TESTING OF HYPOTHESES ON A MULTIVARIATE
POPULATION, SOME OF THE VARIATES BEING CONTINUOUS
AND THE REST CATEGORICAL
by
M. D. Moustafa
This research was supported by the United
States Air Force '~hrough the Air Force
Office of Scientific Research of the Air
Research and Development Command, under
Contract No. AF 18(600) - 83. Reproduction in whole or in part is permitted
for any purpose Qf the United States
Government.
Institute of Statistics
Mimeograph Series No. 179
July, 1957
,
....;.
3.
AFOSR-TN
57-440
4.
ASTIA-AD
136 430
UNCLASSIFIED
SECURITY INFORJvIATION
BIBLIOGRAPHICAL CON/fROL
SHEET
/
1.
University of North Carolina, Chapel Rill, N. O.
2.
Mathematics Division, Air Force Office of Soientific Researoh
5.
TESTING OF HYPOTHESES ON A MULTIVARIATE POPULATION, SOME OF
THE VARIATES BEING CONTINUOUS AND THE REST CATEGORICAL
6.
M. D. Moustafa
7.
July, 1957
8.
AF 18(600)-83
9.
File 3.3
TESTING OF HYPOTHESES ON A i1ULTIVARIATE
POPULATION; SOME OF THE
VARIATES BEING CONTINUOUS
AND THE REST CATEGORICAL
*
by
f'lj.
Summary:
D. Moustafa
We consider a (k+:).)-variate distribution in which k
variates are continuous and
i variates are categorical. The k
variates are assumed to have a conditional multivariate normal
•
distribution with respect to the
i categorical variates which are
assumed to have a multinomial distribution.
Appropriate hypotheses
are framed in this situation, analogous to the customary hypotheses
on a single lTDlHivariate normal distribution, large sample tests of
such hypotheses are developed and some of their properties studied.
Next, instead of assuming a single multinomial distribution on the
i categorical variates, a product of multinomial distributions is
assumed (in case some of the i-categorical are ways of classification) and hypotheses are framed in this situation analogous to the
customary ones for several multivariate normal distributions, and
large sample tests of such hypotheses and some of their properties
are studied.
•
*
This research was supported by the United States Air Force
through the Air Force Office of Scientific Research of the Air
Research and Development Command.
2
Introduction:
Many contributions have been made to the multivariate analysis
of variance and covariance of large masses of data on either continuous
or categorical variates.
In this paper, we are applying the technique of the multivariate
analysis of variance and covariance to a multi-way table such that
certain ways refer to continuous variates and the other ways are
categorical.
For certain problems all the categorical ways refer to
variates; for certain other problems all of them refer to ways of
classification; and for some problems some of the ways refer to
variates and the rest to ways of classification.
We use the fact that under certain broad conditions, the
•
-2 log A statistic, for large samples and on the null hypothesis,
has asymptotically the1l2-distribution with certain degrees of
freedom (Wilkes, Wald).
Also we use the fact that, for categorical
data, the -2 log A and the 1....2-statistics are both equivalent (Anderson,
Ogawa) •
In section 2, we oncsider the two way or (X,Z) table in which
we take X continuous and Z categorical.
Z may be a random variate
or a way of classification.
In section 3, we consider the case of a three way (X,Y,Z)
table, in which X is a continuous variate, Y and Z are both categorical variates or one of them a categorical variate and the other
a way of classification, or both are ways of classification.
In section
4,
we consider the case of a three way (X,Y,Z)
table in which X and Yare continuous variates and Z a categorical
variate or a way of classification.
3
Section 5 points out the possibility of an extension
to a
(k+l)-way 'table where thek's are continuous and the l's categorical.
In all these cases we formulate all possible hypotheses to be
tested concerning conditional independence, joint independence,
and total independence; and we give the -2 log A statistic used,
The likelihood ratio A is defined
together with its distribution.
as:
max P(sample , H )
o
A '"' -------~-.;...
max P(sample H)
__
P(x ,·· . ,x,
'-,'1
l .
'"'-"._._
-
p
i
...
r Plo,
p(x~,
... ,x
a
-.L.
-n I
,
PI'
where, if the null hypothesis is such that Pi '"' p~ for i
•
••• , v, and
H rHo,
= 1,
2,
then the numerator of A is the likelihood
function after subs'ljituting for Pi' i
= v+l, .•• ,
m, the maximum
'" 0
.
likelihood estimates Pi
subJect
to the null hypothesis Ho; and
the denominator of A is the likelihood filllction after substituting
the maximum likelihood estimates Pi' i
= 1,2, .•• ,m,
of Pi subject to
the alternative H.
In all these cases, we assume that the conditional distribution of the continuous variates, given\the categorical variates, is
a multivariate normal whose parameters might depend upon the values
of the categorical variates (at which the conditional probability
is being considered).
In case some of the directions along the
categorical part of the table refer to "ways of classification,"
the parameters of the conditional normal dlstribution might also
depend upon the~paI'ticular cell of the part of the table constituted
,
.".
4
by the ways of classification.
In every case it can be shown
(and has, in fact, been shown in another paper) that the conditional
distribution of the continuous variates, given the categorical ones,
satisfy Doob's conditions, from which, by using certain other theorems,
it has been proved in the other paper that -2 log A for the joint
distribution of the continuous and categorical variates has asymp2
totically a1l -distribtition with proper degrees of freedom.
Actually,
however, in each case, it is not -2 log A but another algebraically
simple and more convenient statistic, proved to be equivalent in
probability to -2 log
~,
that is used.
A method of obtaining such
a statistic is outlined in section 6) the mathematical proof being
reserved for another paper.
•
2•
The case of the two way (X,Z) table, X continuous and Z categorical:
Z may be a random variate or a,way of classification, which may
belong to r exhaustive and exclusive categories i • 1,2, ••. ,r, say.
We assume that the distribution function of X is normal for the different categories of Z but that the parameters, viz., the mean and
the variance of the distribution funotion of X depend on the category.
Suppose we have a sample of n inidviduals; everyone belongs to
one or another of the r categories, and also has a measurement x.
Let n , i = 1,2, ••• ,r, be the number of individuals belonging to, the
i
th
r
~
i
category such that Z n. a n is fixed from sample to sample.
i=l J.
If Z is a way of classification, then n., i • 1,2, ••. ,r, is fixed
J.
I.
from sample to sample; but if Z is a random variate, then suppose
th
that Pi is the probability that an individual belongs to the i
category.
5
~ ~ ~ ~ ~ ~
2.1
According to our assumption,
variate:
the model of the conditional likelihood function is
(2.1.1)
x1,···,xnl
~}
=
!c
J.=1
x
2n~i)
(
exp
-
n
i
)'2
nzi
r
Z
(x .. -1J.(i» _2
,
_...;J.;;,wJ
i=l j=l
2V{ i)
where x .. is the jth measurement on X of an observation belonging
J.J
'
to the ith category, and ~I
= (n1 ,
n
2
••• n ).
Therefore, the
r
likelihood function is:
r
~
(2.1.2)
•
1\
i=l
p.
n.J.
•
J.
We are interes'ted in testing the composite hypothesis that
the distribution of X is the same in the r-different categories, i.e.,
H : lJ.(r) = !-L(r-l)
o
and
= ... = !-L(l) = !-L
,
= v (1) = v
,
V<r)'= v(r-l) =
against the alternative H" Ho ' where !-L and v are nuisance parameters.
n.
For n very large, and
= a constant, the -2 log A s'tatistic is
r1:
(2.1.3)
-2 log A =
r
h
Z n. log
i=l
J.
v
V( i)
~
We shall prove in another paper, that this st<:ltistic is asymptotically
equivalent, in probability, to
(2.1.4)
r n. (~(i) _ ~)2
Z J.
i=l
1\
v
+
n ( v~( J.. )
") 2
.. V
i
Z -.;-------
r
i=1
2.02
,
6
1
ni
n.
ZJ. x ..
j=l J.J
1
J.
n.
Zl. (x .. _ ~(i»2
j=l J.J
~(i) = -
where
~ (i) = n.
and
,
/I
I.l. =
r
1\
1 Z
n.
n i=l J. lJ.(i)
r
n
1
v= ...
Z Zi (xij - "")2
IJ.
n i=l j=l
,
,.
A
and that it has the~2-distribution with 2(r-l) degrees of freedom.
~ ~ ~ ~ ~
2.2
wa~ ~
!
classification:
Here, we have r independent normal populations; and the
hypothesis to be tested is that X(i) , i
= 1,2, .•. ,r,
have identical
distributions.
If n.J. is very large, for i
....
•
(2.1.)
= 1,2, .• ",r, then the statistics
and (2.1.4) hold and have the same 112-distribution with
2(r-l) degrees of freedom, though the
asyn~totic
power of this
test is different from that of the previous section.
).
The case of a three way (X,Y,Z) table, X continuous, and
I, Z categorical:
Suppose
"~"hat
Y
can belong to the categories i = 1,2, ••. ,r,
and Z can belong to the categories k
= 1,2, •.. ,s.
We assume "that, for given cell (i,k), the probability density
function of X is normal with mean lJ.(i,k) and variance v(i,k).
Suppose we have a random sample of size n, where n is fixed
from sample to sample, such that n individuals belong to the
ik
th
r
s
(i,k)
cell and E Z nik = n. Also every individual observation
i=l k=l
has the measurement x ikj
j = 1,2, ... ,n
ik for i = 1,2, ... ,r and k =
1,2, .•. ,8.
We have
s
E n' k
k=l J.
=:
n
io
,
and
r
l:
1=1
n. k = n k'
J.
0
We
7
could have nio ' i = 1,2, ..• ,r, or nok ' k = 1,2, .•• ,5, or n ik fixed
in advance in which case either Y or Z or both would be ways of
classific3tion.
Bo'th - and - are random vari ate
--
3.1
Let
Y
--.
Z
E' = (nIl n12
s:
-.;;;,;;,;;,,;;;.;~ ~.;.-.;;;;,,;;.;..;;.;;..;.
nls ' n21 n 22 ••• n?s' .•• , nrl nr2 ..•
and Pik is the probability that an individual observation belongs
II'
Urs )'
to the (i,k)th cell, where
s
r
= p.1.0
E P'k
k-1 1.
and
s
E p. ,. 1
i"'l k=l 1.k
~
The conditional likelihood function will be
(3.1.1)
•
.'
P { xl,···,xnl
.
x
exp
B}
r
s
E
E
r-IT (
...
1
)
i, k . 2nv(i,k)
n
Eik
(x
~
ik J. - IJ.( i,k»2
i=l k=l j=l
,
2v(i,k)
and therefore the likelihood function in this case is
(J.1.2)
P { xl'''' ,Xn
'!!} = P { xl""
,xn
\!!}, J\
::k!
ZVi~ik
i,k
We consider the following kinds of composite hypotheses:
(3. 1 ,a)
Conditional independence between! ~ 1, siven~:
Given that Z belongs to the kth category, then
for i . 1,2, ... ,r
,
,
and
v(i,k)'" v(o,k)
and
k
= 1,2, ... ,8
agains't H :f Ho ,
where lJ.(o,k) and v(o,k) are arbitrary nuisance parameters.
8
The -2 log
~
statistic is
s
r
"
~ n, log v(o,k)
k=l i=1 ~k
~(i,k)
E
(J.1.3)
,.
and this, except for a quantity that converges to zero in probability,
is equal to
Further, it has the J(2-distribution with 2s(r-l) degrees
of freedom.
We note that
1
~(i,k) '" ~
nik
•
n ik
~ x'k'
j=l ~ J
,
~(o,k)
~(i,k)
r
~ n'k ~(i,k)
i"'l ~
,
~'o,k)
Likewise, we can test the conditional independence between
X and Z,
giv~n
Y, in which case the -2 log X statistic has a
~-distribution with 2r(s-1) degrees of freedom.
(3.1,b) Independence between ~ jointll ~ ~:
The likelihood function in this case can be written in the
form
n
(J.1.6)
x
p
i
exp -
Xl".' ,xn,E
s
r
2:
2:
I E2]
n' k
ZJ.
j=l
k=l r=l
1T (;
:It
1
2nv( i,k)
i, k
2
(x ikj - lJ.(i,k»
9
ik
2
)
1
n k 'JT
r p,* n'kJ
.JL
J.
k=l r
i=l J.k
11" n'k 1
i=l J.
s {
"IT
2v( i,k)
,
f j
and
P !!2
nJ
:It
6
"IT
k=l
n kl
"'sIT
p ok
n
k-l
ok
0
From (3.1.8), we can write the null hypothesis as:
H:
o
= lJ.(i,o)
lJ.(i,k)
and P'k
J. = p,J.O P0 k
H •
against H
o
,
v(i,k)
'
for i = l, .•• ,r, k = 1, ••. ,2 ,
=
v(i,o)
,
r
The -2 log A statistic in this
~
case
~ n. log V(i,?) ...
i=l k=l
J.k
v(i,k)
is
2
~
i=l
and it is asymptotically equivalent, in probability, to:
r
(3.1.8)
2:
s
Z
L
n'k(a(i,k) J.
i=l k=l
~(i,0»2
n'k(~(i,k) - ~(i,0»2
2v 2(i,0)
-to -:;.;J.:--.._....,.~----
v(i,o)
Gnik
+
-
k)2
n.J.O no.
n
_7.
nion ok
n
further, this statistic has the "'X.2-distribution with 2r(s-1)
(r-l)(s-l)
= (s-1)(3r-l)
degrees of freedom.
+
,
10
Likewise, we can test the independence
and Y.
between (X,Z) jointly
Here, the -2 log ~ statistic has asymptotically the)l2-dis -
tribution with (r-l)(3s-1) degrees of freedom; and it will be
asymptotically equivalent, in probability, to:
n'k(~(i,k) - ~(O,k»2
r
s
1:
(J.1.9)
1:[":1
k=l
••
i=l .
.
;(o,k)
n. n
( n ik _ :1;0n 0
n.10n0 k
n
+
betwe~~! ~
(3.1,°) Inddpendence
k) 2
(Y,Z) jointly:
Th6 hypothesis to be tested is
H:
o
,
lJ.(i,k) = IJ.
v( i,k) = v
,
for all i, k ,
where IJ., v are nuisance parmneters, against H ~ H •
°
The -2 log X statistic is
(J .1.10)
r
s
1:
1:
n
i=l k=l
v'"
~(i,k)
s
....
1 r
1:
v=-n 1:
i=l k=l
where
and
log
ik
1
A
=-
I-L
r
s
1:
1:
n i=1 k=-l
n' k
1:J.
j=l
,
n'1k
(X· k · - IJ.
j=l
:1 J
"l ,
1:
x ikj
. -n1 1:r
i=l
s
Z n.;(i,k)
k=l J.
This stntistic is asymptotically equivalent, in probability, to
( 3 • 1.11)
~
~l
~,- nik(lJ.(i~k) 1\
~1
J....
v
")2
I-L
+
,,(.)
-"\)2
:J~k(~::~ - v A
2
~.
_7 ;
and it has the "X,2-distribution with 2(rs-l) dei5rees of freedom.
11
(3;1,d' Total independence between ~ ~ variates:
In view of (3.1.2), the hypothesis to be tested is
H:
o
lJ.(i,k)
= IJ. ,
v(i,k)
:It
v ,
and Pik .. Pio P ok for all i, k ,
against H ~ Ho ' where IJ., v, Pio and Pok are all
nuisance parameters.
The -2 log A statistic is
r
i:
(J.1.12)
S
i:
i=l k=l
£"
A
log _ v
~k
v(i,k)
n'
and this is asymptotically equivalent, in probability, to
r
(3.1.13)
.
s
i:
~
i=l k=l
£"
nik(~(i,k) - ~)2
+
A
nik(;(i,k) _ ~)2
2v 2
V
(n
+
n
n
_ 10 ok )2
ik
n
It has the~2-distribution with 2(rs-l) + (r-l)(s-l) = 3rs - r - s - 1
degrees of freedom.
3.2
!
~ ~
random
varia~~, ~ ~ ~
way
£!
~~assification:
In this case, the hypothesis of (3.l,a) would be replaoed
by the hypothesis that X and Yare independent for each Z separately,
that of (3.l,b) would be replaced by the hypothesis that the joint
distribution of (X,I) is the same for all Z, and that of (3.1,0)
does not seem to have an immediate analogue.
The hypothesis of
(3.l,d) would be replaced by the hypothesis that X and Yare independent, each having a distribution which is the same for all Z.
12
For each of these problems the statistic and the distribution
(on the null hypothesis) are the same as for the corresponding
problem of the previous cases, although the asymptotic power of the
test for any problem in this case would be quite different from that
for the corresponding problem of the previous case.
3.3
~! ~ ~ ~
Here n
.21
=:
(n lo
ik
ways
£! classification:
, for each i,k, could be fixed or the marginals
n 20
nos) could
be fixed.
If nik , i =: l, ••• ,r, k =: l, ••• ,s, are fixed from sample to
sample, we should have rs independent variates A(i,k); on each
(i)
we have a sample of n observations.
ik
The hypothesis (3.l,a) should be replaced by the hypothesis that
the distribution of X(i,k) will be independent of Z for each Z.
The hypothesis (3.I,b) and (].l,d) have no analogues.
But the
hypothesis (3.l,c) should be replaced by the hypothesis that X(i,k)
will have the same distribution for eachY and Z.
For each of
these two problems, the statistic and the distribution (on the
null hypothesis) are the
same as for the corresponding problems
in section 3.1, except that the asymptotic power of the test for any
problem here would be quite different from that for the corresponding
problem in section 3.1.
(ii)
...
If the marginals -nI' .,. (nl 0 n 20 • •• nro ) and -n 2' => (n'l
n"2
0
0
nos) are fixed, the likelihood function of the sample will be
.. (J.].l)
P
{Xl ,x 2 ,· .. ,xn ,!!1
!!l andl!2 fixed]
= P L xl'x 2 ,· .. ,xn ll!
x P
, such thdt
~l
1.2 ) !!l and l!2 fixed J
and.22 fixed
J
13
where
I
in which case p{!! !!l and !!2 fixed
J
could have been obtained from
nJ
= -~_. 1T
nik
Pik
j'f n'kJ i,k
, k J.
J.,
by putting Pik = PioP ok and, under this condition of independence,
finding P { !!
subject to !!l and !!2 being fixed. This makes it
J
difficult to write down P { !! \ 121 and !!2 fixed \
under a distinct
assumption other than Pik" PioPok (Roy and Mitra). If the nik's
are very large, the hypotheses (3.1,a) and (3.1,c) would be the same
in this case and each will have the same statistic with the same
1l2-distribution under the null hypothesis. For the hypotheses
corresponding to (3.1,b) and (3.1,d), the statistics could not be
obtained directly, due to the form of P { !! \!!l and !!2 fixed
may be taken over from (3.1,b) and (3.1,d) by analogy.
1; but
The power of
each of these two tests cannot be obtained.
4.
The case of a three way (K,Y,Z) table, X and Y continuous, and
Z categorical:
Suppose that Z belongs to the i th category, where i
= 1,2, .•. ,r;
and for given i, we assume X and Y to have bivariate normal distribution
with parameters given by the vector ~'(i) .. (~l(i)
and
v (1)
for variance-covariance matrix.
~2(i»
for means,
14
Suppose we have a sample of n observations, where n is fixed
from sample to sample, such that n. individual observations belong
l.
r
th
to tho:.; i
category, i • 1~2, ... ,r, and ~ n = n. Also every
iel i
individual observation has two measurements (x .. , Yi') , j = 1,2, .•. ,n ..
l.J
J
l.
n
We could have n: as random numbers subject to .2: n = n being fixed
l.
l.=l i
in which case Z would be a random variate, or n. fixed for i = 1,2, ••. ,r,
l.
in which case Z would be a way of classification.
4.1
~ ~
categorical variate:
Let Pi be the probability that an observation belongs to the
th
category,
£
P. = 1; and put Xl • (x y).
i=l 1.
The likelihood function, in this caso, is
i
•
=
r(
)ni
i ""1 \ 2n 1veil t .
"IT I
1
-
172
x--r
nl
"IT
n.!
exp -
r n2: i
2:
i=l j=1
r
n.
p.l.
i=1 l.
T[
i=l l.
We consider the following problems in testing of hypotheses.
(4.1,a)
.Q£.nd!~~ lEdependence between! and
If Z belongs to the i
th
(4.1.2)
r
2: n. log
i=l l.
r H.
o
1
.. 2
(1 - r (i»
given~:
Ho : v (i) = 0 ,
12
The -2 log A statistic is
category, we set
for i = 1,2, ••. ,r, against H
1,
,
1( I
-2 x ..
-l.J
1$
A
where rei) is the sstimate of the ordinary correlation coefficient
calculated from the i th category; and, except for a quantity that
converges to zero in probability, is asymptotically equivalent to
r
~
i=l
,.2
n. r (i)
l.
Further, it has the 1l2-distribution with r degrees of freedom.
(4.l,b) Independenc~ between (X,Y) jointly ~!'
The hypothesis to be tested is
1: 1 (i)
H I
o
v (i)
•
against H
rH
o
=
1: '
for i '" 1,2, ••• ,r,
'" V
, where i!: and v consist of five arbitrary parameters.
The -2 log A statistic is
(4.1.4)
~ n.
~
i=l
~
log '-V,- -
log,. ( .) '" (. ) ( ... 2( .) ;
vII ~ v22 1. l-r i)
I V( i)f
and this is asymptotically equivalent, in probability, to
") +
( "()
Uli!:
i
- 1!:
x
where
VI
=
(vII
(
U1
=
,,-1
~-l
11
v12
A-I
.. -1
vl2
•
v22
v22
v12 )
,
)
,
l:r
i=l
n. ("(
v i) - v")1
1.-
u 2(Y( i) -
,y) ,
16
2
'21("-1)2
vn
1(,,-1)2
2 12
v~ -1 ,,-1
v
ll 12
1("·1)2
2 v12
1('-1)2
2 v22
'v* -1 •v-1
hl
v--1
ll v12
A·l
••1 v12
v =-
",-1 ",-1
v22 v12
..
22
,
Iv t
.4-1
v22 + (,,-1)2
v12
vll .11-1
-v'"12
~-l
v ..
12
,
22 12
tv 1
,
and 'r" is the correlation coefficient estimated from all the categories
pooled. This statistic has asymptotically the1l2-distribution with
5(r-l) degrees of freedom.
(4.1,c) Independence between (X,Z) jointly ~
A
1:
necessary and sufficient set of conditions for this is that
v
22
(i) .. v 22
v12 (i)" 0
'
,
for i .. 1,2, ••• ,r ,
against H " Ho
That this set of conditions is sufficient is obvious; but the necessity
we prove in the appendix.
The -2 log
~
statistic is
r
2:
(4.1.6)
n. log A
i=l ~
v
v 22
.
This statistic, except for a quantity that converges to zero in probability, is equivalent to
.
r
2:
L
1
~=
n.(a (i) - ~2)2
2
....;~-......;--------
vIt 22
"2 -
22 (i)(1-r (i»
.
17
2
and has the X -distribution with (Jr-2) degrees of freedom.
~
4.I,d
independence:
The hypothesis we are interested in is
H:
o
j;!:( i) =
u
J;;;
,
V(i)
=
V
for i = 1,2, .•• ,r , against H
,
and v " 0
12
,
r Ho
Here we shall have four arbitrary nuisance parameters, viz.,
~l' ~2'
v ll ' v22
The -2 log X statistic is given by
(4.1.8)
r
l:
n log
i=l i
and this is asymptotically equivalent, in probability, to
2
2
r { ni(~k(i) - ~k)2 + ni(vkk(i) .. Vkk ) }
l:
k=l ial
V
2v
kk
kk
EL
#0
_7
,.
+
r
l:
n (v"
i 12 (i»
2
i=l
Further, it has the~2..distribution with 4(r-l) + r
= 5r
- 4
degrees of freedom.
There is a case which we could not test, viz.,
4.l,e
Conditional independence between!
~~,
given
1:
Owing to some difficulties about establishing for this situation
the asymPtotic~-distributionthat has been established for the other
situations by using a line of proof discussed in another paper, we
•
could not test this case •
18
4.2
~,~
way
~
classification:
If Z may belong to r categories, then we shall be dealing with
r independent bivariate noraml populations.
The hypothesis in (4.l,a)
will be that X and Yare independent in all the r different bivariate
populations, that in (4.1,b) will be that (X,Y) will have the same
distribution in all different r populations, that in (4.1,c) will
have no analogue here, and finally that of (4.l,d) will be that X
and Yare independent and have the same distribution in all the
r different populations.
For each of these problarns, the statistic and the distribution
(on the null hypotheses) are the same for the corresponding problems
in section 4.1, although the asymptotic power of the test for any
problem here would differ from the corresponding problem in section 4.1.
5.
From the line of proof of these results, which will be given in
another place, similar test problems to these could be considered
generally for a (k+})-way table in which k ways refer to continuous
vari3tes and the
~
ways are categorical.
The i categoricnl ways
may refer to all random variates or all ways of classification or
some of them refer to random variates and the remaining to ways of
classification; and the hypotheses have to be phrased accordingly.
6.
In every case we consider, we give, besides the -2 log A statistic
which we calculate
directly from (1.1) according to the null hypo-
thesis Ho and the alternative H, a statistic that differs from the
-2 log A by a quantity that converges to zero in probability. Such
•
a statistic will be called one whiCh is equivalent in probability to
19
-2 leg A.
This statistic has the same 'pattern for all thd cases,
and could be written down easily once the null hypothesis for any
p[jrticulP.Jr
case
is laid down.
We give here an outline, without mathematical proofs, of
the method we used to obtain such a statistic (which is asymptotically equivalent, in probability, to the -2 log
The
mathematical proofs are left to
Having satisfied
be
~
statistic).
given in another paper.
ourselves that the probability density
function of the variates considered, in every case, comply with
Doobls requirements, we can use the fact that, for large samples,
the -2 log ~ has the 1l2-distribution with certain degrees of freedom.
If the null
•
hypoth(~sis
H is no·t a simple hypothesis, as happens in
o
simple hypothesis Ho* and
consider a st3tistic equivalent in probability to -2 log ~ . where
311 the
onsaa W8
consider, we define
a
l'
I
max P (sample H:)
A
1
=------~-
max
I
P (sample Ho )
Alsu, together with the alternative hypothesis H, we define a simple
hypothesis H*
=H:
and consider
[l
statistic equivalent, in probability,
to -2 log A2 , wher8
A
2
=
max P (sample \ H*)
0
max P (sample tHo)
From -2 log ~l based on H* against Hand -2 log A2 based on H*
0 0 0
against H, we g8t -2 log A based on H against H by the following
o
form
(6.1)
-2 log A
=
-2 log A2 + 2 log Al
20
Now, if on the right side of (6.1), we replace, as indicated above,
-2 log A and -2 log Al by statistics equivalent to them im probability,
2
the right side of (6.1) becomes now a statistic equivalent in probabil-
ity to -2 log A, and involving H*.
It is possible to replace this by
o
another statistic, equivalent in probability, and to show that this
latter equivalent statistic does not involve the unknown true values
. .~n H~~ • Furthermore, since this is equivaofth e parame t ers occurr1Dg
o
lent in probability to -2 log A, therefore this has asymptotically
the central1l2-distribution with appropriate degrees of freedom, if:
(a)
H: is true
and (b) the,maximum likelihood estimates of the para-
meters are in the neighborhood of H*.
Ccmbining these two results
o
we observe that this equivalent statistic involves only H (and H)
o
2
and has a limiting central1l -distribution (a) for all H* contained
~.
o
in Ho and (b) for all maximum likelihood estimates of the parameters
that lie in the neighborhood of the unknown true H*o contained in H0 •
The feoture (b), otherwise called the property of consistency, is
proved for each case separately in the other paper.
W6 give us an example, the problem in section 2.1 in which
we h-nve:
and
v(i)
=v
for i
= 1,2, ••. ,r,
and v are 8rbitrary nuisance parameters, against H f rl o •
We dafine H* as:
o
(i)
0
, i = 1,2, .. . ,r,
H*:
and v(i) = v • vo
where
wher,=-
~
~
.
~
o
~
=IJ.=IJ.
and v o ore the true values of the paramdters iJ., v, and we
consider H~ a~ainst Ho '
-2 log Al
We get:
~
"
n ( IJ.
0)2
- IJ. _
o
V
+
("
n v - v
2v
o2
0)2
21
. der~ng
.
cons~
Also,
H~~ ! H*
0
. t
aga~ns
n.(~(i) _ ~o)2
r
-2 log 11. ::::
2
From Al and
11.
H,we ge t :
~
Z
+
i=l
2 ' we get:
(6.2)
a
-2 log
~
r
Z
-2 log A
f
i=l
2 + 2 log Xl
n.(~(i) - ~o)2
11.
"
0 2
~~--o---- _ n(p: ~ l+ )
v
J
v
Now, if ~ and ~ are in the neighborhood of the true values ~o
and
~
V
O
(irrespective of these values),
and this follows from the
consistency of ~ and ; which is proved in the other paper --
then
we can show that, except for a quantity that converges to zero in
probability, the -2 log A given in (6.2) is equal to
r
ni(~(i) - ~)2
Z ......- - - - -
i=l
v
r
+
E
i=l
n.(~(i) _ v)2
~
,
2v?
which is the statistic we suggest for this case; it does not involve
the true values
0 0 2
~ and v , and has a limiting"1C -distribution
2(r-l) degrees of freedom, no matter what ~o,
O
V
might be.
with
22
APPENDIX
(~ and~)
To prove that for the independence of
jointly and
1,
a set of necessary and sufficient conditions is that
~2(i)
= ~2
'
v22 (i) = v 22
for i
'
v12 (i) = 0
= 1,2, ••• ,r
We proceed as follows
x
•
nS
j'( n. t
. 1
1=
If
(~
and
~)
J.
r
Tr Pi~
i=l
jointly are to be independent of 1, then
be independent, and
~
~
1
and
E
and 1 should
For necessity
are independent, and next that
and 1 are independent.
(i)
and we require this to be independent of E'
We have
~
and 1 should also be independent.
therefore, we prove that (i)
(ii)
n.
r
= 'IT (
i=l
23
n.~
1
r
)2" exp _
2nv22 (i)
(Yi.(~2(i»2
n.
1:~
Z
i=l
2v (i)
22
j=l
.
x
J
therefore
r
r
i=l
i-I
'IT (
( Yij -IJ. 2( i ) )
Z
2v (i)
22
In order that this should be independent of categories i, given
r
n (a vector of random nnmberssubject to 'Zl ni ... n fiXed), we
~=
require that :
a function of 1 only for any £
•
Suppose that
n
and let
n
(1) ,
... (n
l + 1, n 2, ••• , ni - 1, •.• , nr )
,
(0) I
then
=
1
This is true for any Yl n +1 ,Yi,n.
, 1
~
and if
1J.2(1) ... 1J.2(1)
for i
•
= 1,2, ••• ,r
This shows that
, are
necess~ry
•
Y1 ,n +1'" y.1.,n ... 0 , then
1
i
v22 (1)'" v22 and 1J.2(i)'" 1J.2
conditions for the independence
'
2
24
of (x and n) jointly and 1. •
-
(ll)
-
For x Bnd 1 to be independent, we have that
-
P(~,B ~ 3.)
=
p(~ \ 3.,E)
P<,~)
P(3. \ !!)
P(I)
r
v
ni
r
22
.. )( (
)2' exp - ~
i=l 2n IV(iH
i=l
nl
x---'r
Jr
•
i=l
and
P(~ \].)
=
nil
r
IT
i=l
n.
P.l.
].
2
I V( i) ~
,
i: P(~,E 11.)
n
this should be independent of 1. , i.e.,
= ¢(~)
, (say) ,
where
x exp -
r
i:
i=l
,
i.e., equals the unconditional distribution of x
If this is to be true for all x .. and y.. (which are random variates),
l.J
l.J
then v12 (i) = 0 for i = l,2, .•. ,r. This therefore, is another necessary condition for (x and n) to be jointly independent of n. That
-
-
-
this set of conditions is sufficient for the independence of (! and E)
25
jointly with 1 is evident.
In conclusion, the author expresses his thanks to Professor
S. N. Ruy for suggesting the problem and the general line of approach,
and to Professor J. Ogawa for his many valuable comments and suggestions.
REFERENCES
1.
Anderson, T. W. and Goodman, L. A. (1957), "Statistical Inference
about Markov Chains,"
2.
,
~. ~'.
Statist. 28, pp. 89 - 110 •
Cramer, H., "MathematicJl Methods of Statistics," Princeton
Univ9rsity Press, 1946.
3.
Doob, J. L. (1934), "Probability and Statistics, If
Math.
.e
4.
~.,
Trans.~.
34, pp. 759 - 775.
Mitra, S. K. (1955), "Contrioutions to the Statistical Analysis
of Categorical data," North Carolina Institute of Statistics,
Mimeo. Series 142.
5.
Ogawa, J. (1957), "A limit theorem of Cramer and its generalizations,II North Carolina Institute of Statistics, Mimeo Series 163.
6.
Ogawa, J. (1957), "On the mathematical principles underlying the
theory of the x.2-test, II North Carolina Institute of Statistics,
Mimeo. Series 162.
7.
Ogawa, J., lVIoustafa, M. D. and 11.oy, S. N., (1957), "On the
Asymptotic Distribution of the Likelihood Ratio in Some Mixed
Variates Populations," North Carolina Institute of Statistics,
Mimeo Series (to be published) •
B.
•
Roy, S. N. and lVlitra, S. K.(1956), "An introduction to some nonparametric generalizations of analysis of variance and multivariate
26
Gnalysis," Biometrika, 43, pp. 361 - 376.
9.
W-31d, A., (1943), "Tests of Statistical Hypotheses concerning
several parameters when the n:umber of observations is large,"
Trans.
~. ~" ~.,
54, 426 - 482.
10. Wilks, S. S., (1938), "The large-sample distribution of the
likelihood ratio for testing composite
Statist., IX, 60 - 62,
e.
hypotheses,"~. ~.