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SOME ASPECTS OF THE STATISTICAL ANALYSIS OF THE "MIXED MODEL"
by
Gary G. Koch and Pranab Kumar Sen
University of North Carolina, Chapel Hill
Institute of Statistics Mimeo Series No. 500
December 1966
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Work partially supported by the National Institute of Health,
Public Health Service, Grant GM-12868.
DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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SOME ASPECTS OF THE STATISTICAL ANALYSIS OF THE "MIXED MODEL"*
by
Gary G. Koch and Pranab Kumar Sen
1
University of North Carolina
Department of Biostatistics
SUMMARY
In this paper, the authors discuss the statistical analysis (both parametric and non-parametric) of "mixed model" experiments.
The general structure
of such experiments involves n randomly chosen subjects who respond once to each
of p distinct treatments.
The hypothesis of no treatment effects is considered
under several different combinations of assumptions concerning the joint distribution of the observations corresponding to each of the particular subjects. For
each situation, an appropriate test procedure is discussed and its properties
studied.
The different methods considered in the paper are illustrated in detail
in two numerical examples.
These examples have been chosen to illustrate the
relative performances of the different test criteria for a situation in which the
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null hypothesis is
essentially true (Example 1) and for a situation in which the
null hypothesis is essentially false (Example 2).
Finally, the section on examples contains algorithms for the efficient
computation of the various test criteria.
A computer program based on these
algorithms has been written and can be made available to any interested persons.
]
*
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Work partially supported by the National Institute of Health, Public Health
Service, Grant GM-12868.
On leave of absence from Calcutta University, India
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INTRODUCTION
1.
In accordance with the familiar "mixed model" experiments (cf. Eisenhart
[1947J and Scheff~ [1959J)? let Y., be the response of the i-th subject to the
1.J
j-th treatment for i=1,2,.... ,n; j=1,.2, ... ,p.
Thus? each subject responds to
each of p distinct treatments exactly once.
The outcome of the experiment may
be represented by the observation matrix
= (Y .. ) composed of n independent
X
II
1.J
nxp
stochastic row vectors
I
Y'
),. i=1,.2, ... ,n .
-i. = (Y'l""'Y'
1.
1.p
(1. 1)
It is assumed that Y. has a continuous p-variate cumulative distribution function
-1.
(c.d.f.)
F.(~),
1.
where
F,(~) = G.(~-m.), m! = (m'l" ... ,.m. );
(1. 2)
moo = b.+t., i=1,2, ... ,.n; j=1,2, ... ,p.
(1. 3)
1.
1.J
1.
1.
1.
~
1.
1.p
J
Together with (1.2) and (1.3), the basic assumption throughout this paper is
A.1.
The joint distribution of any linearly independent set of contrasts among the observations on any particular subject is
diagonally symmetric.
Two additional assumptions which mayor may not be imposed are
A.2.
The "additivity" of subject effects;
A.3.
The "compound synnnetry" of the error vectors.
The assumptions A.1,. A.2, and A.3 will be explained more fully in what follows.
In any event, four cases of interest arise; and these may be described by the
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following table
><
Not A.2
Case I
Case III
A.3
Case II
Case IV
Not A.3
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A.2
In each of the above cases, the hypothesis of no treatment effects, i.e.,
= t p = 0,
is considered, and appropriate test procedures are given.
(1. 4)
Various properties
of these tests are also studied.
2.
THE NORMAL (PARAMETRIC) CASE
In this section, the basic assumptions are A.l, A.2, and
A.4.
Ql
= G2 =...
5
Go
= G is
a multinormal c.d.f.; i.e., the error
vectors have a common p-variate normal distribution with a null
mean vector and a positive definite dispersion matrix
Note that A.4 implies that the
~i'
i= 1,2, ... , n, are independently dis tribu ted
according to the multivariate normal distributions
N(~i'~)'
i=1,2, ... ,n, where
To test H in (1.4), we may proceed
the m.'s are defined by (1.2) and (1.3).
-~
as follows.
~.
0
Let ".;
C be a (p xp) matrix of the following structure
J
where
i'
= (1, .. :, 1) and £11 =
o.
(2.1)
l
I f we let
(2.2)
.- =
§u
n
~
i=l
(U. -U)(U. -U)' ,
-~
-
-~-
(2.3)
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then the test statistic
.-I
~
-
-1-
T2 = n(n-1) U'.§U !!
(2.4)
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Under Ho' (n-p+1)T2 /(n-1)(p-1) has the F-(variance ratio) distribution with
[(p-1), (n-p+1)] degrees of freedom (d.f.), where it is necessary to presuppose
that n
~
Case III.
The test procedure based on T2 represents a parametric solution to
p.
Since T2 is invariant under any (non-singular) linear transformation
on the U., i=1,2, •.. ,n, some computational convenience may be gained by basing
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the test on the contrasts
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(2.6)
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(2.5)
Now, a sufficient condition (also necessary, if the distribution is
mu1tinorma1) for A.3 is
where J is the p X P matrix of l's and I is the p x p identity matrix.
"
Since
N
(2.6) implies that the variability of a subject's response is independent of the
treatment given, we shall say that there is no interaction between treatments
and subjects in this case.
The test of H in (1.4) may now be based on the
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statistic
Ft = (n-1)S~/S~
(2.7)
where
p
S2
t
= n L.
j= 1
(Y .-Y )2, S2
.J
e
••
=
n
L.
p
_
_
_
2
L. (Y .. -Y. -Y .+Y )
i= 1 j= 1 ~ J ~. • J ••
with
Y.
~.
= P
-1 P
L. Y ..,
j=l ~J
Y. j
= n
-1 n
_
L. Y .., Y
i=l
~J
••
= (np)
-1 n
p
L. Y..
i=l j=l ~J
L.
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-...
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Under H , F
o
based on F
t
t
has the F-distribution with [(p-1), (n-1)(p-1)J d. f..
The test
provides a parametric solution to Case IV.
One may note that the validity of (2.6) may be checked by means of a
likelihood ratio test given by Votaw [1948J which is a generalization of one
originally due to Wilks [1946J (for the particular case b = b = ... = b = 0).
n
1
2
For a further discussion of the parametric case, the reader is referred to
Wilks [1946J, Scheffe [1959J, Imhof [1960J~ and Geisser [1963J.
3.
A NON-PARAMETRIC TREATMENT OF CASE I
First, let us clarify the statement of the basic assumption A.1.
Suppose
U. is defined as in (2.2), and m. as in (1.2); let t' = (t , ••. , t ) where the
-1.
-I.
p
1
t.1. 's are defined by (1.3).
Finally, let 8 be a (p-1)
-e
8
-
where £1 is defined by (2.1).
= C1
m.
N-I.
= C1
)(
1 vector defined by
(3.1)
t
N-
Then A.1 asserts that for each i=1,2, ... ,n,
(U.-8) and (8-U.) have the same distribution.
-I. - -1.
Under H ' _8 = _0; and hence, in
o
this situation, A.1 implies that U. and - U. have the same distribution.
--I.
-1.
One
may note that this is much less restrictive than the usual assumption of mu1tino rma l i ty of the U..
-1.
To test H , we may proceed as follows.
o
Let
R. . = Rank (Y..: Y. l' ... , Y. }
1.J
1.J
1.
1.p
j=1,2, •.. ,p
i=l, 2, ... , n
(3.2)
where ties are to be handled by the mid-rank method; i.e.,
= 1 + {The number of Yi ., which} + (.1.) { The number of
are less thanJy..
2
Y.. ' equal to Y.. }
1.J
1.J
1.J
where j'=fj; also j,j'=1,2, ... ,p; i=1,2, .•. ,n.
(3.2 t)
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Let
T
. =
n, J
1
n
for
l: R,.
~J
n i=l
j= 1,2, •.. , p;
(3.3)
(3.4)
T' = (Tn, 1'" ., Tn, p )
--n
~J~'
R, , = Rank {Z,,: Z. l' • • •, z,
~J
~J
~
~p
J
j=1,2, ••• ,p
i=1,2, ••. ,n •
(3.5)
0
-~
~
and - -~
Z,' = (-Z'l"'"
-Z.~p ) have the same distribution by assumption A.l.
~
sign-invariance generates a set of 2
n
~p
11
II
This
conditionally equally likely realizations;
and as a result, the rank vectors
R! = (R.I,
... ,R.~p ) and (P+l)j,'-R!
= (P+l-R'l,
•.• ,p+l-R.~p )
~
-~
~
~
(3.6_
are (conditionally) equally likely, each occurring with conditional probability
t
for i=I,2, •.• ,n.
Thus, under this conditional probability law (say ~n)'
E{T
.11PJ
n
n, J
.J ~n J
Cov{T ., T
n, J n, J
for all j,j'=1,2, •.• ,p.
L:
n
2
i=l
n
Let us define the matrix
v
= (v
... ) .
n, JJ
,'s in (3.3) satisfy the constraint
n, J
=
.E:!:!.
(3.7)
2
1
1
= {. l: (R,.- E:!::!.2 HR, .• - £!!2)
n i= I ~ J
~J
Nll
Since the T
Rij - [(P+l)-Rij ]
n
I
= -
=vn, J..J ,
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But the Z~j'S are contrasts; hence, under H, the vectors Z! = (Z'l""'Z, )
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We next observe that if Z, . = y .. -Y, , j=l, 2, ••• , p, then
~J
II
(3.8)
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II
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eli
(3.9)
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P
L: T
.
j==l n,.1
~
=
p(p+1)!2,
is essentially singular and of rank at most (p-1).
~n
is similar to that given in Puri and Sen [1966J).
,e
.
Thus, if ~ is defined as in
(2.1), the test statistic is
W = T' ~ [C V g~ J
n
--n H.L ,."1""n--.L
Under the permutation model ~, W can have 2
n
n
-1
C
(3.11)
T
""'-n
1
(not necessarily distinct) equally
likely (conditionally) realizations; and this may be used as a basis of a permutationally distribution free test of H.
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Moreover, under
of averages over n independent random variables.
1
n2 T
--n
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~
space (Rp ), one may show that n V has rank (p-1) in probability (the argument
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However, if G. (x.) has a
"scatter" not confined to any lower dimensional space of the real p-dimensiona1
'IiiiIt.
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(3. 10)
lP,
n
T is the vector
-n
Hence, by routine methods,
can be shown to have asymptotically (under ~ ) a mu1tinorma1 distribution
n
of rank (p-1).
with (p-1) d.f.
Thus, under
~, W has asymptotically a chi-square distribution
n
n
From the above remarks, we have the following test procedure
reject Ho if and only if Wn -> X2(1 -E )(p-1)
(3.12)
where p{X2 (p-1) ~ X(1_E)(p-1)} = E, O<E<l, the desired significance level.
If we now define a kernel
(3.13)
where c(u) is 1,
t,
or 0 according as u is >, =, or < 0, then it may easily be
verified that
1
T
.
n, J
=n
n
L: ¢(Y .. ;Y. l' ... , Y. ) for j= 1, ... , p .
i=l
~J
~
~p
(3.14)
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Using the well-known results on U-statistics (cf. Hoeffding [1948J, Fraser [1947J)
and following some essentially simple steps, one may show that the test in (3.12)
is consistent against any heterogeneity of t , •.. ,t '
p
1
Thus, for any fixed
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t 1, .•. ,t p (not all equal), the power of the test will be asymptotically equal to
one.
Therefore, to study the asymptotic power of the test, we shall consider a
sequence of alternatives for which the power lies in the open interval (0,1).
This is specified by
t. = n
-.1.
2
P
A., j=1,2, ... ,p; L: A. = 0
J
where A , ... ,A are all real and finite.
1
p
G
1
=G2 ;;;;
... == G
n
(3.15)
1 . J
J
= G and
Here, let us also assume
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G absolutely continuous .
(3.16)
J, J
If the elements
of the matrix V = (V •• ,) are defined by
JJ
;oJ
00
00
=3
(3.17)
then it may be shown that n V
converges in probability to V
and Wn has asympto~
~
tica11y a non-central
x
2
-distribution with (p-1) d.f. and non-centrality param-
eter
(3.18)
when (3;15) holds - again the reader is referred to Puri and Sen [1966J~ Theorem 4.2.
Finally, one may note that given the permutation covariance matrix V ,
n
we may always select £1 in such a way that ~1~ C! = D = diagonal (d
nN].
In such a case, W reduces to
n
"'n
n,.
1,···,d
(
n, p-
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Let the marginal c.d.f.ls of G be denoted byG[jJ(x), j=1,2, ••• ,p; and the joint
marginal c.d.f.ls (of order 2), by G[. 'IJ(x,.y), jtjl=1,2, ... ,p.
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p-l
W
n
= k=l
I:
.I
/d
n, k
(3.19)
n, k
where the Wn,k are the elements of the vector
91 lri.
4. A NON-PARAMETRIC TREATMENT OF CASE II
As has been explained in section 2, A.3 really means no interaction
Under this condition, Y. is composed of p
between treatments and subjects.
....~
interchangeable random variables (under H in (1.4) ) for all i=l, •.. ,n; and
o
..
hence, the distribution of Y. remains invariant under any permutation of its
coordinates, for i=l, ... ,n.
~
This leads to a set of (PI)n equally likely
(conditionally) permutations and the associated permutational model is denoted
by ~.
n
In this case, we shall also work with the statistic T
~
(3.4), but under ~.
./85)
n
n, J
cov(T
in (3.3) and
Thus, we have
n
E(T
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W2
.,1
n, J
.,T
n, J
~n) =
(4.1)
= (P+l)!2
(0 .. p-l)
~J
(p-l)n
-2
OR'
(4.2)
j"j'=l, •.. ,p
where
(4.3)
R.. 's being defined as in (3.2) or (3.2') and 0 .. is the usual Kronecker delta.
~J
~J
Hence, we may use the test statistic
p
W = [n(p-l)/p cr 2 J I: (T . - (P+l)/2)2 •
R ~l ~J
n
~
1
The asymptotic distribution of n"2(T
(4.4)
-
. - (p+l)/2), j=l, ... ,p, (under (f»
~J
n
can
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easily be shown to be multinormal (with the aid of Berry-Essen theorem [cf.
IV
Loeve (1962, p. 288)J), and hence, W will have asymptotically a chi square
n
distribution with (p-l) d.f.
For small values of n, the exact permutation
distribution of W can be traced with the aid of (P1)n equally likely intran
block rank permutations (under ~), and the exact permutation test can be based
n
....
on W (using the right hand tail as the critical region); while for large n,
n
we may proceed as in (3.12).
In the particular case, when ties are ignored,
cr~ reduces to (p2-l)/12, and our test statistic reduces to Friedman's [1937J
test statistic.
In this case, the permutation distribution and the unconditional
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N
null distribution of W agree with each other, and a tabulation of the exact
n
N
null distribution of W for certain specific small values of nand p is conn
tained in Owen [1962J.
Moreover, for the sequence of alternatives in (3.15),
it follows precisely by the same technique as in Elteren and Noether [1959J
,....
that W has asymptotically a non-central X2 distribution with (p-l) d.f. and
n
the non-centrality parameter
TlW
p
00
= [12p/(p+l)]( L: ,,";)( J g2(x)dx)2
j=l J -00
where g(x) is the density function of Z.., defined just after (3.4).
1J
(4.4)
The
efficiency of this test with respect to the classical analysis of variance
test (which is incidentally valid for Case IV when normality and homoscedasticity
hold) is therefore
(4.5)
where p is the common correlation between Zij and Zij"
If
G(~)
is a multinormal c.d.f., (4.5) reduces to
for jrj'=l, ••• ,p.
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II
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(4.6)
which is the same as in the case of uncorre1ated errors considered by E1teren
and Noether [1959J and Hodges and Lehmann [1962J.
5.
A NON-PARAMETRIC TREATMENT OF CASE III
We define a set of random variables U. . .
1.;JJ'
= Y.1.J. -Y1.J
..
by
j,j'=1,2,
,p
i=1,2,
,n
t
where one should note that U. . . is identically O.
1.;JJ
(5.1)
Under H in (1.4), each
0
Ui;jj' (jrj') is distributed symmetrically about 0 and (Ui ;12"",Ui ;lp"'"
U.
1.;p
l""'U, (
1.; p-
1»
P is diagonally symmetric about 0 (although the distribu-
tion is singular and of rank (p-1».
Let
1
n
~ S.
",
n i=l 1.; JJ
where
(5.2)
S. .,
•
'Ii!'
1.;JJ'
= {Sign(U. ..,)} {Rank[ Iu.
for jrjl=1,2, ... ,p.
1.;JJ
", I:IU1:JJ
", I, ... , IU .. I]}
n:JJ'
1.;JJ
In the above definition, ties are handled by the mid-rank
method and zero is assigned to zero values.
signed-rank statistic; also, W. , . = - W..
J J
JJ'
Thus, W"
JJ I
is the Wilcoxon [1949J
for all jrjl=1,2, ... ,p.
Finally,
either by convention or the definition in (5.2), we write W.. = 0 for j=1,2, ••. ,p.
JJ
Now let
p
T* .
n, J
=
~
j 1=1
W. .,
JJ
j=l, 2, ... , p
'1'*1= ('1'*
n, 1"'" '1'*
n, p )
-n
(5.3)
(5.4)
\,
-12-
From the definition of the W
jj"
it follows that
P
L:Tk.=O
j=l n, J
and hence at most (p-l) of the Tk . are linearly independent.
n, J
If we define
the scores Sij by
S ..
~J
p
L: S.
= j '=1
",
~; J J
i=1,2, ..• ,n
j=1,2, •.• ,p
(5.6)
L
[
then Tk . may be alternatively written as
n, J
1 n
Tk. = - E S ..
n, J
n i=l ~J
(5.7)
Under the permutation model ~ of diagonal symmetry, we have when H holds
n
E{Tk ./
n, J
.,1
E{Tk .Tk
n, J n, J
If V*= (v* 'j') and
Nh
n;J
0
lP } = 0
n
1
'6'}
n
= '2
n
j= 1, 2, •.• , p
(5.8)
n
(5.9)
L: S .. S .. , == V* ..
i=l
~J ~J
n;JJ'
9/1 is as defined in (2.1), a test of H0 may be based on
'VL
(5.10)
Given the permutational model, the ranks are invariant; only the signs are
equally likely and jointly symmetrically distributed.
shown that permutationally W* has sensibly a
n
Hence, it can be easily
x2 -distribution
with (p-l) degrees
For small values of n, the exact permutation distribution of W*
n
n
can be computed by reference to the 2 conditionally equally likely realizations.
of freedom.
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Again, let us denote the marginal c.d.f. of Di;jjl by GfjjlJ(x); the
bivariate c.d.f. of D.. "I' D'. olll by G'I""I IIlllJ(X,y).
1.,JJ
,XIXI
Then, proceeding in
'J,* = (V~t) by
p
V*J'£ =
--
00
p
L. v" 1 Ul'
j 1= 1 £ 1= 1 J J ;
(5.11)
L.
j
•
•
LJJ
a way similar to that followed in Case I, we define elements of a matrix
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1.,XI~
'f j
£
1ft
00
=3
(5.12)
It can be shown that under (3.15), W* has asymptotically a non-central X2 -distrin
bution with (p-1) d.f. and non-centrality parameter
(5.13)
(5.14)
g'l" .. J being the probability density function corresponding to the c.d.f. GF .. J'
LJJI
-LJJ
A NON-PARAMETRIC TREATMENT OF CASE IV
6.
Here we assume that A.1, A.2, and A.3 hold.
R~.
1.J
=
Rank (Z. .: Z11' ... , Z }
1.J
np
Let
i=l, ... , n
j=l, ••. ,p
as in the previous sections, ties are to be handled by the mid-rank method.
If we then define
'
(6.1)
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(6.2)_
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1
~
. =n
n, J
n
j= 1,2, ... , p
L. R~.
~J
i=l
and
1
(
n
n p-
1)
p
1
P
L. (R~. - - L. R~.)2,
i=l j=l ~J
P j=l ~J
L.
(6.3)
then it follows from the results of Sen [1966J that a test of H can be based
o
on
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For small values of n, the permutation distribution of ""
W* can be traced by
n
reference to the (pl)n conditionally equally likely intra-subject rank per,..
mutations. For large n, W has sensibly a X2 -distribution with (p-1) d.f.
n
Asymptotic power properties of this test also follow from the results of Sen
[ 1966].
7.
The data are from experiments undertaken at the Depart-
ment of Pathology" Duke University Medical Center, Durham, North Carolina.
Example 1:
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EXAMPLES
In this section, we shall consider the statistical analysis of two
numerical examples.
I
I
Sixteen animals were randomly placed into one of two groups - an
I,
Ii
1,1
experimental group receiving ethionine in their diets and a pair-fed control
group.
The data for each animal consisted of a measurement of the amount of
radioactive iron among various sub-cellular fractions from liver cells.
The·
cell fractions used were nuclei (N), mitochondria (Mit), microsomes (Mic), and
supernatant (S).
One question of interest to the experimenters was whether
the ratio of the measurements for the experimental group to those for the
I,
I'
I"
-.
I"
I
3.
1
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-15 -
control group was the same for all cell fractions.
are regarded as subjects and cell fractions are regarded as treatments, then
the above question may be attacked by the methods previously considered in
this paper.
The data are as follows:
]
]
Mic
S
Mean
1. 73
1. 08
2.60
1. 67
1.77
2
2.50
2.55
2.51
1. 80
2.34
3
1. 17
1.47
1. 49
1. 47
1.40
4
1. 54
1. 75
1. 55
1.72
1. 64
5
1. 53
2.71
2.51
2.25
2.25
6
2.61
1.37
1. 15
1. 67
1. 70
7
1. 86
2.13
2.47
2.50
2.24
8
2.21
1. 06
0.95
0.98
1.30
Mean
1. 89
1.77
1.90
1. 76
1. 83
N
1
-I
1
Je
Mit
Pair
1
--I
If matched pairs of animals
First, let us obtain the test statistics appropriate for the situations
in which the error vector may be assumed to be normally distributed.
-I
1
J
J
I
]
I·
I
The com-
putations leading to the Hote11ing T2 statistic are listed below.
Pair
N - Mit
N - Mic
N - S
1
2
3
4
5
6
7
8
0.65
-0.05
-0.30
-0.21
-1.18
1.24
-0.27
1. 15
-0.87
-0.01
-0.32
-0.01
-0.98
1.46
-0.61
1. 26
0.06
O. 70
-0.30
-0.18
-0.72
0.94
-0.64
1. 23
Mean
0.13
-0.01
0.14
T2 = 8 [0.13
-0.01
0.14]
(n-1)
-1
S =
""'\l
[-0.82
5.34
-4.15
-0.82
4.56
-4.02
-4.15]
[-0.01
0.13]
-4.02
10.08
(n-p+1)
T2 = (5/21)(1.16) = 0.28;
(n-1)(p-1)
-0.82
4.56
-4.02
-4.15
u t = [0.13
[-0.82
5.34
0.59
0.84
0.58
0.51
-1
(n-1)S=
""u
[0.68
0.59
-0.01
= 1. 16
0.14
d. f. = (3,5)
.
0.51]
0.58
0.54
-4.15]
-4.02
10.08 .
0.14]
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-16-
Since 0.28
~
FO. 75(3~5), we may conclude that this test indicates the data
to be consistent with H (the hypothesis that the ratio is the same for all
o
cell fractions).
The estimated covariance matrix of the cell fractions is
0.26
(n-1)-l S = -0.01
N
[ -0.06
-0.03
-0.01
0.41
0.27
0.22
-0.06
0.27
0.47
0.23
-0.03]
0.22
0.23
0.21
Since the structure of this matrix does not greatly contradict the assumption
that the error vector is symmetrically distributed, the analysis of variance
--I
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I~
test of H is of interest.
o
Source of Variation
d. f.
Sum of Squares
Mean Square
Var. Ratio
Cell Fractions
Pairs
Error
3
7
21
0.15
4.51
4.95
0.05
0.64
0.24
0.21
2.67
Total
31
9.61
Noting that 0.21
~
_I
F . (3,21), we may again conclude that the data tend to be
O 75
consistent with H .
o
Let us now turn to the analysis of the data in terms of the non-parametric
statistics associated with each of the Cases I - IV.
For Case
I~
we first com-
pute the within block ranks corresponding to each of the observations where ties
are handled by the mid-rank method.
These are as follows
Pair
N
Mit
Mic
1
2
3
4
5
6
7
8
Total
3
2
1
1
1
4
1
4
17
1
4
2.5
4
4
2
2
3
22.5
4
3
4
2
3
1
3
1
21
S
2
1
2.5
3
2
3
4
2
19.5
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II
I;
I
II
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-.
I:
I
I.
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-17-
The statistic W is obtained by carrying out three steps
n
i.
fractions; then, within each block, transform the ranks into these
contrasts.
ii.
Eg., we have used N - Mit, N - Mic, and N - S.
Compute the sums and sums of products of deviations associated with
the contrasts defined in (i).
iii.
Compute the matrix product of the row vector of sums, the matrix of
sums of products of deviations, and the column vector of sums, all
of which were obtained in (ii).
This is W •
n
Applying the above procedure to the data, we obtain
Pair
N - Mit
N - Mic
N - S
8
2
-2
-1. 5
-3
-3
2
-1
1
-1
-1
-3
-1
-2
3
-2
3
1
1
-1. 5
-2
-1
1
-3
2
Total
-5.5
-4.0
-2.5
1
2
3
4
5
6
7
..-
•
...
Select (p-l) = 3 linearly independent contrasts among the cell
W
n
=
[-5.5
-4.0
O. 75(3),
Since 1.01 ~ X
-2.5] [ 0.064
-0.024
-0.024
n2 V
N'Il.,U
n
47
= eO.
21. 75
16.53
16.53]
20.25
22.47
-0.024
0.066
-0.041
[ 0.064
-2 -1
V
= -0.024
Nn,u
-0.024
= [-5.5
nT
-n,u
-0.024
0.066
-0.041
21. 75
36.00
20.25
-4.0
-0.024J
-0.041
0.100
-2.5]
-0. 024J[ -5. 5] = 1. 01;
-0.041 -4.0
0.100 -2.5
d.f.
= 3.
we again conclude the data to be consistent with Ro .
The statistic appropriate to Case II is most simply obtained by performing
a type of analysis of variance on the set of within-block ranks.
. denoted by R.., we proceed as follows.
~J
P
L
Compute S~ = (lIn)
n
L:
j=l
~
L: R..
{
i=l ~J
2
If these
ar~
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-18-
Compute 8~ = (npcr~)
ii.
Compute
iii.
W
=
n
=
p
L. R;;. - np(p+1)2/4 and
i=l j=l ~J
n
L.
8~/n(p-l) .
82/S 2
t
e
The results for this method are
82 = 202.06 - 200.00 =
2.06
t
82 = 239.50 - 200.00 = 39.50
e
s~
~
O.75(3)
We note that 1.25 ~ X
d.f. = 3.
and draw the same conclusions as before.
Case III requires the most extensive computations.
However, these may
be performed efficiently by proceeding as follows.
i.
,
Within each block compute the (~) differences associated with each
possible pair of treatments where the treatment to the right is
subtracted from the one to the left.
Eg., N - Mit, N - Mic,. N - 8,
ii.
N - Mit
0.65
-0.05
-0.30
-0.21
-1. 18
1. 24
-0.27
1. 15
N - Mic
-0.87
-0.01
-0.32
-0.01
-0,98
1.46
-0.61
1.26
N - 8
0.06
O. 70
-0.30
-0.18
-0.72
0.94
-0.64
1.23
••II
-IJ
I
i
Mit - Mic, Mit - 8, Mic - 8
Pair
1
2
3
4
5
6
7
8
I
I
I
1
I
I
Ij
1
= (39.50)/(24) = 1.65
Wn = (2.06)/(1.65) = 1.25;
.1
Mit - Mic
-1. 52
0.04
-0.02
0.20
0.20
0.22
-0.34
0.11
Mit - 8 Mic - 8
-0.59
0.93
O. 75
0.71
0.00
0.02
0.03
-0.17
0.46
0.26
-0.30
-0.52
-0.37
-0.03
0.08
-0.03
Each of these new variables has a particular value within each block.
Let signed ranks be associated with each of the values associated
with a given variable (where ties are handled by the mid-rank method
and zero is assigned to zero values).
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i
I;
1
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I.
IA
-.
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-19-
Pair
1
2
3
4
5
6
7
8
iii.
I
-
N - Mic
-5
-1. 5
-3
-1. 5
-6
8
-4
7
N - S Mit - Mic
1
-8
2
5
-1
-3
-2
4.5
4.5
-6
6
7
-4
-7
8
3
Mit - S Mic - S
8
-7
8
7
0
1
2
-4
6
5
-4
-6
-2.5
-5
3
-2.5
Within each block, scores are obtained for each cell fraction by
adding the sum of the signed ranks associated with variables in
which it is the minuend to the negative of the sum of the signed
ranks associated with variables in which it is the subtrahend.
Pair
1
2
3
4
5
6
7
8
Total
I_
I
I
I
I
N - Mit
5
-1
-4
-2
-7
8
-3
6
iv.
N
1
2.5
-10
-5.5
-19
23
-11
21
2
Mit
-20
11
3
8.5
17.5
-6
-9
0
5
Mic
21
6.5
5
-7
6.5
-20
8.5
-12.5
8
S
-2
-20
2
4
-5
3
11.5
-8.5
-15
To the scores obtained in (iii), apply the steps of the computational
procedure given for Case I.
Pair
1
2
3
4
5
6
7
8
Total
N - Mit
21
-8.5
-13
-14
-36.5
29
-2
21
-3
N - Mic
-20
-4
-15
1.5
-25.5
43
-19.5
33.5
-6
N - S
3
22.5
-12
-9.5
-14
20
-22.5
29.5
17
n
n 2 V*
N'n,u
=
-2 *-1
V
""n,u
=
nT*
-n,.u
W*
n
495
2706
1923
e
[ 5.55
-2.22
-1. 76
= [
-3
=
0.39 ;
1923]
2706
4640
2672
2672
2686
-2.22
5.94
-4.32
-6
-1.
76]
-4.32
9.28
17]
d.f.
= 3.
x 10- 4
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-20-
Finally, let us consider Case IV.
The test statistic may be computed
most simply by performing the following steps.
i.
ii.
iii.
iv.
For each observation in a given block, subtract the block mean.
Rank the (np) "residuals" obtained in (i) where ties are handled by
the mid-rank method. Let these be denoted by R* .
ij
*2
p
n *.2
Compute St = (lIn) 2: { 2:; R.. J - np(np+l)2/4
j=l i=l 1.)
*
n
p *2
Compute S*2 = n(p-l)cr 2 (R ) = 2:;
2:; R..
e
i=l j=l 1.)
*2
*2
se = Se In(p-l).
The results associated with this test procedure are given below
Pair
1
2
3
4
5
6
7
8
N - Mean
-0.04
0.16
-0.23
-0.10
-0.72
0.91
-0.38
0.91
Mit-Mean
-0.69
0.21
0.07
0.11
0.46
-0.33
-0.11
-0.24
Pair
1
2
3
4
5
6
7
8
Mean
N
Mit
2
25
18.5
22
29
7
11
9
15.44
15
23
10
12.5
1
31.5
5
31.5
16.19
Mic-Mean
0.83
0.17
0.09
-0.09
0.26
-0.55
0.23
-0.35
Mic
30
24
21
14
27.5
3
26
6
18.94
S - Mean
-0.10
-0.54
0.07
0.08
0.00
-0.03
0.26
-0.32
S
12.5
4
18.5
20
17
16
27.5
8
15.44
_.
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I
I
I
I
I
-.
I
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I.
I
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*2
St = 66.38
*2
S = 2615.62
e
s
N*
W
d. f. = 3.
n
= 0.61;
*2 = 108.98
e
From the previous remarks, one can see that all the test procedures lead us
to conclude that the data are consistent with H.
o
Also, the values of the
computed test statistics tend to be more or less similar to one another.
Example 2:
Sixteen animals were randomly placed into one of two groups - an
experimental group receiving ethionine in their diets and a pair-fed control
group.
The liver of each animal was split into two parts one of which was
treated with radioactive iron and oxygen and the other, with radioactive iron
I_
and nitrogen.
I
I
the combinations ethionine-oxygen (EO), ethionine-nitrogen (EN), control-oxygen
•
treated liver halves.
~
-31
.e
If matched pairs of animals are regarded as subjects and
(CO), and control-nitrogen (CN) are regarded as treatments, then the hypothesis
that neither diet nor gas has any effect may be tested by the methods discussed
in this paper.*
Pair
1
2
3
4
5
6
7
8
Mean
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•-
The data consist of the amounts of iron absorbed by the variously
* Note
EO
38.43
36.09
34.49
37.44
35.53
32.35
31.54
33.37
34.90
EN
31. 47
29.89
34.50
38.86
32.69
32.69
31. 89
33.26
33.16
CO
36.09
34.01
36.54
39.87
33.38
36.07
35.88
34.17
35.75
CN
32.53
27.73
29.51
33.03
29.88
29.29
31.53
30.16
30.46
Mean
34.63
31. 93
33.76
37.30
32.87
32.60
32.71
32. 74
33.57
that the structure of the data allows other tests to be considered; eg.,
equality of diet effects, equality of gas effects, etc. However, we are
interested here only in the hypothesis of equality of all treatment effects •
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After computations similar to those given in Example 1, we find
T2
=
(n-p+1)
(n-1)(p-1)
and
17.14
2
T
= 4.08.
Since 4.08 :;; F . (3,5), the test does not reject Ho '
O 95
However, T2 is large
enough to recommend additional experimental results to study the differences
among the treatment effects.
-'I
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I
II
The estimated covariance matrix for the treatment vector is
5,86
( n _1)-1 ~S = 1.31
1. 29
[
1.47
1.31
7.13
4.35
2.60
1. 29
4.35
4.14
2.40
47
2.60
1.
2.40
3.17
1
Again, the analysis of variance test of H is of interest.
0
Source of Variation
d.f.
Sum of Squares
Mean Square
Var. Ratio
Treatments
3
131.19
43.73
15.40
Pairs
7
82.38
11.77
4.14
Error
21
59. 70
2.84
Total
31
273.27
Since 15.40
~
F . (3,21), we reject Ho and conclude that there are significant
O 99
differences among the treatment effects.
The non-parametric statistics for Case I and Case II are based on the
intra-block rank matrix
Pair
1
2
3
4
5
6
7
8
Total
EO
4
4
2
2
4
2
2
3
23
EN
1
2
3
3
2
3
3
2
19
CO
3
3
4
4
3
4
4
4
29
CN
2
1
1
1
1
1
1
1
9
I
II
Ii
_II
I:
I::
I~
I,
1,1
Case I:
= 600.00
=3
W = 15.90
n
d. f. = 3
W
n
d.f.
N
Case II:
I:
-,
1<'
I!
I
I.
I
I
I
I
~
-23 -
In Case III, the score matrix for the treatments is
Pair
1
2
3
4
5
6
7
8
Total
EO
20
18
2
-7
16
-8
-11
4
34
EN
-18
-11
4
10
-3
4
-1
1
-14
CO
5
9
14
16
-2
18
18
6
84
CN
-7
-16
-20
-19
Case III:
*n = 64.27
W
-11
d. f.
=3
-14
-6
-11
-104
Finally, for Case IV, the matrix of intra-inter-b1ock ranks of residuals is
Pair
1
2
3
4
5
6
7
8
. Total
-e
EO
31
32
20
16
27
13
11
19
169
EN
5
9
21
24
14
15
12
18
118
CO
23
25
28
26
17
30
29
22
200
CN
8
3
2
1
6
4
10
7
41
Case IV:
,..*
W
n
=
16.02
d. f.
=3
Since the test statistics obtained above all exceed the 99th percentile point
of the chi-sqpare distribution with three degrees of freedom, we reject H for
a
each of the Cases I-IV.
One may note that the statistic W is considerably larger than the others.
n
One reason for this is that with W the particular arrangement of the ranks
n
within the blocks not only affects the mean scores associated with a treatment
but also the estimated variance-covariance matrix.
As a result, for data
configurations in which the intra-block ranks show consistent treatment differences, the statistic W is likely to be very large.
n
the case n
(4
3
2
= 8,
p
= 4,
For example, suppose for
we observed a rank matrix in which the arrangement
1) occurred four times and the arrangement (3
4
2
1) occurred four
times; the contrast obtained by subtracting the score assigned to the fourth
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treatment from that assigned to the third treatment has a mean value of one but
an estimated variance of zero.
Strictly speaking, W cannot be computed, but we
n
.-i
can argue that its value is infinite.
..
""*
On the other hand, when we compute Wand W, the estimated variability
n
n
of the treatment mean scores depends only on the values of the ranks assigned
to a block and not on the particular arrangement of them within the blocks.
'Ie
For reasons similar to those given previously for W, the statistic W
may also be noticeably larger than
'"Wand ...*
W.
n
n
n
n
I
I
I:
Since this extreme-type behavior
of Wand W* does not cause us to wrongly accept false hypotheses, we can argue
n
n
that there is nothing to worry about.
On the other hand, when such large values
occur, the chi-square distribution may not provide an idea of the exact significance
level at which the hypothesis would be rejected.
If this were of interest, then
one should perform the associated permutation test.
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•
i
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[1959J.
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Hw
Votaw, D. F., Jr. [1948J. "Testing compound symmetry in a normal multivariate
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