SOLUTIONS TO THE MULTIVARIATE GSAMPLE
BEHRENSFISHER PROBLEM BASED UPON GENERALIZATIONS
OF THE BROWNFORSYTHE 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
sincerelythank
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
BrownForsythe
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
BrownForsythe
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
BEHRENSFISHER
Presented
of Florida
Degree
TO THE
PROBLEM
the
Partial I
of Doctor
MULTIVARIATE
BASED
UPON
Graduate
School
Fulfillment of
of Philosophy
GSAMPLE
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
BrownForsythe
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
(variancecovariance)
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,
secondordei
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,
firstorder
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
nonnormality
(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 nonnormality 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
secondorder
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
BrownForsythe
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
firstorder
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 nonnormality 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
secondorder 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
zeroorder
term
critical
Table
Critical
Value
Terms
Welch's
(1947)
Zero
F First
. and
SecondOrder 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
secondorder
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
Gsample
case
and
multivariate
cases
respectively
Consequently,
tests
using
the
series
solution
are
referred
as James
s firstorder and secondorder tests.
The
zeroorder
test
often
referred
the
asymptotic
test.
Aspin
(1948)
reported
the
third
and
fourthorder
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
2241
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
twosample
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
NeymanPearson
school
thought.
Scheffe
concluded
Welch
test
, which
requires
only
the
easily
accessible
ttable,
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
TaO
was
smaller
the
WelchAspin
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
ttable
carry
out
the
Welch
APDF
test
without
much
loss
accuracy
However,
the
Welch
APDF
test
becomes
conservative
with
very
longtailed
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
nonnormality
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.
NG)
has
an F
distribution
with
G1 and
NG
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
i1
W
w
G
=1
2=1
wix
and
=i2
=l ,...,G
The
statistic
approximately
distributed
with
and
G3
G2l 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
firstorder
test,
James
found
order
1 the
1
critical
value
2
 XGI
2(G2
W
 )
Ff
null
hypothesis
hypothesis
> 2h(
rejected
James
favor
the
provided
alternative
ondorder
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
G1 degrees
freedom.
The
literature
suggests
the
following
conclusions
about
BrownForsythe,
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
(JI itt C.
(198
studied
the
ANOVA
. Welch
, or
J7,
U i IV
r.
when
equal
unequal
sized
sample
were
selected
from
normal
stributions,
chisquare
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,
BrownForsythe
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
BrownForsythe
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
secondorder
test.
Simulated equal
and unequal
sized samples
were
selected where
distributions
were
either
normal
, light
tailed
symmetric,
heavytailed
symmetric,
mediumtailed
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
distributionsnormal,
uniform,
beta(1.5,8
, and
exponential.
The
James
second
order
test
and
Wilcox
test
were
both
affected
nonnormality
When
samples
were
selected
from symmetric
non
normal
distributions
both James'
secondorder test and
Wilcox'
test maintained
r near
When
the
tests
were
applied
to data
sampled
from
asymmetric
distributions,
Ta
increased.
Further,
degree
of asymmetry
increase
ed, I
va
tended
increase.
The
Brown
Forsythe
test
outperformed
the
Wilcox
test
James'
secondorder
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
BrownForsythe
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
secondorder
, 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 si
X2C
where
1),2
Hotelling
demonstrated
transformation
ng +n2
pi
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
4In 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
secondorder
(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
secondorder tests
Yao'
test,
and
Johansen'
test
Yao'
test
, James'
second
order test
and Johansen'
test are
superior to James'
firstorder
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
nonnormality
(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
secondorder
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
1wi
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
consistentmodest
departures from a
for minor degrees of heteroscedasticity and
more
pronounced
departures
with
greater
heteroscedasticity
Korin
(1972)
studied
Roy's
largest
root
criterion
the
HotellingLawley
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
equalsized
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(pd2p+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
secondorder
critical
values
parall
those
developed
James
(1951).
Johansen
(1980)
generalized
Welch
(1951
test
proposed
using
the
James
(1954)
test
statistic
divided
 p(G1)
+ 2A
Gl) +
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
secondorder
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
Ta
I
increased
the
variation
the
sample
sizes
degree
heteroscedasticity
and
number
dependent
variables
increased,
whereas
ra
decreased
the
total
sample
size
increase
Tang
James'
(1989)
first
studied
and
Pillai
secondorder
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 secondorder
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
secondorder
test
are
more
effective
than
either
the Pillai
Bartlett
trace
criterion
or James
firstorder
test
Finally,
Johansen
test
computationally
practical
intensive
than
advantage
James
of being
secondorder
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
pdimensional
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
LawleyHotelling
PillaiBartlett
(V)1
and
Wilks
Define
r
X)
and
 r
Then
the
test
stati
HotellingLawley
trace
criterion,
the
PillaiBartlett
trace
criterion,
the
Wilks
likelihood
ratio
criterion
are,
respectively
trace
trace
flEi
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 HotellingLawley
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
a2
 F
Sph,a
where
= 4
ph +
and
+ h)
 1)
2n + p)
2n +
For
the
PillaiBartlett
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
BrownForsythe
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,
BrownForsythe 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
HotellingLawley,
PillaiBartlett,
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)MI=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.
LawleyHotelling 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
NG,
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
i1,
S=G
G
iSi3 2)
For the modified HotellingLawley
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 PillaiBartlett 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
Xi
 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.
BrownForsvthe
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
PillaiBartlett
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'] lnis1
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
i1
i=1
Using
results
14,
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 
41,a
4, 14
.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
exponentialwere
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
1n 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 nonnormal
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
llingLawley
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
stributionnormal
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 4a ~.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
ma4 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
HotellingLawley
(U,*) ,
second
modified
HotellingLawley
cn~*
modified
PillaiBartlett
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
BrownForsythe
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 BrownForsythe 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
HotellincLawlev
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
ellinqLawley
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
PillaiBartlett
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) .
BrownForsvthe 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
twoway
interactions,
threeway
interactions,
fourway
interactions.
Because R2 was
for the model
with
fourway interactions,
more complex models were not examined.
models
are
shown
in Table
The
model
with
main
effects
and
twoway
through
fourway
interactions
was
selected.
Variance components were computed for each main effect,
twoway,
threeway,
fourway
interaction.
The
variance
component
i=1,...,255)
for each effect was computed using
the formula 106(MSEFMSE)/(2x),
where MSEF was the mean square
for that given effect,
MSE was
the mean square error
for the
fnuirfar;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 FourWay Interaction Models when using
Way Interaction, and FourWay Interaction Models when using
the
Four
Brown
Forsvtne
General
zations
HighestOrder Terms R2
Main Effects 0.52
TwoWay Interactions 0.77
ThreeWay Interactions 0.89
FourWay 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
andin
contrast to
factors
such as d
DTdo 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
ii,
Table
Variance
Comnon F.m t
Fi rst
Mnr i fi sd'
HotallingLawle
Second Modified HotellincgLawley, Modified PillaiBartlett.
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
10continued.
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
threeway
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
HotellingLawley
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 HotellinqLawlev Test
1(U ) and Second Modified HotellinaLawley Test (U ) for
Combinations of Ratio of Total Sample Size to Number of
Dependent Variables (N:p) and Number of Populations Sampled
JGI
(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
PillaiBartlett
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
BrownForsythe
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
BrownForsythe
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
twoway
interactions,
threeway
interactions,
fourway interactions.
Because R2 was
.997
for the model with
fourway
interactions
, more complex models were not examined.
The
main
models
effects
are
twoway
shown
through
in Table
fourway
model
interactions
with
was
selected.
Variance components were computed for each main effect
twoway,
threeway
and
fourway
interaction.
variance
component
i=1,
..,255
for each effect was computed using
the formula 104(MSEFMSE)
, where MSEF was the mean square
for that
given
effect,
MSE was
the mean
square
error
for the
fourfactor 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 FourWay Interaction Models when usinQ
a a
t~hw
Johan
sen
Test
and
WiI
cox
General
action
HighestOrder Terms R2
Main Effects 0.767
TwoWay Interactions 0.963
ThreeWay Interactions 0.988
FourWay 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
andin
contrast
to factors
such
as d and
DTdo
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

PAGE 2
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7DEOH (VWLPDWHG 7\SH (UURU 5DWHV :KHQ D '7 3 1 S 15) 15 6 G XLr 9 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
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7DEOH FRQWLQXHG '7 3 1S 15) 15 6 G 9r /r + r P ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
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7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G 8r 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
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7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
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7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G 9 8r 9r /r + r P ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
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7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 V G 8r 9r /r K ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
PAGE 116
7DEOH fÂ§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
PAGE 117
7DEOH fÂ§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
PAGE 118
7DEOH fÂ§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
PAGE 119
,OO 7DEOH fÂ§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
PAGE 120
7DEOH fÂ§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
PAGE 121
7DEOH fÂ§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
PAGE 122
7DEOH fÂ§FRQWLQXHG '7 S 1 S 15) 15 6 G 8r 8r 9r /r + r / L Â 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 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
PAGE 125
7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G mr 9r /r K ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
PAGE 126
7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
PAGE 127
7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G 9 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
PAGE 128
7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G 9r /r K ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
PAGE 129
7DEOH fÂ§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
PAGE 130
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 fÂ§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 fÂ§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 fÂ§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 fÂ§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 fÂ§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 fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G 9 9 9r /r +Q ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
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7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G XU 8r 9r /r +Pr ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
PAGE 139
7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 V G 8r 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
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7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G XU 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
PAGE 141
7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 V G 8r 8r 9r /r ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
PAGE 142
7DEOH fÂ§FRQWLQXHG '7 S 1 S 15) 15 V G 9 9r /r ( ( ( 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
PAGE 143
7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 V 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
PAGE 144
7DEOH FRQWLQXHG '7 S 1 S 15) 15 6 G XLr !9 9r /r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
PAGE 145
7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 V G 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
PAGE 146
7DEOH fÂ§FRQWLQXHG '7 S 1 S 15) 15 6 G 9 9r /r +Pr 1 L 1 L 1 L 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
PAGE 147
7DEOH fÂ§FRQWLQXHG '7 3 1 S 15) 15 6 G 8r 9r /r K 1 L 1 L 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
PAGE 148
7DEOH fÂ§FRQWLQXHG '7 S 1 S 15) 15 6 G 8r 8r 9r /r + r U U P 1 1 1 1 1
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INGEST IEID E5JELM403_8NKMUW INGEST_TIME 20110913T12:30:30Z PACKAGE AA00002084_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
SOLUTIONS TO THE MULTIVARIATE GSAMPLE
BEHRENSFISHER PROBLEM BASED UPON GENERALIZATIONS
OF THE BROWNFORSYTHE 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 sincerelyâ€”thank
you.
TABLE OF CONTENTS
ACKNOWLEDGEMENTS Ãœ
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
BrownForsythe 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
BrownForsythe Generalizations 51
Wilcox Generalization 52
Design 54
Simulation Procedure 62
Summary 65
IV
4 RESULTS AND DISCUSSION 67
BrownForsythe 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 GSAMPLE
BEHRENSFISHER PROBLEM BASED UPON GENERALIZATIONS
OF THE BROWNFORSYTHE F* AND WILCOX H TESTS
â€” â€”m
By
William Thomas Coombs
August, 1992
Chairperson: James J. Algina
Major Department: Foundations of Education
The BrownForsythe 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 BrownForsythe 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 BrownForsythe 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 twosample BehrensFisher 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 Gsample BehrensFisher 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 (variancecovariance) 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 twosample
BehrensFisher 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 firstorder and secondorder series solutions.
Johansen generalized the Welch (1951) procedure to form an
APDF solution to this problem.
3
The Problem
To date, neither the BrownForsythe (1974) nor the Wilcox
(1989) procedure has been extended to the multivariate
setting. To test Hq: /x, = m2 = â€¢â€¢â€¢ = MG Brown and Forsythe
(1974) proposed the statistic
 *..)2
F*
1=1
n.
E (1 ' N)Sâ€˜
Â¿=1
where nÂ¡ 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 Gl and f degrees of freedom, where
n
Ã =
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
= Â£ 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
Â£*.
2=1
2=1
In the equation for xi , xirii denotes the last observation in
the ith sample. The statistic is approximately distributed
as chisquare with Gl 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 secondorder, 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 firstorder 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 nonnormal symmetric
distributions such as the uniform, t, and Laplace. The
performance of Yao's test, James's secondorder test, and
Johansen's test was slightly superior to the performance of
James's firstorder test (Algina, Oshima, & Tang, 1991).
MANOVA criteria are relatively robust to nonnormality
(Olson, 1974, 1976) but are sensitive to violations of
homoscedasticity (Korin, 1972; Olson, 1974, 1979; Pillai &
Sudjana, 1975; Stevens, 1979). The PillaiBartlett trace
criterion is the most robust of the four basic MANOVA criteria
for protection against nonnormality 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 secondorder 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
secondorder test outperform the PillaiBartlett trace
criterion and James's firstorder test (Tang, 1989).
In the univariate case, the BrownForsythe F* test and
Wilcox test do not require the equality of population
variances for the G groups. Hence, the BrownForsythe 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, BrownForsythe F* test, Welch APDF
test, and James firstorder 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
BrownForsythe test is less sensitive to nonnormality 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 secondorder test, Welch's test, or
Wilcox's H test. In other conditions the F* test has
â€”m â€”
substantially worse control. Oshima and Algina concluded that
James's secondorder test should be used with symmetric
distributions and Wilcox's H test should be used with
â€”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 secondorder
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
secondorder method. Wilcox (1990) indicated that the H test
is more robust to nonnormality 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 BrownForsythe 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 ( â€”+â€”)
p " ru
where
2 _ (^1) sÂ± + (n21) si
Sp ni+n2~2
has a t distribution with n1+n22 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 equalsized samples t is asymptotically
standard normal, even though the two populations are nonÂ¬
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 equalsized
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 = /Â¿2 in situations in which a 2 f a2. Welch
(1947) reported several tests in which the test statistic is
% =
Xix2
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
^21
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 (nÂ¡  l)â€™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 zeroorder term as the critical
value is appropriate with large samples. The firstorder
critical value is the sum of the zero and firstorder terms,
12
Table 1
Critical Value Terms for Welch's (1947) Zero, First, and
SecondOrder Series Solutions
Power of
(n,  1)â€˜1 Term
Zero
z
One
s?
a i^r)2
ni
z[ 1 + z2 * ^_1 ]*
4 2 2
[ Eâ€” ]2
4â€”' n .
i=l
Two
2 S?
y ( 2 )2
â€ž r 1+z2 h. ni(nil)
2 2 a2
â€”i2
4â€”/ n .
2=1 ii2
2 3
r i 1 i2
3+5z2 + z4 fri i^l
3 2 2
[ Eâ€” ]3
H ni
a (Si)2
T [ nâ€˜ l2
15+32zz+9z4 Â£Ã 1 1
32 2 2
tr â€”i4
h ni
Note. Z denotes a fractile of the normal distribution.
13
whereas the secondorder 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
Gsample case and multivariate cases respectively.
Consequently, tests using the series solution are referred to
as James's firstorder and secondorder tests. The zeroorder
test is often referred to as the asymptotic test. Aspin
(1948) reported the third and fourthorder terms, and
investigated, for equalsized samples, variation in the first
through fourthorder 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 xÂ¡ (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
twosample 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 secondorder
14
series tests, BrownForsythe test, and Wilcox test: (a)
the performance of the Welch test and BrownForsythe 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 NeymanPearson school of
thought. Scheffe concluded the Welch test, which requires
only the easily accessible ttable, is a satisfactory
practical solution to the BehrensFisher problem. Wang (1971)
examined the BehrensFisher test , Welch APDF test, and Welch
Aspin series test (Aspin, 1948; Welch, 1947). Wang found the
Welch APDF test to be superior to the BehrensFisher test when
combining over all the experimental conditions considered.
Wang found jra] was smaller for the WelchAspin series test
than for the Welch APDF test. Wang noted, however, that the
WelchAspin 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 ttable to
carry out the Welch APDF test without much loss of accuracy.
However, the Welch APDF test becomes conservative with very
longtailed symmetric distributions (Yuen, 1974). Wilcox
(1990) investigated the effects of nonnormality 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 nonnormality.
In summary, the independent samples t test is generally
acceptable in terms of controlling Type I error rates provided
there are sufficiently large equalsized samples, even when
the assumption of homoscedasticity is violated. For unequalÂ¬
sized samples, however, an alternative that does not assume
equal population variances such as the Wilcox test or the
James secondorder 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/ (Gl)
F = â€”
G
Y (nrl) s\/ (NG)
i = 1
has an F distribution with Gl and NG 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 equalsized 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
Â£ (Xix) V (Gl)
F = â€”
^2(02) A A(1j^)2
G21 fj w
where
w =
7=1
ni i=1
xij
1=1,...,G
17
* = E
7=1
WjXj
w
and
ft = ^1 i = l, . . â€¢ , 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 = Â£ W^X^X)2
i=l
where
w,
i= 1, . . . ,G
and
i= 1 G
* = E
7=1
w
with Gl
series
In the asymptotic test the critical value of the statistic J
is a fractile of a chisquare distribution with Gl 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 chisquare distribution with Gl
degrees of freedom. James (1951) derived a series expression
which is a function of the sample variances such that
G
P [^2 w1(xix)2 i: 2h(sÂ¡) ] = a .
i =1
James found approximations to 2h(Sj2) of orders 1/fj and 1/fj2
(fj = nj1). In the firstorder test, James found to order
1/fj the critical value is
2 h{sj)
Xgl;a t 1 +
3 Xg1 ; a + G * 1
2(G2l)
Et(i)2] â€¢
^ r w
H fÂ±
The null hypothesis is rejected in favor of the alternative
hypothesis if J > 2h(Sj2). James also provided a secondorder
solution which approximates 2h(Sj2) to order 1/fj2. James
noted that this secondorder test is very computationally
intensive.
Brown and Forsythe (1974) proposed the test statistic
G
F*
Y, ni (xÂ¿.
2=1
G
E (1 
i =1
x.)
The statistic F* is approximately distributed as F with Gl
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
BrownForsythe 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?
= [ â€” ] 1
Â«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 chisquare
with Gâ€”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 secondorder,
20
BrownForsythe, 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 BrownForsythe tests are
generally competitive with one another, however, the Welch
test is preferred with data sampled from normal distributions
while the BrownForsythe test is preferred with data sampled
from skewed distributions; and (d) the Wilcox and James
secondorder 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, BrownForsythe F*, Welch APDF, and James zeroorder
procedures when (a) equal and unequalsized 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 BrownForsythe 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 unequalsized
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 unequalsized 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,
chisquare, 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 BrownForsythe tests when (a) equal
and unequalsized 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 secondorder 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 BrownForsythe F* tests using Monte Carlo methods
22
when (a) equal and unequalsized samples were selected from
normal distributions, chisquare 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 BrownForsythe 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, BrownForsythe test, and Welch APDF test
when (a) equal and unequalsized 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 BrownForsythe 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, BrownForsythe F*,
and the Welch APDF test when (a) equal and unequalsized
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 unequalsized samples, the
Welch test should be avoided in favor of the F* test but for
unequalsized 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 secondorder test.
Simulated equal and unequalsized samples were selected where
(a) distributions were either normal, lighttailed symmetric,
heavytailed symmetric, mediumtailed asymmetric, or
exponentiallike; (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 secondorder test and Wilcox's test
have r much closer to a than the Welch or BrownForsythe
tests. The Wilcox test gave conservative results provided nÂ¡
> 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 secondorder 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 secondorder 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 secondorder 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 BrownForsythe test, James's secondorder 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 distributionsâ€”normal, uniform, t(5),
beta(1.5,8.5), and exponential. The James secondorder test
and Wilcox test were both affected by nonnormality. When
samples were selected from symmetric nonnormal distributions
both James's secondorder test and Wilcox's H test maintained
â€”m
r near a. When the tests were applied to data sampled from
asymmetric distributions, Â¡ r â€”o; Â¡ increased. Further, as the
degree of asymmetry increased, Â¡Taj tended to increase. The
BrownForsythe test outperformed the Wilcox test and
James's secondorder test under some conditions, however, the
reverse held under other conditions. Oshima and Algina
concluded (a) the Wilcox H test and James's secondorder test
were preferable to the BrownForsythe test, (b) James's
secondorder 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 secondorder 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 BrownForsythe 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– i2  (5q  x2)' S1 (5?1  x2)
ni+n2
where
g = (nxl) S1 + (n2l)S2
ni+n22
Hotelling demonstrated the transformation
nx+n2 p1 t2
(nx+n22)p
has an F distribution with p and n.,+n2pl 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 equalÂ¬
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
623 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 nÂ¡ > 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 unequalsized 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,
â€¢5,
28
eigenvalues of Â£2Z'1 were 3 or 10. Holloway and Dunn found
equalsized samples help in keeping r close to a. Further,
Holloway and Dunn found that for large equalsized 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
unequalsized 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 equalsized samples, the T2
procedure is generally robust. With unequalsized 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 equalsized 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 secondorder 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) [ â€”
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 secondorder tests,
Yao's test, and Johansen's test: (a) Yao's test, James's
secondorder test, and Johansen's test are superior to James's
firstorder 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 firstorder test and the Yao
test when (a) equal and unequalsized 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 firstorder 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 secondorder, Yao's, and
Johansen's tests for various conditions defined by the degree
of heteroscedasticity and nonnormality (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 firstorder test. Algina, Oshima, and
Tang indicate that Yao's test is also sensitive to skewness.
In summary Yao's test, James's secondorder 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,Gl). Those criteria are:
1.Roy's (1945) largest root criterion
2. HotellingLawley trace criterion (Hotelling, 1951;
Lawley, 1938)
U = traceiHE'1) = Y zi ;
i =1
3. PillaiBartlett trace criterion (Pillai, 1955;
Bartlett, 1939)
S T â–
V = trace[H{H+E)1] = Y 1 ;
+f 1 +T .
2=1 â€˜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 equalsized samples selected
from two populations with unclear degrees of
34
heteroscedasticity, the results were consistentâ€”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 HotellingLawley trace criterion (U), and Wilks's
likelihood ratio criterion (L) when (a) equal and unequalÂ¬
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) equalsized
samples were selected; (b) p was 2, 3, 6, or 10; (c) G was 2,
3, 6, or 10; (d) nÂ¡ 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 {pd2p+l, 1,1, . . . , 1) (d = 2,
3, or 6). Results indicated that for protection against nonÂ¬
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 = (XiX) 'Wi
2=1
where
= [**] i = l G
ni
* = iwi
i= 1
xi = ir i=1 G
â– ^2 j=l
and
x = FT1 Â¿ Vx,
2=1
The James (1954) zero, first, and secondorder 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(Gl) + 2A 
P(G1) +2
where
37
A
E
7=1
trace (IW~1Wi)2 + trace2 (IW~1Wi)
2(nil)
The critical value for the Johansen test is a fractile of an
F distribution with p(Gl) and p(Gl)[p(Gl)+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 secondorder
tests, and Johansen's test: (a) the PillaiBartlett trace
criterion is the most robust of the four basic MANOVA
criteria; and (b) with unequalsized samples, Johansen's test
and James's secondorder test outperform the PillaiBartlett
trace criterion and James's firstorder test.
Ito (1969) analytically examined Type I error rates for
James's zeroorder test and showed r > a. Ito showed jraj
increased as the variation in the sample sizes, degree of
heteroscedasticity, and number of dependent variables
increased, whereas j r â€” or J decreased as the total sample size
increased.
Tang (1989) studied the PillaiBartlett trace criterion,
James's first and secondorder tests, and Johansen's test
when (a) equal and unequalsized 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 secondorder test perform better
than the PillaiBartlett trace criterion and James firstorder
test. While both Johansen's test and James's secondorder
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 PillaiBartlett 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
secondorder test are more effective than either the Pillai
Bartlett trace criterion or James firstorder test. Finally,
the Johansen test has the practical advantage of being less
computationally intensive than the James secondorder 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
BrownForsvthe Generalizations
To test Hq: /i1 = )li2 = ... = MG Brown and Forsythe (1974)
proposed the statistic
F*
Â£ ni (xi.  x..)
2=1
2 = 1
â€”) s?
N)Sl
The statistic F* is approximately distributed as F with Gl
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 pdimensional sample mean vectors
and S,, . ..,SG are pdimensional dispersion matrices of
independent random samples of sizes n1f...,nG, respectively,
from G multivariate normal distributions
, â€¢ â€¢ â€¢ ,Np(^G,ZG) . To extend the BrownForsythe 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
Â£ _ 2=1 2=1
g 
Y' [trace2 [c^A + trace [c,E,] 2}
hf n7  l
where
41
CÂ±
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
LawleyHotelling (U), PillaiBartlett (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 HotellingLawley trace
criterion, the PillaiBartlett 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 (Â¡ p â€” h {  1)
n = . 5 (e  p  1)
For the HotellingLawley 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
a2 ph pha
where
4 + P,h +. 2
b  1
and
b = (2n + h) (2n + p)
2 (n  1) (2n + 1)
For the PillaiBartlett 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) â€¢
For the Wilks criterion, the Rao (1952, p.262)
transformation is given by
_i
â€ž _ 1  L c rt  2 q
ph
ph,rt  2q
1
t
43
where
q = PPâ€” 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 BrownForsythe 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
BrownForsythe 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 HotellingLawley,
PillaiBartlett, 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
LawleyHotelling 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 Â©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 PillaiBartlett trace
45
(Bartlett, 1939 ; Pillai, 1955) V* = trace{H[H+ (f/h)M] â€˜1} = S 0Â¡
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 NG, 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
Â£ _ i=l i1
G 1
{trace2 [c^J + trace 2}
Â¿=1 ~ ^
where
i= I,
. =G
N
Eni
7=1
m = â€¢ 5 ( Â¡ p  h j 1)
n*= . 5 (f  p  1) .
46
For the modified HotellingLawley criterion, the Hughes
and Saw (1972) and McKeon (1974) transformations respectively
are now given by
â€ž (!) = 2 ( SJ2 * + 1) U* _ â€ž
U* S ( 2/n+S + l) S s (2in+s+l) , 2 (sJ2*+!)
and
F .(2)
rW
2nâ€™ a* ,
a *2
where
a* = 4 + Ph_L_2
b*  l
and
_ (2n* + A) (2nâ€™ + p)
2 (n*  1) (2n* + 1)
For the modified PillaiBartlett criterion the SAS (1985,
p.12) transformation is now given by
= 2n * + s + i V _ â€ž
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 _ â€ž
L' ,l ph phâ€™r' t2Q
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 BrownForsythe 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: =...= Â¿tG is true) the following
results are useful:
1. E(xÂ±) = Hi = H
2. E(x) = ii
3. E(x x') = var (x) + i i;
4. E[(xi  ji) (xÂ±  jx>7] = variXj) = j Ej
5 . Â£[ (x  ji) (x  ji)'] = var (x)
Using results 15, E(M) is given by
Â¿ (1  ^Â±)E[ 3X ]
E(M)
48
â– E<*
7=1
Similarly, using results 15, E(H) is given by
G
E(H) = E[ Yni(*i  x) {xÂ±  x)' ]
2â€”1
G G
= Y, ni E(*l  i)  i) ' + Y ni E Ji) (X  n) ' ,
i=l Â¿=1
G G
= Yni var^i> + Y nÃ 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.'] } ,
iâ€”1
= E^(
7=1
EM 4EÂ»A 1
2=1
72 2 i=l
n Â£"[ 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 2Â¿ + â€” E nÂ£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 BrownForsythe 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 â€™ j?) 2 '
i =1
where is approximately distributed chisquare with Gl
degrees of freedom. To extend this to the multivariate
setting, replace wi with WÂ¡, x{ with x, x with x , and define
50
 X) 'Wi ~ x)
2=1
where
W;
ni3lX
ni(ni + 1)
n<
1 _
x,
hi (rÂ¡i + 1)
and
x = [ Â¿ Wi 1 1 Â¿ wÃ*Ã â–
2=1 2=1
The statistic H^* is approximately distributed as chisquare
with p(Gl) 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.
BrownForsvthe Generalizations
Let Yjj = TXj j. Let , SÂ¡* 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 HotellingLawley 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 PillaiBartlett trace criterion
V* = trace{ H* [H* + (f)nr] L}
h
= trace[T H 1?[T H I* + (Â£) t M T/] 'x}
= trace{H[H+ â€” 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
rMr,
r I if + â€”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^^T1
2. Â¿ vfi = (r7) T'1 = (rO [Â¿ wj r1
2=1
2=1
2=1
3. yÂ± = TxÂ± .
4. y = (Â¿WiÃ'ÃÃ‰^i) ,
2=1 2=1
= [ (r') {Â¿ Ã¼r^r1](r/)"1rÂ¡i^'i1r12Â«i ,
21
2=1
= r(J2 Wj) ^(t') 't'Y, nisÂ¡1sti ,
2=1
2=1
= r(Â¿ WÂ¿) _1Â¿ = ric
2=1 2=1
Using results 14, is shown to be invariant as follows:
G
H*m = Â£ (Txi  Tx)'Wi{Txi  Tx) ,
2=1
= E [r^  x) ] '(r7) '1wiir 1 [T{xi  x) ]
2=1
= E (^i â€¢ 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 distributionsâ€”
normal and exponentialwere 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 nonnormal. 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 samplesize ratios between 1:1 and 2:1
(Lin, 1991). Olson (1974) examined only the case of equalÂ¬
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 ÃNRF) . 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 firstorder 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 HotellingLawley test (U^,
U2*) , the modified PillaiBartlett 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 pseudorandom observations was generated (using
PROC IML in SAS) from the target distributionâ€”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 PillaiBartlett 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 HotellingLawley 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 HotellingLawley (U^) , second modified
HotellingLawley (U2*) , modified PillaiBartlett (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 BrownForsythe 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 BrownForsythe 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 HotellinqLawlev 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 HotellinqLawley 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 PillaiBartlett 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).
BrownForsvthe 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
twoway interactions, all threeway interactions, and all
fourway interactions. Because R2 was .96 for the model with
fourway interactions, more complex models were not examined.
The R2 for all models are shown in Table 9. The model with
main effects and twoway through fourway interactions was
selected.
Variance components were computed for each main effect,
twoway, threeway, and fourway interaction. The variance
component (Â©j, i=l,...,255) for each effect was computed using
the formula 106(MSEFMSE)/(2X), where MSEF was the mean square
for that given effect, MSE was the mean square error for the
fourfactor 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. TwoWay Interaction. Three
Way Interaction, and FourWav Interaction Models when usina
the Four BrownForsvthe Generalizations
HighestOrder Terms
R2
Main Effects
0.52
TwoWay Interactions
0.77
ThreeWay Interactions
0.89
FourWay 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,+. . . +Â©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
andâ€”in contrast to factors such as d and DTdo 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 HotellingLavlev.
Second Modified HotellinqLawlev. Modified PillaiBartlett.
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 10â€”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 threeway 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 U2â€™ 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 BrownForsythe generalizations: (a) the
first modified HotellingLawley 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 HotellingLawley 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 HotellingLawley 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 PillaiBartlett test (V*) was adequate
when N:p was 20 and G was 3; (e) the modified PillaiBartlett
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
ÃU,*) and Second
Modified
Hotelling
Lawlev Test
(U,â€™)
for
Combinations of
Ratio of
Total SamÃ³le 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
PillaiBartlett
Test
(V*)
and Modified Wilks Test (L*] for
Combinations of
Ratio
of
Total SamÃ³le 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 BrownForsythe 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 BrownForsythe 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 BrownForsythe 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 BrownForsythe generalizations
tended to be conservative.
Effect of p. Shown in Figure 12, mean f was near a for
the BrownForsythe 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 BrownForsvthe
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
twoway interactions, all threeway interactions, and all
fourway interactions. Because R2 was .997 for the model with
fourway interactions, more complex models were not examined.
The R2 for all models are shown in Table 13. The model with
main effects and twoway through fourway interactions was
selected.
Variance components were computed for each main effect,
twoway, threeway, and fourway interaction. The variance
component (Â©j, i=l,...,255) for each effect was computed using
the formula 104 (MSEFMSE)/ (2*), where MSEF was the mean sguare
for that given effect, MSE was the mean square error for the
fourfactor 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,/[(Â©,+ .. .+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. TwoWay Interaction. Three
Way Interaction, and FourWav Interaction Models when usina
the Johansen Test and Wilcox Generalization
HighestOrder Terms
R2
Main Effects
0.767
TwoWay Interactions
0.963
ThreeWay Interactions
0.988
FourWay 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 andâ€”in contrast to factors such as d and DTâ€”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 * â€” â€”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 BrownForsythe 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 fiftysix 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 HotellingLawley test (U^), second modified
HotellingLawley test (U2*) , modified PillaiBartlett 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 BrownForsythe 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 HotellingLawley test is most effective at
maintaining acceptable Type I error rates.
Conclusion 3. If one can tolerate a somewhat liberal
test the first modified HotellingLawley 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 BrownForsythe
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 BrownForsythe
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 BrownForsythe
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 BrownForsythe
generalizations.
It is clear that the power of the BrownForsythe
generalizations needs to be investigated. It would be prudent
to compare the power of the first modified HotellingLawley
test, second modified HotellingLawley test, modified Wilks
test, and modified PillaiBartlett test with that of James's
(1954) secondorder 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 HotellingLawley Test
(U,*) , Second Modified HotellingLawley Test (U2*) , Modified
PillaiBartlett 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 HotellingLawley 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 16continued.
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 16continued.
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 16â€”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 16continued.
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 16â€”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 16â€”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 16â€”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 16â€”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 16â€”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 16â€”continued.
DT
P
G
N: p
NRF
NR
S
d2
u/
U2*
V*
L*
J
Â«;
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 16â€”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 16â€”continued.
DT p G N: p NRF NR S d2 U,* U * V* L* J H *
L 1 Â¿ 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 17continued.
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 17â€”continued.
DT
P
G
N: p
NRF
NR
S
d2
V
Â«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 17continued.
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 17â€”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 17â€”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 17â€”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 17continued.
DT
P
G
N: p
NRF
NR
S
d2
ur
V
V*
L*
J
Â«;
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 17â€”continued.
DT
P
G
N: p
NRF
NR
S
d2
V
U2*
V*
L*
J
Â«;
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 17â€”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 17â€”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 17â€”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 17â€”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 18â€”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 18continued.
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 18â€”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 18â€”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 18â€”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 18â€”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 18â€”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 18continued.
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 18â€”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 18â€”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 18â€”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 18â€”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
REFERENCES
Algina, J., & Oshima, T.C. (1990). Robustness of the
independent sample Hotelling's T2 to variancecovariance
heteroscedasticity when sample sizes are unequal or in
small ratios. Psychological Bulletin. 108. 308313.
Algina, J., Oshima, T.C., & Tang, K.L. (1991). Robustness of
Yao's, James', and Johansen's tests under variance
covariance heteroscedasticity and nonnormality. Journal
of Educational Statistics. 16(2), 125139.
Algina, J., & Tang, K.L. (1988). Type I error rates for Yao's
and James' tests of equality of mean vectors under
variancecovariance heteroscedasticity. Journal of
Educational Statistics. 13, 281290.
Anderson, T.W. (1958). An introduction to multivariate
statistical analysis. New York: Wiley.
Aspin, A.A. (1948). An examination and further development of
formula arising in the problem of comparing two mean
values. Biometrika. 35. 8896.
Aspin, A.A. (1949). Tables for use in comparisons whose
accuracy involves two variances, separately estimated
(with appendix by B.L. Welch). Biometrika. 36. 290296.
Bartlett, M.S. (1939). A note on tests of significance in
multivariate analysis. Proceedings of the Cambridge
Philosophical Society. 35. 180185.
Bennet, B.M. (1951). Note on a solution of the generalized
BehrensFisher problem. Annals of the Institute of
Statistical Mathematics. 25. 290302.
Boneau, C.A. (1960). The effects of violations of assumptions
underlying the t test. Psychological Bulletin. 57.(1) , 49
64 .
Box, G.E.P. (1954). Some theorems on quadratic forms applied
in the study of analysis of variance problems, I. Effect
of inequality of variance in the oneway classification.
Annals of Mathematical Statistics. 25. 290302.
141
142
Bradley, J.V. (1978). Robustness? British Journal of
Mathematical and Statistical Psychology. 31. 144152.
Bray, J.H., & Maxwell, S.E. (1989). Multivariate analysis
of variance (2nd ed.). Newbury Park, CA : Sage.
Brown, M.B., & Forsythe, A.B. (1974). The small sample
behavior of some statistics which test the equality of
several means. Technometrics. 16(1), 129132.
Clinch, J.J., & Keselman, H.J. (1982). Parametric
alternatives to the analysis of variance. Journal of
Educational Statistics. 7, 207214.
Cochran, W.G. (1947). Some consequences when the assumptions
for the analysis of variance are not satisfied.
Biometrics. 2, 2238.
Dijkstra, J.B., & Werter, S.P.J. (1981). Testing the
equality for several means when the population variances
are unequal. Communication in StatisticsSimulation and
Computation. BIO. 557569.
Everitt, B.S. (1979). A Monte Carlo investigation of the
robustness of Hotelling's one and twosample T2 tests.
Journal of the American Statistical Association. 74. 48
51.
Fisher, R.A. (1935). The fiducial argument in statistical
inference. Annals of Eugenics. 6, 391398.
Fisher, R.A. (1939). The comparison of samples with possible
unequal variances. Annals of Eugenics. 9, 174180.
Glass, G.V., Peckham, P.D., & Sanders, J.R. (1972).
Consequences of failure to meet assumptions underlying
the fixed effects analysis of variance. Review of
Educational Research. 42.(3), 237288.
Hakstian, A.R., Roed, J.C., & Lind, J.C. (1979). Twosample
T2 procedure and the assumption of homogeneous covariance
matrices. Psychological Bulletin. 86, 12551263.
Holloway, L.N., & Dunn, O.J. (1967). The robustness of
Hotelling's T2. Journal of the American Statistical
Association. 62., 124136.
Hopkins, J.W., & Clay, P.P.F. (1963). Some empirical
distributions of bivariate T2 and homoscedasticity
criterion M under unequal variance and leptokurtosis.
Journal of the American Statistical Association. 58.
10481053.
143
Horsnell, G. (1953). The effect of unequal group variances on
the Ftest for the homogeneity of group means.
Biometrika. 40. 128136.
Hotelling, H. (1931). The generalization of Student's ratio.
Annals of Mathematical Statistics. 2, 360378.
Hotelling, H. (1951). A generalized T test and measure o f
multivariate dispersion. In Jerzy Neyman (Ed.),
Proceedings of the Second Berkley Symposium on
Mathematical Statistics and Probability (pp. 2341).
Berkley : University of California Press.
Hsu, P.L. (1938). Contributions to the theory of 'Student's'
ttest as applied to the problem of two samples.
Statistical Research Memoirs. 2, 124.
Hughes, D.T., & Saw, J.G. (1972). Approximating the percentage
points of Hotelling's generalized To2 statistic.
Biometrika. 59. 224226.
Ito, K. (1969). On the effect of heteroscedasticity and nonÂ¬
normality upon some multivariate test procedures. In P.R.
Krishnaiah (Ed.), Multivariate AnalvsisII. (pp. 87120).
New York : Academic Press.
Ito, K. , & Schull, W.J. (1964). On the robustness of the T2
test in multivariate analysis of variance when variance
covariance matrices are not equal. Biometrika. 38. 324
329.
James, G.S. (1951). The comparison of several groups of
observations when the ratios of population variances are
unknown. Biometrika. 38. 324329.
James, G.S. (1954). Tests of linear hypotheses in univariate
and multivariate analysis when the ratios of the
population variances are unknown. Biometrika. 41. 1943.
Johansen, S. (1980). The WelchJames approximation to the
distribution of the residual sum of squares in a weighted
linear regression. Biometrika. 67, 8592.
Kendall, M.G., & Stuart, A. (1969). The advanced theory of
statistics. Vol. 1. New York: Hafner.
Kohr, R.L., & Games, P.A. (1974). Robustness of the analysis
of variance, the Welch procedure and a Box procedure to
heterogeneous variables. Journal of Experimental
Education. 43. 6169.
144
Korin, B.P. (1972). Some comments of the homoscedasticity
criterion M and the multivariate analysis of variance
tests T, W, and R. Biometrika. 59, 215216.
Lawley, D.N. (1938). A generalization of Fisher's ztest.
Biometrika. 30. 180187.
Levy, K.J. (1978). An empirical comparison of the ANOVA F
test with alternatives which are more robust against
heterogeneity of variance. Journal of Statistical
Computing and Simulation. 8, 4957.
Lin, W. (1992). Robustness of two multivariate tests to
variancecovariance heteroscedasticity and nonnormality
when totalsamplesizetovariable ratio is small.
Dissertation Abstracts International. 52.(8), 2899A.
Lindquist, E.F. (1953). Design and analysis of experiments in
education and psychology. Boston: Houghton Mifflin.
McKeon, J.J. (1974). F approximations to the distributions
of Hotelling's T02. Biometrika. 61. 381383.
Micceri, T. (1989). The unicorn, the normal curve, and other
improbable creatures. Psychological Bulletin. 105(1),
156166.
Nath, R., & Duran, B.S. (1983). A robust test in the
multivariate twosample location problem. American
Journal of Mathematical and Management Sciences. 3, 225
249.
Nel, D.G., & van der Merwe, C.A. (1986). A solution to the
multivariate BehrensFisher problem. Communications in
Statistics  Theory and Methodology. 15(2), 37193735.
Novick, M.R., & Jackson, P.H. (1974). Statistical methods
for educational and psychological research. New York:
McGraw Hill.
Olson, C.L. (1974). Comparative robustness of six tests in
multivariate analysis of variance. Journal of the
American Statistical Association. 69. 894908.
Olson, C.L. (1976). On choosing a test statistic in
multivariate analysis of variance. Psychological
Bulletin. 83(4), 579586.
Olson, C.L. (1979). Practical considerations in choosing a
MANOVA test statistic: A rejoinder to Stevens.
Psychological Bulletin. 8j6(6), 13501352.
145
Oshima, T.C., & Algina, J. (in press). Type I error rates for
James's secondorder test and Wilcox's test under
heteroscedasticity and nonnormality. British Journal of
Mathematical and Statistical Psychology.
Pearson, E.S. (1929). The distribution of frequency constants
in small samples from nonnormal symmetrical and skewed
populations. Biometrika. 21. 259286.
Pearson, E.S. (1931). The analysis of variance in cases of
nonnormal variation. Biometrika. 23. 114133.
Pearson, E.S., & Please, N.W. (1975). Relation between the
shape of population distributions and the robustness of
four simple test statistics. Biometrika. 62. 223241.
Pillai, K.C.S. (1955). Some new test criteria in
multivariate analysis. Annals of Mathematical Statistics.
26, 117121.
Pillai, K.C.S., & Sampson, P. (1959). On Hotelling's
generalization of T . Biometrika. 46, 160165.
Pillai, K.C.S., & Sudjana. (1975). Exact robustness studies
of tests of two multivariate hypotheses based on four
criteria and their distribution problems under
violations. The Annals of Statistics. 3, 617636.
Pratt, J.W. (1964). Robustness of some procedures for the two
sample location problem. Journal of the American
Statistical Association. 59, 665680.
Ramsey, P.H. (1980). Exact Type I error rates for robustness
of Student's t test with unequal variances. Journal of
Educational Statistics. 5, 337349.
Rao, C.R. (1952). Advanced statistical methods in biometric
research. New York: Wiley.
Ratcliffe, J.F. (1968). The effect on the t distribution of
nonnormality in the sampled population. Applied
Statistics. 17(1), 4248.
Rider, P.R. (1929). On the distribution of the ratio of mean
to standard deviation in small samples from nonnormal
populations. Biometrika. 21. 124143.
Rogan, J.C., & Keselman, H.J. (1977). Is then ANOVA Ftest
robust to variance heterogeneity when sample sizes are
equal?: An investigation via coefficient of variation.
American Educational Research Journal. 14, 493498.
146
Roy, S.N. (1945). The individual sampling distribution of
the maximum, the minimum, and any intermediate of the p
statistics on the nullhypothesis. Sankhva. 7, Part 2.
133158.
SAS Institute Inc. (1985). SAS user's guide, 1985 edition.
Raleigh, NC: Author.
Satterthwaite, F.E. (1941). Synthesis of variance.
Psvchometrika. 6, 309316.
Satterthwaite, F.E. (1946). An approximate distribution
of estimates of variance components. Biometrics. 2.(6),
110114.
Scheffe, H. (1943). On solutions of the BehrensFisher
problem based on the tdistribution. Annals of
Mathematical Statistics. 14. 3544.
Scheffe, H. (1959). Analysis of variance. New York: John
Wiley and Sons.
Scheffe, H. (1970). Practical solutions to the Behrens
Fisher problem. Journal of the American Statistical
Association. 65, 10511058.
Stevens, J. (1979). Comment on Olson: Choosing a test
statistic in multivariate analysis of variance.
Psychological Bulletin. 86(2), 355360.
Stevens, J. (1986). Applied multivariate statistics for the
social sciences. Hillsdale, NJ : Lawrence Erlbaum
Associates.
Tang, K.L. (1990). Robustness of four multivariate tests under
variancecovariance heteroscedasticity. Dissertation
Abstracts International. 5.1(5) , 2444B.
Tomarken, A., & Serlin, R. (1986). Comparison of ANOVA
alternatives under variance heterogeneity and specific
noncentrality structures. Psychological Bulletin. 99. 90
99.
Wang, Y.Y. (1971). Probabilities of the type I errors of the
Welch tests for the BehrensFisher problem. Journal of
the American Statistical Association. 66, 605608.
Welch, B.L. (1938). The significance of the difference
between two means when the population variances are
unequal. Biometrika. 29. 350362.
147
Welch, B.L. (1947). The generalization of 'Students' problem
when several different population variances are involved.
Biometrika. 34. 2335.
Welch, B.L. (1951). On the comparison of several mean
values: an alternative approach. Biometrika. 38. 330336.
Wilcox, R.R. (1988). A new alternative to the ANOVA F and
new results on James' secondorder method. British
Journal of Mathematical and Statistical Psychology. 41.
109117.
Wilcox, R.R. (1989). Adjusting for unequal variances when
comparing means in oneway and twoway fixed effects
ANOVA models. Journal of Educational Statistics. 14(3).
269278.
Wilcox, R.R. (1990). Comparing the means of two independent
groups. Biometric Journal. 32. 771780.
Wilcox, R.R., Charlin, & Thompson, (1986). New Monte Carlo
results on the robustness of the ANOVA F, W, and F*
statistics. Communications in StatisticsSimulation and
Computation. 15., 933944.
Wilks, S.S. (1932) Certain generalizations in the analysis
of variance. Biometrika. 24., 471494.
Yao, Y. (1965). An approximate degrees of freedom solution
to the multivariate BehrensFisher problem. Biometrika.
52, 139147.
Young, R.K., & Veldman, D.J. (1965). Introductory statistics
for the behavioral sciences. New York: Holt, Rinehart &
Winston.
Yuen, K.K. (1974). The twosample trimmed t for unequal
population variances. Biometrika. 61. 165176.
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
GkÂ¿Sl
Chairperson, Foundations of
Education
Dean, College of Education
Dean, Graduate School
UNIVERSITY OF FLORIDA
3 1262 08556 8235
