1
2
0130
0100
0060
0085
0330
0405
E
3
3
20
2
2
1
9
0430
0370
0220
0340
0460
0570
E
3
6
10
1
1
0
2
0060
0020
0025
0030
1140
3530
E
3
6
10
1
1
0
9
0260
0115
0050
0095
1760
4205
E
3
6
10
1
1
1
2
0145
0060
0035
0060
1745
4010
E
3
6
10
1
1
1
9
0085
0015
0020
0020
2780
5280
E
3
6
10
1
2
0
2
0090
0025
0015
0015
2815
6335
E
3
6
10
1
2
0
9
0170
0055
0015
0040
2575
6350
E
3
6
10
1
2
1
2
0090
0020
0005
0015
3235
6835
E
3
6
10
1
2
1
9
0060
0010
0005
0010
4895
7700
E
3
6
10
2
1
0
2
0060
0020
0005
0015
2360
5140

104
Table 16--continued.
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
<
E
3
6
10
2
1
0
9
0100
0020
0005
0020
3105
5825
E
3
6
10
2
1
1
2
0085
0045
0005
0035
2840
5630
E
3
6
10
2
1
1
9
0200
0055
0020
0055
3975
6460
E
3
6
10
2
2
0
2
0090
0015
0015
0025
1570
3855
E
3
6
10
2
2
0
9
0105
0035
0030
0035
2040
4430
E
3
6
10
2
2
1
2
0090
0005
0025
0025
2345
4550
E
3
6
10
2
2
1
9
0275
0100
0050
0095
3810
5870
E
3
6
20
1
1
0
2
0125
0070
0035
0060
0340
1230
E
3
6
20
1
1
0
9
0335
0180
0115
0165
0555
1655
E
3
6
20
1
1
1
2
0125
0060
0045
0070
0590
1515
E
3
6
20
1
1
1
9
0135
0085
0040
0055
1230
2400
E
3
6
20
1
2
0
2
0120
0060
0035
0060
0555
3495
E
3
6
20
1
2
0
9
0265
0170
0100
0140
0585
3795
E
3
6
20
1
2
1
2
0155
0080
0070
0075
1200
3800
E
3
6
20
1
2
1
9
0125
0050
0025
0050
1990
4770
E
3
6
20
2
1
0
2
0170
0075
0040
0065
0415
0930
E
3
6
20
2
1
0
9
0175
0095
0080
0090
0740
1470
E
3
6
20
2
1
1
2
0130
0065
0065
0060
0665
1295
E
3
6
20
2
1
1
9
0345
0165
0105
0145
1035
1630
E
3
6
20
2
2
0
2
0125
0060
0055
0065
0625
1190
E
3
6
20
2
2
0
9
0165
0085
0055
0065
0635
1345

105
Table 16—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
V
V*
L*
J
E
3
6
20
2
2
1
2
0090
0055
0035
0050
1145
1775
E
3
6
20
2
2
1
9
0380
0140
0070
0135
1770
2455
E
6
3
10
1
1
0
2
0090
0085
0060
0090
0140
0445
E
6
3
10
1
1
0
9
0355
0310
0155
0255
0290
0740
E
6
3
10
1
1
1
2
0090
0085
0050
0080
0240
0565
E
6
3
10
1
1
1
9
0085
0075
0045
0045
0560
1305
E
6
3
10
1
2
0
2
0070
0055
0050
0060
0225
0550
E
6
3
10
1
2
0
9
0305
0280
0145
0235
0195
0630
E
6
3
10
1
2
1
2
0075
0050
0060
0055
0315
0980
E
6
3
10
1
2
1
9
0130
0090
0095
0100
0815
1965
E
6
3
10
2
1
0
2
0080
0075
0070
0070
0135
0465
E
6
3
10
2
1
0
9
0125
0115
0085
0100
0395
0890
E
6
3
10
2
1
1
2
0080
0075
0055
0065
0210
0670
E
6
3
10
2
1
1
9
0320
0290
0095
0240
0475
1005
E
6
3
10
2
2
0
2
0080
0075
0070
0075
0200
0540
E
6
3
10
2
2
0
9
0135
0135
0095
0125
0270
0710
E
6
3
10
2
2
1
2
0105
0090
0105
0110
0435
1010
E
6
3
10
2
2
1
9
0235
0185
0020
0105
0885
1595
E
6
3
20
1
1
0
2
0110
0110
0100
0105
0165
0285
E
6
3
20
1
1
0
9
0255
0255
0190
0235
0145
0275
E
6
3
20
1
1
1
2
0125
0110
0075
0105
0160
0340

106
Table 16--continued.
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
H *
m
E
6
3
20
1
1
1
9
0125
0120
0105
0110
0280
0595
E
6
3
20
1
2
0
2
0085
0075
0080
0085
0110
0210
E
6
3
20
1
2
0
9
0295
0260
0215
0240
0160
0285
E
6
3
20
1
2
1
2
0085
0080
0065
0080
0230
0395
E
6
3
20
1
2
1
9
0135
0130
0120
0130
0490
0910
E
6
3
20
2
1
0
2
0125
0120
0110
0115
0105
0255
E
6
3
20
2
1
0
9
0155
0140
0100
0135
0240
0435
E
6
3
20
2
1
1
2
0105
0100
0085
0085
0165
0330
E
6
3
20
2
1
1
9
0315
0300
0180
0245
0235
0440
E
6
3
20
2
2
0
2
0105
0105
0090
0105
0155
0300
E
6
3
20
2
2
0
9
0105
0095
0070
0095
0150
0275
E
6
3
20
2
2
1
2
0110
0100
0105
0100
0255
0475
E
6
3
20
2
2
1
9
0465
0405
0190
0330
0460
0740
E
6
6
10
1
1
0
2
0160
0055
0045
0055
2380
6865
E
6
6
10
1
1
0
9
0255
0120
0015
0070
2640
7210
E
6
6
10
1
1
1
2
0105
0050
0030
0055
2995
7305
E
6
6
10
1
1
1
9
0065
0015
0005
0015
4445
8010
E
6
6
10
1
2
0
2
0100
0040
0025
0035
5255
9405
E
6
6
10
1
2
0
9
0330
0160
0025
0100
4840
9445
E
6
6
10
1
2
1
2
0100
0035
0030
0040
6650
9485
E
6
6
10
1
2
1
9
0085
0015
0010
0020
8185
9810

107
Table 16—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
E
6
6
10
2
1
0
2
0080
0020
0015
0025
2410
6075
E
6
6
10
2
1
0
9
0150
0065
0015
0040
3000
6510
E
6
6
10
2
1
1
2
0145
0030
0030
0035
3435
6860
E
6
6
10
2
1
1
9
0230
0065
0015
0030
4275
7320
E
6
6
10
2
2
0
2
0070
0035
0015
0030
3100
6460
E
6
6
10
2
2
0
9
0140
0045
0020
0045
2885
6435
E
6
6
10
2
2
1
2
0120
0035
0020
0040
4565
7520
E
6
6
10
2
2
1
9
0145
0035
0010
0015
6075
8315
E
6
6
20
1
1
0
2
0120
0085
0055
0075
0445
2545
E
6
6
20
1
1
0
9
0320
0210
0110
0180
0565
2930
E
6
6
20
1
1
1
2
0075
0055
0045
0050
0620
2495
E
6
6
20
1
1
1
9
0130
0100
0070
0095
1160
3300
E
6
6
20
1
2
0
2
0190
0110
0085
0115
0715
5320
E
6
6
20
1
2
0
9
0330
0250
0100
0195
0680
5945
E
6
6
20
1
2
1
2
0110
0060
0045
0060
0865
5250
E
6
6
20
1
2
1
9
0145
0070
0040
0070
1725
6075
E
6
6
20
2
1
0
2
0155
0115
0095
0125
0500
1685
E
6
6
20
2
1
0
9
0210
0140
0080
0120
0735
2055
E
6
6
20
2
1
1
2
0135
0070
0045
0060
0575
1925
E
6
6
20
2
1
1
9
0390
0240
0070
0175
0965
2205
E
6
6
20
2
2
0
2
0130
0100
0070
0100
0780
2110

108
Table 16—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
V
V*
L*
J
H*
m
E
6
6
20
2
2
0
9
0160
0120
0095
0120
0650
2040
E
6
6
20
2
2
1
2
0145
0075
0055
0085
1270
2715
E
6
6
20
2
2
1
9
0380
0205
0045
0110
1810
3325
N
3
3
10
1
1
0
2
0090
0065
0045
0075
0130
0300
N
3
3
10
1
1
0
9
0150
0115
0065
0090
0080
0170
N
3
3
10
1
1
1
2
0105
0070
0065
0085
0130
0240
N
3
3
10
1
1
1
9
0080
0050
0030
0040
0145
0365
N
3
3
10
1
2
0
2
0115
0085
0070
0090
0165
0370
N
3
3
10
1
2
0
9
0195
0160
0065
0150
0130
0270
N
3
3
10
1
2
1
2
0100
0055
0030
0060
0190
0550
N
3
3
10
1
2
1
9
0080
0050
0020
0040
0230
0775
N
3
3
10
2
1
0
2
0120
0080
0055
0075
0135
0260
N
3
3
10
2
1
0
9
0140
0120
0080
0115
0125
0270
N
3
3
10
2
1
1
2
0090
0075
0050
0065
0125
0310
N
3
3
10
2
1
1
9
0200
0135
0050
0120
0225
0420
N
3
3
10
2
2
0
2
0115
0100
0060
0090
0110
0240
N
3
3
10
2
2
0
9
0145
0120
0075
0100
0120
0265
N
3
3
10
2
2
1
2
0090
0050
0040
0050
0260
0525
N
3
3
10
2
2
1
9
0055
0035
0010
0025
0350
0620
N
3
3
20
1
1
0
2
0100
0090
0070
0090
0090
0115
N
3
3
20
1
1
0
9
0200
0185
0110
0155
0085
0140

109
Table 16—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
V
V*
L*
J
H *
m
N
3
3
20
1
1
1
2
0095
0085
0085
0095
0080
0125
N
3
3
20
1
1
1
9
0115
0090
0065
0095
0100
0210
N
3
3
20
1
2
0
2
0050
0050
0040
0045
0075
0100
N
3
3
20
1
2
0
9
0210
0185
0150
0180
0115
0175
N
3
3
20
1
2
1
2
0100
0080
0050
0075
0100
0190
N
3
3
20
1
2
1
9
0105
0090
0070
0095
0115
0205
N
3
3
20
2
1
0
2
0095
0085
0075
0080
0080
0120
N
3
3
20
2
1
0
9
0140
0115
0100
0125
0120
0150
N
3
3
20
2
1
1
2
0145
0105
0090
0115
0115
0190
N
3
3
20
2
1
1
9
0125
0105
0060
0095
0095
0135
N
3
3
20
2
2
0
2
0140
0120
0110
0135
0095
0145
N
3
3
20
2
2
0
9
0165
0130
0095
0125
0095
0135
N
3
3
20
2
2
1
2
0110
0095
0070
0090
0070
0160
N
3
3
20
2
2
1
9
0175
0115
0050
0085
0145
0230
N
3
6
10
1
1
0
2
0150
0030
0015
0030
1135
3300
N
3
6
10
1
1
0
9
0250
0070
0035
0095
1135
3325
N
3
6
10
1
1
1
2
0140
0050
0015
0045
1345
3470
N
3
6
10
1
1
1
9
0165
0025
0025
0040
1595
3935
N
3
6
10
1
2
0
2
0175
0060
0010
0040
2380
5805
N
3
6
10
1
2
0
9
0255
0080
0025
0065
2165
5515
N
3
6
10
1
2
1
2
0155
0025
0035
0040
2905
6195

110
Table 16—continued.
DT
P
G
N: p
NRF
NR
S
d2
ur
U2*
V*
L*
J
h;
N
3
6
10
1
2
1
9
0080
0025
0015
0015
3835
6935
N
3
6
10
2
1
0
2
0140
0050
0020
0045
1950
4605
N
3
6
10
2
1
0
9
0170
0055
0020
0065
1895
4670
N
3
6
10
2
1
1
2
0185
0075
0040
0075
2195
4910
N
3
6
10
2
1
1
9
0160
0065
0030
0060
2795
5430
N
3
6
10
2
2
0
2
0155
0055
0035
0055
1335
3385
N
3
6
10
2
2
0
9
0205
0050
0035
0070
1260
3365
N
3
6
10
2
2
1
2
0130
0050
0020
0035
1915
4105
N
3
6
10
2
2
1
9
0190
0050
0035
0065
2710
4725
N
3
6
20
1
1
0
2
0115
0060
0045
0055
0260
0820
N
3
6
20
1
1
0
9
0290
0180
0080
0165
0175
0870
N
3
6
20
1
1
1
2
0135
0085
0075
0090
0235
0835
N
3
6
20
1
1
1
9
0145
0095
0065
0095
0315
0865
N
3
6
20
1
2
0
2
0165
0085
0035
0070
0365
2825
N
3
6
20
1
2
0
9
0355
0260
0165
0235
0370
2975
N
3
6
20
1
2
1
2
0160
0080
0030
0085
0525
2965
N
3
6
20
1
2
1
9
0170
0065
0030
0070
0785
2995
N
3
6
20
2
1
0
2
0215
0125
0080
0105
0260
0630
N
3
6
20
2
1
0
9
0220
0130
0065
0105
0220
0610
N
3
6
20
2
1
1
2
0150
0090
0085
0100
0295
0745
N
3
6
20
2
1
1
9
0250
0125
0045
0095
0355
0770

Ill
Table 16—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
V
V*
L*
J
h;
N
3
6
20
2
2
0
2
0210
0120
0095
0120
0415
0930
N
3
6
20
2
2
0
9
0255
0140
0110
0140
0345
0735
N
3
6
20
2
2
1
2
0125
0045
0025
0030
0705
1130
N
3
6
20
2
2
1
9
0280
0110
0040
0095
0815
1305
N
6
3
10
1
1
0
2
0135
0135
0100
0100
0135
0435
N
6
3
10
1
1
0
9
0195
0180
0095
0120
0145
0410
N
6
3
10
1
1
1
2
0130
0120
0100
0130
0170
0530
N
6
3
10
1
1
1
9
0115
0095
0075
0085
0160
0685
N
6
3
10
1
2
0
2
0070
0050
0065
0045
0105
0485
N
6
3
10
1
2
0
9
0130
0115
0065
0100
0120
0360
N
6
3
10
1
2
1
2
0090
0070
0070
0075
0185
0695
N
6
3
10
1
2
1
9
0055
0045
0035
0040
0205
0930
N
6
3
10
2
1
0
2
0105
0080
0045
0070
0085
0360
N
6
3
10
2
1
0
9
0180
0170
0075
0130
0125
0405
N
6
3
10
2
1
1
2
0105
0095
0095
0100
0155
0510
N
6
3
10
2
1
1
9
0115
0100
0020
0060
0225
0585
N
6
3
10
2
2
0
2
0100
0075
0070
0080
0150
0500
N
6
3
10
2
2
0
9
0190
0175
0090
0130
0145
0415
N
6
3
10
2
2
1
2
0130
0125
0085
0105
0310
0880
N
6
3
10
2
2
1
9
0060
0045
0005
0020
0370
0965
N
6
3
20
1
1
0
2
0160
0150
0150
0150
0090
0200

112
Table 16—continued.
DT
P
G
N: p
NRF
NR
S
d2
u/
U2*
V*
L*
J
«;
N
6
3
20
1
1
0
9
0175
0145
0120
0135
0115
0220
N
6
3
20
1
1
1
2
0095
0090
0080
0085
0100
0245
N
6
3
20
1
1
1
9
0175
0170
0115
0160
0130
0285
N
6
3
20
1
2
0
2
0115
0110
0105
0110
0085
0210
N
6
3
20
1
2
0
9
0250
0230
0195
0225
0095
0185
N
6
3
20
1
2
1
2
0120
0110
0095
0100
0110
0280
N
6
3
20
1
2
1
9
0055
0045
0040
0055
0080
0285
N
6
3
20
2
1
0
2
0125
0125
0120
0125
0115
0250
N
6
3
20
2
1
0
9
0210
0195
0145
0190
0175
0295
N
6
3
20
2
1
1
2
0080
0080
0060
0070
0085
0190
N
6
3
20
2
1
1
9
0155
0145
0055
0100
0115
0220
N
6
3
20
2
2
0
2
0135
0125
0110
0120
0140
0285
N
6
3
20
2
2
0
9
0160
0155
0140
0145
0115
0205
N
6
3
20
2
2
1
2
0085
0065
0050
0060
0130
0295
N
6
3
20
2
2
1
9
0125
0105
0050
0080
0095
0300
N
6
6
10
1
1
0
2
0195
0090
0040
0080
1865
6015
N
6
6
10
1
1
0
9
0300
0135
0015
0080
1605
5895
N
6
6
10
1
1
1
2
0150
0055
0050
0060
2115
6375
N
6
6
10
1
1
1
9
0235
0070
0010
0060
2735
6835
N
6
6
10
1
2
0
2
0185
0090
0070
0095
4590
9115
N
6
6
10
1
2
0
9
0395
0190
0050
0140
4455
9185

113
Table 16—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
V
V*
L*
J
h;
N
6
6
10
1
2
1
2
0135
0045
0015
0030
5475
9350
N
6
6
10
1
2
1
9
0205
0030
0015
0045
7065
9600
N
6
6
10
2
1
0
2
0135
0075
0045
0070
1740
5250
N
6
6
10
2
1
0
9
0220
0105
0025
0085
1630
5105
N
6
6
10
2
1
1
2
0190
0070
0040
0085
2365
5670
N
6
6
10
2
1
1
9
0305
0095
0000
0045
3085
6370
N
6
6
10
2
2
0
2
0190
0090
0090
0105
2530
5610
N
6
6
10
2
2
0
9
0295
0165
0070
0130
1625
4905
N
6
6
10
2
2
1
2
0180
0065
0025
0050
3745
6710
N
6
6
10
2
2
1
9
0175
0035
0000
0020
4865
7510
N
6
6
20
1
1
0
2
0185
0115
0090
0125
0220
1815
N
6
6
20
1
1
0
9
0355
0290
0125
0235
0270
1860
N
6
6
20
1
1
1
2
0175
0115
0105
0110
0240
1785
N
6
6
20
1
1
1
9
0180
0130
0075
0115
0310
1820
N
6
6
20
1
2
0
2
0155
0125
0095
0125
0390
4525
N
6
6
20
1
2
0
9
0310
0245
0100
0185
0365
4615
N
6
6
20
1
2
1
2
0195
0150
0070
0120
0420
4355
N
6
6
20
1
2
1
9
0180
0140
0075
0115
0615
4335
N
6
6
20
2
1
0
2
0140
0085
0065
0085
0135
0940
N
6
6
20
2
1
0
9
0200
0135
0085
0115
0285
1090
N
6
6
20
2
1
1
2
0210
0170
0135
0170
0405
1200

114
Table 16—continued.
DT p G N: p NRF NR S d2 U,* U * V* L* J H *
L 1 ¿ m
N 6 6 20
N 6 6 20
N 6 6 20
N 6 6 20
N 6 6 20
2 119
2 2 0 2
2 2 0 9
2 2 12
2 2 19
0290 0165
0145 0130
0205 0140
0165 0130
0285 0145
0050 0115
0100 0120
0100 0130
0065 0095
0025 0090
0365 1200
0385 1335
0280 1170
0525 1685
0735 2005

115
Table 17
Estimated Type I Error Rates When g=.05
DT
P
G
N: p
NRF
NR
S
d2
V
V
V*
L*
J
V
E
3
3
10
1
1
0
2
0395
0365
0275
0355
0530
0745
E
3
3
10
1
1
0
9
0810
0705
0555
0675
0830
1000
E
3
3
10
1
1
1
2
0375
0325
0285
0320
0715
0870
E
3
3
10
1
1
1
9
0450
0405
0355
0430
1470
1835
E
3
3
10
1
2
0
2
0345
0290
0240
0310
0635
0775
E
3
3
10
1
2
0
9
0795
0730
0550
0710
0770
0995
E
3
3
10
1
2
1
2
0350
0325
0275
0315
0980
1310
E
3
3
10
1
2
1
9
0465
0390
0310
0405
2045
2545
E
3
3
10
2
1
0
2
0390
0355
0295
0360
0640
0795
E
3
3
10
2
1
0
9
0460
0385
0330
0420
1205
1455
E
3
3
10
2
1
1
2
0430
0385
0330
0390
0675
0805
E
3
3
10
2
1
1
9
0930
0815
0590
0805
1245
1435
E
3
3
10
2
2
0
2
0375
0315
0255
0340
0665
0860
E
3
3
10
2
2
0
9
0530
0480
0345
0475
1010
1255
E
3
3
10
2
2
1
2
0500
0420
0350
0425
1165
1380
E
3
3
10
2
2
1
9
0710
0625
0450
0640
1700
1935
E
3
3
20
1
1
0
2
0565
0530
0450
0520
0560
0575
E
3
3
20
1
1
0
9
0840
0800
0660
0745
0645
0710
E
3
3
20
1
1
1
2
0465
0440
0430
0455
0680
0715
E
3
3
20
1
1
1
9
0515
0485
0415
0440
1060
1125

116
Table 17--continued.
DT
P
G
N: p
NRF
NR
S
d2
U1*
U2*
V*
L*
J
h;
E
3
3
20
1
2
0
2
0400
0385
0355
0360
0505
0580
E
3
3
20
1
2
0
9
0785
0745
0690
0735
0695
0745
E
3
3
20
1
2
1
2
0440
0425
0380
0415
0880
0880
E
3
3
20
1
2
1
9
0520
0470
0465
0485
1250
1475
E
3
3
20
2
1
0
2
0475
0445
0415
0440
0575
0605
E
3
3
20
2
1
0
9
0505
0470
0460
0490
0845
0895
E
3
3
20
2
1
1
2
0495
0465
0415
0450
0620
0640
E
3
3
20
2
1
1
9
1000
0950
0795
0915
0915
0955
E
3
3
20
2
2
0
2
0400
0390
0335
0405
0615
0635
E
3
3
20
2
2
0
9
0515
0485
0445
0475
0740
0810
E
3
3
20
2
2
1
2
0590
0535
0495
0560
0890
0915
E
3
3
20
2
2
1
9
0950
0890
0665
0815
1085
1165
E
3
6
10
1
1
0
2
0290
0130
0130
0150
2780
4735
E
3
6
10
1
1
0
9
0580
0345
0240
0350
3400
5400
E
3
6
10
1
1
1
2
0360
0200
0205
0215
3420
5195
E
3
6
10
1
1
1
9
0295
0175
0120
0155
4645
6335
E
3
6
10
1
2
0
2
0310
0140
0165
0215
4555
7335
E
3
6
10
1
2
0
9
0600
0305
0175
0330
4445
7250
E
3
6
10
1
2
1
2
0310
0140
0135
0180
5310
7790
E
3
6
10
1
2
1
9
0185
0075
0095
0085
6690
8510
E
3
6
10
2
1
0
2
0330
0135
0115
0155
4175
6235

117
Table 17—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
«2*
V*
L*
J
h;
E
3
6
10
2
1
0
9
0315
0155
0115
0180
4960
6665
E
3
6
10
2
1
1
2
0295
0140
0140
0180
4795
6695
E
3
6
10
2
1
1
9
0465
0260
0155
0245
5635
7300
E
3
6
10
2
2
0
2
0340
0200
0145
0200
3345
5135
E
3
6
10
2
2
0
9
0375
0195
0150
0220
3740
5545
E
3
6
10
2
2
1
2
0325
0165
0145
0180
4085
5780
E
3
6
10
2
2
1
9
0565
0365
0215
0360
5450
6825
E
3
6
20
1
1
0
2
0505
0355
0300
0355
1135
2420
E
3
6
20
1
1
0
9
0815
0585
0430
0540
1515
2890
E
3
6
20
1
1
1
2
0520
0350
0315
0330
1670
2645
E
3
6
20
1
1
1
9
0545
0360
0280
0370
2610
3530
E
3
6
20
1
2
0
2
0465
0310
0275
0315
1650
4965
E
3
6
20
1
2
0
9
0785
0585
0425
0520
1625
5195
E
3
6
20
1
2
1
2
0540
0360
0295
0360
2390
5290
E
3
6
20
1
2
1
9
0480
0265
0210
0280
3680
6140
E
3
6
20
2
1
0
2
0515
0365
0335
0375
1355
1890
E
3
6
20
2
1
0
9
0535
0400
0340
0395
1795
2425
E
3
6
20
2
1
1
2
0540
0370
0330
0375
1710
2240
E
3
6
20
2
1
1
9
0830
0530
0405
0525
2110
2740
E
3
6
20
2
2
0
2
0500
0345
0300
0345
1630
2315
E
3
6
20
2
2
0
9
0600
0450
0335
0440
1815
2545

118
Table 17--continued.
DT
P
G
N: p
NRF
NR
S
d2
V
V
V*
L*
J
h;
E
3
6
20
2
2
1
2
0495
0315
0280
0315
2380
2960
E
3
6
20
2
2
1
9
0795
0550
0315
0475
2950
3470
E
6
3
10
1
1
0
2
0445
0430
0405
0430
0700
1215
E
6
3
10
1
1
0
9
0925
0890
0670
0815
1050
1720
E
6
3
10
1
1
1
2
0375
0375
0355
0375
0760
1375
E
6
3
10
1
1
1
9
0440
0400
0370
0405
1485
2260
E
6
3
10
1
2
0
2
0455
0430
0400
0430
0720
1335
E
6
3
10
1
2
0
9
0790
0775
0595
0705
0890
1420
E
6
3
10
1
2
1
2
0405
0385
0340
0400
1235
2005
E
6
3
10
1
2
1
9
0530
0465
0430
0460
1970
3130
E
6
3
10
2
1
0
2
0410
0365
0385
0405
0680
1190
E
6
3
10
2
1
0
9
0555
0520
0445
0530
1200
1930
E
6
3
10
2
1
1
2
0390
0370
0335
0345
0865
1405
E
6
3
10
2
1
1
9
0780
0715
0455
0635
1225
1955
E
6
3
10
2
2
0
2
0345
0335
0305
0330
0780
1280
E
6
3
10
2
2
0
9
0555
0525
0370
0475
1010
1590
E
6
3
10
2
2
1
2
0480
0440
0365
0425
1260
2030
E
6
3
10
2
2
1
9
0595
0530
0235
0440
1860
2720
E
6
3
20
1
1
0
2
0550
0535
0525
0535
0640
0865
E
6
3
20
1
1
0
9
0780
0770
0620
0715
0575
0840
E
6
3
20
1
1
1
2
0510
0485
0460
0470
0695
0930

119
Table 17—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
E
6
3
20
1
1
1
9
0525
0510
0475
0500
1025
1400
E
6
3
20
1
2
0
2
0490
0470
0460
0470
0610
0890
E
6
3
20
1
2
0
9
0805
0785
0705
0770
0615
0885
E
6
3
20
1
2
1
2
0470
0445
0440
0460
0815
1225
E
6
3
20
1
2
1
9
0570
0525
0480
0515
1280
1830
E
6
3
20
2
1
0
2
0500
0485
0495
0500
0570
0805
E
6
3
20
2
1
0
9
0610
0585
0560
0600
0885
1085
E
6
3
20
2
1
1
2
0570
0565
0535
0555
0760
1030
E
6
3
20
2
1
1
9
0855
0815
0630
0750
0805
1165
E
6
3
20
2
2
0
2
0525
0520
0500
0520
0665
0895
E
6
3
20
2
2
0
9
0560
0545
0495
0530
0725
0935
E
6
3
20
2
2
1
2
0570
0545
0505
0545
0845
1125
E
6
3
20
2
2
1
9
1045
1005
0725
0905
1220
1585
E
6
6
10
1
1
0
2
0545
0360
0285
0340
4255
7975
E
6
6
10
1
1
0
9
0745
0445
0180
0335
4665
8250
E
6
6
10
1
1
1
2
0420
0250
0195
0255
5070
8340
E
6
6
10
1
1
1
9
0355
0160
0115
0160
6405
8875
E
6
6
10
1
2
0
2
0465
0265
0210
0285
7075
9675
E
6
6
10
1
2
0
9
0885
0600
0270
0475
6785
9690
E
6
6
10
1
2
1
2
0390
0180
0190
0205
8115
9755
E
6
6
10
1
2
1
9
0315
0100
0090
0120
9105
9900

120
Table 17—continued.
DT
P
G
N: p
NRF
NR
S
d2
U2*
V*
L*
J
E
6
6
10
2
1
0
2
0485
0300
0230
0280
4430
7290
E
6
6
10
2
1
0
9
0560
0400
0210
0355
4880
7620
E
6
6
10
2
1
1
2
0470
0300
0230
0290
5480
7975
E
6
6
10
2
1
1
9
0570
0305
0060
0230
6050
8265
E
6
6
10
2
2
0
2
0475
0285
0195
0270
5005
7590
E
6
6
10
2
2
0
9
0495
0305
0230
0290
5090
7600
E
6
6
10
2
2
1
2
0405
0240
0190
0270
6480
8355
E
6
6
10
2
2
1
9
0430
0160
0045
0130
7535
8945
E
6
6
20
1
1
0
2
0485
0375
0340
0375
1420
4235
E
6
6
20
1
1
0
9
0860
0700
0475
0610
1685
4570
E
6
6
20
1
1
1
2
0465
0330
0315
0360
1625
4215
E
6
6
20
1
1
1
9
0625
0460
0385
0435
2520
5060
E
6
6
20
1
2
0
2
0615
0500
0455
0505
1760
7035
E
6
6
20
1
2
0
9
0925
0795
0580
0685
1930
7225
E
6
6
20
1
2
1
2
0485
0360
0335
0370
2210
6795
E
6
6
20
1
2
1
9
0625
0455
0375
0425
3370
7465
E
6
6
20
2
1
0
2
0595
0540
0485
0545
1565
3050
E
6
6
20
2
1
0
9
0700
0575
0500
0560
1965
3480
E
6
6
20
2
1
1
2
0485
0385
0305
0360
1745
3200
E
6
6
20
2
1
1
9
0885
0675
0390
0565
2065
3645
E
6
6
20
2
2
0
2
0560
0475
0380
0450
2025
3675

121
Table 17—continued.
DT
P
G
N: p
NRF
NR
S
d2
U2*
V*
L*
J
E
6
6
20
2
2
0
9
0600
0490
0415
0480
1885
3370
E
6
6
20
2
2
1
2
0545
0430
0380
0420
2585
4330
E
6
6
20
2
2
1
9
0835
0575
0235
0465
3135
4855
N
3
3
10
1
1
0
2
0520
0505
0410
0505
0620
0730
N
3
3
10
1
1
0
9
0555
0470
0360
0470
0435
0560
N
3
3
10
1
1
1
2
0455
0425
0370
0430
0580
0770
N
3
3
10
1
1
1
9
0420
0345
0300
0370
0660
0920
N
3
3
10
1
2
0
2
0525
0475
0370
0485
0680
0880
N
3
3
10
1
2
0
9
0570
0540
0430
0515
0580
0760
N
3
3
10
1
2
1
2
0540
0450
0335
0440
0815
1195
N
3
3
10
1
2
1
9
0285
0260
0200
0250
0935
1450
N
3
3
10
2
1
0
2
0550
0465
0420
0465
0650
0760
N
3
3
10
2
1
0
9
0505
0460
0375
0445
0555
0735
N
3
3
10
2
1
1
2
0465
0415
0320
0390
0550
0740
N
3
3
10
2
1
1
9
0515
0430
0215
0435
0740
0930
N
3
3
10
2
2
0
2
0550
0500
0380
0475
0610
0775
N
3
3
10
2
2
0
9
0505
0465
0380
0450
0535
0710
N
3
3
10
2
2
1
2
0445
0385
0330
0385
0890
1075
N
3
3
10
2
2
1
9
0290
0225
0100
0200
0975
1250
N
3
3
20
1
1
0
2
0495
0460
0445
0450
0485
0515
N
3
3
20
1
1
0
9
0610
0560
0520
0580
0515
0555

122
Table 17--continued.
DT
P
G
N: p
NRF
NR
S
d2
ur
V
V*
L*
J
«;
N
3
3
20
1
1
1
2
0425
0405
0400
0425
0460
0510
N
3
3
20
1
1
1
9
0545
0510
0455
0480
0575
0700
N
3
3
20
1
2
0
2
0435
0415
0385
0410
0450
0460
N
3
3
20
1
2
0
9
0655
0620
0565
0610
0490
0490
N
3
3
20
1
2
1
2
0460
0430
0385
0420
0495
0600
N
3
3
20
1
2
1
9
0435
0400
0345
0400
0440
0595
N
3
3
20
2
1
0
2
0530
0515
0480
0505
0465
0480
N
3
3
20
2
1
0
9
0520
0505
0500
0515
0460
0505
N
3
3
20
2
1
1
2
0610
0585
0530
0580
0605
0675
N
3
3
20
2
1
1
9
0515
0490
0360
0470
0435
0485
N
3
3
20
2
2
0
2
0545
0500
0490
0525
0525
0560
N
3
3
20
2
2
0
9
0545
0510
0490
0535
0485
0520
N
3
3
20
2
2
1
2
0485
0450
0415
0455
0535
0625
N
3
3
20
2
2
1
9
0555
0485
0320
0440
0605
0700
N
3
6
10
1
1
0
2
0575
0315
0235
0315
2505
4470
N
3
6
10
1
1
0
9
0595
0355
0260
0380
2635
4515
N
3
6
10
1
1
1
2
0590
0320
0255
0355
2825
4590
N
3
6
10
1
1
1
9
0625
0295
0205
0315
3245
5150
N
3
6
10
1
2
0
2
0485
0265
0245
0290
4095
6860
N
3
6
10
1
2
0
9
0770
0450
0225
0385
3755
6760
N
3
6
10
1
2
1
2
0570
0280
0215
0330
4660
7200

123
Table 17—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
«;
N
3
6
10
1
2
1
9
0360
0140
0180
0200
5650
7795
N
3
6
10
2
1
0
2
0485
0260
0210
0270
3745
5670
N
3
6
10
2
1
0
9
0565
0315
0275
0350
3635
5730
N
3
6
10
2
1
1
2
0475
0280
0215
0260
4020
5810
N
3
6
10
2
1
1
9
0445
0210
0110
0215
4540
6430
N
3
6
10
2
2
0
2
0595
0380
0280
0360
2955
4540
N
3
6
10
2
2
0
9
0660
0430
0305
0410
2735
4575
N
3
6
10
2
2
1
2
0530
0285
0215
0300
3615
5270
N
3
6
10
2
2
1
9
0485
0255
0150
0245
4215
5830
N
3
6
20
1
1
0
2
0620
0475
0395
0480
0865
1775
N
3
6
20
1
1
0
9
0805
0650
0405
0580
0830
1835
N
3
6
20
1
1
1
2
0545
0390
0345
0395
0915
1725
N
3
6
20
1
1
1
9
0570
0360
0310
0375
0930
1865
N
3
6
20
1
2
0
2
0625
0455
0380
0435
1180
4285
N
3
6
20
1
2
0
9
0890
0710
0545
0645
1105
4390
N
3
6
20
1
2
1
2
0655
0450
0360
0445
1360
4270
N
3
6
20
1
2
1
9
0655
0435
0350
0405
1725
4480
N
3
6
20
2
1
0
2
0625
0510
0430
0495
0925
1350
N
3
6
20
2
1
0
9
0700
0510
0430
0505
0835
1350
N
3
6
20
2
1
1
2
0715
0495
0435
0495
1045
1540
N
3
6
20
2
1
1
9
0690
0455
0295
0450
1100
1535

124
Table 17—continued.
DT
P
G
N: p
NRF
NR
S
d2
ur
u2*
V*
L*
J
N
3
6
20
2
2
0
2
0685
0490
0470
0505
1345
1775
N
3
6
20
2
2
0
9
0690
0550
0475
0530
1080
1505
N
3
6
20
2
2
1
2
0490
0375
0260
0345
1570
2055
N
3
6
20
2
2
1
9
0610
0390
0225
0325
1700
2285
N
6
3
10
1
1
0
2
0555
0510
0495
0520
0645
1140
N
6
3
10
1
1
0
9
0565
0530
0370
0455
0630
1025
N
6
3
10
1
1
1
2
0560
0515
0505
0520
0725
1265
N
6
3
10
1
1
1
9
0530
0490
0435
0480
0780
1605
N
6
3
10
1
2
0
2
0490
0450
0455
0465
0680
1195
N
6
3
10
1
2
0
9
0540
0530
0400
0490
0590
1165
N
6
3
10
1
2
1
2
0460
0435
0380
0435
0830
1530
N
6
3
10
1
2
1
9
0430
0350
0315
0345
0890
1985
N
6
3
10
2
1
0
2
0495
0470
0430
0445
0595
1020
N
6
3
10
2
1
0
9
0545
0530
0415
0475
0600
1045
N
6
3
10
2
1
1
2
0510
0455
0405
0455
0710
1265
N
6
3
10
2
1
1
9
0570
0470
0190
0335
0800
1445
N
6
3
10
2
2
0
2
0505
0495
0460
0470
0750
1235
N
6
3
10
2
2
0
9
0570
0550
0445
0515
0675
1100
N
6
3
10
2
2
1
2
0550
0505
0430
0500
1115
1740
N
6
3
10
2
2
1
9
0245
0205
0075
0140
1065
1805
N
6
3
20
1
1
0
2
0515
0515
0480
0510
0485
0710

125
Table 17—continued.
DT
P
G
N:p
NRF
NR
S
d2
U1*
U2*
V*
L*
J
N
6
3
20
1
1
0
9
0640
0620
0505
0575
0525
0760
N
6
3
20
1
1
1
2
0545
0515
0485
0545
0550
0780
N
6
3
20
1
1
1
9
0465
0460
0435
0460
0585
0845
N
6
3
20
1
2
0
2
0575
0565
0530
0555
0590
0835
N
6
3
20
1
2
0
9
0795
0790
0695
0780
0620
0820
N
6
3
20
1
2
1
2
0520
0505
0490
0505
0535
0790
N
6
3
20
1
2
1
9
0460
0450
0365
0405
0545
0830
N
6
3
20
2
1
0
2
0600
0595
0575
0575
0575
0730
N
6
3
20
2
1
0
9
0650
0645
0575
0625
0540
0740
N
6
3
20
2
1
1
2
0515
0485
0420
0455
0480
0700
N
6
3
20
2
1
1
9
0650
0615
0450
0545
0520
0805
N
6
3
20
2
2
0
2
0630
0605
0625
0625
0610
0795
N
6
3
20
2
2
0
9
0605
0600
0535
0565
0590
0800
N
6
3
20
2
2
1
2
0490
0480
0460
0470
0595
0825
N
6
3
20
2
2
1
9
0490
0480
0260
0385
0595
0950
N
6
6
10
1
1
0
2
0565
0420
0310
0400
3625
7260
N
6
6
10
1
1
0
9
0735
0510
0215
0410
3440
7230
N
6
6
10
1
1
1
2
0505
0345
0270
0355
4005
7450
N
6
6
10
1
1
1
9
0675
0420
0245
0380
4725
7895
N
6
6
10
1
2
0
2
0680
0460
0355
0465
6275
9555
N
6
6
10
1
2
0
9
1005
0770
0370
0595
6110
9520

126
Table 17—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
<
N
6
6
10
1
2
1
2
0520
0265
0210
0280
7230
9650
N
6
6
10
1
2
1
9
0485
0235
0135
0240
8140
9795
N
6
6
10
2
1
0
2
0580
0395
0290
0370
3805
6485
N
6
6
10
2
1
0
9
0680
0470
0330
0425
3490
6450
N
6
6
10
2
1
1
2
0585
0400
0365
0415
4165
6945
N
6
6
10
2
1
1
9
0675
0370
0070
0235
4920
7560
N
6
6
10
2
2
0
2
0715
0495
0385
0460
4430
6870
N
6
6
10
2
2
0
9
0785
0605
0410
0555
3375
6260
N
6
6
10
2
2
1
2
0640
0430
0300
0435
5545
7775
N
6
6
10
2
2
1
9
0405
0195
0020
0110
6455
8330
N
6
6
20
1
1
0
2
0665
0590
0485
0560
0930
3255
N
6
6
20
1
1
0
9
0910
0770
0495
0665
0985
3470
N
6
6
20
1
1
1
2
0660
0565
0500
0555
0985
3270
N
6
6
20
1
1
1
9
0615
0505
0435
0490
1165
3315
N
6
6
20
1
2
0
2
0650
0565
0470
0535
1220
6320
N
6
6
20
1
2
0
9
0850
0710
0475
0615
1150
6285
N
6
6
20
1
2
1
2
0570
0490
0455
0510
1340
6170
N
6
6
20
1
2
1
9
0580
0415
0355
0415
1590
5860
N
6
6
20
2
1
0
2
0520
0405
0350
0385
0825
2110
N
6
6
20
2
1
0
9
0730
0600
0510
0595
1035
2190
N
6
6
20
2
1
1
2
0640
0550
0505
0580
1105
2320

127
Table 17—continued.
DT p G N: p NRF NR S d2 U,* U2* V* L*
J H *
m
N 66 20 2119 0770 0570 0255 0425
1065 2420
N 66 20 2202 0530 0465 0375 0445
1225 2530
N 66 20 2209 0720 0625 0515 0580
1050 2285
N 66 20 2212 0655 0545 0455 0525
1490 3100
N 66 20 2219 0660 0530 0170 0340
1745 3350

128
Table 18
Estimated Type I Error Rates When a=.10
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
Hm*
E
3
3
10
1
1
0
2
0790
0740
0655
0705
1155
1230
E
3
3
10
1
1
0
9
1330
1255
1105
1255
1545
1610
E
3
3
10
1
1
1
2
0745
0705
0650
0750
1250
1285
E
3
3
10
1
1
1
9
0850
0825
0760
0830
2275
2430
E
3
3
10
1
2
0
2
0755
0700
0600
0685
1260
1325
E
3
3
10
1
2
0
9
1205
1145
1060
1140
1395
1515
E
3
3
10
1
2
1
2
0755
0695
0680
0740
1820
1935
E
3
3
10
1
2
1
9
0775
0745
0745
0815
2870
3180
E
3
3
10
2
1
0
2
0800
0750
0715
0770
1165
1225
E
3
3
10
2
1
0
9
0910
0870
0715
0795
1930
2060
E
3
3
10
2
1
1
2
0815
0775
0770
0830
1375
1355
E
3
3
10
2
1
1
9
1310
1265
1015
1225
1855
1960
E
3
3
10
2
2
0
2
0785
0745
0685
0735
1250
1345
E
3
3
10
2
2
0
9
0990
0945
0830
0925
1640
1800
E
3
3
10
2
2
1
2
0950
0880
0740
0895
1945
1995
E
3
3
10
2
2
1
9
1115
1075
0810
1035
2450
2560
E
3
3
20
1
1
0
2
1060
1045
1060
1055
1180
1110
E
3
3
20
1
1
0
9
1375
1350
1250
1315
1175
1160
E
3
3
20
1
1
1
2
0890
0850
0860
0870
1250
1190
E
3
3
20
1
1
1
9
1060
1030
0935
1005
1610
1665

129
Table 18—continued.
DT
P
G
N: p
NRF
NR
S
d2
U1*
V
V*
L*
J
<
E
3
3
20
1
2
0
2
0835
0795
0795
0815
1045
1045
E
3
3
20
1
2
0
9
1280
1260
1160
1215
1290
1205
E
3
3
20
1
2
1
2
0925
0895
0890
0915
1410
1380
E
3
3
20
1
2
1
9
1015
0940
0920
0970
2015
2080
E
3
3
20
2
1
0
2
0955
0935
0905
0940
1090
1060
E
3
3
20
2
1
0
9
1030
0990
0945
0965
1455
1425
E
3
3
20
2
1
1
2
0935
0910
0905
0925
1195
1145
E
3
3
20
2
1
1
9
1505
1460
1340
1440
1510
1465
E
3
3
20
2
2
0
2
0890
0870
0860
0875
1140
1120
E
3
3
20
2
2
0
9
0955
0935
0895
0950
1355
1335
E
3
3
20
2
2
1
2
1065
1040
1015
1040
1410
1420
E
3
3
20
2
2
1
9
1420
1365
1215
1345
1745
1750
E
3
6
10
1
1
0
2
0620
0355
0410
0415
3875
5460
E
3
6
10
1
1
0
9
1000
0700
0515
0655
4395
6010
E
3
6
10
1
1
1
2
0675
0445
0475
0480
4470
5960
E
3
6
10
1
1
1
9
0600
0340
0315
0385
5640
6885
E
3
6
10
1
2
0
2
0650
0440
0430
0460
5685
7920
E
3
6
10
1
2
0
9
1010
0705
0530
0735
5490
7830
E
3
6
10
1
2
1
2
0595
0345
0360
0415
6380
8210
E
3
6
10
1
2
1
9
0420
0220
0295
0300
7610
8815
E
3
6
10
2
1
0
2
0545
0355
0340
0370
5285
6835

130
Table 18--continued.
DT
P
G
N: p
NRF
NR
S
d2
ur
U2*
V*
L*
J
<
E
3
6
10
2
1
0
9
0620
0375
0340
0380
5960
7215
E
3
6
10
2
1
1
2
0535
0365
0325
0380
5855
7315
E
3
6
10
2
1
1
9
0705
0530
0310
0490
6585
7725
E
3
6
10
2
2
0
2
0590
0450
0410
0470
4440
5905
E
3
6
10
2
2
0
9
0690
0420
0415
0470
4860
6225
E
3
6
10
2
2
1
2
0565
0395
0370
0455
5165
6320
E
3
6
10
2
2
1
9
0770
0630
0515
0590
6365
7300
E
3
6
20
1
1
0
2
0910
0730
0650
0745
1955
3115
E
3
6
20
1
1
0
9
1330
1130
0890
1070
2400
3650
E
3
6
20
1
1
1
2
1025
0775
0735
0800
2455
3345
E
3
6
20
1
1
1
9
0950
0755
0690
0775
3495
4265
E
3
6
20
1
2
0
2
0915
0740
0655
0735
2615
5975
E
3
6
20
1
2
0
9
1325
1100
0905
1060
2500
6005
E
3
6
20
1
2
1
2
0905
0710
0650
0700
3420
6100
E
3
6
20
1
2
1
9
0775
0585
0520
0620
4675
6810
E
3
6
20
2
1
0
2
0970
0760
0695
0785
2150
2555
E
3
6
20
2
1
0
9
0960
0785
0705
0770
2670
3155
E
3
6
20
2
1
1
2
1010
0820
0715
0805
2615
3000
E
3
6
20
2
1
1
9
1230
0955
0745
0880
3065
3415
E
3
6
20
2
2
0
2
1015
0820
0705
0790
2685
3050
E
3
6
20
2
2
0
9
1040
0865
0730
0845
2750
3255

131
Table 18—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
<
E
3
6
20
2
2
1
2
0960
0715
0690
0780
3380
3735
E
3
6
20
2
2
1
9
1135
0895
0680
0860
3840
4180
E
6
3
10
1
1
0
2
0970
0910
0850
0915
1330
1920
E
6
3
10
1
1
0
9
1445
1400
1095
1275
1775
2280
E
6
3
10
1
1
1
2
0865
0835
0785
0820
1425
2175
E
6
3
10
1
1
1
9
0880
0830
0755
0805
2220
3090
E
6
3
10
1
2
0
2
1005
0965
0915
0940
1370
2035
E
6
3
10
1
2
0
9
1290
1270
1100
1240
1510
2010
E
6
3
10
1
2
1
2
0905
0855
0870
0910
1990
2830
E
6
3
10
1
2
1
9
0925
0860
0810
0880
2890
4025
E
6
3
10
2
1
0
2
0890
0860
0780
0850
1295
1920
E
6
3
10
2
1
0
9
1070
1050
0885
1000
2015
2555
E
6
3
10
2
1
1
2
0760
0715
0745
0800
1455
2085
E
6
3
10
2
1
1
9
1270
1200
0890
1075
1955
2635
E
6
3
10
2
2
0
2
0750
0725
0645
0715
1410
2050
E
6
3
10
2
2
0
9
1035
1000
0895
0995
1690
2375
E
6
3
10
2
2
1
2
0850
0820
0870
0850
2060
2770
E
6
3
10
2
2
1
9
1050
0965
0490
0840
2650
3520
E
6
3
20
1
1
0
2
1080
1045
1010
1045
1240
1460
E
6
3
20
1
1
0
9
1295
1280
1155
1240
1220
1480
E
6
3
20
1
1
1
2
0925
0910
0920
0915
1190
1495

132
Table 18—continued.
DT
P
G
N: p
NRF
NR
S
d2
ur
V
V*
L*
J
h;
E
6
3
20
1
1
1
9
1005
0950
0900
0945
1690
2050
E
6
3
20
1
2
0
2
1000
0985
0955
0975
1200
1530
E
6
3
20
1
2
0
9
1370
1340
1260
1300
1185
1440
E
6
3
20
1
2
1
2
1025
1005
0940
1000
1510
1910
E
6
3
20
1
2
1
9
1130
1095
1030
1095
2050
2585
E
6
3
20
2
1
0
2
0970
0960
0930
0960
1085
1330
E
6
3
20
2
1
0
9
1080
1065
0995
1050
1440
1685
E
6
3
20
2
1
1
2
1190
1170
1135
1185
1395
1545
E
6
3
20
2
1
1
9
1380
1350
1135
1220
1420
1735
E
6
3
20
2
2
0
2
1020
1000
1000
1005
1325
1600
E
6
3
20
2
2
0
9
1010
1000
0955
0975
1330
1625
E
6
3
20
2
2
1
2
1100
1075
1055
1090
1460
1860
E
6
3
20
2
2
1
9
1570
1515
1250
1450
1860
2320
E
6
6
10
1
1
0
2
0955
0740
0625
0760
5525
8460
E
6
6
10
1
1
0
9
1195
0880
0445
0685
5980
8700
E
6
6
10
1
1
1
2
0795
0585
0485
0570
6345
8815
E
6
6
10
1
1
1
9
0680
0425
0320
0425
7370
9155
E
6
6
10
1
2
0
2
0865
0615
0565
0660
7895
9790
E
6
6
10
1
2
0
9
1310
1120
0630
0920
7710
9775
E
6
6
10
1
2
1
2
0710
0485
0460
0545
8740
9845
E
6
6
10
1
2
1
9
0615
0310
0210
0310
9435
9950

133
Table 18—continued.
DT
P
G
N: p
NRF
NR
S
d2
U1*
U2*
V*
L*
J
E
6
6
10
2
1
0
2
0860
0670
0545
0685
5640
7845
E
6
6
10
2
1
0
9
0965
0670
0555
0695
6030
8155
E
6
6
10
2
1
1
2
0745
0575
0500
0550
6490
8455
E
6
6
10
2
1
1
9
0930
0600
0280
0455
7015
8630
E
6
6
10
2
2
0
2
0895
0680
0560
0670
6295
8130
E
6
6
10
2
2
0
9
0980
0710
0530
0690
6115
8165
E
6
6
10
2
2
1
2
0720
0505
0420
0515
7380
8750
E
6
6
10
2
2
1
9
0640
0355
0145
0285
8185
9245
E
6
6
20
1
1
0
2
1025
0875
0805
0900
2340
5260
E
6
6
20
1
1
0
9
1375
1205
0835
1060
2670
5510
E
6
6
20
1
1
1
2
0885
0790
0765
0780
2485
5155
E
6
6
20
1
1
1
9
1085
0950
0775
0915
3475
5970
E
6
6
20
1
2
0
2
1105
0975
0925
0970
2620
7905
E
6
6
20
1
2
0
9
1430
1315
1060
1215
2935
8000
E
6
6
20
1
2
1
2
0910
0805
0745
0785
3245
7665
E
6
6
20
1
2
1
9
1055
0885
0750
0870
4360
8135
E
6
6
20
2
1
0
2
1070
0970
0920
0965
2405
4050
E
6
6
20
2
1
0
9
1100
1000
0880
0995
2850
4365
E
6
6
20
2
1
1
2
1010
0880
0805
0865
2620
4165
E
6
6
20
2
1
1
9
1360
1115
0715
0975
3025
4670
E
6
6
20
2
2
0
2
1110
0995
0915
0965
2975
4570

134
Table 18—continued.
DT
P
G
N: p
NRF
NR
S
d2
Ul*
U2*
V*
L*
J
<
E
6
6
20
2
2
0
9
1095
0970
0865
0935
2745
4395
E
6
6
20
2
2
1
2
0990
0855
0765
0860
3640
5245
E
6
6
20
2
2
1
9
1255
0955
0545
0815
4125
5620
N
3
3
10
1
1
0
2
0970
0935
0885
0920
1095
1125
N
3
3
10
1
1
0
9
0940
0890
0785
0865
0945
0975
N
3
3
10
1
1
1
2
0900
0865
0805
0890
1205
1220
N
3
3
10
1
1
1
9
0835
0795
0685
0785
1295
1485
N
3
3
10
1
2
0
2
0960
0920
0820
0915
1240
1330
N
3
3
10
1
2
0
9
0990
0940
0845
0915
1155
1210
N
3
3
10
1
2
1
2
0925
0865
0765
0875
1535
1710
N
3
3
10
1
2
1
9
0665
0585
0580
0655
1575
2015
N
3
3
10
2
1
0
2
1030
0975
0910
0980
1105
1180
N
3
3
10
2
1
0
9
0990
0905
0830
0945
1095
1155
N
3
3
10
2
1
1
2
0905
0820
0790
0855
1205
1280
N
3
3
10
2
1
1
9
0910
0865
0630
0785
1330
1415
N
3
3
10
2
2
0
2
0990
0945
0915
0970
1230
1290
N
3
3
10
2
2
0
9
0955
0905
0815
0870
1065
1155
N
3
3
10
2
2
1
2
0860
0795
0705
0835
1550
1645
N
3
3
10
2
2
1
9
0570
0525
0295
0525
1605
1870
N
3
3
20
1
1
0
2
1005
0970
0965
0980
1015
0955
N
3
3
20
1
1
0
9
1120
1100
1015
1065
1035
1010

135
Table 18—continued.
DT
P
G
N: p
NRF
NR
S
d2
Ul*
V
V*
L*
J
N
3
3
20
1
1
1
2
1010
0985
0940
0980
1020
1010
N
3
3
20
1
1
1
9
0995
0980
0930
0960
1045
1110
N
3
3
20
1
2
0
2
0975
0970
0940
0975
1000
0965
N
3
3
20
1
2
0
9
1165
1150
1065
1140
0945
0910
N
3
3
20
1
2
1
2
0885
0855
0800
0835
1015
1025
N
3
3
20
1
2
1
9
0880
0840
0795
0855
0920
1035
N
3
3
20
2
1
0
2
1020
0990
0935
0955
0990
0945
N
3
3
20
2
1
0
9
1085
1055
0990
1030
1020
1005
N
3
3
20
2
1
1
2
1185
1160
1125
1155
1165
1120
N
3
3
20
2
1
1
9
0980
0920
0720
0875
0865
0850
N
3
3
20
2
2
0
2
1095
1085
1035
1085
1110
1045
N
3
3
20
2
2
0
9
1015
0985
0975
1000
1035
0945
N
3
3
20
2
2
1
2
0885
0855
0855
0870
1090
1120
N
3
3
20
2
2
1
9
0975
0940
0785
0870
1120
1150
N
3
6
10
1
1
0
2
1120
0775
0570
0755
3630
5155
N
3
6
10
1
1
0
9
0945
0670
0510
0650
3645
5190
N
3
6
10
1
1
1
2
1030
0745
0675
0785
3845
5290
N
3
6
10
1
1
1
9
0990
0700
0585
0715
4310
5875
N
3
6
10
1
2
0
2
0960
0620
0645
0695
5110
7435
N
3
6
10
1
2
0
9
1200
0895
0630
0865
4745
7390
N
3
6
10
1
2
1
2
0940
0640
0625
0750
5640
7720

136
Table 18--continued.
DT
P
G
N: p
NRF
NR
S
d2
U1*
V*
L*
J
<
N
3
6
10
1
2
1
9
0630
0365
0405
0495
6655
8155
N
3
6
10
2
1
0
2
0895
0595
0500
0610
4790
6265
N
3
6
10
2
1
0
9
1020
0665
0635
0725
4820
6405
N
3
6
10
2
1
1
2
0840
0560
0505
0610
5100
6355
N
3
6
10
2
1
1
9
0730
0450
0345
0445
5610
7015
N
3
6
10
2
2
0
2
0935
0715
0575
0755
3970
5210
N
3
6
10
2
2
0
9
1105
0790
0605
0790
3855
5270
N
3
6
10
2
2
1
2
0910
0645
0650
0680
4620
5915
N
3
6
10
2
2
1
9
0715
0470
0365
0465
5245
6390
N
3
6
20
1
1
0
2
1125
0970
0875
0960
1530
2545
N
3
6
20
1
1
0
9
1270
1050
0860
0985
1525
2445
N
3
6
20
1
1
1
2
1085
0860
0765
0850
1585
2470
N
3
6
20
1
1
1
9
1050
0840
0710
0860
1785
2665
N
3
6
20
1
2
0
2
1090
0905
0860
0895
1965
5160
N
3
6
20
1
2
0
9
1405
1235
1060
1160
1810
5305
N
3
6
20
1
2
1
2
1150
0925
0815
0950
2245
5080
N
3
6
20
1
2
1
9
1040
0810
0700
0850
2600
5210
N
3
6
20
2
1
0
2
1155
0930
0920
0960
1630
1930
N
3
6
20
2
1
0
9
1170
0980
0910
0985
1625
1920
N
3
6
20
2
1
1
2
1235
1050
0930
1025
1830
2145
N
3
6
20
2
1
1
9
1105
0850
0630
0805
1750
2090

137
Table 18—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
V
V*
L*
J
h;
N
3
6
20
2
2
0
2
1160
0985
0950
1010
2040
2355
N
3
6
20
2
2
0
9
1180
1030
0910
0970
1800
2155
N
3
6
20
2
2
1
2
1025
0770
0725
0810
2355
2705
N
3
6
20
2
2
1
9
0965
0675
0445
0645
2495
2925
N
6
3
10
1
1
0
2
1160
1130
1025
1085
1285
1880
N
6
3
10
1
1
0
9
0950
0920
0720
0850
1140
1635
N
6
3
10
1
1
1
2
1065
1005
0985
1000
1330
1960
N
6
3
10
1
1
1
9
1085
1060
0925
1040
1510
2390
N
6
3
10
1
2
0
2
0955
0920
0880
0910
1250
1820
N
6
3
10
1
2
0
9
1055
1010
0855
0955
1225
1740
N
6
3
10
1
2
1
2
0905
0880
0820
0880
1455
2220
N
6
3
10
1
2
1
9
0830
0775
0700
0795
1670
2865
N
6
3
10
2
1
0
2
0945
0930
0870
0905
1090
1605
N
6
3
10
2
1
0
9
0950
0925
0755
0865
1125
1855
N
6
3
10
2
1
1
2
0970
0935
0855
0925
1360
1945
N
6
3
10
2
1
1
9
0925
0880
0530
0805
1430
2145
N
6
3
10
2
2
0
2
1025
0995
0960
0995
1320
1895
N
6
3
10
2
2
0
9
1195
1135
0950
1080
1220
1740
N
6
3
10
2
2
1
2
1060
1015
0930
0995
1775
2450
N
6
3
10
2
2
1
9
0550
0515
0210
0355
1720
2445
N
6
3
20
1
1
0
2
1030
1020
1005
1005
1050
1260

138
Table 18—continued.
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
N
6
3
20
1
1
0
9
1165
1155
0990
1120
1045
1285
N
6
3
20
1
1
1
2
1055
1025
0990
1020
1115
1350
N
6
3
20
1
1
1
9
0955
0920
0890
0910
1090
1440
N
6
3
20
1
2
0
2
1155
1130
1115
1135
1220
1460
N
6
3
20
1
2
0
9
1295
1295
1185
1240
1125
1340
N
6
3
20
1
2
1
2
0955
0930
0925
0940
1025
1435
N
6
3
20
1
2
1
9
0925
0915
0855
0900
0960
1515
N
6
3
20
2
1
0
2
1025
1015
1000
1020
1040
1295
N
6
3
20
2
1
0
9
1085
1070
1035
1055
0965
1215
N
6
3
20
2
1
1
2
0965
0940
0905
0925
1020
1280
N
6
3
20
2
1
1
9
1060
1020
0870
0975
1065
1430
N
6
3
20
2
2
0
2
1175
1165
1125
1150
1105
1370
N
6
3
20
2
2
0
9
1155
1145
1055
1105
1105
1375
N
6
3
20
2
2
1
2
0935
0920
0880
0915
1035
1350
N
6
3
20
2
2
1
9
0950
0905
0620
0795
1130
1580
N
6
6
10
1
1
0
2
0985
0835
0690
0795
4775
7870
N
6
6
10
1
1
0
9
1165
0895
0460
0700
4575
7910
N
6
6
10
1
1
1
2
0985
0755
0645
0755
5210
7975
N
6
6
10
1
1
1
9
1065
0780
0620
0785
5860
8390
N
6
6
10
1
2
0
2
1080
0880
0735
0850
7175
9695
N
6
6
10
1
2
0
9
1610
1260
0765
1110
7055
9695

139
Table 18—continued.
DT
P
G
N: p
NRF
NR
S
d2
U2*
V*
L*
J
N
6
6
10
1
2
1
2
0945
0645
0525
0650
8050
9745
N
6
6
10
1
2
1
9
0835
0495
0395
0530
8835
9860
N
6
6
10
2
1
0
2
1135
0905
0730
0900
4880
7275
N
6
6
10
2
1
0
9
1145
0890
0685
0885
4640
7230
N
6
6
10
2
1
1
2
1035
0790
0730
0800
5295
7595
N
6
6
10
2
1
1
9
0990
0675
0210
0540
5995
8085
N
6
6
10
2
2
0
2
1230
1040
0910
1045
5520
7500
N
6
6
10
2
2
0
9
1200
1000
0825
0955
4545
6940
N
6
6
10
2
2
1
2
1090
0835
0710
0815
6520
8295
N
6
6
10
2
2
1
9
0660
0355
0110
0260
7310
8675
N
6
6
20
1
1
0
2
1100
1030
0985
1030
1585
4330
N
6
6
20
1
1
0
9
1400
1275
0945
1125
1700
4585
N
6
6
20
1
1
1
2
1185
1090
1000
1090
1755
4265
N
6
6
20
1
1
1
9
1070
0930
0810
0890
1785
4320
N
6
6
20
1
2
0
2
1145
1045
0990
1040
1900
7235
N
6
6
20
1
2
0
9
1370
1290
0960
1140
1805
7150
N
6
6
20
1
2
1
2
1045
0915
0860
0925
2140
7175
N
6
6
20
1
2
1
9
1095
0900
0740
0870
2415
6830
N
6
6
20
2
1
0
2
1000
0925
0845
0900
1625
2975
N
6
6
20
2
1
0
9
1270
1165
1010
1130
1765
3095
N
6
6
20
2
1
1
2
1090
1000
0930
1020
1845
3170

140
Table 18—continued.
DT p G N: p NRF NR S d2 U,* U,* V* L* J H *
r r 12 m
N 6 6 20
N 6 6 20
N 6 6 20
N 6 6 20
N 6 6 20
2 119
2 2 0 2
2 2 0 9
2 2 12
2 2 19
1275 1030
1030 0920
1350 1225
1115 0945
1070 0795
0610 0845
0820 0890
1045 1145
0880 0945
0385 0685
1845 3320
1990 3370
1815 3210
2400 4050
2595 4215

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BIOGRAPHICAL SKETCH
William Thomas Coombs was born September 30, 1954. He
received two bachelor's degrees with majors in history (1976)
and psychology (1979), both from the University of Tennessee.
He next received three master's degrees with majors in human
relations (1981), mathematics (1987), and statistics (1989),
from Shippensburg State College, Bowling Green State
University, and the University of Florida, respectively.
In the fall of 1989, he began studying for the Ph.D.
degree in the Foundations of Education Department at the
University of Florida, majoring in research and evaluation
methodology. He will graduate with the Ph.D. degree in
August, 1992, and begin his career as an Assistant Professor
in Applied Behavioral Studies at Oklahoma State University.
148

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Jdirtes J. Algina'X Chair
Projfessor of Foundations of
Education
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Linda M. Crocker
Professor of Foundations of
Education
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
U
M. David Miller
Associate Professor of
Foundations of Education
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
&A H
Ronald H.Randles
Professor of Statistics

This dissertation was submitted to the Graduate Faculty
of the College of Education and to the Graduate School and was
accepted as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
August, 1992
Gk¿Sl
Chairperson, Foundations of
Education
Dean, College of Education
Dean, Graduate School

UNIVERSITY OF FLORIDA
3 1262 08556 8235