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HYPOTHESES OF NO INTERACTION IN FOUR-DIMENSIONAL
CONTINGENCY TABLES
v. P. Bhapkar
University of North Carolina
University of Poona
and
Gary G. Koch
University of North Carolina
Institute of Statistics Mimeo Series No. 449
September 1965
Hypotheses of no interaction of various orders in fourdimensional contingency tables are considered appropriate to
the underlying probability models. General and computationally
simple test procedures based on Wald's criterion (1943) are
offered and illustrated.
This paper is a follow-up of the earlier expository
paper for the three-dimensional contingency tables.
This research was supported by the National Institute of Health
Institute of General Medical Sciences Grant No. GM-12868-0l.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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HYPOTHESES OF NO INTERACTION IN FOUR-DIMENSIONAL
CONTINGENCY TABLES
1.
Introduction
In a previous expository paper, the authors have discussed hypotheses
of "no interaction" in three-dimensional contingency tables (See Bhapkar
and "Koch [1965] )." There the importance of specifying the underlying model
and of formulating hypotheses appropriate to it was strongly emphasized.
In addition, it was demonstrated that for any given formulation, a
computationally simple statistical test (based upon a criterion due to
Wald [1943] ) may be constructed.
In this paper, hypotheses of "no interaction" in four-dimensional
contingency tables are considered from the same point of view as outlined
above.
First of all, there are four principal types of four-dimensional
contingency tables of experimental interest.
(i) •
(ii) •
(iii).
(iv) •
These are
the "four response, no factor" tables,
the IIthree response, one factor" tables,
the II two response, two factor ll tables,
the II one response, three factor" tables.
Given any particular one of the above situations, our object is to obtain
a clearer understanding of the behavior of experimental units by
investigating whether the structure of the underlying model can be
simplified in a meaningful manner.
We shall see that one way of
accomplishing this is by considering hypotheses of "no interaction ll •
Some of the problems arising here are natural extensions of those discussed
for the three-dimensional case; others, however, have no such analogue.
Before proceeding any further, we recall that in multi-dimensional
contingency tables, the term II no interaction" has been given two
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interpretations.
One is related to the nature of the association among
responses; the other, to the nature of the way in which factors affect
(the distribution of) responses and functions thereof.
This distinction
will be emphasized throughout this paper since it is important to the
formulation of appropriate hypotheses of "no interaction" for the models
cited previously.
Finally, it may be pointed out here that only the
case (i) has received an adequate consideration in the literature so far;
reference may be made to Darroch [1962J, Good [1963], Goodman [1964].
NOTATION:
The subscript "0" denotes that the corresponding suffix has
been summed ov·er.
The subscript * denotes that a quantity is independent of the
corresponding suffix.
The subscript "." denotes that the definition of a quantity
does not involve the corresponding suffix.
2.
2.1.
Formulations of Hypotheses of "No Interaction ll
The IIFour Response, No Factor" Model
Let Phijk ( where we assume Phijk
>0
denote the probability that
an experimental unit belongs to the response (h, i, j, k) - i.e., to the
h th category of the first response, the i th category of the second
response, the j th category of the third response, and the k th category of
the fourth response - in an r x s x t x u contingency table; thus, the
pIS are subject to the constraint
Z
Phijk
h,i,j,k
We shall say that there is
II
=1
no third order interaction" among the
four responses if some measure of association among any three of the
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responses is constant over levels of the fourth.
Before proceeding any
further, however, we need to clarify what we mean by a measure of
association among three responses.
2 x 2 x 2 x 2 table.
Let us consider for a moment a
If
lllljk
=
Pllj~22jk
P2ljkP12jk
j
= 1,
2
k
= 1,
2
is taken as a measure of association between the first and second response
within the j th category of the third response and the k th category of the
fourth response, then
\llk
m~
k
=
= 1,
2
be interpreted as a measure of the extent to which the data within
the k th category of the fourth response tend to deviate from the
hypothesis of no interaction among the first three responses.
reason
~llk
For this
is a natural measure of association among the first three
responses within the k th level of the fourth.
following formulation of the hypothesis of
II no
Hence, we obtain the
third order interaction ll
among four responses
~lllk
=
A1ll*
k
= 1,
2;
and this is readily seen to be a straightforward generalization of
Bartlett's formulation of the hypothesis of no interaction (among three
responses) in 2 x 2 x 2 tables
(See Bartlett [1935]).
to the r x s x t x u tables, we let
3
Returning now
I
(2.1)
Li
j
for
k
hijk
= 1,
= 1,
=
2,
2,
PhiikPrsjk
h
= 1,
2,
PrijkPhsjk
i
=1,
2,
·.. , t
·.. ,
;
u
then
~jk
=
=
·.. , (r-1)
·.. , (s-l)
h
=1,
2,
i = 1, 2,
j = 1, 2,
El
hijk
El
hitk
PhijkPrsjk
PritkPhstk
PrijkPhsjk
PhitkPrstk
·..
, (r-l)
··.... ,, (s-l)
( t-l)
can be regarded as a measure of association among the first three responses
within the k th category of the fourth response where k
= 1,
2, ••• , u.
The hypothesis of "no third order interaction" among four responses may
then be written as
(2.3)
}"hijk
for
k
= 1,
h
i
j
=
}"h"lJ"*
2,
·.. , u
= 1,
= 1,
=1,
2,
2,
2,
·..
, (r-l)
·..
, (s-l)
·.. , (t-l)
This formulation is essentially identical to that originally given by
Darroch [1962].
At this point, one should observe that (2.3) is symmetric
in the sense of Simpson [1951]; i.e., if the given measure of association
among (say) the FIRST three responses is independent of the level of the
fourth response, then the measure of association among ANY three responses
is independent of the level of the excluded response.
Finally, note that
a sufficient condition for (2.3) to hold is that the P
satisfy
hijk
PhijoPhiokPhojkPoijkPhoooPoiooPoojoPoook
PhiooPhojoPhookPoijoPoiokPoojk
as noticed by Darroch [1962].
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Hith the above discussion of "third order interaction" in mind, the
question naturally arises as to how lower order interactions may be
meaningfully defined.
(say) the fourth response and consider the marginal probability Ph' .
~Jo
of the marginal response ( h, i, j, • ) .
,
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Thus, we are in essentially
a three response situation and the corresponding hypothesis of no interaction is
· .•
Ah~J
for
j
PhiJoPrsio
-=
PrijoPhsjo
2,
·.. , t.
=
Phiio
Prijo
= 1, 2,
= 1, 2,
·.. , s
·.. , t
= 1,
=
Ah~·*
.
h
=1,
2,
i = 1, 2,
... , (r-l)
... , (s-l)
h
= 1,2,
••• , (r-l)
If we let
(2.6)
6 h~J.
·.
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Suppose we ignore classifications according to
for
j
then note that
Ah · .
~J •
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.
hSJ.
He call (2.5) the hypothesis of "no second order interaction ll among the
first three responses.
This definition is consistent in the sense that
it is valid when there is no fourth response (i.e., u
= 1).
It is to be
distinguished from the formulation giv'en by Goodman [1964] since it is
expressed in terms of sums of probabilities as opposed to sums of
natural logarithms of probabilities.
Now observe that a sufficient
condition for (2.5) to hold is that the Ph"
~Jo
=
satisfy
PhojoPoii oPhioo
PhoooPoiooPoojo
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Now if (2.4) and (2.7) hold, then
Phijk
Poook
(2.8)
PhiokPhojkPoijk
PhookPoiokPoojk
=
holds; and if (2.8) holds, then
(2.9)
L:-.
hijk
= 1,
k = 1,
j
for
where 6
is defined by (2.1).
hijk
=
2,
2,
h
L:-.hi*k
i
= 1,
= 1,
2,
2,
... , (r-l)
... ,
(s-l)
... , t
... , u
But (2.9) is a formulation of the
hypothesis that within each level of the fourth response, there is no
interaction among the first three responses.
The reader should observe
here that (2.9) is really a special case of (2.3) in which ~hij* = 1.
Carrying the above ideas one step further, suppose we ignore
classifications according to (say) the third and fourth responses and
consider the marginal probability Ph'
100
(h, i, • , .).
of the marginal response
If we have
(2.10)
~ ,
h1 ••
for
i
= 1,
Phioo
Prioo
=
~h* ••
h
=1,
2, ••• , (r-l)
2, ••• , s
then we shall say that there is "no first order interaction" between the
first and second response;
note (2.10) holds if and only if
(2.11)
i.e., if and only if the first two responses are independent.
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Next,
observe that (2.7) and (2.11) imply
PhojoPoiio
Poojo
Phijo
=
l'.h ~J.
°
=
l'.h*
2,
... , s
(2.12)
and if (2.12) holds, then
(2.13)
0
i
for
j
= 1,
= 1,
2,
h
°
= 1,
2, ••• , (r-l)
J•
... , t
where l'.hij. is defined by (2.6); that is, within each level of the third
response (with the fourth response summed over), there is no interaction
between the first two responses.
Finally, we note that (2.13) is a
=
special case of (2.5) in which Aho*
1 •
1.
Thus, we see that the various hypotheses of no interaction are all
related to one another in a particular way and that they impose conditions
on the Phijk which
2.2.
m~
be interpreted in a meaningful fashion.
The 'Three Response, One Factor" Model
Let Phijk
(where we assume Phijk
>0
) denote the probability that
an experimental unit from the k th category of the factor belongs to the
response (h, i, j) in an r x s x t x u contingency table; thus, the pis are
subject to the u constraints
k
L:
h,i,j
= 1,
2, ••• , u •
If we regard A
( as defined in (2.2) ) as a measure of association among
hijk
the three responses within the k th category of the factor, then one
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possible formulation of the hypothesis of" no third order interaction"
among three responses and one factor is given by (2.3) where now, however,
the Phijk appearing there also satisfy the constraints (2.14).
In the
'rour response, no facto! case, we observed that this formulation was
symmetric in the four responses.
For the giv'en situation, this symmetry
may be interpreted as saying that the nature of dependence of a measure
of association between any two of the responses upon the factor level is
independent of the third response; for example, if we let
II.
(2.15)
-_
~hijk
r ·
hijk
~h"
"
~Ju
then (2.3) holds if and only if
(2.16)
~hijk
= ~hi*k
Next we note that (2.4) as such has no meaningful interpretation here since
quantities like Phi"
JO
are not probabilities.
A more appropriate sufficient condition for (2.3)
by *'s; in doing this, we understand that quantities like ~h""* (which
~J
~Jo
) are not necessarily probabilities.
Now let us consider the nature of second order interactions in this
model.
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_,
involve summation ov'er levels of the factor and hence
to hold for this model is obtained by replacing p's by ~'s and I~" subscripts
replaces Ph""
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We first observe that (2.5), (2.6), (2.7) are meaningless here
because they involve quantities like Ph""
~Jo
•
On the other hand, if we
ignore (say) the first response and consider the marginal probability Poijk
of the marginal response (., i, j) within the k th category of the factor,
then
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(2.17)
)-.
for
k
.i jk
= 1,
=
2,
= 1,
PoijkPostk
i
PosjkPoitk
j = 1, 2,
2,
·.. , (s-l)
·.. ,
(t-l)
·.. , u
may be taken as a measure of association between the second and third
response (ignoring the first response) within the k th category of the
factor.
Hence,
(2.18)
)-.
for
k
.ijk
= 1,
i
= A. .1· J.*
2,
·.. ,
j
= 1,
= 1,
2,
2,
·.. ,
·.. ,
(s-l)
(t-l )
u
may be regard ed as a formulation of the hypothesis of "no second order
jnteractiod' between the second and third responses and the factor.
Another type of hypothesis of interest is (2.9).
Here, howev'er, it is
interpreted as saying that within each category of the factor, the measure
I_
( defined by (2.1)) between the first two responses
hijk
does not depend on the level of the third response; i.e., within each category
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of the factor, there is no interaction among the three responses.
,.
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of association 6
that (2.9) is a special case of (2.3) in which A , .*
hIJ
case of (2.9) which may be of additional interest is
6
(2.19)
for
j
k
hijk
= 1,
= 1,
2,
Finally a special
h = 1, 2,
= 6 hi **
2,
= 1.
Note again
i
·.. , t
·.. , u
= 1,
2,
·.. , (r-l)
·.. , (s-l)
i.e., this measure of association between the first two responses is
independent of the third response and the factor.
To carry these ideas one step further, we want to deal with first order
interactions.
Again (2.10), (2.11), (2.12), (2.13) are all meaningless
here because they involve quantities like Ph"
•
been summed ov'er.
IJO
in which the factor has
However, if we ignore (say) the first two responses and
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consider the marginal probability p
ok of the marginal response (., ., j)
oOJ
within the k th level of the factor, then the statement
(2.20)
j
Poojk is independent of k
= 1,
2,
·.. , (t-l)
k ;:: 1, 2, • •• , u
may be regarded as a formulation of the hypothesis of "no first order
interactiorr' between the third response and the factor; the hypothesis
(2.20) may also be interpreted as saying that the marginal distribution
of the third response does not depend on the factor.
If we let
(2.21)
!J.
po1.J
. Ok
=
.ijk
Posjk
i = 1, 2,
j = 1, 2,
... ,
k = 1, 2,
... , u
for
·.. , (s-l)
t
for
j
k
= 1,
= 1,
= !J. .1. *k
i = 1, 2 , ••• , (8-1 )
°
2, ••. , t
2, ••• , u
is a formulation of the hypothesis that within each of the factor categories,
there is no (first order) association between the second and third responses;
i.e., within each factor category, the second and third responses are
independent.
Note that (2.22) is equivalent to the statement
= 1,
2,
j = 1, 2,
i
(2.23)
for
k
=1,
2, ••• , u
and is also a special case of (2.18) in which A . °
'1.J*
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-,
then
!J. .1.J
° Ok
.'I
=1.
·.. , (8-1)
·.. , (t-l)
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2.3.
The "Two Response, Two Factor" Model
Let Phijk ( where we assume Phijk
>0
) denote the probability that an
experimental unit from the (j, k) th combination of factor categories
belongs to the response (h, i) in an r x s x t x u contingency table;
thus, the pIS are subject to the (tu) constraints
L:
h,i
Let us regard ~
°
=
Phi ok
J
j
1
k
= 1,
= 1,
2, ... , t
2, ••• , u
Ok (as defined in (2.1)) as a measure of association
h~J
between the two responses within the (j, k) th combination of factor
categories.
We will say that there is "no third order interaction"
between two responses and two factors if the nature of the dependence of
~hijk
first factor is constant over levels of the second factor. A multiplicative
formulation of such a hypothesis is
(2.25)
~hijk
j
for
k
= 't"hij*
= 1,
= 1,
2,
2,
"C'hi*k
... , t
... , u
h
i
= 1,
= 1,
2, ••• ,(r-l)
2, ••• ,(s-l)
;
howev'er, (2.25) is equivalent to (2.3) where here the Phijk also satisfy (2.24)
and where Ahijk is interpreted as a measure of the dependence of ~hijk on the
first factor within the k th level of the second factor.
Also of interest is the formulation (2.9). Here, this is interpreted
as saying that within each category of the second factor, the measure of
association
factor.
~hijk
between the two responses is independent of the first
Similarly, we could consider (2.19) which here means that the
measure of association
both of the factors.
~hijk
between the two responses is independent of
Each of these two hypotheses are, of course, special
cases of (2.25).
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on the
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We could have formulated the hypothesis of "no third order
interaction" in terms of another measure of association between the two
responses within the (j, k) th factor combination; for example,
h
Phiik
~hijk = PhojkPoijk
(2.26)
j
for
k
= 1,
= 1,
2,
2,
i
= 1,
= 1,
2,
2,
·.. , (r-l)
·.. , (s-l)
·.. , t
·.. , u
The corresponding hypothesis of " no third order interaction" is
h
~hijk = ~hij*~hi*k
(2.27)
for
j
= 1,
2, ••• , t
k
= 1,
2, ••• , u
i
= 1,
= 1,
2,
2,
·.. , (r-l)
·.. , (s-l)
Note that (2.27) implies (2.3)where here,of course,the pIS satisfy the constraints(2.24).
Let us now consider the nature of lower order interactions in this
'model.
As before (2.5), (2.6), (2.7) are meaningless here for the same
reasons as given in Sub-section 2.2.
Suppose we ignore (say) the first
response and consider the marginal probability p . 'k of the marginal response
01J
(., i) within the (j, k) th combination of factor categories.
We are in
essentially a " one response, two factor" situation; hence, an "additive"
formulation of the hypothesis of "no second order interaction" between two
factors and the (marginal distribution of the) second response is
(2.28)
=~
Poijk
for
.1'**
j = 1, 2,
k
= 1,
2,
+~
, '*
.1J
+
1:
.i*k
i = 1, 2, .•• , ( 5-1 )
·.. , t
·.. , u
or equi v'alently
p olJ
"k
for
l
-
k
= 1,
t1
2, ... , u
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"*
.1J'
i
j
= 1,
= 1,
2,
2,
·.. , (s-l)
·.. , (t-l)
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similarly, a "multiplicative" formulation would be
(2.30)
i ::: 1, 2, ••• , (8-1)
P01J
"k:::'1" . l......
J R'1." 1·~fk
"
for
j ::: 1, 2, ••• , t
k ::: 1, 2, ••• , u
or equivalently
Poiik
(2.31)
Poitk
for
:::
i ::: 1, 2,
o,.ij*
j
= 1,
2,
·.. ,
·.. ,
(s-l)
(t-l)
k = 1, 2, ••• , u
Also, hypotheses of the above type could be considered for both the marginal
distribuUon of the first response and the marginal distribution of the
second response together.
For example, a "multiplicative" formulation of
such a hypothesis is given by
(2.32)
for
Phojk
= '1"h.j*
'1"h.*k
h ::: 1, 2,
Poijk
= '1".ij*
'1".i*k
i
j
k
= 1,
= 1,
= 1,
2,
·.. , (r-l)
·.. ,
(8-1)
2, •.• , t
2, ••• , u
Note that this is a hypothesis of simultaneous vanishing of two second
order interactions.
For this situation, the definitions of first order interaction given
in the previous sub-sections hav"e no meaningful interpretation. "He can,
however, consider the statement
i = 1, 2,
independent
of
k
j ::: 1, 2,
(2.33)
p01J
. 'k is
k = 1, 2,
·..
, (s-l)
··.... ,, tu
i. e., within each category of the first factor, the marginal distribution
of the second response does not depend on the second factor.
(2.33) is a special case of (2.28) with '1" '*k
.1
(2.30) with '1" '*k
.J.
= 1,
= 0,
Note that
a special case of
or more generally a special case of any reasonable
forrn.ulation of the hypothesis of "no second order interaction"
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between the two factors and the second response.
Let us now asswne that there is"no second order interacUon" between
the two factors and the second response; for example, let us asswne that
(2.28)( or (2.30)) holds.
Then we shall say that there is"no first order
interaction" between the second factor and the second response if
the conditional formulation
(2.33) given (2.28)
(or
(2.35)
holds.
(2.33) given (2.30))
This definition is consistent with those given in the previous
sub-sections since the relationship between the formulations
(2.28)( or (2.30)), (2.34)( or (2.35)), and (2.33) is similar to the relationship
formulations (2.18), (2.20), and (2.23).
The "One Response, Three Factor" Model
Let Phijk ( where we assume Phijk
>0
denote the probability that
an experimental unit from the (i, j, k) th combination of factor categories
belongs to the h th category of the response in an r x s x t x u
contingency table; thus the pIS are subject to the (stu) constraints
i
j
k
(2.36)
= 1,
= 1,
= 1,
2,
2,
2,
·..
, s
·.. , ut
·.. ,
We shall say that there is "no third order interaction" between one response
and three factors if the dependence of the (distribution of the) response on
any two of the factors is constant over the levels of the remaining factor.
An "additive" formulation of such a hypothesis is
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-,
between the formulations (2.5), (2.11), and (2.12) and between the
2.4.
.'I
I
I
I
I
I
I
I
.,
I
I
I.
I
I
I,
I
I
I
I
I_
I
I
I
I
I
I
I
.e
I
(2•.37)
= ~hij*
Phijk
=
=
1, 2,
j = 1, 2,
k 1, 2,
i
for
=1,
2, •.. , (r-1)
= 1,
=1,
=1,
2,
2,
2,
h
=1,
2, ••• , (r-l)
h
= 1,
= 1,
=1,
2,
2,
2,
h
+ ~hi*k + ~h*jk
·..
,s
·.. , ut
·.. ,
which is equivalent to
(2 • .38)
h
Phijk - Phsjk - Phitk + Phstk
for
k
= 1,
= ~hij*
i
j
··.... ,, (r-l)
(s-l)
·.. , (t-l)
2, ••• , u •
Similarly, a "multiplicative" formulation is
= ~hij*
Phijk
for
= 1,
= 1,
i
j
k
=
~hi*k ~h*jk
2, • •• , s
2,
,t
1, 2, • •• , u
·..
which is equivalent to
Phijk Phstk
Phsjk Phitk
for
k
= 1,
i
j
·.. ,
·..
·.. ,,
(r-l)
(s-l)
(t-l)
2, ••• , u •
Note that the formulations of hypotheses of "no third order interaction" for this model
are different from (2 • .3) which was appropriate for each of the prev'ious ones;
in fact, (2.40) is a special case of (2 • .3) in which
For this situation, the definitions of second order interaction given in
the previous sub-sections have no meaningful interpretation.
can consider formulations of the following type
Phijk
=~hij*
+ ~hi*k
15
However, we
or
(2.42)
Phijk
=~hij* ~hi*k
Each of these is interpreted as saying that within each category of the
first factor, there is "no second order interaction" between the response
and the second and third factors.
Also, (2.41) is a special case of
(2.37) with ~h*jk = 0 while (2.42) is a special case of (2.39) with
~h*jk = 1.
Let us now assume that there is "no third order interaction" between
the response and the three factors; for example, let us assume that
(2.37)( or (2.39)) holds.
Then, we shall say that there is "no second order
interaction" between the response and the second and third factors if
the conditional formulation
(2.41) given (2.37)
(or
(2.44)
holds.
(2.42) given (2.39))
This definition is consistent with those given in the previous
sub-sections since the relationship between the formulations
(2.37) br (2.39)), (2.43)( or (2.44)), and (2.41)( or (2.42)) is similar to the
relationship vetween the formulations (2.3), (2.5), and (2.9) and
between (2.3), (2.18), and (2.9).
Hence, giv-en that there is "no third
order interaction" between the response and the three factors, there
is "no second order interaction" between the response and the second and
third factors if those factors affect the (distribution of the ) response
ind epend ently ( within each lev'el of the first factor) •
Continuing in this manner, we can consider the hypothesis
16
I,
I,
e
II
II
II
II
II
I!
I
_,:
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
p "k is independent of k
h~J
i.e., within each combination of categories of the first two factors, the
distribution of the response dges not depend on the third factor.
that (2.45) is a special case of (2.41) with ~hi*k
of (2.42) with ~hi*k
= 1.
(i).
there is "no third order interaction" between the response and
the three factors,
(ii).
there is "no second order interaction" between the response
and the second and third factors;
for example, let us assume
that
(2.41)(or (2.42))holds.
Then, we shall
say that there is "no first order interaction" between the response and
the third factor if the conditional formulation
(2.45) given (2.41)
(or
(2.47)
holds.
(2.45) given (2.42))
As has been noted previously, this type of definition is consistent
with previous formulations of the hypothesis of "no first order interaction".
,.
I
and a special case
Let us assume that
I_
I
I
I
I
I
I
I
=0
Note
17
I
3.
Test Criteria
Here, we shall use a criterion due to Wald [1943] to obtain test
statistics for some of the hypotheses considered in the previous section.
For a more complete discussion of Wald's procedure as it applies to multidimensional contingency tables, the reader is referred to Bhapkar [1965],
Bhapkar and Koch [1965].
3.1.
The "Four Response, No Factor" Model
Let n hijk denote the frequency of response (h, i, j, k); let
qhijk
= (nhijk/N),
where N
= h,i~j,k
n hijk , denote the unrestricted maximum
likelihood estimator of Phijk.
a.
~othesis
(2.3) of no third order interaction among four responses
(with pIS subject to
~
h,i,j,k
Ph' 'k
J.J
=1):
-'I
I
I
I
I
I
I
_I
k = 1, 2, ••• , u, let
For
=
=
and let V
...QTk
h
i
j
qhijk qrsjk ~itk qhstk
~ijk qhsjk qhitk qrstk
= 1,
= 1,
= 1,
2, ••• , (r-1)
2,
(8-1)
2,
... ,
... , ( t-l)
n hijk nrsjk nritk n hstk
n
n
n
n
rijk hsjk hitk rstk
denote the "sample covariance matrix" of l.k (that is, the
consistent estimate of the covariance matrix of l.k obtained by replacing
p'S by qls in the covariance matrix of the linear parts of the Taylor
expansions corresponding to their components).
It can be easily verified
that to the first order of approximation for large N,
uncorrelated while we have for elements of V
...cvk
18
~
and lk
'
terms of the type
are
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
A
2
1
1
1
1
1
i l l
Var(Yhl"J"k) = YhiJk
"" (-----+ -----+ -----+ -----+ -----+ ------)
n
nrsjk + -----n
nhstk + -----n
n
n
n
hijk
ritk
rijk
hsjk
hitk
rstk
h
Cov(y
Y
hijk' hij'k
with j'f j, i
l
1
1
= yhijrhij'k
.v
(----n
)
ritk
i and hI
*
h and
A
1
1
1
+ ----- + ----- + -----)
n
hstk
n
hitk
n
rstk
denoting the estimate.
The test statistic
X2 is given by
Y
I_
I
I
I
I
I
I
I
••
I
d.f. = (r-1)(s-1)(t-1)(u-1)
where
'/., = ( ~u
-1)-1
u -1
)
( ~ V Xk •
k=!'Y k
k=l ~k
V
Special cases:
(i) r
=s =t =2
Let Yk = Y111k and
.
V
Yk
2'
k n111k
1
1
1
+ ------ + -----nl12k
n121k n 211k
= Y (------- + -
Then
19
1 + -----1
+ -----n
n
122k
212k
1
n221k
1)
n222k
+ ------ + ------ •
I
-'I
d.f.
(ii)
= u-l
•
r = s = t = u = 2
d.f.
=1
•
Note here that the test of (2.3) can be carried out alternatively by
using the Wald statistic for the formulation obtained by taking natural
logarithms of both sides of (2.3).
ghijk
Let
= log
I
I
Yhijk '
and let V be the "sample covariance matrix" of .Ik; it can be verified easily
....gk
that V
can be obtained from the expressions given above for elements
....gk
of V by delating the yts.
-Yk
The alternative test statistic is then
d.f.
= (r-l)(s-l)(t-l)(u-l)
where
for the special cases mentioned above we simply replace y's by gts and take
xy2
and X2 , though not identical, are asymptotically
g
equivalent.
20
I
I
I:
II
_I,
I
I
I
I
I
I
I
-,
I
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
••
I
b.
i~pothesis (2.5) of no second order interaction among the first
three responses:
This hypothesis is really associated wi t.h the "three response, no
factor" situation in a three-dimensional contingency table.
are given, for example, in Brlapkar and Koch [1965J.
Test statistics
These statistics have
(r-1)(s-1)(t-l) degrees of freedom.
c.
Hypothesis (2.9) of no second order interaction among the first three
responses for each level of the fourth response:
For j = 1, 2, ••• , t and k .::: 1, 2 ,
d. ,
'1
OlJK
qhi,ik qrs,ik
, 0]{ qhsjk
rlJ
=q
d
'
....J"k = [d 11-J' 1
K
and 18t. V
.... d jk
••• ,
... , u,
.:::
let
nhi,ik n rsjk
n rijk n, 'k
nSJ
h
= 1,
i
.:::
1,
... ,
... ,
d ( r-.1 ) 1 J".'
k ••• , d ( r- 1 ) ( s- 1)'J K
1
denote the "sample covadance matrix" of d'k.
.... J
(1'-1)
(s-t)
]
Again it can
be easily verified that to the first order of appruximatjon for large N
(~ '.k' s n.re
uncorrelated while for th(; elements of V
.... d jk
....v
the
we have terms of
type
'"
'Iar(l~ ,
'1
C;ov (d, ,
'1 ,
IllJK
i1lJK
)
.::: d 2 '" (1+ --1- t- _1_ + .-L.)
h1JK n
n
n
D
hsjk
hijk
rsjk
rijk
d, "J'k)= d, , 'k d, "'k(
rn
[llJ
'11
J
.-L+ --1-)
n ..
n "
rSJK
h SJK
Ccv(dJ..., .. , d. ""k) .::: d"" 'k d, " I ' ] (-1...-)
!l 1 J
ill J
n 1 J {: n
ilJ J K
rSJ'k
't'n 1"
Hl
+'
T 1 an d"n
+
in.
I
2
The test sta·t1' Stl' c ..,
Yd l'S gl'ven by
21
,
I
_I
d.f. = (r-l)(s-l)(t-l)u
where
Special cases:
(i) r = s
Let d
jk
=2
= dlljk and
_I
Then
2
X
d
t
u
~ ~
~ v~l
k=l j=l
u
d2Jok -
jk
t
k=l j=l
2
t_l
d'k) /( ~ Vd )
jk J
j=l jk
-1
~ (~Vd
d • f. = (t-l) u.
(ii)
I
I
I
I
I
I
I
r
=s =t = 2
As in (a), the tests can be carried out alternatively in terms of logarithms
by letting
e hijk
= log
d hijk
'
~'jk = [e lljk , ••• , e(r-l)(s-l)jk]
22
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
••
I
and cbtaining V
""e
X; are
jk
from Vd by deleting the d's.
"" jk
The alternate test statistics
defined by expressions similar to those of
replacing d's bye's and
particular, note that e
jk
V
,JlJ'k
by V
. ""e
X;
obtained from
For the special case r=s=2, in
jk
2
= log do, and V
e jk
JK
= Vd /d'k
jk J
Again X2 and X2 , though not identical, sre asymptotically equivalent.
e
d
Statistics for hypotheses of type (2.11) and (2.13) involving lower
order interactions are fairly well-known and hence are deleted.
3.2.
The "Three Response, One Factor" Model
Let n
hijk
denote the frequency of response (h,i,j) by subjects from
=h
the factor category k; let qh' 'k = (n l , " In
k)' where n oook
~J
llJk' 000
~ n , 'k'
' , hlJ
,l,J
denote the unrestricted maximum likelihood estimator of Phijk.
a.
Hypothesis (2.3) of no third order interaction among three responses
and one factor (with p's here subject to
'h
~
.
,
,~,J
Ph' 'k
~J
= 1):
The statistics defined in (a) of Sub-section 3.1 can be used.
b.
r~pothesis
(2.18) of no second order interaction between the second
and third responses and the factor:
This hypothesis is really associated with the "two response, one
factor ll situation in a three-dimensional contingency table.
Test statistics
are giv'en, for example, in Bhapkar and Koch [1965]; these have (s-l)(t-l)(u-l)
degrees of freedom.
c.
~wpothesis
at
each factor level:
(2.9) of no second order interaction among the three responses
Statistics defined in (c) of Sub-section 3.1 can be used •
d.
Hypothesis (2.19) that the association between the first two responses
23
I
is independent of the third response and the factor:
Let 2jk and Vd
be defined as in (c) of Sub-section 3.1.
,.., jk
statistic
The test
X~ is giv"en by
o
d.f.
= (r-l)(s-l)(tu-l)
where
Comments similar to those at the end of (c) of Sub-section 3.1 apply here.
Statistics for hypotheses of type (2.,20) and (2.22) involving lower
order interactions are fairly well-known and hence are deleted.
3.3. The "Two Response, Two Factor" Model
Let n hijk denote the frequency of response (h,i) by subjects from the
factor combination (j,k); let qhijk
= (nhij~noojk)'
where
denote the unrestricted maximum likelihood estimator of
a.
n oojk
= h~.nhijk'
,J.
~ijk'
lwpothesis (2.25) of no third order interaction between the two
responses and the two factors, or equivalently, (2.3) with pIS subject to
~ Ph'J.J'k == 1:
h , J.·
The statistics in (a) of Sub-section 3.1 can be used.
b.
~othesis
(2.9) that within each category of the second factor, the
association between the two responses is independent of the first factor:
The statistics in (c) of Sub-section 3.1 can be used.
24
--I
I
I
I
I
I'
I
-,
I
I
I
I
I
I
I
-,
I
I
I.
I
I
I
I
I
I
I
c.
Hypothesis (2.19) that the association between the two responses is
independent of the factor combination:
The statistics in (d) of Sub-section 3.2 can be used.
d.
Hypothesis (2.30)(or (2.28)) of no second order interaction between the
two factors and the second response:
This has been discussed in the
••
I
0ne response, two factor" situation in
the three-dimensional contingency table. Test statistic, say X2 , is given
m
in Bhapkar and Koch [1965]; this has (s-l)(t-l)(u-l) degrees of freedom.
e.
Hypothesis (2.33) that the marginal distribution of the second response
is independent of the second factor (within each category of the first
factor) :
2
The Pearson - X statistic is easily seen to be
u
I_
I
I
I
I
I
I
I
1I
s
t
L:
L:
n
L: (n
k=l j=l i=l
Ok OlJ
0
n Ok ')
n
n Ok
01,10 oOJ )""/( OlJO 00.1
non
oOJo
oOJo
0
0
0
0
0
d.f. = (s-l)t(u-l) ;
this is simpler to obtain than the
Neyman -
xi
~.Jald
statistic (or, equivalently, the
statistic as shown by Bhapkar [1965]) which is, say
2
f
X =
u
L:
t
L:
s
L: (n
k=l j=l i=l
A
2
Ok - n OOJOk p OlJ*
.. ) /n OlJ
° Ok
OlJ
0
where
n
and
°
....QQJ.Q.
q
=
U
L:
n
'k
~
k=l qoijk
00
.1J
x; is seen to simplify to
25
X;,
given by
I
--I
tn.
~
OOJO
j=l
t
where N =
~
j=l
f.
n
.
oOJo
x;
q•0J.
- N
also has (s-l)t(u-l) degrees of freedom.
Hypothesis (2.35)(or (2.34)) of no first order interaction between the
second factor and the second response:
The statistic is
X; - X~
x; mentioned in (d) and
x~,
with (s-l)(u-l) degrees of freedom (with
(e) above).
3.4. The "One Response, Three Factor" Model
Let n hijk denote the frequency of response h by subjects from the
factor combination (i, j, k); let qhijk = (nhij~noijk)' where n oijk = ~ nhijk ,
denote the unrestricted maximum likelihood estimator of Phijk.
a.
Additive formulation (2.37) of the hypothesis of no third order
interaction between one response and three factors (with
to 2:
h
!hi j k
piS
subject
= 1):
For k = 1, 2, ••• , u, let
b hijk = qhijk - qhsjk - qhitk + qhstk
h
i
j
ane
=1,
=1,
= 1,
let .Y:b kdenote the "sample covariance matrix" of .ek.
are mutually uncorrelated and elements of Vb
.... k
26
·.. , (r-l)
·.. , (s-l)
·.. , ( t-l)
Note that .9k's
are of the type
I
I
I
I
I
I
-,
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
••
I
"
Gov'( b hijk , b h I ijk)
"
COV'(b
hijk , b hi 'jk)
= qhsjk(l-qhsjk)
nos j k
"
(
GOV(bhijk,bhli'jk)
t
h, i '
f
i
nos t k
qh S.l°kqh' S.lok + qhstkqh'stk )
nos j k
nos t k
qhstkqh'stk
nos t k
"
Gov(bhijk,bh'i'j'k)
with h'
+ qhstk(l-qhstk)
and j'
The test statistic is
t
j.
~ given by
d.f.
where
27
= (r-l)(s-l)(t-l)(u-l)
I
••I
Special cases:
= s = t =2
Let bk = blllk
(i) r
= qlllkq2llk
v
b
q
q
q
o12k
n o22k
I
I
I
I
I
I
t Q12lk 2,lk t ql12k 2i2k t Q122k 222k
n ollk
k
and
n
n
o2lk
Then
d.f. = u-l
(ii)
b.
r
= s =t =u =2
-,
!'-Iultiplicative formulation (2.39) of the hypothesis of no third order
I
I
I
interaction between one response and three factors:
For k
a hijk
= 1,
2, ••• , u, let
h
qhijkqhstk
= qhsjkqhitk
i
j
and let V
roJa
Note that
= 1, ·.. ,
= 1, ·.. ,
= 1, ·.. ,
(r-l)
(s-l)
(t-l)
denote the "sample covariance matrix" of
k
~'s
are mutually uncorrelated and elements of Va
roJ
2
1
1
1
= ahiJok ( - t - t - t
n hijk
n hstk
n hsjk
28
---Ln hi tk
1
n oijk
k
are of the type
1
nos tk
1
n osjk
1
noi tk
--- - - - - -
I
I
I
) I
-.
I
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
II
A
Cov(a,rl1J.
. 'k' a n I"lJ'k)
n 01J
"k
=
n hstk
A
a,n:lJ
' 'k a !1
, 1 1" 1 J'k
" 'k,ah'"1 'k)
Cov(al,1lJ
1 J
.-
Cov(ahijk,ahiljlk)
n os tk
n OSJ'k.
n,
01 tk
I I I
1
(-+ - - ---=-)
A
Cov(a\,11" J"K ,ah'l
'k)
1 J
( --L- + ---L- + _1_ + .-1.-)
a h1" J' k 8 h 1 l' J' k
= ahijkahilj'k
n hsjk
nos t k
nos j k
(---L- + ----l.-)
nos t k
n osjk
(_1__ --L-)
nhstk
nos t k
.-
Cov(a,!1J-J
" 'k' a hl 1' I J' I k)
with hi
4 h,
i l =f: i and jl 4= j
2
The test statistic Xa with (r-l)(s-l)(t-l)(u-l)
d.f. is defined as
X;
above with bls replaced by a1s and !b
replaced by
k
v • ] n particular, for the special case r
M1k
= s = t
= 2,
we have
and
An
asymptotically equivalent statistic corresponding to the logarithmic
2
fonuulation of (2.39) is, say X , with
c
c hijk
2~
= log
= [c lllk '
a hijk
••• , c(r-l)(s-l)(t-l)k ]
29
I
x2c
and V obtained from V by deleting a's
tvC k
tv~
":k and !C
k
just as
~
is in terms of 2k and !b •
k
is defined in terms of
Special cases follow
along similar lines.
C.
Additive formulation (2.41) of the hypothesis of no second order
interaction between the response and the last two factors within each category
of the first factor:
For i = 1, 2, ••. , s and k = 1, 2, ••• , u, let
h
j
= 1,
= 1,
... , (r-l)
... , ( t-l)
and let V
be the "sample covariance matrix" of w. k •
tvWi*k
tv~*
Note again that w'*kIS are mutually uncorrelated and elements of V
tv~
tvW. k
~*
are of the type
= qhijk(l-qhijk)
n oijk
+ qhitk(l-qhitk)
n
oitk
Cov(whijk'Wh'ijk)
A
= qhitk(l-qhitk)
n oitk
,.
qhitkqh'itk
n oitk
COV(Whijk,Wh'ijlk)
I
I
I
I
I
I
-,
I
I
I
I
I
I
I
A
Cov·( whijk ,whij ' k)
••I
-.
2
The statistic is X given by
W
30
I
I
I.
I
I
I
I
I
I
I
d.f.
where
~i**
Special cases:
(i)
r
=t =2
qli2k q2i2k
n oi2k
I_
(ii)
I
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= (r-l)s(t-l)(u-l)
d.
r
= t =u =2
Multiplicative fonnulation (2.42) of the hypothesis of no second-order
interaction between the response and the last two factors within each
category of the first factor:
This is very similar to the case (c) abov'e with
Wi S
defined as
Zhijk
= qhijk/qhitk
~*k = [Zlilk'
and with elements in V
••• , Z(r-l)i(t-l)k]
of the type
,."zi *k
31
replaced by ZIs
I
_.
1
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n oijk
,.,
Cov(zhijk' Zh'ijk)
=
,.,
Cov(zhijk,Zhij'k)
= ZhO 0kzh o o'k(_ln hitk ~J
,.,
~J
-L)
n oitk
I
1
zho~J°kzh' ~J
° o'k ( n
)
oitk
Cov·( Zhijk' zh' ij 'k)
2
The statistic X is defined as X2 and has (r-l)s(t-l)(u-l) degrees of
w
Z
freedom.
Special cases follow along the same lines.
An asymptotically equivalent statistic X2 corresponding to the
E
logarithmic formulation of (2.42) is also immediate if we take
Ehijk = log Zhijk
and define V
from V
by deleting z's.
....Ei*k
.... ~i*k
Again special cases follow in
the same manner.
e.
Additive formulation (2.43) of no second order interaction among the
response and the last two factors:
Statistic is
xe - ~ with (r-l)(t-l)(u-l) d.f. where x~ and ~ are
defined in (c) and (a) abov·e.
f.
Multiplicative formulation (2.44) of no second order interaction among
the response and the last two factors:
2
2
Statistic is X2 - X with (r-l)(t-l)(u-l) d.f. where X may be used in
Z
a
E
2
2
2
place of Xz and similarly Xc in place of Xa ; these are defined in (d) and
(b) above.
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Hypotheses involving lower order interactions can be tested along
similar lines and the details are omitted.
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33
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4. Numerical Illustrations
In this section, we consider four contingency tables from the
unpublished data of Lessler.
The experiment was conducted to study the
nature of sexual symbolism with the basic observation being a subject's
classification of an object
as being either masculine or feminine when
that object is shown to him at the following exposure rates:
1/1000 second, 1/500 second, 1/100 second, 1/50 second, 1/5 second.
The subjects involved in the study were classified according to
(a)
sex (males and females)
(b)
those who were not told the purpose of the experiment and those
who were told the purpose of the experiment (Group A and Group C
respectively)
while the objects inv·olved in it had been (by a previous study Lessler (1962))
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assigned
(i)
(ii )
a cultural meaning (M or F) related to which sex used it
an anatomical meaning (H or F) related to which sex it inately
evoked
(iii)
an intensity (W or S) related to the strength of the degree to
which the object was representative of its classification
according to (i) and (ii).
Thus, as can be seen, the experiment is of the "five response, five factor
type ll with the factors being (a), (b), (i), (ii), (iii) above and the
responses being the classifications at 1/1000, 1/500, 1/100, 1/50, 1/5
second.
The tables below are presented only to illustrate tests of the
hypotheses of "no interaction" in various situations; no complete analysis
34
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is attempted here.
given here appears in Bhapkar and Koch [1965].
I.
A "Three Response, One Factor" Situation
Table 1:
Subject Type - Males
Object Type
- Culturally Masculine
Anatomically Feminine
Weak Intensity
I~k) Subject Group
(.
(h) Response
+> J
(i
Q) Q) :tespons
0)(1)
I:l o 9.t 1/100
°0
Sec.
0.
til
•
t)
at 1/5 Sec.
M
F
Total
C
A
Sub-Total
H
F
Sub-Total
0
O)...-l
Q)
~~
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Also, an earlier analysis of parts of the same data
M
171
7
178
184
7
191
369
6
7
13
10
20
30
/+3
177 14
191
194
27
221
412
F
H
Sub-Total
W
M
18
7
25
38
7
45
70
F
12
56
()8
14 114
128
196
30 63
93
52 121
173
266
77
284
246 148
394
678
Sub-Total
207
rrOTAI
d
rna
= 28.5000
erna
= 3.3499
V
- 0.458229
erna
V
drna
= 372.1965
d
mc
= 52.5714
emC
= 3.9621
V
= 0.298292
Vd
mc
= 824.4051
d fa
= 12.0000
efa
:=
2.4849
'I
::: 0.299603
Vd
=
d fc
= 44.2041
e
= 3.7887
V
fc
ernc
era
e rc
= 0.249373
35
Vd
fa
fc
43.1428
= 487.2755
I
y a = 2.375000
ga = 0.8650
V = 0.757832
ga
V = 4.274646
Ya
Yc = 1.189288
gc = 0.1734
V = 0.547665
gc
V
= 0.774621
Ya-Y c = 1.185712
g -g = 0.6916
a c
V
+V
= 5.0/l-9267
2
=1
2 _
X - 0.278
Y
1.
X = 0.366
g
d.f.
+V
ga
gc
= 1.305497
Yc
V
Ya
Yc
Hence, the data are consistent with the hypothesis of "no third order
interaction" between three responses and one factor.
d -d
= 16.5000
ma fa
d
mc
2.
-d
fc
=
8.3673
Xd2 = 0.709
emc-e fc = 0.1734
2 _
X - 1.042
e
d.f.
Ve +V e = 0.757832
ma
fa
Vd +Vd
= 415.3393
ma
fa
Ve +V e = 0.547665
mc
fc
Vd
+Vd
mc
=131.1. 6806
fc
=2
flenee, the data are consistent with the hypothesis of "no second order
interaction" between the three responses within each subject group.
2.
d
2
Xd
= 17.4798
=
o
3.982
e = 3.4217
2
Xe = 4.459
d.f.
=3
o
Hence, the data are consistent with the hypothesis that the association
between response at 1/5 sec. and response at 1/100 sec. is independent of
response at 1/1000 sec. and subject group.
4.
If we ignore classifications according to response at 1/100 second and
consider the marginal frequencies obtained by summing over this response
(Le., over i), we obtain Table 3 as given in Bhapkar and Koch [1965].
There, the test statistic for the hypothesis of "no second order interaction"
between two responses
X~ = 0.89
and one factor was
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d.f. = 1
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fl enco , the data are consistent with the hypothesis of "ne second order interaction"
betMeen the response at 1/1000 3ec. and the response at 1/5 sec. and the
factor subject group.
II.
Table 2:
Subject Type - Males
Object Type
- Culturally Masculine
l'!eak Tntensi ty
~ (k)
Subject Group
.)
(
(h) Response
at 1/5 Sec.
(i)
{)
.rj till Hesponse
f .. ~
o 'rl at 1/1000
+' >:.:
ro ro
Sec.
.~
1'1
F
Sub-Total
H
F
Sub-Total
202
256
42
298
500
34
62
96
178
.~
,?~
18'7 15
l'~
..
1:
42
40
82
229
55
284
290 104
394
678
H
177 14
191
194
27
221
412
F
30 63
93
52 121
173
266
77
284
246 148
394
678
436 132
568
536 252
788
1356
F
Sub-Total
F
207
Sub-Total
TOTAl
d
Total
C
~
~J
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"Two Response, Two Factor" Situation
il.
ma
::: 11.8730
e
ma
:::
2.474
Ve
= 0.120822
Vd
ma
= 17.0321
ma
me
= 11.1148
e
me
= 2.408
V
eme
= 0.073255
V
dme
=
dfa
= 26.5500
e fa
= 3 • 27°/
V
e fa
= 0.126283
V
dfa
= 89.0172
d fe
= 16.7194
ere
= 2.816
V
e
= 0.069685
V
::: 19.4796
d fe
d
fe
37
9.0498
I
Ya
= O. 4472
ga
= -0.805
V
= 0.247105
V
= 0.049418
Yc
= 0.6648
gc
= -0.408
V
= 0.142940
V
= 0.06.3174
Yc-Y a
= 0.2176
gc-g a
V +V
= 0•.390045
V +V
= 0.112592
1.
X2
= 0.421
Y
2
X =
g
ga
gc
= 0•.397
0.404
ga
d.f.
gc
Ya
Yc
Ya
Yc
--I
=1
Hence, the data are consistent with the hypothesis of "no third order
interaction" between two responses and two factors.
dma-d fa
= -14.6770
d -d
me fc
=
2.
Xd2
-5.6046
= .3.1.32
eme -e fc
2
Xe
= -0.408
= .3.787
d.f.
Ve +V e
ma
fa
= 0.247105
V
= 0.142940
e
me
Vd +Vd
ma
fa
= 106.049.3
=
28.5294
=2
Hence, the data are consistent with the hypothesis of "no second order interaction"
between the two responses and anatomical meaning within each subject group;
i.e., within each subject group the (6) association between the two responses
is independent of anatomical meaning.
.3.
d
2
X
d
o
= 1.3.2962
e
= .3.220
Xe
= 2.7089
2
o
= 4.4.31
= .3
d.L
Hence, the data are consistent with the hypothesis that the
(~)
association
between the responses is independent of the levels of each of the factors
anatomical meaning and subject group.
4. If we ignore classifications according to response at 1/5 second and
consider the marginal frequencies obtained by summing over this response (i.e.,
over h), we obtain Table 4 as given in Bhapkar and Koch [1965].
There, the
test statistic for the "multiplicative" formulation of the hypothesis of
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"no second order interaction" between one response and two factors was
2
xm
= 9.82, d.f. = 1 •
~Ience,
the "multip1icative ll formulation of the hypothesis of
II
no second order
interaction" between the response at 1/1000 second and the factors sUbject
group and anatomical meaning is rejected.
5.
-q
·mm
= 0.736786
q.fm
= 0.260704
-
-q.om = 0.997490
p ofm*-- 0.261360
-q.rnf = 0.602823
Pomf*-- 0.61073.3
-q.of
~
j
n
.
~
<:i. oj
= 1366.6021
= 0.987049
X2f = 10.6021
Poff*= 0.389267
d.f.
=2
Hence, the hypothesis that Poijk is independent of k is rejected; that is,
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within each category of anatomical meaning of an object, the response of a
subject at 1/1000 sec. is not independent of the SUbject's group.
39
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Table 3:
Subject Type - Males
Object Type
~
-'I
A "0 ne Response, Three Factor ll Situation
1] I.
- Weak Intensity
{k)Subject Group
C
A
(h) Response at
(i)
1/1000 Sec.
~
t)
Cultural
-g ~ l>1eaning
(j J
M
F
Sub-Total
M
F
Sub-Total
o§
1i1
:a~
M
M
202
82
284
298
96
394
F
124 160
284
182 212
394
326
242
568
480 308
788
191
93
284
221 173
394
60 224
284
65
329
394
568
286
502
788
Sub-Total
101
F
F
Sub-Total
1.
251
= -.1866
Vb
b c = -.1015
Vb
ba
a
317
= 0.002951
= 0.002073
c
Vb +Vb = 0.005024
a
c
Hence, the data are consistent with the
hypothesis of
II
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,.
= 0.27465
Var{ w )
m*a
W
m*c
= 0.29442
Wf
= 0.46127
Var{ W )
m*c
,.
Var{ w )
f *a
,.
Var{ wr*c)
*c
=1
no third order interaction" between one response and three
2. W
m*a
Wf
d.f.
additive ll formulation of the
factors.
*a
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2
JCb=l.44
= 0.0851
bc -b a
"
= 0.39594
,.
= 0.001589
= 0.001099
= 0.001362
= 0.00897
40
,.
,.
Var{ wm*a )+Var{ wf *a )
,.
Var{ "tn*c )+Var( Wf*C)
= 0.002951
= 0.002073
,.
Var{ W* )+Var{ W )
ma
m*c
,.
,.
Var{wr * )+Var{W * )
arc
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:=
0.002688
= 0.002336
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II
x2w,gr =
11.802 + 4.972 = 16.774
2
xw,ana
t = 0.145 +
1.827
= 1.972
d.f. -:: 2
n.f. = 2
Hence, the data are consistent with the "additive" formulation of the
hypothesis of fino second order interaction" between the response and the
factors cultural meaning of object and subject group within each category
of anatomical meaning of object;
i.e., within each category of anatomical
meaning, the factors cultural meaning and subject group affect the Phijk
jn an independent additive way.
The "additive" formulation of the
hypothesis of fino second order interaction" between the response and the
factors cultural meaning and anatomical meaning within each sUbject group
is rejected;
i.e., the factors anatomical meaning and cultural meaning
do not affect the Phijk in an independent additive way within each
subject group.
2
xW,f!,
r
xb2 = 15.334,
d.f.
= 1;
2
Xw,ana t
x~
= .532, d. f. = 1
•
!lence, the data are consistent with the "additive formulation" of the
hypothesis of "no second order interaction"between the response and the
factors cultural meaning and group.
The interaction between cultural
meaning and anatomical meaning, however, is significant.
41
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IV.
Table 4:
~(k)
'~ ~
Subject Type - Group C Males
Weak
Intensity
(j)
~()
_.
A "One Response, Three Factor" Situation
(i)
Cultura
Neaning
Strong
(h) Response at
1/1000 Sec.
M
F Sub-Total
M
F
Sub-Total
~ '§
:a~
M
F
M
298
96
.394
.375
19
.394
F
182
212
.394
120
274
.394
Sub-Total
480
.308
788
495
293
788
1'1
221
173
394
279
115
394
F
65
329
394
29
365
394
286
502
788
308
480
788
"1 = 0.1142
x~
Sub-Total
1. a.
b
b
vI
s
= -.1015
Vb
=
Vb = 0.001352
Vb +V b = 0.003452
ssw
Since the n
.0127
= .05
w
= 0.002073
b -b
s
= 3.78
d.f.
= 1.
critical yalue of the chi-square distribution with one
degree of freedom is 3.84, we do not reject the "additive" formulation of
IIno third order interaction"between the response and the three factors;
42
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:Iowever, the data are not. completely consistent ""ith this hypothesis either.
a w .- 0.481577
1. b.
.,
[1,..
--
0.324821
cw ::: -0.7308
c
s
= -1.1245
aw
-a = 0.156756
c -c s
sw
TV
= 0.038915
V
V
cs
= O.3~37
VC +V C
w s
2
a
= 0.057522
s
= 0.004106
Va +v a
w s
= 0.008421
=1
d.f.
Xa = 2.92
a w = 0.004315
V = 0.018607
cw
;rence, the data are consistent with the "mul tiplicativeil formu1ati on of the
hypot~18sis
of "no third orcl er interacti on ''between the response and the
t.hree factors.
Since X~
< x~,
we feel the multiplicative model (2.39)
provides a better "fit" to the data.
Thus, the subsequent analysis w:il1
be based on mul tiplicati ve formulations.
2.
,.,n
z
m~:·w
= 1.6374 tmi~ w = 0.49.31
milS
= 3.1250
.
Var(E m*)
.
.
= .3.4000 Ef*w
= 1.2238
Var(E
~
= 9.6207
= 2.2639
Var(E
n
L i~S
Ef*s
..
f *w
f *s
)
= 0.01483.3
Var(zf
)
= 0.032991
Var(zf
= 17.109
+ 13.561
= 30.670,
d.f.
=2
2
X
= 28.695
+ 32.494
= 61.189,
cl.f.
=2
X2Z ,anat = 32.558 + 11.999
= 44.557,
d.f.
=2
xe2 ,anat =-43.071
= 65.692,
d.f.
=2
+ 22.621
2
- Xa = 27.75
int
2
X2
e , int Xc = 58.50
2
2
Xz ,anat
Xa = 41.64
2
X2
Xc = 63.00
e ,anat
X2
Z,
d. f.
=1
d.f.
=1
d .f.
=1
d.f.
=1
4.3
..
Var(z
x2Z , In
. t
e , int
..
)
0.00.3774 Var( z
m*w
= 0.005924
Em*s = 1.1394 Var(Emi~s )
z·fi~W
-
..
m*s
..
*w
*8
= 0.010118
)
= 0.057852
)
= 0.171469
)
= .3.053577
I
Hence, the second order interaction between the response and the factors
anatomical meaning and cultural meaning is significant; it is also
significant at each level of intensity.
In addition, the second order interaction between the response and
the factors intensity and cultural meaning is significant; it is also
significant at each level of anatomical meaning.
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5.
Remarks
It would be noted that the test criteria for the corresponding
hypotheses in the first three situations are identical while those for the
last situation are, in general, quite different.
Moreover, even when the
test criteria are identical the meaningful interpretations of the corresponding
hypotheses are quite different for the first three probability models
depending on the adopted experimental scheme.
This point has already been
stressed in our earlier paper [1965] to which reference may be made for a
more detailed discussion.
In particular, it may be pointed out that in
many experimental situations, it is more or less obvious whether a particular
dimension is a "response" or a "factor ll , so that the problem belongs to one
of the four situations discussed in a unique manner; on the other hand, in
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some instances a particular dimension may be viewed either as a "response"
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of view.
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or a "fa.ctor" in which case the problem can be tackled from different points
Note also that the hypotheses for the first three situations are
formulated in terms of specific measures of association between two or
three responses as the case may be.
The criteria are offered only for those
measures which have appeared in the literature so far and which seem to be
satisfactory.
An alternative measure of association between two responses
has been offered in our earlier paper [1965] and is mentioned briefly in
2 •.3 (see (2.26)).
Formula.tions corresponding to hypotheses (2.5) or (2.17)
can be offered in terms of this measure and test criteria can be constructed
for these,but this has not been pursued further here.
The methods of this paper can be now immediately generalized to multidimensional (multi-response and/or multi-factor) contingency tables.
45
Note
I
t~at
the four-dimensional tables, rather than the three-dimensional ones,
point out the different problems that may arise in the higher-dimensional
tables;
this is mainly because a multi-response,multi-factor situation
cannot arise in a three-dimensional table.
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BIBLIOGH.APHY
Bartlett, M. s. [1935), "Contingency table interactions", Journal of the
Roval Statistical Society, Supplement, ~, 248-52.
Bhapkar, V. P. [1965], "A note on the equivalence of some test criteria",
Institute of Statistics, University of North Carolina, Mimeo Series
No. 421.
Bhapkar, V. P. and Koch, Gary G., [1965), "On the hypothesis of I no
interaction' in three-dimensional contingency tables", Institute of
Statistics, University of North Carolina, ~limeo Series No. 440.
Darroch, J. N. [1962], "Interactions in multi-factor contingency tables",
Journal of the Royal Statistical Society. Series B. 24, 251-63.
Good, I. J. [1963], "Maximum entropy for hypothesis formulation, especially
for multidimensional contingency tables", Annals of Mathematical
Statistics, 34, 911-35.
Goodman, L. A. [1964], "Interactions in multidimensional contingency
tables", Annals of Mathematical Statistics, 35, 632-47.
Lessler, K. J. [1962), "The anatomical and cultural dimensions of sexual
symbols", Unpublished Ph.D. Thesis, Michigan State University.
Lewis, B. N. [1962), "On the analysis of interaction in multi-dimensional
contingency tables", Journal of the Royal Statistical Society.
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