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SOME ASPECTS OF THE STATISTICAL ANALYSIS OF
"SPLIT PLOT" EXPERIMENTS
PART I: COMPLETELY RANDOMIZED LAYOUTS
by
Gary G. Koch
University of North Carolina
Institute of Statistics Mimeo Series No. 527
May 1967
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This work was 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 "Split Plot" Experiments
Part I:
Completely Randomized Layouts *
by
Gary G. Koch
University of North Carolina (Chapel Hill) and Research Triangle Institute
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SUMMARY
In this expository paper, the statistical analysis (both parametric and nonparametric) of completely randomized "split-plot"Aexperiments is discussed
from the point of view of the underlying multivariate model.
The general
structure of such experiments involves N randomly chosen subjects to whom
treatments have been assigned according to a completely randomized design
and from each of whom is obtained an observation vector, the components of
which represent the responses of the subject to each one of several conditions.
The different conditions correspond to the "split-plot" treatments in agricultural experiments while the different treatments correspond to the "whole
plot" treatments.
In such experiments, a number of hypotheses are of interest -
the hypothesis of no treatment effects, the hypothesis of no condition effects,
the hypothesis of no interaction between treatments and conditions.
Various
formulations of these hypotheses are considered under several different combinations of assumptions concerning the joint distribution of the components
of the observation vector.
In each case considered, appropriate parametric
or non-parametric test procedures are discussed.
Some of the methods considered in the paper are illustrated in a
numerical example.
The example is representative of a situation in which
some of the standard assumptions regarding normality and variance homo-
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geneity do not hold.
In this part of the paper,
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*This work was partially supported by the National Institute of Health,
Public Health Service, Grant GM-12868.
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algorithms for the efficient computation of the various test criteria are
given.
Finally, a computer program based on the algorithms has been written
and can be made available to any interested persons.
1.
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,
has been assigned at random to one of v treatment groups and an observation
has been taken on his response to each of p conditions.
k
= 1,2, ••. , p.
N=
In this framework,
Y~~) denote the response of the j-th individual in the i-th treatment
group to the k-th condition where i = 1,2, ""
v
}:
.1
1=
n.
1
= 1,2,
""
~j
function (c.d.f.)
... ,
y~~»
1J
j
= 1,2,
••• , v
= 1,2,
... , n.
1
has a continuous p-variate cumulative distribution
F.~),
1
and that the different
where
~j
are statistically independent.
The parameters
m. are indicative of the locations of the distributions and the functions
-}.
G characterize their shapes.
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Together with (2), the basic assumption
throughout this paper is
A.1.
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n i ; and
Let
,
We assume that
v; j
Since there are N subjects in all, we have the relation
i
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INTRODUCTION
Let us consider an experimental situation in which each of N subjects
let
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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
(1)
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3
A.2.
No interaction between treatments and conditions.
A.3.
The "compound symmetry" of the error vectors.
The assumptions A.l, A.2, and A.3 have been discussed in Koch and Sen
[1967].
Their application to the problems considered in this paper are
explained more fully in what follows.
In any event, four cases of interest
arise; and these may be described by the following table.
A.2 is not known
to hold
A.2 is known to
hold
Case I
Case II
Case III
Case IV
A.3 is not known
to hold
A.3 is known to
hold
In each of the above cases, the hypothesis of no treatment effects
.
ot·
G
G = G
Z
1
v
H
E!l
= m
-2
=
(3)
= m
-v
and the hypothesis of no condition effects
H
oc
(1)
m.
=
l.
(2)
ID.
l.
= m(.p)
l.
for
l..
= 1 ,2,
... ,
v
(4)
are considered, and appropriate test procedures are given.
2.
THE NORMAL (PARAMETRIC CASE).
Thenorrnal (parametric) case is characterized by an additional assumption.
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A.4. G
l
= G2 = ... = Gv
--I
is a multinormal c.d.f.; i.e., the error vectors
have a common p-variate normal distribution with a null mean vector and an
unknown positive definite covariance matrix
~
.
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Now A.4 implies that the Y.. are independently distributed according to
-:1.J
multivariate normal distributions
j
= 1,2,
""
n , the
i
~
N~
~)
,
= 1,2,
where i
being defined by (2).
T~e
... , v
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and
statistical analysis of
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this model by the use of the methods of multivariate analysis has been discussed recently in a paper by Cole and Grizzle [1966].
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The results cited
here are similar to the ones given there; but for a more complete e1aborat ion of the motivation behind the basic approach, the reader is referred
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to their paper.
To test Hot in (3), we may app1y'une-way multivariate analysis of
variance"(MANOVA) as described in Smith, Gnanadesikan
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and Hughes [1962].
For this procedure, we obtain the matrix
(5)
with
v
SB =
~
i=l
n
V
Sw =
- Y )(Y.1. - Y )
ni(Y.
-:1..
i
,
-
Y. ) ,
Y. )<.:4. - ~.
<.:4j - -:1..
J
i=l j=l
~
~
n.
where Y.
-1.
1
(6)
1
~
n .. 1
1 J=
L.
-.!oj
1
i = 1,2, ""
v; Y
="N
v
~
n.
1
~
Y ...
i=l j=l -:1.J
Three test criteria based on the characteristic roots of R have received
t
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extensive consideration in the literature.
These are the largest character-
istic root of R (Roy [1953]), the sum of the roots of R (Hotelling [1951]),
t
t
and the likelihood ratio criterion which equals the product of the reciprocals
of the sum of unity and the respective characteristic roots (Wilks [1932]).
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Tables which provide critical values for tests based on these criteria are
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Schatzoff [1966G. *
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available.
The reader is referred to Pillai [1960], [1964], [1965], and
However, the actual choice of which criterion to use
remains largely an open question left to the tastes of the experimenter
(see Schatzoff
[196~).
As a final comment, one may note that if L is
known, a test of H may be based on
ot
v
2
X = L n. (1t - y ) , L-l(y. - y
~.
~
i=l
.
(7)
)
2
When (3) holds, X has the X2-distribution with p(v-l) degrees of freedom.
Also, for large N, (7) may be used as the basis of an approximate test of
Hot for the case when L is not known; however, other approximate tests are
also available for which the reader is referred to Anderson [1958], Box
[1949], Rao [1952], [1965].
Next we shall consider three approaches to the construction of test
criteria for H as given in (4).
oc
In each of these, we first make the
transformation of variables
i = 1,2,
= C y ..
l ~J
J4j
j = 1,2,
... ,
... ,
v
(8)
n.
~
where C is any «p-l) x p) matrix whose rows are linearly independent
l
contrasts; i.e., C
l
1 = Q where i
is a (p x 1) vector of ones.
true, then each of the vectors J4j has the normal distribution
*The
If H
oc
N~_l
is
' CIL Cl )
parameters of the test are the design matrix rank v, the hypothesis
rank (v-I), the variate rank p, and the design error (N-v). One should note
that the tests simplify if p = 1,2 or v = 1,2,3; for details here, see
Anderson [1958].
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where 0 1 is a «p-1) x 1) vector of zeros.
-pF
c
U'
= N(N-p+1)
(p-1)
8- 1
u
~I
As a result
U
(9)
-
n.
v
ni
v
l.
where U = 1:. E
E U•• and 8 = E
L (U ..
u
N i=l j=l ~J
i=l j=l ~J
, has the
(d. f.) - where it is necessary to presuppose that N ~ p.
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The test of H
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based on (9) has been previously discussed in Danford, Hughes, and McNee
[1960] Greenhouse and Geisser [1959], and Cole and Grizzle [1966].
Although
F is a valid statistic for testing H , i t is not sensitive to alternatives
oc
in which certain conditions have a positive effect in some treatment groups
In these cases,
one might use the following type of procedure which has been discussed,
for example, by Bi~nbaum [1954].
ni(ni-p+l)
Fl.' c
=
n
where U.
~.
i
1
=- L
n. j=l
l.
=
(10)
n.
l.
.!!.t.
and 8
J
1,2, ••• , v
for i
-1-
--==---=--u!
s.
U.
p-l
~.
Ul.~.
ui
=
L
j=l
U. )(U ..
(Q .. - -l..
~J
l.J
-
-U.
)
,
-1..
Let
> F. }
Pr{U(p-1,n.-p+1)
l.C
1.
(11)
and
v
A =-2
c
When H
L
i=l
R.n(A.1.C ) •
is true, A has the X2-distribution with 2v degrees of freedom.
c
Note that A is sensitive to departures from H occurring within each of
c
oc
OC
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but a negative effect in other treatment groups (i.e., situations in which
there is interaction between treatments and conditions).
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U-(variance ratio) distribution with [(p-1), (N-p+1)] degrees of freedom
c
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(12)
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7
the separate treatment groups.
However, in order for a test based on A
c
to be possible, one must assume that n
i
~
p for i = 1,2, •.• , v; if this
condition does not hold, the statistic A can be modified to
c
* = -2
c
A
v
I:
O(n.-p) tn(A. )
i=l
I
1
(13)
1C
1ifu>O
where 6 (u)
c
Oifu<O
and A
is taken to be unity if n < p
ic
i
·
. h 2
ut10n
w1t
h as t h e X2- d·1str1· b
true.
v
I:
i=l
A
*c
o(ni-p) degrees of freedom when H is
oc
The disadvantage to A* ,of course, is its lack of sensitivity to
c
departure from H occurring within the treatment groups for which n. is
oc
1
less than p.
Finally, a multivariate test of H may be based on the characteristic
oc
roots of
(14)
where
v
8 = I: n U U'
M i=l i~.~.
8
WU
=
v
ni
I:
I:
(15)
i=l j=l
If this is done, comments similar to those following (5) and (6) may be
made regarding the actual choice of a test criterion. *
We shall say that there is no interaction between treatments and condition
*The
parameters of the test are the design matrix rank v, the hypothesis
rank v, the variate rank (p-l), and the design error rank (N-v).
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(assumption A.2) if the m(k) satisfy the following additive structure
i=1,2, ... ,v
(16)
k = 1,2, ... , p
v
where
P
1:
T.
i=l
~
mean effect,
= 0 and
1:
k=l
T
i
Yk
=o •
In this case,
~
represents an overall
represents the effect due to the i-th treatment, and Y
k
represents the effect due to the k-th condition.
If (16) holds, then Hot
is equivalent to the hypothesis
(17)
while H is equivalent to the hypothesis
oc
(18)
* may be readily obtained by performing
A test for the hypothesis Hot
univariate"one-way analysis of variance"(ANOVA) on the subject means
e
Yij =1.
P k:l y(k)
ij for which E(Y ij ) = ~ +
Ti
an d var (Y)
ij = ("~
'"t.. J../ p 2) •
The test statistic is
*
*2
*2
Ft = (N-v) St /(v-l) Se
(19)
where
*2
St = p
*2 .
Se = P
v
I:
i=l
2
n. (Y. - Y.. )
~
1.
v
n.
1
1:
I:
i=l j=l
(20)
(Y.
1j
- -Yi. )2
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1
i
=L Y•. and Y
n i j=l ~J
1
= -
v
When (16) and (17) both
L
N i=l
hold, F* has the u-distribution with [(v-I), (N-v)] d.f.
t
is a well-known one and is widely used.
This procedure
However, when (16) does not hold,
the test in (19) is directed at the hypothesis
= ev
e1 =
h
El-i
were
=1 ~
m(k)
i
~
p k=l
(21)
The test is not sensitive to departures from H
ot
* is true (i.e., it is not sensitive to interaction in the sense
in which Hot
of the effects of treatments on contrasts among the conditions).
*
When (16) holds, the hypothesis H may be tested by using the statistic
oc
F
F
c
c
in (9).
The condition of "no interaction" eliminates the dra'l7backs to
mentioned earlier.
Finally, the validity of Assumption A.2, as stated in (16), may be
tested by applying one-way MANOVA to the vectors U.. given in (8).
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In
this case, the matrix
(22)
where
v
SBD =
L
i=l
v
Swu =
is obtained.
L
n.~-,.
(D. - D)(D.
- -,.
n
- U)'
(23)
i
L (D .• - D.) (D ..
-J. -J.J
-~J
i=l j=l
-D. ) ,
- -J..
The actual test statistic will be one of the previously
mentioned functions of the characteristic roots of R
tc
This test has
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been previously considered by Greenhouse and Geisser [1959] and Cole and
Grizzle [1966] to whom the reader is referred for further details.*
We now turn to a discussion of test statistics for the hypotheses (3),
(4), (16), (17), and (18) when Assumption A.3 holds.
A sufficient condition
for the "compound symmetry" of the error vectors (in the normal case) is
(24)
where J is a (p x p) matrix of l's and I is the (p x p) identity matrix.
Given that (24) is satisfied, the following ANQVA may be performed.
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*The
parameters of the test are the design matrix rank v, the hypothesis
rank (v-I), the variate rank (p-l) and the design error rank (N-v)
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......
- - - ..... .. e_..'-- .. - - ..e - e
Source of Variation
d. f.
Variance Ratio
Sum of Squares
Treatments
(v-I)
*2
Q1 = St
F
Subjects in Trmt. Groups
(N-v)
*2
Q2 = Se
Fc
Conditions
(p-1)
-1Q = NU'(C C')
U
3
1 1
-
F
Conditions x Treatments
(v-I) (p-1)
Q = En.fU -U)'(C C,)-l fU -:Q)
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1 1~.
i=l ~~.
(N-v) (p-1)
v
~
QS = i:1 j:1
*
t
*
= (N-v)Q/QS
(N-v)Q4
*
=
tc
(v-1)QS
v
n.
Error (Subj. x Condo in Trmt. Groups)
'l4 j
_
- ~.)' (C1 Ci)
-1
(~j
-
.!Lt.)
!-'
!-'
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In the preceding, one may also write
~ (y(k) _ y)2
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k=l
p
~ n. (y~k) - Y.~. _ y(k) + y)2
Q4 = L
.
k=l ~= 1 ~ ~.
n
v
Q5 =
L
L
i
where y~k) = ~.
n
i
i
L
j=l
(25)
~ (Y (k) _ y~k) - Y + -Y. )2
ij
~.
i=l j=l k=l
n
1
ij
~.
1
y~~) , -(k)
Y•• = ~J
N
v
L
n.
~
L
i=1 j=l
y(k)
ij
are as defined in connection with (19) and (20).
and Y..
~J
Y.
Y
~.
From this ANOVA the
* may be used to test the hypothesis of no treatment x condition
statistic Ftc
If one assumes that this condition holds in the sense of (16) ,
interaction.
then F* may be used to test H ,and F* may be used to test H
t
ot
c
oc
However,
when (16) is not known to hold, then
Fc
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=
may be used to test H while F* is directed at H* in (21).
oc
t
ot
(26)
If p
~
0
then there is no error term for a sum of squares appropriate for H
ot
However, a test may be constructed by obtaining
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A* = Pr{tl«v-l)(p-1), (N-v)(p-1)) > F* }
tc
- tc
(27)
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Then a test statistic for Hot is
At
= - 2tn At*
- 21n At*c
(28)
which has the X2-distribution with 4 degrees of freedom when (3) holds.
On the other hand, if one assumes p
~
Ft
=
=0
, then
(Ql + Q4)(N-v)
(29)
(Q5 + Q2)(v-l)
may be used to test Hot when (16) is not known to hold
(30)
may be used when (16) does hold; also when p
=0
~ * may be similarly changed from
F* ,F * ,and F
c
tc
c
*
When (24) does not hold, F* and Ftc
c
under the corresponding null hypotheses.
, the error terms of
Q5/(N-v) (p-l)
to
no longer have u-distributions
Greenhouse and Geisser [1959]
consider the approximate sampling distributions of F* when
c
L is general
and indicate procedures for the construction of conservative tests.
However, this author prefers the multivariate analysis approach previously
indicated.
Finally, we consider an interpretation of the "compound symmetry"
condition (24).
When Assumption A.3 holds, we shall say that there is no
interaction between conditions and subjects in the sense that the variability
of a subject's response is independent of the condition employed.
This
definition may be motivated, to some extent, by considering the additive
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model
Y"
1J k
= m~k) +
1
(k)
where m
i
s(k) + x~~)
ij
1J
(31)
is the mean response for subjects in the i-th treatment group
under the k-th condition,
S~~)is
1J
a random subject effect corresponding to
the j-th subject in this situation with
E{Si~)} =
0 , and
x~~) is a technical
1J
error reflecting the variability in the response of the j-th subject under
repeated exposure to the i-th treatment and k-th condition with
E{X~~)} = 0 .
1J
The quantity
E(k) = s(k) + X~~)
ij
ij
1J
(32)
represents the total error to which the Assumption A.3 applies.
represents the variance matrix of the vector --1J
E.. where
X~~)
If we assume that the
1J
2
with a common variance 0
x
~J'
_U.L
=
The matrix L
(E(l)
ij ,
... ,
_I
are statistically independent random variables
and that the
X~~)
1J
are statistically independent
(k)
of the S .. , then we may say that the variance matrix of the vector
1J
~j where ~j
L
s
=
... ,
= (S~:),
1J
L-
0
2
x
I
is
(33)
P
The matrix L has the structure (24) if and only if the matrix L has the
s
structure (24).
In particular, if we have a variance components model in the
sense that S~~) = Sij
1J
0
2
J
s
2
+ 0x I
,
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then L
S
may be written as
which satisfies (24).
0
2
J
s
and L as
However, s~~)= S .. means the effect
1J
1J
of the j-th subject is independent of the condition of measurement; in other
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words, there is no interaction between subjects and conditions.
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addition
as = 0 , then all the error degrees of freedom may be used to
estimate
ax (i.e., Q and Q may be pooled).
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If, in
2
This completes the discussion of the parametric case.
As a final
note, we mention that the validity of (24) may be tested by means of
likelihood ratio tests.
The yeader is referred here to Box [1949],
[1950], Votaw [1948], and Wilks [1946].
For other discussions of the
parametric case, the reader is directed to all the references previously
mentioned and also
Scheff~
3.
[1959] and Geisser [1963].
A NON-PARAMETRIC APPROACH TO CASE I.
First, we consider the hypothesis Hot in (3).
The proposed test is
discussed in Chatterjee and Sen [1966] and Puri and Sen [1966] and represents a multivariate version of the well-known Kruskal-Wallis [1952] test.
The test statistic is constructed in the following manner.
y(k)
11 '
... , y(k)
In '
1
Let
(k)
y(k)}
... , YvI ... , vn
v
(34)
for i = 1,2, ••• , v; j = 1,2, ••• , n.; and k = 1,2, ••• , p; here ties are
1
handled by the mid-rank method in the sense that
R(k)
iJ'
=1 +{
The number of
are less than
" , J")
were
h
(1
J
T
( •• )
1, J
and k = 1,2, •.• , p.
which }
The number of
+ (1/2)
(k)
{ Y.,.,
equal to
1
J
. .1 , = 1 , 2 , ••• , v; J.,.,
; a 1 so 1,
J = 1, 2,
We next form the average ranks
... ,
y~~)
1J
n ..
1J
} (34')
16
n.
-(k)
1
=R.
1.
n.
1
1
i = 1,2,
R~~)
I:
(35)
1J
j==l
••• ,V
1,2, ••• , p
k
and note that these satisfy the constraints
~
-(k)
<.
n R.
i
i=l
1.
N(N+l)
2
k
1,2, ""
(36)
p •
Y.. has the same distribution.
When H is true, each of the . vectors ~J
ot
Hence,
the joint distribution of all these N vectors is invariant under the
different possible assignments of the vectors to treatmer.t groups.
This
invariance generates a set of N! conditionally equally likely realizations
,
-lJ
for the rank vectors R.. where R..
~J
=
(1)
1J
(2)
1J
Thus, under
(R .. , R.. , . '"
this conditional probability law (say P ) , we have
Nt
{
I
E ~j P Nt
Var{R ..
~J
N+l.
} = -2~
IP N }
t
(37)
n.1
1 v
N+1
_ N+1 ')'
I:
- N I:
~j - -2- i)(~j
2 ~
- VN
i=l j=l
(38)
(39)
where (i' ,j') f (i,j) and
.. ,
1,1
= 1,2, ' ' ' ' v; j,j'
1,2, ""
n. ; k=1,2, ... , p .
1
Hence,
E{R
-i,
IPNt }
= N+ 1 .
2
(40)
~
N-n.
Var{~. IP Nt } = n.(N~l)
1
VN
(41)
(42)
where R.
~.
The test statistic for this procedure is
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17
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L
N1
N, t
= ( -N)
N+1
1
v
Ln. (R. 2
j)' V-N (R.
i=l 1~.
~.
(43)
-1
where V is assumed to be non-singular (If V is singular, V may be replaced
N
N
N
by a conditional inverse of V ; however, as long as G(y) has a "scatter" not
N
confined to any lower dimensional space of the real p-dimensiona1 space R , V
N
p
may be shown to be non-singular with very high probability for large N; for
further details, the reader is referred to Chatterjee and Sen [1966]).
v
' L t has (N!I IT n.!) - not necessarily
Under the permutation model P
Nt
N,
i=l 1
distinct - equally likely (conditionally) realizations; and this may be used
as a basis of a permutationa11y distribution-free test of Hot.
sample sizes are large, the computations associated with
become prohibitively large.
t~is
If the
exact test
On the other hand, when the sample sizes are
large the permutation distribution of L t is asymptotically a x2-distribution
N,
with p(v-1) degrees of freedom.
Finally, this test is consistent against
location alternatives where the vectors m. are different.
~
Again, the reader is
referred to Chatterjee and Sen [1966] for a proof of the statements as well as a
discussion of power properties and other aspects of the test of Hot based
L
Nt
•
011
The basic point, however, is that L
represents a non-parametric
Nt
analogue of the one-way MANOVA statistic based on (the trace of) R in (5).
t
A non-parametric approach to the hypothesis H has been considered in Koch
oc
and Sen [1966].
The construction of test statistics here depends upon the
basic Assumption A.I.
Suppose U.. is defined as in (8), and m. as in (2).
~J
-"l.
Let
O. be the (p-1) x 1 vector defined by
-"l.
(44)
i = 1,2, ... , v
where C is as in (8).
1
Then A.1 asserts that for each i
1,2, ... , v and
18
j = 1,2, ... , n.
1
distribution.
_I
, the vectors (U" - i.;) and (6. - U.,) have the same
-1J
~
~
--LJ
Under H
6
oc ' -i.
implies that U.. and -U..
-:l.J
-:l.J
=0 .
'
and hence, in this situation, A.l
have the same distribution.
Such an assumption
is less restrictive than the usual assumptions of multinormality of the
U ' .
- iJ
As for the parametric case, there are several ways in which to construct
test statistics for H
oc
If each of the n
is not too small (i.e., n
i
i
at least exceeds p), one procedure is to compute for each treatment group
the statistic associated with Case III of Koch and Sen [1966].
These have
the form
(45)
where
n
T
(k)*
ni
1
=-
ni
i
(46)
L
j=l
n,
l.
L
j=l
S~~)
the quantities
1.
= (v
ij
Let
*
n
ijkk
,
(47)
)
represent scores obtained in the following manner.
First all possible differences U(kk')
where
2.
1J
(k)
(k')
S ij Sij
=
Y (,k,) _ y(.k,') are formed
1J
1J
u~~k) is identically zero.
1J
S~~k') = {sign (U~~k'»}{Rank [IU~~k')I: lu~klk')I, ••. , lu~kk')I]}
1J
1J
1J
1
1n,
1
for k I k' where ties are handled by the mid-rank method and zero is
assigned to zero values; hence S(kk') is a signed rank statistic (see
ij
Wilcoxon [1949]).
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Then
3.
s~~)
1J
~
=
k' =1
(kk')
s ..
1J
2
For large values of n. ,W* has permutationally a X -distribution with
1
n.
1
(p-l) degrees of freedom when there are no condition effects within the
n.
i-th group. The underlying permutation model P
is due to the 2 1
Nc
equally likely (conditionally)realizations associated with sign invariance
imposed by the condition of diagonal symmetry when
of the preceding remarks, we can form the
v
*
N,v
=
W
~i
followi~g
= Q. *
As a result
test statistic
*
(48)
I: W
n.
1=1
1
2
When H is true, W*
has the X -distribution with v(p-l) degrees of
oc
Ntv
freedom provided all the n. are sufficiently large.
1
are particularly small, W*
N,v
only those n
i
If some of the n.
1
can be modified to a sum of the W*
n
for
i
which are suitably large; of course, the degrees of freedom
2
for the approximating X -distribution would have to be correspondingly
modified.
*
Finally, the test of H
based on W
is a non-parametric
N,v
oc
analogue to both the statistic A in (12) and the MANOVA statistic based
c
on (the trace of)R in (14).
c
Alternatively, in the situations where all of the n. are small, the
1
approach previously indicated is not entirely appropriate because of
singularities in the V* and difficulties in the
n.
x2-approximation.
For
1
this case, we develop a non-parametric analogue to the statistic F in (9).
c
There are two cases of interest here.
G1 = G2 =
=
G
v
If in (2),we have
- G, a test of H may be based on the statistic W*
N
oc
which is computed in a manner analogous to that used to obtain W*
n.
but
1
*This
fact may be used as the basis of an exact permutation test for small n. •
1
I
20
I
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applied to the sample of individuals obtained by pooling treatment groups.
·
. h ( p- 1) d .f. b y arguments
· true, W*N h as t h e X2- d·1str1· b
1S
ut10n
W1t
Wh en H
oc
similar to those referred to previously.
On the other hand, if one cannot assume that the distributions of the
vectors from the different treatment groups are identical except for
location (i.e. if the G. are different), then one may use the statistic
1
associated with Case I of Koch and Sen [1966].
This has the form
(49)
where
n.
v
L:
1
1
- (k)
N i=l j=l 1J
=-
v
1
N;kk'
= -
L:
v
L:
- (k)
1
V
p+l)
L:
= Rank {Y
by the mid-rank method.
N
n.
N2 i=l j=l
the quantities R
ij
to the 2
(50)
R .•
2
(k)
(1)
'
N
=
(p)
: Y
, ••• , Y
} where ties are handled
ij
ij
ij
An exact test may be based on W by reference
N
equally likely (conditionally) realizations associated with the
permutation model P
when (4) holds.
Nc
Alternatively for large N, W
N
asymptotically has the x2-distribution with (p-l) degrees of freedom under
H
oc
Finally, one should note that W and W* have the same weakness as Fe
N
N
Namely, they are not sensitive to some
condition interaction is present.
alte~natives
in which treatment x
However, for the cases here where W
N
and W* are recommended (i.e., small n ), this deficiency must be
N
i
tolerated.
(51)
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4.
A NON-PARAMETRIC APPROACH TO CASE II.
As was indicated previously, when there is no treatment x condition
interaction, the hypothesis Hot is equivalent to
H* .
ot·
G
l
= G2 =
= Gv
1"1
= 1"2 =
= 1" V
(52)
while H is equivalent to the hypothesis H* in (18).
oc
oc
*
A test of Hot
may be constructed by applying the Kruskal-Wallis·test to the random
variables Yo ° defined prior to (19).
J.J
The statistic is
(53)
where
(54)
N+l) 2
(55)
2
the R = Rank {y °
ij
J.J
0
Yll '
... , Yl n ' ... , Yvl , ... , Yvn
handled by the mid-rank method.
} where ties are
l
v
When (52) holds, the statistic H has the
t
°
° h (v-I) degrees of freedom when the n. are large.
X2- dOJ.strJ.°b utJ.on
WJ.t
J.
The
* based on Ht is a non-parametric analogue to the test based on
test of Hot
F*
t in (19).
Next, we note that the statistic W in (49) may be used to test H*
N
oc
even when the G. represent different functional forms.
J.
if G
l
= ...
=
G
v
=G
, then the test may be based on W*
N
are similar to those used in connection with F
c
at the main effects of conditions.
On the other hand,
These remarks
as a test statistic directed
22
_I
The validity of the assumption of no interaction may be tested by obtaining a statistic similar to L
defined in (43) based on scores obtained
Nt
in the following manner.
1-
2.
Let
; (~~')
~
k' where ties are
(also S(kk) =
ij
4.
- (k)
Let S"
1J
Since
=
P
B±l)
E
k'=l
-(k) =
E S, ,
k=l 1J
handled
by the mid-rank method
- (kk')
Sij
2
(N+l) for i = 1,2, •.. , v; j = 1,2, ••• , n, ,
2
P
1
we form the vectors
5.
1
... ,
note S(kk') = N+1 _ S(k'k)
ij
ij
2
P
(kk' )
,
U
vl
)
... , U(kk'
In
'
U(kk' )
= Rank {U(kk').
ij
• 11
'
1J
for k
3.
(k' )
are formed
Yij
(kk' )
First all possible differences U.,
= y(k)
ij
1J
where U(kk) is identically zero.
ij
~j
(S~:~
1J
= C S., where S ~ , =
l -"1J
~J
and C is defined as in (8)
1
1
v
Form 0, = EO. for i
~.
n i j=l ~J
... ,
1,2, •.• , v and
n
i
v
VN=N E
E ~'SLi.,
i=l j=l
J
J
-*
6.
1
The test statistic
1'S
N-1
v
L
- --- l
~
n
."
N,tc -N i=l
1
In the above, V-*N is assumed to be non-singular.
.9.i
-*-1 -
•
VN
S4'.
The properties of the
statistic LN,tc are similar to those of L
in (43) and the reader should
Nt
I
I
(kk' )
}
U
vn
v
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23
refer to Chatterjee and Sen [1966] for a more complete discussion of this type of
statistic.
We do mention, however, that in large samples L
has the
N,tc
X2-distribution with (v-I) (p-l) degrees of freedom when condition A.2
holds in the sense of (16) and G
l
based on L
is a
N, tc
on (the trace of)R
5.
non~parametric
tc
= G2 = ... = Gv = G combined.
The test
analogue of the MANOVA statistic based
in (22).
A NON-PARAMETRIC APPROACH TO
CAS~
III
The test statistic for H
in this situation is similar to that given
ot
for Case 1.
The computations involved are similar to those given for L
N,t
except V is replaced by V where
NS
N
V ::: a I + b J
NS
with b =
1
p(p-l)
(56)
and a
-.,=---,...
1
= -p.
{tr(VN )} - b
i. e. ,
is determined from V in such a way as to have the compound symmetry
N
property in its variance matrix.
Again there are several approaches to H
oc
useful when the n
and Sen [1966].
i
One procedure which is
are not too large is associated with Case IV of Koch
The test statistic has the form
v
l:
i=l
=
(57)
I
I
where
n,
1.
-(k)*
1
*(k)
T , =L R..
n,1.
n.
1.
1. j=l 1.J
(58)
n,
o
2
*
(R.)
1.
=
1
1.
L
n. (p-l)
1.
j=l
where Z(k) =
ij
y~~)
1.J
- Y.•
1.J
P
(k) 2
L R.,
)
(59)
P k=l 1.J
... ,
(k)*
the R
ij
*
1
... ,
... ,
Z(l)
in, '
1.
For large values of
th~
2
X -distribution with v(p-l) degrees of freedom.
n
-* has the
W
N,v
i
In small samples, an exact
permutation test can be based on the (p!)N conditionally equally likely
intra-subject rank permutations.
For a further discussion of the properties
of this test, the reader is referred to Sen [1966].
however, that the procedure based on the ranks of the
We do point out here,
Z~~) lead to a test
1.J
in which the random subject effects have been eliminated (by subtraction of
the subject mean).
1
i
P k=l
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.-I
If these subject effects are not considered important
(similar to the case p = 0 in (24», we can replace the Z(k) with
ij
and
_I
R(k) with (n, p+l)/2
iJ'
1.
1.J
in the above computations.
As with Case I, the above approach is not
enti~ely
the n, are small because of difficulties in the
1.
indicate other non-parametric analogues to F
c
satisfactory when
x2-approximation.
If in (2), we have G = G = •.. = G
l
2
v
of H may be based on the statistic
oc
Thus, we
in (9) but directed to the
case when the error vector has the "compound symmetry" property.
two cases of interest.
y~~)
=G
There are
, a test
-* which is computed in a manner
W
N
-* but applied to the sample of
similar to that used to obtain the W
n.
1.
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25
individuals obtained by pooling treatment groups; L e. , the R~~)* are
1J
ranks assigned to the entire set of
i
Z~~)
1J
and range from 1 to Np and
(T~k)* _ NP;1)2/ 0 2(R*)
(60)
k=l
-(k)
where TN
1
= -
v
E
N 1=
. 1
ni
p
E
E
j=l k=l
-* has the X2-distribution with (p-1) d.f. for large N by
When H is true W
N
oc
arguments similar to those referred to previously.
When the G are different, the statistic W in (49) may be employed.
N
i
However, when Assumption A.3 holds, V simplifies because of the permutation
N
model of interchangeability.
As a result, in this case, we use
(61)
_ p+1)2
2
and 0kk' is the usual Kronecker delta.
Because of the special form v . ' takes here, W simplifies to
N
N, kk
WN
= N(p-1)
-2
P oR
i
(62)
k=l
As before, W has the x2-distribution with (p-1) d.f. for large N under H
•
oc
N
In the case of no ties, the W defined above reduces to the Friedman [1937]
N
test statistic.
-* have the same weaknesses as F mentioned
Finally, W and W
N
N
c
previously since they are directed at the main effects of conditions.
26
6.
_I
A NON-PARAMETRIC APPROACH TO CASE IV.
When there is no treatment x condition interaction, the condition of
"compound symmetry" does not introduce too many alterations to the test
* and Hot
The statistic H is again appropriate for Hot
t
procedures.
-k
*
Also, W in (60) and W in (62) may be used to test Hand
H
oc
oc
N
N
depending on whether the G. represent different functional forms.
1
Some changes may be used in the construction of the statistic L
Ntc
for checking the validity of the hypothesis of no'treatment x condition
-*
The major alteration occurs in the variance matrix V
N
interaction.
This arises as follows.
0
2
s
=
L
~
Let
n.
P
L (S~~)-
1
L
Np i=l j=l k=l
-*
then we use V
N
=
C C'
(63)
1J
n.
v
1
p
L (S ~~)
i-I j=l k;lk' 1J
1
°cp = N(p) (p-l)
L
(02
s
-
L
0
cp
(N+;) p) (S ~~ , )
1J
) in t h e computation of L
N, tc
N+l)p).
2
'
The same
remarks about L
given previously apply equally well now.
N,tc
7.
S~WARY
TABLE OF TEST STATISTICS.
Below is a table of the statistics discussed in this section.
procedures are possible.
Other
However, the ones cited here represent a set of
methodologies appropriate to the different situations considered.
they may be readily computed by straightforward methods.
Also,
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Parametric
Non-Parametric
Hot
R
t
L
H
oc
F , A , Rc
c
c
-*
*,v ' W
WN
N,v
*
Hot
F*
t
H
t
H*
oc
F , F*
c
c
A.2
*
Rtc ' Ftc
N,t
-*
*, W
WN,WN, WN
N
L
N,tc
28
8.
EXAMPLE
I
I
The data for this example are from a study involving the calibration
of audiometers undertaken at the University of North Carolina Memorial Hospital.
The basic measurements recorded represent
the calibration errors for a given
audiometer (subject) at each of five frequencies (conditions).
correspond to the ownership
background of the audiometers.
The treatments
The data were as
fo11O\"s:
See Table I
From an inspection of the data, one can see that there are a number of observations which are quite large positively or quite large negatively when compared
to the majority of the other observations.
These numbers are not outliers which
should be thrown out of an analysis, but rather they are important indicators
of the great extent to which the corresponding audiometers are out of calibration.
Hence, they must be retained in a statistical analysis.
On the other hand, when
this is done, assumptions concerning multinormality or homogeneity of variance
matrices may he questionable.
I,n such cases, the use of some non-parametric
methods may be considered to be of practical interest.
For the sake of completeness, however, we should indicate some of the
results of a parametric analysis.
First the correlation matrices for the
groups are
County Boards of Health
* o.t~2
~I,
0.38
0.72
..,'(
0.70
0.50
0.72
~"
0.40
0.66
0.67
0.56
*
School Boards of Education
0.73
0.77
0.93
0.80
0.91
0.90
*
0.20
0.10
O. t~o
0.20
separa~e
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29
Private Physicians
University Clinics
0.91
*
0.',9 0.15
0.38 0.28
0.82 . 0.18
'Ok
-0.).7
0.77
0.75
~k
*
0.69
"k
0.86
0.89
'":I':
0.80 0.21
0.89 0.63
0.96 0.39
-/:
0.32
~.\
*
:From the abovc, one can see that the correlation matrices for the diffcrent
groups tend not to be the same.
Also, none of the above matrices shO',·} a tendency
tOHard the compound synnnctry property.
The results of the ulUltivariate parametric analysis (not assuming
compound synnnctry) arc
1.
Hypothesis of no treatment effects Hot
a.
Hotelling Statistic;: llt.86 (For approx.x2~ d.L ;:.15)
b.
Roy Statistic;: 0.24 (See Heck's Charts [1960J with (3, 0.5) 13) )
c.
Likelihood Ratio Statistic;: 0.66 (For approx. x2 ;: 12.66, d.f. ;: 15)
lIenee, the data do not contradict Hoe' and we
~y
conclude that there are no
stat{stica11y significant differences between the four groups.
2.
Hypothesis of no treatment x condition interaction
ll~.18 (For approx.
2
X , d.L ;: 12)
a.
Hotelling Statistic;:
b.
Roy Statistic;: 0.24 (Sec Heck's Charts
c.
Likelihood Ratio;: 0.67 (For approx. X ;: 12.25, d.f. ;: 12)
[1960J \vith (3) 0.0) 13) )
2
Hence, the data do not contradict the hypothesis of no treatment x condition
interaction.
3.
Hypothesis of no condition effects H
oc
a.
Hote11ing Statistic (groups ignored) - 15.54; F ;: 3.55, d.f. ;: (4) 32)
. c
b.
Within group Hote11ing Statistics
i.
County Boards of Health;: 26.52; F
ii.
School Boards of Education;: 19.89; F
;: 3.48, d.f.
2c
A ;: .08
2.C
ic
;: 4.14) d.f. ;: (4,5); A1c='OS
=
(4,7);
30
iii.
University Clinics
4.05; F :: 0.58,
3c
c
~.f.
:: (4)4); A - .70
3c
Private Physicians::
F,~c :: 11.36, d.L :: (1IJI);
A4c :: .02
. 79.53)'
.
.
Hence, Ac = 18.64, d.f. = 8.
Multivariate A n a l y s i s '
iv.
ew
.
2
1.
Hotelling Statistic :: 29.22 (For approx. X , d.f.
ii.
Roy Statistic
ii.L
Likelihood Ratio
= 0.3l~
= 16)
(Sec Heck's Charts [1960J h7ith (1+, -0.5,13) )
= 0.46 (For approx . .; :: 21~.37, d.L ::: 16)
.
llence, there appear to be significant differences bet\-leen the errors of calibration associated Hith the differertt frequencies.
The most nota11le difference
.
arises from the errors for the frequencies 250, 500, '1000, 2000 being on the
order of -2.0 Hhile that for the frequency 4000 being on the order of +0.5.
The results associated Hith the above tests of significance \vcre
obtained from standard computer programs for analysis of variance and multivariat~ analysis of variance.
will not be elaborated.
Thus, the numerical aspects of these procedurcs
The reader is referred to Cole and Grizzle (1966J for
additional details.
Next, let us calculate some of the non-parametric statistics considered
in the paper.
1.
Hypothesis of no treatment effects H
1.
at
First fonn the matrix of cOElbined group ranks for each frequency
(condition)
. . . . . . .' . . . . . . . . .
See Tahle II
.....
ii.
Compute the suo of cross products of deviations
set of cOEiliined group ranks
iii.
Compute the within group sums of deviations from (N + 1)/2
associated \-lith each group and condition
D~trix
for the
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31
The results here are
3970.5 1670.0 2296.0 1935.2 1304.2
[
4026.0 3121.0 2596.8 1891.8
Q21=(E~.
3882.0 2641.5 1494.8
Q3'=(B3 .
8(37/2)j)
I
629.8J Q4'=(B4 .
3881.0
8 (37/2) j)
I
3882.5
I
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I.
I
I
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I
I
fN*
I
Ie
I
Q1!=(B 1; - 9(37/2)j)' = (-20.0,9.0,16.0,-1.5,-20.0)
iv.
- 11(37/2)j), = (18.0,-23.0,-9.5,-47.0,8.5)
(1l.5,23.0,6.5,26.0,13.5)
= (- 6 . 5 , -1.
°,-13 . °,22.5 , - 2 . 0)
4
-1
Compute LN,t = (N - 1) 2: (Q.' (NV )
D./n.)
N
i=l
~
-~
~
LN,t = 10.25, d.f.
~
15.
The conclusion here is the same as that for the parametric case.
2.
Hypothesis of no treatment x condition interaction
i.
.....(k)
First obtain scores S..
~J
given below.
as described in Section 4.
These are
See Table III
ii.
Form a new set of scores by subtracting each of the last four
columns from the first column; i.e., (250) - (500), (250) (1000), (250) - (2000), (250) - (4000).
See Table IV
iii.
Compute the sum of cross products of deviations matrix for the
set of combined group scores obtained in (ii); compute the vector
of within group sums for each variable and group.
The results here are
.,
f77 ,992 l~6,913 44,321 40 ,47 l f!,, n 1.Q 1 .' = (-1l9.5, 248.0, -1l6.5, -46.0)
(193.0, 121.5, 298.5, 24.5)
75,551 50,814 39,970' n2.Q2.'
I
(-125.5, -30.0, -184.0, -38.0)
95,577 19,755; n .Q '
I 3 3·
93,157 n 4.Q .' = (52.0, 156.5, 2.0, 59.5)
1
4
4
~,
-1
(n ..Q . ) J/n.
iv.
Compute ~,tc = (N -1).2: [ (n.Q. ) '(NV~)
,.J
~=1
~-~.
,.J
LN, tc = 12.04; d. f. = 12
1
~
~,
~
32
The conclusion here is the same as that for the parometric case; namely, no
I
I
interaction appears to be present.
Before proceeding further, one should note that the tests based on
,.J
LN,t and L
N, tc
are not strictly applicable here because the distributions for
the different groups appear to be different in the sense of having different
variance matrices.
Hence, the conclusions based on them can only be valued
to a limited extent.
Finally, we indicate the results of some of the tests for H
Oc
each of the groups.
within
The computational steps are not given since the algorithms
involved appear in Koch and Sen [1966J.
County Boards of Health
W
N
School Boards of Education
W
N
University Clinics
W
N
Private Physicians
W
N
=
117.1, d.£.
=
187.1, d.f.
--
7.4, d.£.
=
82.1, d. £.
=
4; ""'
W
N
=
15.19, d. £.
=4
~
4; W = 17.09, d.£. = 4
N
'"
4; W = 8.00, d. f. = 4
N
=
.-J
4; W
N
=
9.30, d. f. = 4
These tests again indicate that some differences exist bet\veen the conditions.
Note that some of the WN's are quite large.
larities in the variance matrices V .
N
This is due to some near singu-
For an explanation of this anomaly,
the reader should refer to Koch and Sen [1966J.
Othenvise, he should remember
that proper probability statements about such large values should be based on
2
the null permutation distributions rather than the associated X distributions.
Before concluding, we should admit that the analysis considered here
has been
confinedtod~ actual
observed errors.
Alternatively, one might also
analyze the absolute values of the errors, perhaps by the use of the rank methods
considered here.
reveal
Some aspects of the data indicate that such an approach Hould
some differences beth een the groups.
1
In any event, Hhat has been
given here does illustrate some of the computations and results ari sing from the
use of the methods described in the paper.
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33
Table I
Frequency
Subject:
_________.__
Group
~
500
250
..__.
~
.~
1
-1.6
2
0.0
3
-4.3
40.5
5
0.0
6
-4.6
7
-2.3
8
-1.5
9
-1.3
County
Boards
of Health
-3.5
-2.9
-5.5
5.1
-4.4
-1.6
-3.0
-3.1
-1.3
•
1000
2000
-3~0
-3.7
-1.8
-6.0
-1.3
-2.2
. -3.6
-1.4
-1.6
-1.0
-6.0
-0.5
-3.9
-1.0
-2.2
-2.2
-1.6
.
...
Subject
Hean _
n_· .__·
. __.
-2.0
0.2
-3.9
4.1
2.5
3.4
-2.1
1.6
1.9
~1.0
-2.76
-1.10
-5.14
1.58
-1.60
-1.48
-2.20
-1.36
-0.66
=--=---_.-._._;=.~~~:_:_-~l!~~~.=~=-~~~~J~i8~~_~:'1~~f=~~:'I:'3
~-=~_--=}~~r _·--_-=-_~:6~--=----==~J-:~l~-__
- ---- .
-5.4
-5.2
4.7
-5.1
-6.0
3.5
1.~
6.7
1.4
2.4
-3.7
-2.9
-3.0
-3.5
-3.9
-1.2
-1.5
-1.1
. tf
0.0
-1.9
-1.5
-0.7
0.1
5
-1.5
-2.7
School
-3.6
-2.8
-2.1
6
-2.7
-4.7
Boards
-0.8
-2.1
-2.1
7
-2.3
-2.5
of
-1.4
-0.6
-0.4
8
-1.5
-1.1
Education
-1.2
-1.8
2.7
9 .
-3.0
-0.5
-3.2
-1.3
-1.7
10
-2.5
-2.0
1.3
0.3
0.9
11
-2.7
0.0
------_._----- -------------1-.5-6----1 . S-6-----·0:- t.y--Mean
-2.13
-2.08
-------- ...
.. __ .
.. .. _- ... --_.-._--, .... ----_.- - "- -----,._---_. --------.----_. __ ._.-_.
1
2
3
---~._-----------
._.
_..
,--_._.-.~-_
1
2
3
4
5
University
Clinics
6
7
8
-------------i---l-lean-----
~
_
._.u
-1.2
-0.3
10.7
-5.1
-8.9
-1.9
-2.3
-3.8
._~
. _,
-4.2
-4.1
24.9
-2.2
-12.9
-5.2
-2.6
1.9
. _ _..
-2.7
-2.0
0.9
-2.0
-6.7
-3.0
0.2
-2.5
..
.__ . _.
-2.0
-2.4
-2.5
-0.3
-16.4
-5.8
-0.3
-8.5
~
__
~_.~.
._.. _.
_
-3.40
3.08
-3.40
-1.14
-1.26
-3.18
-1.96
-1.00
-0.76
-2.14
-0.04
-- ---1.-=-3-=-8---_._. --_._---
-0.1
1.1
1.8
-6.5
-2.2
-2.7
1.4
12.7
-2.04
-1.54
7.16
-3.22
-9.42
-3.72
-0.72
-0.04
-0.1
-0.5
1.4
0.3
2.4
-2.2
1.9
:-0_·_2 __
_ 0.38
-2.40
-1.90
-1.00
-2.22
5.62
-4.24
-3.00
-.0_.~_Q_
-1.18
69- ---- ':1.69---
~---=--f-.-6o_-·--:-(f.-55------~-2:-22-------:if-:i8--------()-:
--j -1
.2
3
4
Private
Physicians
______.
5
6
7
~____.
Hean
Pooled
}lean
Pooled
Error
Variance
-1.7
-1.4
-1.1
-3.7
0.7
-4.1
-3.3
-3.4
-3.1
-3.2
-1.1
3.7
-5.5
-3.1
-1:.} ____'":0. 9__
-1.99
-2.08
-2.8
-0.2
0.0
-2.6
10.5
-4.6
-4.1
Q_~.Q
-0.48
-1.87
-1.78
-1.67
9.33
32.53
8.82
-4.0
-4.3
-2.1
-4.0
10.8
-4.8
-6.4
Q.9
-1.7!f
-2.55
- ----- _._--
H
_4
__
~--
0.52
12.37
8. !t5
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34
Table I I
Subject
Group
1
2
3
County
Boards
of Health
4
5
6
7
8
9
_ _ _ _•._
.. 0._.•_ ••.- . - •.
-=::::-:..._ ... _
U
:..cc
..
- - ---
.._ .•__ ."_.__
I total_.
'::'.1 .. ':.:: ,.. .
I
4
5
6
7
8
9
10
11
_ _ _ _ _ _ _ _.•
'
. . . . . . _ .. _ . _ _
••.
~
_..
total
..
•• __ . _ . •
-==:;:.===-.-.~~:.=::=:::::::.:::.:-:::.~
1
2
3
University:
Clinics
500
17
6
32
4
6
33
21
16
11.5
26
19
33.5
2
30
12
20
22
11
26.5
11.5
35
9
31
11.5
20.5
20.5
17
. - ------- -
-- - - -------------
4
5
6
7
8
I
.
I
1
2
3
Private
Physicians
4
I
5
6
II
7
8
I
I
total
I
27
18
35
4
7
5
29
12
9.5
- ------- -- ---
.-
165.0
146.5
34.5
2
28
6
16
24.5
21
16
26
23
24.5
35
4
27
13
18
31
16
9.5
7
14
6
34
2
30
13.5
16
29
10
15
13.5
28
3
31
2
24
13
8
23
18.5
7
15.5
10.5
4
3
2
34
25
19
29
29
23
6
26
16
' _ ••
._.
108
1
34.5
36
19
21
.• _ _ •
._.
194.0
29
28
1
15
36
32
17
1L~1.
5
25
22
24
9.5
3
33.5
22
8
147.0
_..
-.-.--'"---
,
156.5
•
,_, _._••-_0., •. __.
212.0
•
24
18.5
4
18.5
36
26.5
5
17
21
22
5.5
36
32
5.5
35
20.5
15
11
36
31.5
33
13.5
1
22
25
8
6.5
23
1
33
32
6.5
135.0
161.5
27.5
29
18.5
27.5
1
30
34
3
170.5
20.5
24
13.5
17
8
31.5
9.5
22
--_. -- -_
146.0
_
-==--==:
.. 'u __'::U:::... :':::=.=:-_ ::::::-:uuU·-··.::C--::::::,:·,::'::-':'--:.:::::-::::.:·:::-u
.__..L._
_ 159.5
.~~ __. .__...~_3._u_.u._
154.5
171.0
i
18
13
9
29
3
31
27
11.5
_I
-=.".-.. -.,.c:...: :::.:=:....::.-:=::.-_..=.:.=:.:.~.- :-:::-:c:==.=.
:::-:-::=•.=-.:=:.:.:-::.::-:.:::::: ..
..
total
.-.
26
15.5
33
10.5
20
25
12
14
9
182.5
:~:"~::::--::::.::'
i
..
4000
175.5
I
221.5
180.5
..::_F::::.:.:..;::.::::::::-- .=::::=--=-.:::==:::: --
I
II
__ _- -
.
2000
146.5
:.:~.·_·=.-'--_r::~:::·'C:··-'·:c::·::::c:~ _U. :-::::'CC::-:-:
1
2
3
School
Boards
of
Education
250
Frequency
1000
.... -------
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35
Table III
Group
County
Boards
of Health
School
Boards
of
Education
University
Clinics
Private
Physicians
Subject
250
500
1000
2000
4000
1
2
3
4
5
6
7
8
9
49.5
54.5
60.5
68.0
53.0
149.0
93.5
90.5
108.0
86.5
122:0
75.0
63.5
133.5
75.5
93.5
117.5
79.0
101.5
82.0
123.0
126.5
145.5
78.5
89.0
110.5
118.0
103.5
95.0
98.5
131.5
82.5
119.5
49.5
79.5
83.5
121.5
109.0
105.5
73.0
48.0
40.0
137.0
64.5
74.0
1
2
3
4
5
6
7
8
9
10
11
108.5
131.0
92.5
53.0
99.5
71.5
100.5
108.0
133.0
102.5
145.0
110.0
88.0
81.0
89.5
115.0
118.5
85.0
66.0
63.0
59.0
77 .0
117.0
72.0
101.5
95.0
96.0
106.5
48.5
103.5
108.5
126.5
48.5
99.0
119.0
58.0
87.0
54.0
56.5
85.0
61.5
102.0
47.5
77 .0
28.0
52.5
129.5
138.0
98.0
109.5
143.5
123.5
56.0
127.0
115.0
1
2
3
4
5
6
7
8
60.5
52.5
57.5
112.5
89.5
40.0
131.0
121.0
136.0
141.5
22.5
67.0
126.0
109.0
126.5
61.5
109.5
103.0
105.5
92.5
56.5
67.0
61.5
99.0
68.0
95.5
150.0
32.0
157.5
129.0
58.0
158.5
88.5
70.0
12, 7.0
158.5
33.0
117.5
85.5
22.5
1
2
3
4
5
6
7
8
65.0
77.0
89.0
120.5
144.0
87.5
90.5
120.5
94.0
102.5
134.0
48.5
97.0
105.5
73.0
87.5
103.0
40.5
58.0
97.5
45.5
97.5
113.5
82.0
124.0
135.5
104.0
122.0
3l~. 5
89.0
145.5
37.5
76.5
107.0
77 .5
74.0
141.5
83.0
40.0
135.0
36
Table IV
Group
County
Boards
of
Health
School
Boards
of
Education
University
Clinics
Private
Physicians
250-500
250-1000
250-2000
250-4000
-37.0
-67.5
-14.5
4.5
-80.5
73.5
0.0
-27.0
29.0
-52.0
-27.5
-62.5
-58.5
-92.5
70.5
4.5
-20.0
-10.0
-54.0
-40.5
-38.0
-63.5
-29.5
29.5
44.0
11.0
24.5
-72 .0
-54.5
-45.0
- 5.0
5.0
109.0
-43.5
26.0
34.0
-
- 8.5
59.0
- 9.0
-42.0
3.5
-35.0
52.0
4.5
24.5
-24.0
96.5
9.5'
12.0
34.5
-34.0
45.5
15.0
15.5
46.5
31.0
55.0
68.0
80.5
78.5
-37.0
-85.0
1.5
-38.0
-43.0
-15.5
77 .0
11
1.5
43.0
11.5
-36.5
-15.5
-47.0
15.5
42.0
70.0
43.5
68.0
1
2
3
4
5
6
7
8
-75.5
-89.0
35.0
45.5
-36.5
-69.0
4.5
59.5
-49.0
-50.5
-48.0
20.0
33.0
-27.0
69.5
22.0
- 7.5
-43.0
-92.5
80.5
-68.0
-89.0
73.0
-37.5
-28.0
-17.5
-69.5
-46.0
56.5
-77 .5
45.5
98.5
1
2
3
4
5
6
7
8
-29.0
-25.5
-45.0
72.0
47.0
-18.0
17.5
33.0
-38.0
36.5
31.0
23.0
98.5
-10.0
-23.0
38.5
-59.0
-58.5
-15.0
- 1.5
109.5
- 1.5
-55.0
83.0
-u.S
Subject
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
10
- 2/+.5
30.0
-30.0
11.5
46.5
2.5
4.5
50.5
-14.5
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BIBLIOGRAPHY
Anderson, T. W. (1958). An Introduction to Multivariate Statistical Analysis,
John Wiley & Sons, Inc., New York.
Birnbaum, A. (1954). "Combining independent tests of significance," Journal
of American Statistical Association, 49, pp. 559-574.
Box, G. E. P. (1949). '~ general distribution theory for a class of likelihood ratio criteria,"Biometrika, 36, pp. 317-346.
Box, G. E. P. (1950). "Problems in the analysis of growth and wear curves,"
Biometrics, 2., pp. 362-389.
Chatterjee, S. K. and Sen, P. K. (1966). "Non-parametric tests for the multivariate mu1tisamp1e location problem." S. N. Roy memorial volume. Edited by
R. C. Bose et a1.
Cole, J. W. L. and Grizzle, James E. (1966). "Applications of multivariate
analysis of variance to repeated measurements experiments," Biometrics, 22,
pp. 810-827.
Danford, M. B., Hughes, H. M., and McNee, R. C. (1960). "On the analysis of
repeated-measurements experiments," Biometrics, 16, pp. 547-565.
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