,..
II
.•
"'--~-
~
UNIVARIATE, MULTIVARIATE AND INCOMPLETE MULTIVARIATE
NON-PARAMETRIC MIXED M:lDELS
IN THE ANALYSIS OF INCOMPLETE AND/OR ORTHOGONAL
TWO-WAY LAYOUT EXPERIMENTS
by
..
.
(
Elsa C. Servy
Institute of Statistics
Mimeograph Series No. 795
Raleigh
1972
iv
TABLE OF CONTENTS
Page
1.
REVIEW OF LITERATURE ~ RANK TESTS AND MIXED MODELS FOR THE
ANALYSIS OF VARIANCE PROBLEM. •
• • . .
1. 1
•
1
Rank Tes ts
1
1.1.1 Introduction
1. 1. 2 Conditional Tests
••••
1.1.3 Permutation and Randomization Tests
1. 1. 4 Invariant Tests • . . • • . . . • •
1. 1. 5 Rank Tests and Rank-Permutation-Type Tests •
0
1.1. 6
•
•
•
•
•
•
•
Rank Tests for the Analysis of Variance
Prob.lem . • •
0
•
•
•
Introduction
Rank Tests for One-way Layout
Experiments . . . • • . • .
1.1.603 Rank Tests for Two-way Layout
Experiments . . . . . . • .
1.106.4 Rank Tests for Complex Experiments
. • • .
1.1.6.1
101.6.2
1.2
Mixed Models
eo.
1.2.3
102.4
2.
Eisenhart Models
Scheff~ls Model.
10
13
19
23
28
Non-parametric Mixed Models
General Remarks and Notation used in
Relation to the Extensions of Nonparametric Mixed Models
. • ••
Introduction
. . . . • . • . .
Mixed Mode.l for the Case of' Several (eventually one
or zero) Results per Treatment. Extensions of
Models I and IV • • • •
0
0
Introduction •
Case UI
Case U IV
9
20
21
MIXED MODElS FOR THE CASE OF SEVERAL (EVENTUALLY ONE OR ZERO)
RESULTS PER TREATMENT. UNIVARIATE RESPONSES .
2.2.1
2.2.2
2.2.3
9
20
e
1.2.1 Introduction
1.2.2 Mixed Linear Models
1.2.2.1
1.2.2.2
1
3
4
6
7
0
•
•
•
•
•
•
33
37
43
43
45
45
46
53
Table of Contents (Continued)
Page
203
Mixed Model for the Case of Several (Different from
zero) Results per Cell. Extension of Model II
(Cas e UII ) 0 . 0
0 '0 • 0 0
0".. . .
2.4 A NumericaloExample
0 .• 0 . . 0 . . .
61
69
MIXED MODELS FOR THE CASE OF SEVERAL (EVENTUALLY ONE OR ZERO)
RESULTS PER TREATMENT AND MULTIVARIATE :8.ESPONSES
o.
72
.d
...
30
0
3.1 Introduction
302 Case MUI
3.3 Case MUII
3.4 Case MUIV
0
4.
•
0
72
74
79
86
0
MULTIVARIATE MIXED MODEL WITH SEVERAL (EVENTUALLY ONE OR ZERO)
RESULTS PER TREATMENT AND INCOMPLETE RESPONSES - CASE lMUII
401 Introduction .
4.2 The Model and The Test . • • • • .
4.3 A Numerical Example . • .
0
5.
•
•
•
•
92
. 93
104
•
ON FURTHER APPLICATIONS OF MIXED MODELS, SUMMARY AND
SUGGESTIONS FOR FUTURE RESEARCH
...
0
Introduction . . . . . .
Some Further Applications of Mixed Models .
5.2.1
5.2.2
..
5.3
••
.. .
Test of the Interaction Terms in a Factorial
Design Laid in Complete Blocks with Unequal Number of Observations in the Cells
One,'Ana-Lys±s Jof Covariance-Ty:pe of Problem
with Unequal Number of Observation Per
Cell • • • • • •
Summary
5.3.1 Underlying Designs and Models
5.3.2 The Tests
5.3.3 Examples . . . • .
5.4 Suggestions for Further Research
92
108
108
108
109
113
115
115
.119
120
121
6.
LIST OF REFERENCES • •
123
7.
APPENDIX..
127
7.1
Limit Theorem.
129
vi
Table of Contents (Continued)
Page
7.1. 1
7.1. 2
7.1.3
7.1.4
7.1.5
7.1.6
7. 1. 7
7.1. 8
7.1. 9
7.1.10
7.1. 11
7.1.12
•
Theorem
Theorem
Theorem.
Theorem.
Theorem.
Theorem.
Theorem
Theorem
Theorem.
Theorem •.
Theorem
Theorem.
. ..
•
iii
•
•
•
"
130
133
134
137
138
144
144
147
147
148
149
153
7.2 Computation of Terms that Enter in the Formulas for the
7.3
Expected Values, Variances and Covariances of Seotion
2.3
153
Computation of Terms that Enter in the Formulas for the
Expected Values, Variances and Covariances of Section
4.2 . . . . .
156
7.3 ..1 Case A
7·3.2 Case B
7·3.3 Case C
159
163
167
1.
REVIEW OF LITERATURE:
RANK TESTS AND MIXED MODELS
FOR TEE ANALYSIS OF VARIANCE PROBLEM
101
1.1.1
Rank. Tests
Introduction
The theory of rank tests has developed in a compact system that
forms, at present, a specialized part of the theory of testing statistical hypothesis.
Rank tests were
rega~ded
rather skeptically at first because
their use was justified only on intuitive grounds by those who
introduced them.
Later on, the works/by Terry (1952), Lehmann
(1953) and others showed that the asymptotic power of these tests
is high when compared with the power of the standard tests used
for solving analogous parametric problemso
"
From then on, many works have been devoted to ,the. search for
new rank tests and to the study of their properties o The bibliography compiled by Savage (1962) includes valuable refe~ences on
rank tests.,
Some chapters on rank tests are included in general books
of statistics (Wilks (1962) and Lehmann (1959»,'and other books
that treat the SUbject in detail have been written by Kendall
(1948), Siegel (1956), Fraser (1957), H~jek and Sid~k (1967),
Hajek (1969) and Puri and Sen (1970).
As recently as June, 1969, the First International Symposium
on Non-parametric Techniques in Statistical Inference was held at
2
Indiana University, Bloomington, Indiana.
Interesting articles on
rank tests can be found in the proceedings of the symposium edited
by M. L. Purl (1970).
Some of the advantages offered by rank tests are:
i)
They can be applied even in those cases in which
only the order 'of the observations are known.
ii)
They are suitable for testing hypothesis that involve
quite general assumptions.
iii)
For many statistical problems., the asymptotic power
of the tests is high as compared with that of the
tests developed for parametric models.
Rank tests, like other non-parametric methods, are now widely
applied to many fields of biometrics, agriculture,economics,
industry, etc.
But to avoid over-elitimation of their value, it has
to be said that:
i)
The asymptotic results are only applicable if the
sample is large enough.
ii)
The tests available at the present assume the existence
of simple experimental designs.
Looking for ways to overcome this second drawback, this thesis
is oriented toward the search for methods that can make possible
the application of rank tests to experiments that exhibit awkward
patterns.
•
3
To this end special attention has been paid to tests known as
randomization-rank tests whose underlying theory is described in
the works of Chatterjee, Puri, Sen, Koch, Gerig and others.
These tests, based on the ord.er properties of the observations,
share common characteristics with the conditional tests, permutation
tests, and invariant tests that are considered. in the next sections.
1.1.2 Conditional Tests
Some hypothesis testing problems are difficult to handle on the
whole sample space, but become simple in a subspace of it.
The
name conditional tests is given to those tests performed in one of
the cells of a partition of the sample space, the cell being determined by the actual sample obtained and the inference being made
by the use of a distribution constructed from the results of the
experiment.
The size of the test inside the cell is chosen by trying to
keep the total size of the test over the whole sample space equal
to some pre-assigned coefficient
~
If the test-function obtained conditionally does not depend
on the cell of the partition, the test is said to be "unconditional"
in relation to the whole sample space.
But, even if the test-
function depends on the partition, the results can be re-interpreted,
considering again the total space and in many situations the optimal
properties of the test inside the cell imply optimal properties of
the test in the general sample space.
4
1.1.3
Permutation and Randomization Tests
Permutation tests can be used for testing a hypothesis (H) that
is invariant under a finite group G of (M) permutations.
Invariance
of H under G means that if y has a distribution that belongs to H,
then y and g(y)(g
€
G) are identically distributed random variables.
Permutation tests are conditional tests in which the cell
where the test is carried out is the set of (M) points g(y )
o
obtained applying to the actual result of the experiment the M
transformations of G.
The cell can be identified as S(y ).
o
Inside that cell, the conditional probability, given S(Y ,)., (un91er
o
the hypothesis is ~ , whatever measure in H is taken into account.
The a-level test of the hypothesis H can be defined as a rule that
. determines, for every possible cell S(y ), a subset S' (y ) of s
o
0
a
points (s
a
~
aM), in such a way that the hypothesis is rejected if,
and only if, the observed sample point lies in S'(y ).
o
If the group G of transformations is the group of n! permutations of the observations of the sample
{ Y , Y , ••• , y }, the
symmetry of the measures of the hypothesis
l
2
~~der
n
G can be obtained
by:
i)
theassumpti.on that,und~r H, the observations at hand
are a random sample from a certain population.
In
this case,H is said to be a hypothesis of randomness;
ii)
by an actual randomization introduced in the experiment.
In this case, it is not necessary to assume that the
5
observations are independent and
identicallydis~ri-
buted and H is called a hypothesis of interchangeability.
Some authors differentiate the tests connected with case i) from
those connected with case ii) by using the names permutation tests
and randomization tests respectively, while for others both types are
called indiscriminantly by one or the other
JL~
denomination~l
example of the application of the permutation (or randomiza-
tion) principle is offered by the two-sample test of Fisher and
Pitman, (Fisher (1925) and Pitman (1937»·
Fisher was the first to emphasize the importance of randomization and to conceive of the permutation tests.
The randomization
model for complete block design was formulated by Neyman (1935»and
randomization models have been formulated for other designs by
Kempthorne (1952 and 1955), Wilk (1955) and Wilk and Kempthorne (1955).
The basic ideas involved in randomization tests have been
formalized by Lehmann and Stein (1949).
The hypothesis under test
is defined as a hypothesis of ,invariance under a certain partition
IT
of the sample space.
Permutation tests are defined as similar
a tests having the Sea) structure.
This last condition means that their test-functions, call
them
, are such that
~
L:
~ (y')
= Ma
Y
E
It •
y'€S(y)
where
a
is the size of the test.
S(y) is the set of points that lie
in the same cell of the partition to which y belongs.
6
1.1.4 Invariant Tests
There are many examples
o~
statistical problems whose essence
does not change if some group of transformations are applied to the
variables involved.
For instance, in testing the hypothesis about
the difference in magnitude of two means, the unit of measurement
used is alien to the substance of the problem.
So, it is logical
that in order to solve it, the search for the correct procedure
has to be restricted to those tests that are independent or
invariant under the changes of the unit of measurement.
In general, the principle of invariance confines attention
to test functions that are invariant with respect to the group of
transformations of interest.
Formally, a test function ~(x) is invariant with respect to a
group G of transformations if
~(g(y»
= ~(y)
for all g
€
G.
Every invariant (re G) statistic can be expressed as a function
of another statistic of the same family (i'~"
say m(y»
invariant under G,
that is called maximal invariant (re G).
So, in looking
for an invariant test only functions of m(y) come into consideration,
and this amounts to restating the original hypothesis problem in
terms of m.
Making this transformation has the advantage of allowing
one to deal directly with invariant functions, and besides this, it
produces in many cases mathematical simplications in the setting up
of the problem.
If this is the situation, invariance is a logical property to
require of a test statistic as well as a means of making the problem
at hand more manageable.
In certain problems, invariance reduces the
7
data so completely that the actual values of the observations are
discarded and only order relations between group of variables are
retained.
The related tests are based on the ranks of the observa-
tions.
Rank Tests and Rank-Permutation-Type Tests
In connection with the invariance principle, rank tests can be
defined as tests that are invariant under the group of transformation
G where g(EG)
is a continuous and strictly increasing function.
In
particular, they are invariant under changes of location and scale
parameters.
If the hypothesis under consideration is the hypothesis of
symmetry under the group G of n! permutations of the sample values
{Y , Y , ••• , Y }, rank tests can be defined as permutation tests
l
n
2
that are independent of the order statistic of the sample (depending,
as a consequence, only
on the ranks of the observations).
For other hypotheses, other definitions of rank tests have
be given.
~o
Ha.jek (1967) gives careful definitions of rank tests for
the hypothesis of interchangeability, randomness, symmetry, independence and random blocks.
In order to avoid technicalities, the term rank tests will be
used when the test-functions are based primarily on order properties
of the observations, depending implicitly or explicitly on some
ranking defined over them.
This work will concentrate on rank tests
that are generalizations of the ones created by Friedman, Wilcoxon
8
and
Kruskal~Wallis
and whose characteristics can be outlined as
follows:
i)
A certain null hypothesis is specified on the
distribu~
tion of a scalar or vector variable y.
ii)
Certain functions, say Z.
J.
=
f.(Y) of the observations
J.
(the observations themselves, their differences, etc.)
are considered.
They are chosen in such a way that
their distributions depend on the pertinent
hypothesis.
iii)
Some rules are given for ranking the values of Z and
accordingly, the original sample space is transformed
into a finite set R whose elements are functions of
those ranks.
'iv)
The testing hypothesis problem is restated in terms
of R and the permutation principle.
v)
When the sample is large, an approximation to the
distribution of the test statistic is found restricted
to the cell of R in which the test has been constructed,
~.
and a conditional large sample test is derived.
If
the asymptotic test does not depend on the partitton
of R, it is an approximate test unconditionally.
These tests will.be identified as.rank randomization. type' tests.
':t'hey are very useful for solvtng analysis of
which we are concerned.
vari~nceproblems
about
In the next section a review of the procedures
9
based on ranks that are suitable for testing analysis of variance
hypotheses will be presented and emphasis will be put on those that
belong to this rank randomization class.
1.1.6
Rank Tests for the Analysis of Variance Problem
1.1.6.1 Introduction.
The most famous tests in the literature
are known by the following names:
1)
Sign test
2)
Wilcoxon's signed rank test
3) Wilcoxon's sum-rank test
4)
Mann-Whitney test
5)
Kruskal-Wallis test
6)
Friedman test.
They were introduced by different statisticians around the
decade of the forties.
As rank tests gained in reputation, a lot of
work was done in order to extend them to more complex or more
general situations.
The extentions have been accomplished in different directions,
in order
i)
ii)
to be able to deal with tied observations;
to consider more complex experimental situations:
for instance, incomplete block and factorial designs;
iii)
to consider test-functions that depend on the ranks
through some
~~ctions
called
II
scores
II
and that can
10
replace the ranks in the algebraic expressions of the
classical test statistics;
iv)
to include multivariate responses.
The review of rank tests for testing analysis of variance type
of hypothesis to be presented in the following section is not intended·
to be complete or exhaustive.
No mention is made to the analysis of
covariance procedures or to the problems of multi-comparisons of
t;reatments.
The tests are classified according to the ty,pe of
experiment for which they are suited, i.e.:
i)
Rank tests for one-way layout experiments;
ii)
Rank tests for two-way layout experiments;·
iii)
Rank tests for complex experiments.
1.1.6.2 R~Dk Tests for One-way Layout Experiments.
Y.. (j
~J
=
Let
1, ••• , N.)represent N. independent identically distributed
~
~
random variables from a continuous cumulative distribution function
F
i
= G(y
- ~).
Let i vary from 1 to v.
Ho·.
for i
=
The hy'pothesis to be tested is
1, 2, ••• , v
.?'
)
and the alternatives are
at least one ~
Ranks are assigned to the observations,
R. .
~J
= Rank
(Y.. : Y.,.,
~J
~
J
Ties are handled by the
i'
f
2:..~.,
= 1,2, ".,
mid-ra..~method.
a. •
v;
j'
= 1, ••• , N.} •
This method can be
~
II
described as a rUle, that, assigns to each observation in a tie, the
average ofrai1ksthat the tied observations would have had if they
would have been distinguishable.
Under H , the distribution of the vector of ran..'k.s is invariant
o
under the N!(N
= ~.)
~
possible allocations of the ranks of the
members of the v populations.
Conditional on the magnitude of the
Y's, the probability attached to any permutation of the ranks is
~i no matter what measure in Ho is taken into consideration.
A
permutation test is performed using this invariance property; its
test function is given by the statistic:
v
L =
I:
(_
i=l
N. R. ~
~
N+l)2
2
with
N.~
1
R.l =
N.~
I:
j=l
R.~j
and
C1
2
1
=-N
NR
v
N.~
I:
I:
i=l
j=l
(R.... N+l )2 •
~J
2
If N and the N.'sare simultaneously made large, L is, under H,
0
~
2
asymptotically distributed as Xv_I' conditionally on the permutation
structure.
Since the X2 distribution does not depend on it, the
approximation is also satisfactory unconditionally.
12
Specialization of the Test
If there are no ties, the test becomes the one proposed by
Kruskal-Wallis (1952).
In this case,
and
L
=:
v
12
N(N+l)
r:
i=l
N.(R.
J.
J.
N+l ) 2 •
2
Also, Wilcoxon's rank sum and the Mann-Whitney test can be seen, in
turn, as special cases of the Kruskal-Wallis test.
The Wilcoxon's
sum rank test (Wilcoxon, (1945»\was created for the case in which
the sUbsamples are of equal size.
it to unequal sUbsample sizes.
Mann and Whitney (1947) extended
The test fUnction used by Wilcoxon
is the minimum sum of the ranks of both samples, and the one for
the Mann-Whitney test is
u
=
where
1,
if Yh < Y "
h
=
When two-sided alternatives are considered, these statistics give the
same critical regions that occur when L is taken for v=2.
13
10106.3
Rank Tests for Two-way Layout Experiments.
The case
in which the number of treatments to be compared is two and the
case in which the number is greater than two will be considered
separately
i)
two treatments (paired samples);
Sign Test
(Dixon and Mood, 1946)
Let (y
il
,Y
i2
);
(i = 1, 2, ••• , b) be stochasti-
cally independent pairs o
Also, let Y and Y be
il
i2
independent with cumulative distribution functions
F , F
il
i2
such that
with
and
14
The hypothesis to be tested is
~
= 0,
test is directed to location alternatives,
6
f
so that the
!.~.,
to
0 (or ~ > 0, or 6 < 0).
Two-sided as well as one-sided tests can be
obtained by transforming the original variables to
i
= 1,
2, ••• , b
1
Restated in terms of the Z's, the hypothesis is a
hypothesis of symmetry of the distributions of the Z.'s.
J.
The test statistic is:
b
S
=
j.L(Z.) •
L:
J.
i=l
where
. l
/
"il,.
1,
if X. > 0
0,
if Xi
J.
=
=0
Under the hypothesis, S follows the binomial law of
1
2
probability with p = - •
Wilcoxon's Signed Rank Tests
Let the vectors
independent for i
{(Yil ,
= 1,
(Wilcoxon, (1945)
Yi2)} be stochastically
2, ••• , b.
Let the members of
each pair be independent with marginals F , F
il
i2
respectively, and such that
Fil = G(y - cr:t)
:1-5
and
with
= j.L
and
•
The hypothesis to be tested is
is directed to location alternatives.
or
fj.
O,and the test
fj. =
(fj.
p 0,
fj.
>
°
< 0).
Let
Zi
= Yil
- Y
i2
(i
= 1,
••• , b).
In
terms of the vector (Zl' Z2' ••• , ~) the hypothesis
becomes a hypothesis of randomness and symmetry.
Let us define
S~....
=
Sign Z~.Il {Rank
Iz. I : IZl"
~
•• - ,I 0z-
11 ,
where ties are handled by the mid-ranks method and
zero is assigned to zero valuesConditional on the magnitude of the ranks assigned
to the
Izlt s ,
the sign of each S. has equal probability
~
of being plus or minus under the hypothesis.
This
b
randomization model bas 2 possible (equ~lly probable)
realizations for sign invariance.
The test statistic used for the test is
b
S
=
E S~
i
....
Under the hypothesis, the statistic S is approximately
16
normally distributed for large b.
One-sided as well
as two-sided tests can be obtained by using this normal
approximation.
More than two treatments (r~~dom blocks)
ii)
Let
{y
.. '.
i
= 1,
2, ••• , b;
= 1,
j
2, ••• , p}
~J
be independent random variables from some continuous
cumulative distribution fUnction F..
~J
= G.(y
~
-
a.).
J
The hypothesis to be tested is
for j
= 1,
2, "',
P
and the test is directed to location alternatives,
provided there is no treatment x block interaction
with respect to location.
\
\
(i stands for blocks, j
stands for treatments).
Let
___
Rij
= Rank
_
0
..
{Yij: Y~~~
...
o •• ,
--
Y. '}
~p
with ties handled by the mid-ranks method.
Conditional on the magnitudes of the Y's, a
randomization-type model can be formulated considering
the (p!)b equally probable matrices of ranks, «R )),
ij
that can be obtained under H when all the possible
o
permutations of the ranks in each block are taken into
account.
A permutation test has been developed using.
17
the conditional uniform distribution (under Ho )
_1__
(p~ )b'
Q
=
and its test statistic is the quadratic form
p
b(p-1)
2
p O"P,R
Z
j=l
with
b
Z
R• J. =
R .
i=l
iJ
b
and
b
1
~ (R , _ P+21)2.
O"P,R = bp i~l
iJ
2
Under H , and conditionally on the actual matrix
o
of ranks ((R, .)), Q is asymptotically distributed as a
lJ
2
X with (p-1) d.f. This approximation applies unconditionally as well.
This test was the result of work by Kendall (1948).
S~ecia1izations
of the Tests
If there are no ties, Q takes the form
12b
p(p+1)
p
~
j=l
that is the statistics of Friedman (1937) for the problem of b rankings.
18
If p=2, the problem of random blocks is reduced to the one
related to the sign test, and the critical region given by Q coincides
with the one afforded by S for two-sided alternatives.
other Tests Related to the Rank Tests for Two-way Layout Experiments
As a generalization of Friedman's procedure, Durbin (1951)
presented a test suitable for the problem of random blocks when the
experimental data form a balanced incomplete block design and, two
years later, Benard and van Elteren (1953) extended Kendall's statistics to a two-way experimental array of data with arbitrary frequencies in the cells.
With respect to incomplete two-way layouts, the methods of paired
comparisons should be mentioned.
These methods are used for the
comparison of p treatments presented in pairs to obtain N.. responses
lJ
in relation with treatments i,j; (i,j = 1,2, • •. , p).
A monograph on the sUbject has been written by David (1963).
It
presents several (univariate) models and suitable procedures for the
problems of hypothesis testing and estimation.
The monograph describes
I
the works of Bradley and Terry (1952), Scheffe (1952) and others.
These methods are based on intra-pair rankings and, for this reason,
they can be seen as extensions of the classical sign-test.
On the
other hand, the test developed by Puri and Sen for the
problem
s~ae
(1969) uses intra and interblock information; when p=2, it coincides
with the Wilcoxon's sign test.
There are several rank tests available for multivariate crossclassified data.
19
The Friedman test has been extended by Gerig (1969) in the case
of multivariate responses.
The same has been done for the sign test:
several tests go by
the name of bivariate sign tests, notably those by Hodges (1955),
Blumen (1968) and Chatterjee (1966).
to Bennet (1962).
A mUltivariate sign test is due
Bickel (1965) has found two statistics for testing
the nullity of a location parameter of a multivariate population;
one is asymptotically equivalent to the sign statistics, and the
other asymptotically equivalent to the Wilcoxon statistic; Sen a.TJ.d
Puri (1967) have constructed a test for the same problem that can
be seen as a multivariate version of the Wilcoxon sign test.
The development of multivariate tests for paired comparisons is
just starting.
The worJ~s by Sen and David (1968) and Davidson and
Bradley (1969) solve the problem by using intra-block ranking, while
two recent papers by Shane and Puri (1969) and Pi.lr:h:andShane (1970)
present tests that also use the inter-block information contained in
the samples.
In this review of
rarL~
tests for two-way classified data,the-
articles by Sen (1968 and 1969) and Koch and Sen (1968) must be
included.
Nevertheless, the discussion of these papers will be post-
poned until the next sections since they need special treatment in
view of their connection to the main sUbject of this research.
1.1.6.4 Rank Tests for Complex Experiments. We include both
factorial designs and split-plot designsQTJ.der the name of complex
experiments because their location parameters under the alternatives
can be expressed as rather complex linear combinations of effects.
I ,
Separate tests on the individual effects or group of effects are of
interest in these experiments, the hypotheses being of the same kind
as the ones of interest in·the analogous parametric problems.
Mehra ~~d Sen (1969) presented a test for interaction in a
factorial experiment replicated in blocks.
A permutation model is
applied to the (aligned) observations, and the resulting test statistic has the same structure, as the one used for testing the similar
parametric hypothesis.
Later on the test was extended (Puri and
Sen, ~970))for ~~alyzing multi-response factorial exp~riments.
The discussion of split plot type designs from the point of view
of non-parametric theory has been done by Koch in two recent papers
(Koch, '~969)and~970»). A certain number of hypotheses are considered.
They are analyzed under several different assumptions concerning the
joint distribution of the components of the observation vector, and
different rank tests are proposed.
1.2 Mixed Models
1.2.1
Introduction
In the preceding sections most of the tests have been classified
primarily according to the design of the experiment and the kind of
j
responses (univariate of multivariate) involved, and no emphas~s has
been put on the model in which they are based.
Oppositely, in this
section special attention will be paid to the discussion of the models.
This study will. be concerned with two-way layout experiment
models whose underlying assumptions are non-parametric counterparts of
21
those involved in the traditional mixed linear models.
Non-parametric
mixed mOdBls are the subject matter of this work, and in order that
their relation with the classic analysis of variance type models be
understood~
a brief review of the parametric mixed models will be
done in the next section.
1.2.2 Mixed Linear Models
According to the traditional point of view, to describe the
data obtained from an experiment by an analysis of Variance model
is to assume that the observations consist of linear combinations
of effects.
Under the term effect what are usually called.I',general
mean", "main effects tl, "interactions" and
stood.
tl
errors II have to be under-
The effects (not directly observable quantities) are idealized
formulations of some properties of interest to the'investigatorin
the phenomena underlying the Observations and in relation to these
properties eac.heffect is an unknown constant or a random variable.
The mod.el is called a fixed effect model if the only random
effect in the model equations is the error term.
It is called a random
effect model if all effects, except the additive constant (if there
is one1 are random effects.
A case which falls into neither of these
categories is called a mixed model.
The main effect of a level of a factor is considered a random
variable if the set of levels actually tested can be thought of as a
sample from a larger set of similar levels and the conclusions are to
be extended. to the whole class of levels and not only to the ones
tested.
Otherwise, if the levels under study are the only ones of
22
interest, their effects are considered parameters of the model.
It
is customary to consider the interactions among random effects or among
fixed and random effects as random.
When an experimental situ.ation can be modeled by a mixed linear
model, the researcher can have in mind two different kinds of
inferences.
Estimation and tests of hypothesis on the population
variances of the random effects of the model;
ii)
estimation and tests of hypothesis on the fixed effects.
If the interest is in the type i) inference, the problem is
identified as a variance component problem.
~m
this point of view,
the fixed effects of the model are nuisance parameters whose influence
on the conclusions should be eliminated.
On the other hand, for problem ii),the variances of the random
effects are taken into account only to the extent that the errors
associated with the estimates of the fixed effect treatments are
functions of them.
The methods of analysis for handling both problems depend on
the model equations and on the assumptions about the distributional
properties of the random effects.
Among the many models that can be generated by the'se means,
only two are going to be considered
here~
Eisenhart and the one given by Schefft.
the one presented Py
,
Solely the related test of
hypothesis will be mentioned, since this thesis is addressed only to
hypothesis testing problems.
23
Both models will be discussed in relation to a two-way layout
e;Kperiment.
In the sequel Y.• will represent the yield or the response
1.J
obtained When the jth treatment is applied to the i~block
(j
= 1,
2, ••• , p; i
1.2.2.1
= 1,
2, ••• , b).
Eisenhart Models.
Eisenhart (1947) defined a mixed
model or model III as a mixture of his models I and II described
below:
Model I
The numbers Y are (observed values of) random variables distriij
buted about the true mean m that i~ a fixed constant and it is
ij
assumed that
B
= ~"1. +
A
+ e. "
J
1.J
at.
(1.2.2.1.1)
(i=l, ... b; j = 1, ... p)
Where
~. ~sthe effect of the block and a~ is the treatment effect
(both considered fixed) and the error terms e's are independent, homoscedastic and normally distributed.
Model II
The random variables Y.. are assumed to be the sum of a fixed
constant
~
1.J
and random variables
a~,
a~J
1.
and e .. that reflect the
1.J
random effects of the blocks, treatments and the experimental error
respectively, that is
24
Y..
~J
(1. 2. 2. 1. 2)
(i = 1, ..• , b;
j = .1, ..• ,
p).
It is assumed that the als, bls and els are jointly independent and
normally distributed random variables with zero means and variances
222
erB, erA and er , respectively.
The assumptions of Eisenhart govern the methods of analysis
described in most of the current textbooks and, only when defining the
interactions, do some divergences occur.
We will say that a model
belongs to the Eisenhart class if all the random components (main
effects, interactions, errors) are jointly normallY distributed with
zero mean and diagonal covariance matrix.
It should be noted that the
observable variables Y.. described by a mixed model are correlated
~J
even if the errors are independent.
Let us consider the covariance
matrix of the Y's when the model. equation is
l
= 1, 2,
Y.. J- =
~J
jJ,
+
A
B
AB
+ a. + a
ij + e.~J'Jj
~
Ql
= 1, 2,
= 1, 2,
••
o
g
,
•• ,
... ,
:)
(1. 2. 2. 1. 3)
where a.AB. represents the (random) effect of the interaction between
~J
the fixed factor (A) and the random one (B).
imposed on the effects are
The usual side conditions
25
~ a~ =
0
J
and
p
~
a ..
j=l lJ
=0
i
= 1,
2, .•• , b ,
Let the yls to be ordered as in yl
"'"
:£'
=
(Ylll' ••• , Yllc'···' Ylpl ' ••. , Ylpc ' "',
;'".
and let
~
"'"
represent its covariance matrix.
It can be expressed as
follows:
~
"'"
= diag(D,
"'"
••• , n)
(b) "'"
with
D
,...,
:::;
I
"'"
!
"""
\\
C
B
.
B· ••• B
,.....
C
,...,
••••• a .••
B
"'"
B
,...,
...
0
"'"
B
~
••••
... "'"C
~
)
where D has dimension (pc x pc) and the B's and
,...,
,
(l~
CIS
""'.
2. 2.1.5)
are (c x c)
matrices as defined below:
(1.2.2.1.6)
with
26
2
Po
O'B
=
2·
2
O'IA. + O'B
and
0'
2
0
=
2
O'A
-n 0'2
B
J
O"-C
,
where
and
. The sYIPtbols
entr~es equa~
J" and I stand for a c :l\: c
er--c
-e
to 1
~nd
for the c :x; c unity
~atri:x;
ma~ri:x;,
with .a1l its
respectively.
When the hypothesis
H:
o
A
a..
J
=;
0
is considered, the classical
t~st
is based on the vector of cell means
wh'¢>'S:e covariance mat.rix, call it ,...,
'f
,
.
exhibit .the following pattern
f = O2i ageD., _•• , 'D)
,...,
(i;; 2- 2.1. 9J
"'"" (0) ,...,
~
(
} (J 2 I + P C'
D,...,
-1
' - 'p"
2'2
""p
2 2
2 +
Cf:e
-..
(J2:
AB
J
~.
+ 0'2
/e
..
:,,'
.-;
expressed. as the .linear G.ornoination
I and ,....
J are called uniform mat.rices-
".,."
The ted of' the' hYPotn,esis H
o
is made' by using the test
~tatis'~
tic
p.,...
E (oY•.
j=1 :T.J.
b
P
I:
t . (Y.. - y . lJ.
• J..
i::;;.1 j=1
where the
yls:
Y.l ••
+y.... ) 2
represent the m~an.s of the Y's with rel3JPeci;; to the
If the assumptions of the Eisenl').art model hold, the test statistic
is d~strib'Uted as
an
F~varia1;Jle w~th'<:p-i) ~nd' (p.• l}l6.:.i)
and it provides an exact test for tJq.e hypothesis H.
o
d. f.
1.2.2. 2Scheffe' s Model.'
assumptions have led to
re~lUlts
The
I i ;
t~euse
the validit¥ of the assumptions
oftnese
C~
Scheffe points out that the
"-,'
obtained ..unde;r-
mode~s
Eis:e~n.art'
s
even in eases in which
hardly be defended.
ass~ptions
of
Eis~nh~rt's
models
are unsatisfactory and M. hoo, and he presents a new mi:x:ed model for
two-way
Glass~fied
data that is less
than Eisenhart's mixed model.
restrictiv~
=
Y
+ e ijk
ijk m..
J.J
= 1,
j
k = 1,
wheremij is the true cell mean and
an error.
CQ~l~te.
The model equations are
i = 1,
r~presents
",
and mOfe
~ijk
, .. , b
... , P
... , c
is a
rand~m
variable that;
m..
£ixed.
levels of a factor
J.J is a function of the .
A ~dthe random levels of a faetorB.
1, ••• " b
i
j
The b x p mat~i~
= 1,
•.~"
:f?
..
is a random matrix,and each row-vector is
..
co~sid:lered'a :l;'andbm.sample
from a normal distribution with the covariance matrix «a j
.,))
J
'J;.'he errors
fe.;l.J'k
Li.
const;Ltu~e. a random sarnJille from a no~l di~tripution W(O,C1 2 ) (;lJld.
.
.
.:
I
,.
..,
;
:c,
...?
29
~re
independent from the means
[m.. :
~J
k = 1, ••• , b;
p} •
j = 1, ""
,
,
The effects of the levels of the factors A,B and their inter~
actions are defined in terms of the basic mrs as follows:
General mean:
= m••
JJ.
A
a.'=m . ';'; m
J
•J
••
Fixed effect (orA):
B
Random effect (or B):
a ...4
m.J,.. - m
r=
AB
Interaction:
a .. = m•• = m.
,~J,
~J
~.
- m • +m
~J
'A dot replacing aj-subscript indicate$ summation on the levels
C
of A- anda>dotsubstituting an i-subscri:pt means e~ecta'bion with'
respect to the levels of B.
~quation
The model
c~n
be represented by the
"
Yijk = JJ.
The
dist~ibutional
+
a~J +
a
B
+
a~~
~J + e ijk
properties of the effects and their side
conditions are a consequence of their functional relations with the
variables
(mIl' ••• , ~p} •
a)
b)
I:
I:
j
j
B
AB
:;L
~J
AB
a ij
=0
the a. and a .. are jointly normally distribu.ted with a
zero mean
~nd
with the
f911ow~ng
covarianGes:
30
Cov(a.B
, a.B
,)
J.,
~
= d.. ,
~~
(J
••
(,
~B
COy ( a .. ,
~J
B
CQV ( a~
AB
, a~'J' )
......
'(
= d.. ,
cr .... cr ••.) ,
~~.J
where d.. , is the Kronecker index and the dots indicate
~~
either summation on the rows or columns of the
~atrix
«cr . ,,)) •
JJ
It can be seen that in this model the interactions and main
effects of the random, factor, as well as the interactions associated
withthe same random level qf B, are or can -be
the basic differE?,nce
be;tweenth~
Gorrela;~ed.']$is
is
earlier model/;! an(l' tn~ one Jproposed
i.' .
by Scheffe.
These ~ssumptions generate a covariance matrix for the o'bserve.Eisen~artfs
tions that differs from the one obtained with
model.
tl;le vector of observations'y' defined as in (1.2.2.1.3); then its
covaJ;':i~nce ~atrixunder.the.Scheffefsmodel
'r; = diag(D.
,...
{o •• ,
D)
ib)""
~2
...
£2p
... .
£pi
is:
D
""p 2
••• D
""1'P
_=(n .. ,))
."
'
JJ
Let
31
where
j :;: j ' .
•
li
J
if
"'j'
J'
o-c
~
Scheff6 constructs
v~ctor
if
Jr:} •. , + I (12
c;>-c JJ
.....c
(
test of the
j ~ j'
hypothes~s
Ho' basrd on the
of'mearj.s
..'. ,
whose covariance
~trix
t
"'"'
is
=
diag(D, D, ,••• , 1)
,.... \0)
"'"'
with
.£ =r(a j j ,»
The
Gorrespondi~g
+
Ik ale
·
test statistic is
.
,
V:>:
«Vj j '»
is a p:~~matri4i; 'Withent:t;'ies ,
,
~nd ,...
C
is a (p-l).:jC p matrix of, linearly
ind~;p,endj.'!nt c~mtrasts.
,
b-;I2+1
'T.('b~_---l~) (p.. i)
If the null ;hy,potnes;i,.s is true
" T.2
•
J.S dist:r~buted
as the Snedecor F variable with (b-V+1)and (b~l)(p~l) d.f.
(It is
aSi;lumed that b ;:> l?)'
Note that if
=
p
=
.-
.
and,' in:' thi/:;' s'ituation, the test' givenby (l~2. 2.1.11) is app:;Licab:J.e.
~efore entering the discussion on non~parametric mi~ed m6dels, let
us r;.ay that mixed linear mOQ-eJ,1'!
dimen$:Lonal
d~tE,l..
:Roy
~d
multivariate linear models,
the $isenhart t s modeJ,s.
studied by
~$in~
can' be
a,.l$o Ci.e.f:Lned fOr mu1ti-
Gnanadesikan
fi~ed
and
(+959) haye defined and studied
~andom,
that are
e~tensions
of
The mu;tt.ivariatl'l mixed linear mode;L can be
methods which
methods gi v~p. for tl1e, s,~l'ara,te
~e,
essentially,
mo~e1s,
I
~d'
acomb~nation of
the
II.
Another )ilaper by Ro;y and Cobb (1960) q,ea;j.s with mixed fllode1s
with
nQn-:t;lqrmal:r~dom:eff~9ts. ~t
their basic
iute~est
\.
lies on the
~,
'
/
-.-;:-.
."
33
variance component problem an do not consider the t\Slst for the fixed
effeots.
1.2~3
\'
Non-par~etric Mixed
i
Mix~d
_
.-
models
_ I
w~re
Models
discussed in connection with linear models,
but their essential characteristics make it possible to extend the
concept 'beyond-this: o:dginal context.
For the tW9-way layout experiments, these basic characteristics
can be summarized as follows:
i)
The observations in the same block are correlated, so
that in spite of their univariate appearance the blocks
have a multivariate nature;
ii)
the location parameters (expectations in the normal
case) of the components of each block vary according
to the levels of a certain fixedfact6r.
The fOllowing definition Of a non-parametric mixed model is free
Of many of the assumptions understood in the
p~rametric
model, but
still keeps the basic characteristics cited in thepreviQus paragraphs.
Given the matrix of observable random variables
... Yip
Y11
Y12
Y21
Y ••• Y2p
22
.... .
Y ... Y
b2
bp
.-) --'lit ,.
Y
bl
..
.
.-
,
34
it. wi1.J, 'be
s~id
that it conforrns toa mixed model if:
':Che row-vectors
.' {(yiI'
Yi2'· •• , Y)
:j..p:
= 1, 2, ••• , b)
i
ate mutually independent random variables frqm some
continuous joint cumulative distribution fullction F.(Y)
J.,...
= G.(e)
J.,....
F.J. (Y) = G.J. (y
- m.)
."",,""J.
("'oJ
with lOcation parameter ~
= (mi~'
~i2~ .•• , m ) such
iP
that
m:i,j
t:;
(i = 1, ••• ,
bi + T j
'b~
= l~
j
... ,,'p).
(1.2.3.3)
'. "I
Koch and Sen (1968) describe fovr types of mixed models, that
-.:
arise
~n
about tbe
relation with the
nat~re
of the
a~sumptions
distributio~
.
that can
b~
functions F..
J.
(or cannot be) made
Tqe basic assump-
tiol;'), that holds in all the cases is;;
~:
The joint distribution of any linear].y independent
set of contrasts among the
observatio~son any
parti-
cular vector _J.
Y. is diagonally symmetric.
1Wo additional assumptions which mayor may not be imposed are:
A :
2
The "additivity" of block effects.
That is to say,
.35
A.3~
The,llconJrpoundsywmei;r.yll of·the errorve.exto!,s.
Th.~t
melms. that G;i. (Xl~ x , ... ,x!) is :symmetl{i~ ill,
2
(xl' •.• , ~:P).
Tqe
p~es~nce
' f :;: 'l,- l o U , b.
or absence of A ,and A.3
~ong
2
the assumptiqns
deter~
lUiri€S -fl;n:l.r <;lases of interest, that are descpiped in the following
tablE).
"
Case I
I
,
Case'III-"
Case-IV
Case XI'
A·
.3
."
-
-:
is a se.t.of (p':'l) independent contrasts w:i-tb locationP.ar.ameter S, 'then
(p".9')and (e-U)'have th-e same distribution.
This·@.,ssumption is le.ss
I1estrie;tJ1,ve thanthefUulti,norrnality of theyt
s.
"",-* ... ;W-'
-,..." ,.,.",--
."~-"'"
Depau:seifY:
is
,.....,
u9rmal.lyq.istribut.ed,£, '''i'£1 !."ha~ ;.a(p-l) ~dime:Q;si9nja.~n?.rmal dis~ri"
. ' - '
.~
~tion·'and, then"notol1_*y (~."9'. . _,an¢!.. '(B.V).h~Yethesamedis
tpibut:Lprl
~ -,~ ~
r-_
;::;;
"
.-
".;
whE;re Krepresents:'some constant'and"Q .stands
the'InUltinormal density function.
¥Ael~
the
er~or
f~ir;the
exppnent of
.,' '
the assumptiono:f normality is
m~d,e,it
is also assumed thai!
vectors e. have the same distribution for any i.
'.
This condi-
,..,,~
tiQ,n is. ~ot required in cases It and II.
has to be equal to G for any i.
Thi~
But 'il.nGas,e~"IIIa.nd IV, Gi
propet'tYi kIfown as tp.e,addittvity
property, does not hold for heteroscedastic experiments.
The cqncept of. compound. symmetry is connecteq. with the concept
9f interaction.
d~nt
of the
Compound symmetry means that' the ,errors are indepe~...
c~J.urnns(tre~tments)
wiyh which th,e;y ar,e associated.
If
oompound ,symmetry e.Xists, all the marginal distribution functions that
can be
obtainedf~om G.(e)
:
:,~
are equal, provided they have the same
,..",
number of coordinates.;
This :hm:\?lies
.
'
I
~
t~at
the covariance matri:x: of
!,;i. 1s Uniform, ~.,!. '. it c~ ,,~e expressed as
aI
-Ir
,.."
(b-a) J • ,
,.."
In accordance with the four types of mixed models,
present four different tests for thenypothesis Ho of
.
','
nO
Ko~h
and Sen
treatment
'-,'
effepts.
In symbols,
H:
o
'1"1
=.,. 2 =. •• : ; l'1". = 0
Allt]:le tests are addressed to
.'
translat~on-tY]>e of
.,~
.,:
€\lternatives and
be~ong to th~ classofraQk~permutationtestsdescribed in the first
In the following chapters
and
IY will
be considered.
somee:x:t~nsiop.s
of the cases I, II
But before discussing these extensions in
37
d13tail, some
~ll
remarkswill'bem~de
in the :next sect:j..on that a,pply to
the chapters of this thesis.
1.2.4 General Remarks and Notation used.: in :Relation to the Extensions
;
of Non-parametric Mixed Models
Theextensionso:f mixed models to be discussed in the next
seG~iQngoin two directions;
i)
The number of results per treatment in each block is
allowed to be different from 1;
ii)
The responses of the treatments can be multivariate.
;;:,
Several extensions are presented; their descriptions require the
involved in -the variousmodels,,i . :But;i.n order to simplifY the writing,
-t~s$lneusymbols-'are-used-for
-sindl-ar' ,ideas in-diff'erentsituations.
So,they"have;tobe int$;rpreteda.ti the light of tbe context in Which
they
appear~
.;
.:. Tne notation, as well as· theiemarks that follow, are valid
for ',aJ.l the cases for which they are applicable.
i)
Notation
i
Ei,ther, th,e terms "block" ,or "sv.bject!'.;is us'ed
indistinctly to designate the set of observations
t;hat constitute an independent replication of the
experiment'. . band, p stand for tne number of blocks
and treatments in the experiment.
c
indica~es
the
"
dimension of each response.
Nij indicates the number
of results obtained under treatment j fbrsubjecth
~
(N..
,
~J
0).
P
I'lL...
- ' N.~ ,
. L, '.
~J'
J=l
•
the :q.umber of results associated with sU:bject
.,
0••
~"
11.11
~
b
'E N.
~
. 1
~:;:
number
:;:
ofresultsi~
N,
the experiment; b. indicates the
J
number of blocks in which treatment j appears
(0
< b. ~ b), and Sb"'.
J
'
set of blocks 'in
J
stands for the
which~ treatment j
i'8 present~':
Rapk totals, their covariance andcorrelationmat~ix
ii)
With only one excevtion, all the test @t,a'l:;istics
'to be, considered in the·'sequel a,rebased'0ntotals"bf
.-._~
.
ranks obtained for eac:tItreatment :and varia,b;Le'
separately.
Withp treatments and c-variate,
re&ponses, the vector of rank
and they are
as~umed
total~
has pcpqordinates,
to be ordered intnefol.lwing
way
where
T~r
•(k:;:' 1" .•• , c;
stands for the
tota~
j ""
1,. ..•) p)
th
that correspond to the k--
39
variable and the j th treatment.
The sUbscript b
means that the total is computed over a number of
blocks that depends on bo
The covariance matrix of
T'- is identified as
"""b
!b
can be partitioned as follows:
00 •
•
0.
=
=
v(cc)
(C2)
V
"""b
0
o.
"""b
where
represents the matrix of covariance of those pairs of
totals in which one total refers to the variable "k"
and the other to the variable "kIll
The correlation coefficient of
T(k'))
bj'
is represented by
(k)
Cov (T.b. ,
=
J
t
'
0
40
The symbol
is also used for
1b* designates the vector obtained from! by
omitting the c coordinates that correspond to the
treatmentiii)
!b* denotes the covariance matrix of
p!£
~*
0
Non-comPared set of treatments
If there exist two or more complementary sets of
treatments such that any two treatments belonging to
different sets do not appear together in
any
block,
II
II
that sets of treatments are called non-compared sets If there is more than one non-compared set of treatments the design is disconnectedFor any two treatments j, j' belonging to different
non-compared sets
b
l:
i=l
No
°
~J
NiJo 1
=
0
0
The number of non-compared sets is called S and
th
the number of treatments in the s-- set is called
41
If there are S
non~compared
sets of treatments !b
!'" .
can be'written by permuting rows and COlum..l1S as
·.
·..
0
"
"'"'
o
V
-b 2
=
V
,..,b
o
where'yb
0
0
"'"'
·.. .
·.. s
0
,...,
V
~
stands for the covariance matrix of the
s
cp -Vector of treatment totals in the set
s'
iV)
(1.2.4.4)
II
S
Ii
0
(.... >
- 1).
;
~
Ties
All the models discussed in this work assume that
the data of the experiment are observed values of
random variables that have a continuous cumulative
distribution
For that reason ties are
~~ctiono
assumed to happen with probability zero, and they are
not considered in the theoretical discussion.
If ties occur in practice, they can be managed
by the
v)
mid~ranksmethod."
~uperfluous Blocks
When the
r~Dking
is made individually for each
block those blocks such that No
0
J..J
=
0 for all j except
at most for only one treatment are omitted from the
experimental data to be analyzed.
for
~l1Y
This implies that
block No must be at least equal to 2.
J..
In practice, the vectors with all tied
observa~
tions are omitted as well.
vi)
Distribution of the Test Statistic
When the
f~re~uencies
of the treatments are
une~ual~
the camputation of the permutation distribution of the
test statistic becomes very difficult beca:llse of the
large number of parameters involved.
in all the types of
~ixed
For this reason,
:modBls that are considered
in this work only the asymptotic (permutational)
distribution of the test criteri.on is considered..
In the discussion of the limit distribution, the
ra.."lk of
lim
b
-+
«
00
(1<".k/))
Pb. 0/
JJ
comes into consideration.
)
The
~uestion
of 'wnder
which conditions the rank of this limit has a given
value is not discussed here.
For practical
¥~oses,
it is assumed that the sample is large enough
the actual rank of
H
(kk')
Pbjj'
))
coincides with the rank of its limit.
SO
that
2.
MIXED MODEIS FOR TEE CASE OF SEVERAL
(EVENTUALLY ONE OR ZERO) RESULTS PER TREMMENTo
UNIVARIATE RESPONSES
2.1
Introduction
Linear mixed models for two-way classified data such that the
number of results per treatment in the same block is (or can be)
different from one arise in relation to randomized block experiments
of several kinds.
If the number of results per treatment takes only the values
zero end one y the design of the experiment is named incomplete
randomized block design.
If the number of results per treatment is different from one
and bigger than zero for at least one treatment J the design
ca~
be
of the type either of a generalized randomized block design or of a
randomized block design with several observations per cello
In the
first kind of experiment"the various results associated with a treatment in the same block come from different experimental units, for
instance different plots of the same block of land or different
trials performed on the same patient.
On the other hand, in the
second type of experiment, the (several) results that correspond to
the same treatment are obtained from the sam.e experimental unit, say,
by selecting several samples from the plot of land or by performing
several determinations after the patient has been submitted to a
unique trial of the treatmento
44
The theory for the analysis of parametric mixed linear models
of the type described above has been fully studied for balanced
designs or designs with a high degree of symmetryo
that illustrate the models worked by Eisenhart and
(The examples
Scheff~
are
applicable to randomized block designs with several, equal in number,
observations per cell.)
But, although a lot of work has been done
for the case of unbalanced designs, a complete and satisfactory answer
is still lacking.
In extending the non-parametric mixed models to the case of
several results per treatment,it is convenient to keep in mind the
type of experiment that lies in the background of each model.
There
is not a one to one correspondence between the models that will be
presented in the next sections and the designs mentioned in the
previous paragraphso
It can be anticipated that the extension of
case I to be given in Section 202.2 can be applied, according to the
circumstances, to any type of experiment.
On the contrary, the
extension of model IV (to be developed in Section 202.3) is only
suitable for either experiments of the generalized block design
type, or incomplete block
~esign
type.
In Section 2 3, two rank tests will be presented that are
0
related to a model, that is the non-parametric counterpart of the
mixed linear model underlying a complete randomized block design
with several observations per cell.
45
2.2 Mixed Models for the Case of Several
(eventually one or zero) Results per Treatment.
Extensions of Models I and IV
2.2.1
Introduction
Let
1'2l\T
.~
i2'
0 .•• ,
Y.~pl'
(i :;:: 1, ""
Y. N )
~p ip
o •• ,
b)
be b stochastically independent vectors where
Y.
'n
~JR
(j:;:: 1, ... , p; J :;:: 1, ""
N.. )
~J
represents the response of the "th trial of the jth treatment in the
. th sub'Jec t or bl oc.
k
~
It is assumed that I'V~
Y. has a continuous N.-variate
cumulative
~
distribution fUnction F.(y)
such that
~I'V
F.(v)
:;:: G.(y
- I'V~
m.),
~ ;t.,
~
I'V
m~ :;:: (m~ll' ••• , m'N
I'V~...
~
il
'
•• 0 ,
m·· N
~p
ip
with
(L= 1, ••• ,. p;,ej.= 1,
no,
p;,;'.,:;::
2,
,.•. , N.. )
~J
)
46
The hypothesis to be tested is the hypothesis of homogeneity of
treatment effects
=
H :
o
o
against
F"1. :
at least one .,. is different from zero.
Together with assumptions (2.2.1.2) and (2 0201.3) some additional
assumptions mayor may not be added.
These optional assumptions are
A :
l
Diagonal symmetry
A :
2
Additivity of SUbject effects
~:
Compound symnetry of the error vectors of the same
block.
The meaning of these assumptions are obvious extensions of the
ones considered in the case N..
J.J
= 1.
Only that' now the number of
coordinates is N. instead of p.
J.
With respect to A ,it has to be added that in this case the
2
eqUality of the cumulative distribution functions G. has to be
J.
understood as equality of the marginal of the same group of variables.
2.2.2 Case
Dr
Together with (2.2.1.2) and (2.2.1.3), it is considered that
assumption A holds.
l
......
In order to test H , let us define
o
and
= rank{Z J.JJCI
.. ,,: Zill' ••• , Z. N 1
J.p ip
(i
= 1, ••• , b;
j
= 1, ••• , p;
t
= 1, ••• ,
N..).
J.J
The Z's are contrasts; hence, under Ho'and assumption AI'
..... Z~
.....J.
(and
(i
= 1, •.• ,.b) ,
where
Z . 2N
J.
'
i2
••• ,
have the same distribution.
R' 2N
J.
i2'
z.J.p l'
,
••• ,
As a consequence, the vectors of ranks
••• , R. l' ••• , R. N ) ,
J.p
~p
ip "
are (conditionally) equally probable, each one occurring with condi.
tional probability 1/2 for any'value of i., Then, given the actual
vectors
fR.:
-J.
i
= 1, ••• , b} , this sign invariance generates a set
b
of 2 equally likely realizations- - LetuscallP the related condib
tional probability law,
(
I
t')
E Rijt u-b
.'
)
(
~'
J
J.
n
= R..
~J
+ (N. + 1 - R.. n)
~
XI
if i
XI
.
b
I
=
N~ + 1
...
2
= i'
~
N. + 1
I
~J XI
2
:r.--: ,~' ",'
E(R.. nR.,.,n/IP )
~JXI
)
(~
~
2
)(
N. / + 1
~
.
2
)
if i ~ i /
and
N.+ 1
~?
~2
(R
ijt
if i ~ i'
o
for all
i, i' = 1,
~
.. , b;
j, j '
= 1,
... , p;
and
t = 1,
N..,
~J
t' = 1,
The test statistic to be
b -_.N ·
ij
u~ed
~j = i~l t~l Rij>t
" ' , N.,,,
~ J
•
to test Ho is based on the totals
(j
= 1,
... , p)
(2.2.2.8)
that have the fo.l,lowing first and second ¥!loments
N. + 1
)
E(Tb .IPb ) = i~l Nij ( J. 2
b
.
J
b
[Pb):c
I:
Nij
[I:
i=l .t=l
the summation (over i) being extended only on those blocks in which
both treatments j and j' are present.
The TIs are suoject to a linear constrain
P
E
j=1
T.
J
=
b
I:
i ..........._
.....;J.;;..~
N. (N
+1)
i==l
2
and also can be seen that
(j :;: 1, "', p)
So that the rank 9f !b is at most (p.. 1)..
ff
the~e
are f;?n,on..compafed
set:!! of treatm.ents the rank is, at most (p ...S).
If the rank of'!o is (p .. 1) the test statistic is taken to be
•
where!o is def:i,nedas in (1.2,4.1) and,S is any (p"l) x p matrix of
linearly independent contrasts.
50
In particular, C can have the form
"'"
C .~ [J
.
. ....,p~l
N
°] ~ ±p
p-~l
J
p~.J...;.p
and in this case,
where lb* is the vector formed by the fi~st (p-l) coordinates of Zb'
and !b* is its covariance matrix.
If there are S(fl) non-compared sets and the· covariance matrix of
the S~ set has rank (ps-l), W still has the form (2,2.2.13) with
b
£ = diag
(~.J,.' C ,
,-
""'2/
... ,
with
C
. ISIS
(s
= 1,
.•. , S)
being a matrix of (ps
-1) linear i~dependent contrasts among the p
s
treatments in the same set.
Also, in this'
~ase
S
Wb =
~~l [!b(S)
where !b(s) (s
= 1,
.
Wb can be written as
- E(!b(S)!f'b)J''s(s)['s(S)Yb(S)£(s)J-
1
... , S) is the vector of the totals that correspond
to the s~ set of non-compared sets.
51
In this situation, W tests the hypothesis of homogeneity of
b
the treatments inside each set and it is insensitive to the differences
of treatments in different sets.
The distribution of W
b
(2.2.2.13) is given by the following
theorem~
Under the hypothesis Ho and assumptions (2.2.l.2),(e.. e.l~3)
and
A.:L'
if
i)
lim
( (Pb ..
JJ
b -tCO
ii)
iii)
,»
=
«p JJ.. ,»
has rank (p-l)
The N. t S are uniformly bounded
1.
r
For each j,
i
\
r
/1 5\ ) >
var(Tb
€
>0
and
the permutation distribution of W , as defined in (2.2.2.13) is
b
asyn:q:>totica11ythe distribution ofaX2 with (p-l) d.f. [The proof
follows from theorems 7.1.1 and 7.1.2 of Chapter 7, taking c=l.]
Also, it can be easily seen that the one of (2.212;17) is the
distribution ofaX2 w~th
S
E
s=l
(p s - 1) d.f.
It follows also frqm the remarks at the end of theorem 7.1.12
of Chapter 7 that if conditions i), ii), and iii) are fulfilled but
52
the, rank of ~ (that' is assum:ed to coincide with the' rank. of
is r < (p-S) ,the distr:Lbution of
«p jj')))
,.'
with C' (r x p) such that £S:£' is non-singular, is asymptotically that
ofaX2-variable with'r degrees of freedom. 'Nevertheless, in this
case the invariance property of W under different choices of the
b
matrix of contrasts £ is lost ,and the value of W will depend on the
b
particular selected matrix.
Condition iii) is-difficult to discuss in a 'general set-up,
since it cannot be asserted without further research that
var(Tbjlrb)
will be bigger than zero for any treatment.
The situation is more cleat whenN .. is chosen to be 0 or 1;
J.J
N. + 1
R..
J.J
=
J.
2
for all the blocks in the experiment and this would imply that some
functional relationship among the set 'Of variables' (y.. ;
.
••• ,' p}
J.~
j
= 1, 2,
hold with probability one, a case that is not of interest
in relation to this ·work.
In pract1cewe
expectthat'var{TbjIPb ) will be bigger than zero;
if it is not that way, the treatment can be eliminated from the data
and the analysis
perform~d
for the rest of treatments-
53
The asymptotic permutation test for H is given by the rule
o
Reject H
o
Accept H
o
2
if W < X
,
r,a.
b
2
r is the rank of Y"k and X
is such that
--u
r, a.
2
p[X
r
~
2
X }
r,a
= l-a.
(0 < a < 1)
for a desired level of significance-
2-2-3 Case U IV
The assumptions that characterize this case are (2-2-1-2),
(2.2-1-3), A2 and
~.
The conditions A
2
and~.
together imply that all the marginals
that can be obtained from G. are equal if they have the same number
J.
of coordinates.
Let'usdefine the aligned yields and errors by
..
with
Y.J.
P
Nij
I:
I:
j=l 1,=1
i
= 1,
••• , b
j :::: 1, ••• , p
1,
= 1,
••• , N..
J.J
Y"
n
J.J)(I
and
- e.
= eij.t
€ij.t
~
with
e.
::;
~
i
1
p N..
~J
E
E
eij.t
N.~ j=l
.t=1
(2.2.3.2)
= 1, ••• , b
j ::; 1, " . , p
.t
= 1,
... ,
N..
~J
From the properties of the ets, it follows that the vectors
(€. ::;
'"'-'J.
(€·11' ••• , €. N ):
~
~p ip
i::; 1, ••• , b}
are independent, identically distributed with a distribution function
that is symmetric in its N
i
argument~.
Under H "it-can be written that
o
i
= 1,
j
= 1,
.t
= 1,
... ,
., .. ,
... ,
-
So, the test of the hypothesis Ho reduces to testing the interchangeability
o~
... ,
z.
N
~p
ip
) •
..
55
The test is based on inter-block ranking.
Ranking is carried
out over the
b
N =
I: N.
. 1 J.
J.=
observations.
Let
r J.JJtI
.. fJ
= rank(Z J.JJtI
..
Zlll'···'
fJ:
' ••• , ~pl'
ZipN.
J.p
.. , ~P bp
~
N
}.
Given the va.lues of
{rij ;.:
i
= 1, ••• , b;
= 1, ••• ,
j
it is assumed that unde.r H , any
o
j
th
t = 1,
p;
••• , N.. }
treatment in the i
J.J
th
block,
could be related to any combination of N.. rankspresen't in the
J.J
.th
J.
bloc'k•
The test is based on intra-block permutation of the ranks of
each block.
The number .of configurations derived from a given set
of data is
b
n (N.)
J.
!
i=l
each configuration having (conditionally) the' same probability
under H •
o
Let us denote with
law.
~\
this uniform (conditional) probability
· The test statist;i.cisa function of the totals
b
Nij
(j
T , = 1:: 1:: r.. n
bJ
i=l t=l J.JoKI
= 1, ... ,
p)
p . Nit
1:: 1:: r'
= r.
t=l h=l J. th
J.
1
E(rJ..J.nIPb) = N
oKI
i
J.' f 'J. = J.., , J.=.J , ,oKIn
__
n'
oKI
J
N·t
p N,t
J.
2
J. 2
- [ 1:: 1::r
- (I: 1:: r i th) .
t=l h=l ith
Ni
t=l h=l
-1
p.
if i = i'
j = j' and
t 1= t'
and
r.r. ,
J. J.
for any i,i' = 1, 2, ••• , b;
if i 1= i'
j,j' = 1, ... , p;
t
= 1, " . , N.. ;
J.J
Proof
UnderP the N orderings
i
b
(rill' ••• , r ipN . }
J.p
are equally likely, and it implies that each r, . n has a probability
J.JoKI
1
of occurrence equal to
~
•
Hence,
J.
E(rJ.~J.nIPb)
XI
l
=N
i
N
it
P
~
~
rJ.~th
t=l h=l
•
If i ~ i ' the random vectors
(rJ.·ll' ••• , r.J.pN. )
J.p
and
(r~lll.' ••• , r.
'"'
~re
I·:TIT
..
J. Poll. I
J. P
)
independent, then
E(r J.JXI
.. nr""/J,IP
'/JIPb)E(r""/J/IP
J. J XI b") = E(r.J.JXI
J. J XI b )
= r.J. r.,
•
J.
If r"/J
J.JXI and r.J.J." bXI ' are different coordinates of
(r·J. ll", ••• , r.J.pN. )
J.p
(!_!_,
j ~ j
I
or j
= j
I
and t ~t I
)
,
p
N·~ t
E(r. ·n r .. ,n,IP ) =!:
!: E(J:' .. n Ir· th )
b
~J~ ~J XJ
t=~ h=l
~J~ ~
Nit
= I: E_
t=l h=l
p
.
p
~
!:
r ith
-N
, i
Nit'
!:
t';'l h'=l
r· t , ,
~ h
N. - 1
~
P Nit 2
I: r'
t=l h=l ~ th
!:
J
.
.-
Under the conditional law P ,
b
b
E(TbJ.lrb )
=
!: N.. r. ,
i=l
~J
~
where d.. , is the KrQneck.erlndex and
.
JJ
~.
~
=
(2-2-,-10)
The Tbj's satisfy the relation
b
=
I: N.r.
. 1 ~ ~
~=
that is a fixed number.
Also, it can be seen that
59
v
j
,
so the rank of ~ will be at most (p-l).
If superfluos rows are omitted, it can be seen that a necess.ary
and sufficient condition for COV(Tbj,Tbj ,) to be different from
2:ero is that
b
E
i
that
impli~s
N..N.. ,
J.J J.J
>0 ,
that at least in one block treatments j and j' appear
together.
Also, it can be shown as in B~nard anA van Elter~n (1953)
that:
"If and only if there are S non-colllJ?ared sets of
treatments the rank of the
matrix~,
is, for any b,
p_S.,1t
The proof is based on the Tausski·s Theore~ (Taus ski (1949».
If the design is connected, the test statistic is taken as
where~,
is the vector of treatment
-
totals,~
its covariance
matri~,
and C is a (p-l) x p matrix of linearly independent contrasts.
If the design consists of
as in (2~2.2.l7)
S( > 1)
non-compared sets,
W
becomes,
b
60
wnere !b(s) and ~(s) have analogous meaning to those in CaSe IThe distribution of W is stated in the following Theorem_
b
Under the hypothesis Ho and assumptions (2-2-1-2), (2-2-1-3),
..
i)
lim
b .... 00
ii)
"
«P bjj ,»
= «Pjj'»
the N.'s are uniformly
~
has rank (p-l)
bounded~
b.
iii)
.2 =h!=O,
lim
b
b .... CD
then the permutation distribution of W defined in (2.2-3.11) is,
b
asymptotically, the di~tribution ofaX2 with (p-l) d.f- [The
proof follows from theorems 7.1.5 and 7.1.6 of Chapter 7 taking
'0=1.,1:
Also, it, can be seen that' Wb,as,defined in' (2-2.3.12), has
2
the distribution ofaX with
s
r:(p . ':"1) '= p-Sd.f.
s=l . s·
The
given by the rule
. ,test for His
,0' .'
.'
asymptoticcpermut~tion
, ,
. c..·
.' Re'jectH'
if W ...~ X2.
b
r, a.
Accept H
if W < X
b
r,a
o
o
2
,
61
r is the rank of V... and X2
is such that
""1.1
r, a.
(0 < a. < l~
for a desired level of significance.
2.3 M.ixed Model for the Case of Several
(Different from zero) Results per Cell.
Extension ofw,Model II (Case U IIal
.,~.'
~'
-
Let the data be represented by a set of stochastically
indep~ndent
vectors
Y.J.=v
"lIT
,
i2
(i == 1, ... , b)
e •• ,
let the set of treatments und.er test be divided in disjoint subsets
•• 9 ,
of Pl' P2' •• ', PU. treatments, respectively
J.
U.
J.
p
( i:
u==l
in the i
~; if
th
bloc~. If treatment
i t belongs to
u
j
== p)
belongs to
S~, let
s~, let Nij be ~ and so on.
In each block the relation
U.J.
i:
u==l
•
•
pJ. MJ.
u
U
==
N.
J.
N.. be equal to
J.J
62
i
is assumed to hold and all the numbers M in the same block are
u
assumed to be bigger than zero
~~d
different from one to each other.
The numbers
as well as
~,
--:L
i
M
2'
O.
0, -u
M~.
~
are fixed by design and they may vary from block to block sUbject
to the relations previously stated.
The y's are considered to arise from a two-stage random process
that can be described as follows.
At the first random stage, b partitions
i
= 1, ••• , b}
of the treatments are selected at random, individually for each
block.
In the i
th
blockjto all the
p.I
.0" , p . 1
U.~
possible
partition~ equal
chances of being selected are given.
~
J!.?,.. ,
After
,
the decision of taking Mi replicated observations
u
i
in all that treatments that form the set 8 is made. 'Bo, in this
the selection of
n.,
~
u
model N. . is a random variable that takes values in the set
~J
63
and whose distribution of probability is determined by the fact that
N..
J.J
==
i
M
u
iff treatment j belongs to the random set Si.
u
The cumulative distribution function of Y. is assumed to be a
"""J.
N.-variate continuous fUnction
J.
F. (y)
(e)
J.,...
G.
J.,...
with
o •• ,
m'1N
J.
il '
.... , mJ.'pl' •• " mipN. .
)
J.p
and
(i
== 1,
u.,
b;
1- == 1, ••• , N.. ·) ~
J.J
The second stage of the random process associated with this model
refers to the selection of the actual yts from the distribution
function F that depends on TI through the variables
i
i
A further assumption on the distribution of the errors ,...
e's
co~pletes
certain
the model:
treatmen~
vectors of
errors~
for any given block the errors related to a
are interchangeable random variables and so are the
•
64
(e,~J'1' e,~J'2' .•••
, e"N )
.
~J~ , ,
~J
and
(e, ,/ , e, "
~J
1
~J
2
••• , e. :I
'
~J
for all j and j' in the same subset 8
i
u
tion will be called
"fragmented
'1"1
=
'1"2
)
of treatments.
compound symmetry" and
The hypothesis to be considered is,
Ho :
Nij'
~s
This assumpsymboli~e~
usual
= ••• = '1"P = 0
•
The procedure for testing H is as follows:
o
Let
R. . n = rank[Y, . n:
:l,.J XI
••• , Y, l' .".,
~J XI
~p
i
= 1,
... ,
b
j
= 1,
... ,
P
L
= 1,
(2,,3· 5)
, .... , lit..
~J
and
... , Ripl',
(i = 1, •• ", b) •
.... ,
R.~pN, )
~p
Under H the vectors of ranks have the property of fragmented
o
compound symmetry.
It will be associated to each set
{R. :
J.
= 1, "., b}
i
the
b
II
i=l
u.
i
J.
II
P
i
(M
u=l
=
1) u
u
U.J.
.. II P
u
=
u=l
equally probable (under H ) results that come into consideration
o
observing that if H is true,
o
i)
the actual ranks in R. could have been obtained under
"""J.
any of .the equally probable partitions II.;
J.
ii)
for a given partition any permutation of the vectors
of ranks corresponding to the treatments in the same
i
set Su are equally probable;
iii)
for any individual treatment, all the permutations of
its associated ranks are equally probable (and this
is true even under hypotheses different from
Conditionally to this set of
b
U
i
II
II
i=l
u
.
i
P
(W!) up!
u
H).
o
66
results, a uniform probability distribution can be definedcall it
6'\
Let us
0
Two statistics for the test of H can be suggestedo
One based
on the totals
b
~j
Nij
'E
'E R.. n
i=l t=l ~JXI
=
(~-3-10)
=
d .. 1 p-1
J'J
x
p(p-1)
d being the Kronecker index and
Nij
R..
~J-
=
'E
t=l
R.. n
~JXI
-
:p
b
t ! t R.
i=l :p t=l J.t
and
x
:p
x
b
-2
R
it
i=l t=l
r:
:p
r:
2
L: R..
t=l J.J
:p
In Chapter 7 (Section 7-2) ~t is shown how these moments have
been obtained-
If ~ and kR' represent the covariance matrices of the Tb
IS
and TbR\flS respectively" the related statistics for testing H are
o
and
(2-3-16)
where
k,
is the vector of the totals defined in" (2-3·7) ancl ~bR'
is
the veotor of the unweighted su,xn of rank averages defined in (2-,-8)C is any (p-l) :x: p matrix Of linearlJT independent contrasts,
"""
68
After some algebra, it can be shown that
and
b
P
I:
I:
i=l j=l
2
t'R .. ~J
P R.· 2
(I: 2.J.) ]
j=l P
It can be seen from (2.3.11) and(2.3.14) that the columns of
either
~bT
or
~bR
add to zero.
So, they have, at most, Jlank
(p-1) •
The distribution of W (2.3.15) is given by the following
b
theorem.
Under the hypothesis H and assumptions (2.3·2), (2.3.3) and
o
i)
ii)
iii')·
lim
b ... eo
«P b JJ.. I))
=
«P JJ../))
has rank (p-1)
the N.'s are uniformly bounded
~
lim
var(~j I~\)
=
00,
then
b "'00
the permutation distribution of W defined in (2.3.15) is asymptotib
2
ca~ly the distribution ofaX variable with (p-1) d.f.
A similar theorem hoi,dsfor W -(2.3.16h The proof's of these
bR
theorems are given in Chapter 7 (theorems 7 1.8 and 7. 1. 9) •
0
The asymptotic permutation for His:
o
Reject Ho
if W·
Accept H
o
r being the rank of
b
k, (Yc,'R)
(W - )
bR
2
< Xrex,
,
and X;ex, chosen such that
(0 < ex, < 1)
for the level ex, of significance.
2.4 A Numerical·Example
To illustrate some of the techlliques of this chapter, a set of
data has been analyzed.
The data represent the results of replicated
blood analysis in a group of six swines submitted to some experimental
diet.
The blood samples were taken monthly from October to February.
At each time several "determinati'b:6.s1 bf the proportion of Calcium in
•
.<
<
,.
,.,
·';c,~');", ,", .'.",
I"h, J:,I~till -
,,",
,
serum (mg per JOO ml,serllin) were:;,fjr~Ormed:' The following table
shows the outcome
.,,
of!!iJ:ieexperiment~\:i,·,
.. :.,:.•...
;
,;~'
=
Blood. Analyses
,
Swine
-
-
Month
Oct.
-
..
1
w
=....
:-,
,.
10~8
lleO
" 11.2
9-8
10.2
10·5
10·3
.10·5
10·1
·10.3
10·7
11.0
11.1.
10.4
10·5
' . :u.1
11.6
_,10·9
12.2
12-'5
3
5
6
.11.1
10·7
9·5
10.8
10.0
9.. 9 . . -- . --- ----: - . . - - . . .
12.0
11.8
2
4
.. .. -- - " .. -Bee. .
Nov.
ll.2
Jan.
Feb•
9·5
9·9
10.Q
10·9
10.4
10.4
10·5
10.0
10·5
IO·3
10·3
Io.6
'1Gr7 -_ ..
f).9
.-
9·.5
10.0
9·9
10·7
10.6
10.4
10.6
10·7
10·5
9·9
10.4
+0.2
10..7
10.1
10.6
10.2
10·5
.1.0.4
9·2
10·3
11.2
u.8
U·9
n.4
-11.0
.11·7
10·5
10.2
IO.6
10.8
10·9
1.0.4
10·7
11.0
10·9
10.4
10·5
10.8
U.2
·11.1
10·5
10.6
10·7
10.8
11.8
13.4
12.2
10.6
10·5
10.0
11.0
10·5
n.4
10.8
9·7
10.2
10.410.8
10·3
9·8
10.1
_9·8
10.4
10.1
11.4
12.0
12·3
12·5
11.9
10·9
-11.0
10·3
10.8
9·8
10.6
10·9
10·5
10.8
11.0
10·9
10.6
10·7
·10.8 .
10·9
10.6
10!t7
-..::I
o
,
\
.,
.,
..
71
Several models were applied to this set of data and their
corresponding tests for the hypotheses of no effects of months were
constructed.
The lists of models and the results of the tests are
shown below:
i)
Case I of Koch and Sen applied to the average of
the cells
ii)
Case U I
(Section 2. 2) •
iii)
W = 6.000
b
W
b
= 5.950
Case U II
(Section 2. 4)
The number of blocks (b=6) is hardly lisufficiently large ll to
justifY 'Qsing a large sample approximation and any conclusion
based on these results has to be taken with reticense.
72
3.
MIXED MODELS FOR THE CASE OF SEVERA.:):, (EVENTUALLY
ONE OR ZE~O) RESULTS PER TREATMENT AND
MULTIVARIATE RESPONSES
3.1 Introduction
Extensions to multivariate responses of Case VI, Case UII (that
underlies the Benard and van Elteren test) and Case UIV are presented
in this chapter.
Case UII
o
has not been extended to multidimensional responses,
but the procedure for obtaining such extension can be done following
the same steps that are used in the univariate situation.
For the type of problems to be considered in this section, the
outcome of the ekperiment can be represented by the set of
stoch~stically independent
a different subject) {Xl'
matrices (each matr:j.,x being associated with
12 ,
.,', !b}' with
Y(1)
·.. ,
1.
""~
=
Y
(2) ,
ill
... ,
Y(c) ,
ill
... ,
(2)
y(2)
i.:l..N.~.l'
(c)
Y' IN
~
Yi21 '
(c) ,
Y
'
i2l
i1
• I")1\T
~t:;.J.'i2
Y' 2N '."'"
~ i2'
·.. ,
(c)
Y' 2N "
~ i2
y(l)
ipN.
~p
(2)
· .. ,
... ,
, ••• ,
.• ,
y(2) .. , ., y(2)
ipl'
iPNip
y(P)
ipl'
... ,
Y( c)
ipN.~p
(3.1. .1)
where
••
stands for the
t!&
result of treatment
to
,
j
taken on the i th subject.
73
;It is assumed that each matrix comes from some (c x N.)-va1ued
~
continuous cumulative distribution function
Fi(Y)
,.... '" G.(y-m.)
~
with
~i
=
((m~~)))
,....,....~
=
(3 ..1. 2)
Gi(e)
,....
"being a (c x Ni ) matrix such that
(k) = b~k) + '1"~k)
mij,t
~
J
(:
=1,
•
ra
_ ,
:;:: 1,
o •
" ,
:;:: 1,
• • '\ J
bj
p
•
N ..
~J
The hypotpesis to be tested is
...
= ,,(k,) = 0
p
for k
= 1,
2, .•• , C'
Tp.e assumptions Al-' A and A that have b~erL de+:JLn:edfor univariate
2
3
mod~ls ce,nbe adapted to the multivariate case in the following way:
Al
Skew symmetry:
The joint distribution of any
linearly indeperndent qet pf contrast$ among the
vectors Y.. , is (diagqna11y) symmetrical-ly dis""l'J.J~
tributed about its location parameter.
In 'other wor&s, let ,....C be a (N.-l)xN.matrix
of corttrasts.
J.
J.
CY!
,..,....1.
have
th~
meter of
and
[e·
- """'J.
CY!]
,....
same distribution for any i, where ,....
e is the location
.sri'
Then
para~
74
A Additivity of
2
~ubjedt
effects:
Any marginal
obtained from G is the same (for the same group
i
of variables) whatever block is considered.
~
Compound symmetry of the error columns:
This
means that
... , e·
G.(e il , e· ,
~ "'"'
"'~ 2
"'"'~ Ni
)
,
= Q.(e·
~ "'J. h1
e· ·, ••• , ei'~_ )
"'~ h 2
."'" -~.
J.
where hI' h , ... ,
2
~.
is any permutation of the
~
subscripts {I, 2, ••. , N.}.
~
The three cases to be presented are identified as cases MUI, MUIl
and MUIV; for
~ll
of them, assumptions (3.1.2) and (3.1.3) are
supposed to hold, and for:
Case MUI:
Al holds (A2 and ~ may not)
Case MUIl:
~ holds CAl and A may not)
Case MUIV:
A and A
2
3
2
hold (A may not).
l
3.2 Case MOl
To test H ' the fQllowing matrices are considered:
o
~i = (~l'll' •.• , ......Z'lN
~
il '
... , "'"'lp.
Z. l' ... ,
Z. N
"'"'~p
where
Y.
~ij£ = Xij £ - "'"'1
(
= 1,
... ,
.. ,. ,
= 1,
,.
= 1,
:)
..
,
N..
~J
ip
),
75
with
1 P Nij
-N J' &1
n &1 Y, . n •
-. .,X>.-.
"'~ J X>
1,1 :;:
oJ,.
,
l
If H holds, the distribution of {Z,: i ::;
o
"'~
.1;'
b} remains in-
variant under the group of transformations.g such that g EGis
defined as
1:
where
Ci.
~
(i::; 1, ••. , b) can take the
~a:..iues 0 or' 1.
As a consequence, the matrices
~'
.'
R,
"'~
and
(N, + l)J - R.,
'"
~
"'~
where
·.. ,
R'(l)
R(l)
ilNil ' i21'
... ,
·.. ,
•
- (2)
,R'lN
.'
(2)
Eil1 ,
(c)
Rill'
(c)
·.. , RilN'1'
l.
(1)
Rill'
:R.
"'l
::;
l
(2) :
' Ri21 , ! .
il
(c)
R
,
i21
e: ,
(1)
(1)
R'l 2Ni2''''JR,lp.l'
(2')1
(2)
R'2N , ... ,R. l'
]. i2
lp
•
0
Cl
,
·.. ,
R(l)
ipNip
R(2)
ipN.
lP
(c)
(c)
R( c)
... , R'2N
, ... ,R. l' · .. , ipN,
i2
lp
lp
~
(3.2.4)
are cqndition~llY equally probable, each one occurring with probability
1/2 for any value of i.
If given the set of matrices (~l' ••• , ~}, we restrict the space
to the set of 2b points generated by g, then, as a consequence of the
76
~i~n
i.qva!'iance under H ' a Wliform
o
~et
on that set,
(conq.itiona~)
ll:\.w can be defined
Pb designate that conditional law. Dnde!' Pb'
~(~) + (N +1 - R(k»
N +1
;.ii
E(Rijll~\) = "l~t
i
i~~ =?
(3.2.5)
(It) (k/)
(k)
)
RUiR:l.j /.t' + (Ni +1-Ri '1, HNfH. - Rij /,t/. .
_ -.
I
.J
- . ,;J.f J. == if,
2
(3.2.6)
.>
(Ni +1)
2'
and
if i 1= ~/
o
tor i, i '
J/ =1,
= 1, •••
f
1{; j" j'
= 1, ,p,
N / jl and;k., k f =1,
i
t~q
p)
.t = 1,
T'"
~+j'
c.
.~.,
The test statistic is based on the rank totals
j
(
Cpnditiona11y,
k =
E(T~~)lrb) = :I.~l
()
j
(
kl)
Tb
J'
1, ••.• ~ ).•
(3.2.8)
1, •.. , c ..
we bavE; that
b
COV(Tb~-'
•
b
'Pb) :::
N.+l
Nij (+-)
.Nij
~~l(.t~l(Rij.t
NitI; . Nij,
-
"1r)J[t~l
(Rijfj,-
Ni +1
~)J.
(3.2.10 )
77
Let us consider the vector :Eb as defined in (L 2 4).
For each k,
p (k)
b Ni (Ni +1) (
j~lTbj =i~l
2
k=l, 2, ... ,c)
(3,2 ..1.1)
and also
(3.2.12 )
for j
= .~, ;E, ••• ,
p and any pair (k, k').
From (3.2.12) it can be seen that the rank of !b is at most c(p-1).
If there 4I'e S non-compared sets, it follows from (L2.4.4) that the
rank of !b is at most c(p-S).
If the rank of !b is c(p-1), the test statistic is taken to be
where C' is a (p-l)c x pc matrix of the form C = diag(D, ... , D), and
,."
'"
'" . (c) '"
D is a (p-l) x :p matrix of linearly independent contrasts. D can be
~
-
-
~.
taken as C in (2.2.2.14), and then,
'"
(3.2.14)
If there are S non-compared sets of treatments and the rank of
each lo(s) is c(ps-l), a formula similar to (2.2.2.17) can be used,
:!:.. ~.,
S
Wb
=
s~.l [!b(s fE(:Eb (s) I~\)] '£(8 )[£(s )Yb (s )£(s) rl's(S )[:Eb(8 fE(!b (s) IPb ) ]
The distribution of Wb is stated in the
fol~owing.theorem:
Under the pypothesis H and assumptions (3.1.2),' (~.,1. J) and A.l' if
o
i)
lim
((pkk'
.. ,))
b-+oo
i1)
iii)
JJ
The N.
~
I
= ((Pkk'
.. ,))
has rank c(p-l)
JJ
are uniform.l.y bounded
S
For eaoh k and j, va.r(T~~)If>b) > € > 0 and
(k)
Var(T
) goes to infinity with b,
bj
b
Ir
then, the permutation distribution of W , as defined in (3.2.13) is,
b
2
asymptotically, the distribution ofaX with c(p-l) d.f.
The proof 1s given in
th~orems
7.1.1 and 7.1.2 of Chapter 7.
If the rank of ((pkk'
jj ,)) isr, then the corresponding Wh is
distributed as a X2 with r degrees of freedom.
As in the univariate case, it can be said with respect to
assun~on iii)
that
var(T~~)If>b) will
be positive for each j and k, if
N.. is taken to be equal to either 0 or 1, unless some functional
~J
relations among the coordinates of the blocks exist.
The asymptotic permutation test for H is given by the rule:
o
Reject H if
o
w.b
1;:0;
X2
r,OI
2
Accept Ho if W,b < Xr,OI
where r is
~he
rank of
Yb
.~
.
2
and Xr,OI is such that
79
3.3
Case MUll
The test is based onintrablock rankings.
Ranking is carried out
over the N. obse.rvations within each block and done individually for
J.
each variable.
Let
y(k)
~p1'
•
0
0-,
and
(1)
Rill'
R.
,..,J.
=
(2)
Rill'
(c)
. Rill'
• e-. ,
R(l)
R(l)
i.lNi l' 121'
• •• J
(2)
(2)
R'J. lNil ' Ri21 ,
·-. --. ,
o •• ,
... ,
.. -,
(1)'
(1)
R' 2N , •.• ,R. l'
J. i2
J.p
·.. , R(l)
ipN
ip
(2)
(2)
R' 2N , ••• ,R. l'
J. 12"
J.p
·.. ,
R(2)
ipN.
J.p
· .. ,
R(C)
(c)
(c)
RJ.. 2Ni2 " • • , RJ.p
. l'
(i
= 1,
0'"
b)
ipN.
J.p
(3,',3.1)
Y.,
B-ecause of A , all the matrices that can be obtained from
• .......:L
2
(3.3.1) by permutation of the Ni columns, are, under Ho ' equally,
probable.
The same is true with respect
to "",J.
R., so that thedistri.
..
bution of R. on the set of N.! matrices originated by the
"",J.
J.
l 'J.nvarJ.ance
.
. (N.1 )1' S 0,
t·
permu t a·J.ona
J.S
J.
b
1
= i~h -,o(N:::.i.....)-J
since the .rankings are independent for different blocks.
'P this uniform (conditional) law.
b
Let us call
80
Under Pb'
N.+1
:I.
=-2
N
P
1
tt (k) (k / )
N.:I. t~l h=.lRithRith
i =i'
if
j = j'
t = t'
N. (N. -1) 2
)J
(:1.:1.
2
(3.3.3)
if
i
=i
j
and
N.+1
N. +1
~
2
:I.'
foranyi, i
t'
=
l, ... ,
:I. 1
=1,2, ... , b; j,
l
i/j/ ; k, k '
N
if
1
= j'
and t
~ t'
{ orj ~ jl
i ~ i
1
=1, ... , p; t =1, ... , N.. ,
:l.J
= 1,2, ... , c.
jl
Proof
Under ~b' the N.: orderings of (~i11' .. ~, ~ipN. ) are equallY
~
:l.p
likely, and hence,
N. (N. +1)
:I.
:I.
2N.
N.+1
:I.
=-2-
:I.
.
(k).
(k / ) belong to the same vector .Biji,'
Also, s:l.nce R
and
R
ijt
ijt
(k) , . (k / )
1
P Nit (k) (k / )
Rij.t IPb ) = if." t~l h~l Ri thRith
~(Rij.t'
:I.
and the resl:\.lt is true ifk and k ' are either equa.l or not.
81
If i fi/~ the .r~ndom matrices Eiand ~il are independent, so
f or MY' j , j',
t,
~d
i/, k
k'.·
If
R.. 'n.. and --;I..
R. 'j~n,
are different because either j
.
"""~Jo'f,I
'"'I
and i,
J 1. ' ,
indevendent
J
jl or j
=
butlJoth vectQrs be10n~ to the same block, they are not
and
~y +ong~r
(k / )
N
(k){k')
E(I1ij~ Rij/i,1 I~b)
it
P
=
=
t~l h~l
~
N~t(
•. t;~l.:~:iRi~:h'
-Ri~~ Ri~~)]
h=1
. (N. -1)
N.
t
t=.L
~
.1
=
(k) (k') Rith
E(Riji, IRith )N. i
.
pNit
.
(k)
p
Nit
~
(k/)
:N.{N.-1)[(t~1
h~l Rith)(t~.1 h~l Rith )
"
~.;L
The last equality
f0~lQWS
from the fact that, for any k,
The test will be based 04 the totals
{j
=
.i>
\k=l,
,
,
The first and se<;:Q.Q.d moments of the T' are, conditional}y,
s
jl
= 1,
j
(
(k)(k' )
COV(T' j , Tb ·,
b
J
If'b) -_ . b~1
.
~-
k ;:: lJ
(kk' )
N,' ,(0 .. ,N, - N, ., )~i
~J
JJ
~
~J'
'
(3.3.6)
where
For each k,
and
for j :;: 1, 2, ~ .. , l' and any pair (k, k').
If !b is defined as in (1.2.4.1), it can be, seen that the rank of
its covariance matr:tx Yb is at most c(P-l)"Yb can be written as
V1-
_u
. where
«
b.
;::
J.' ~l
-,
'n, , ., )) ®
'I~JJ
«6Y,'..(kk' ) ) )
~
J'
(3.3.8)
83
Let 211* be the matrix obtained from
and column.
b..
J.·~l «Tj.J.J
. "J .
"'*» 0
th~ v~riance~covariance matrix
If
by amittiilg, the last row
Then,
V1...*
=
~
is
1k
. (kk')
«~.J.
»
of !b*.
(71 ijj ,4f.-» and «~~kk'») are non-singular for each 1,
is positive definite and so is Yb*.
Tawski's theorem can be applied to the matrix 21 * and it can be
i
co.tlcluded that:
"If for any block i, «&y'~'»
is non-singular and
J.
the matrix ]i* cannot be transformed into a matrix
of the form
by the same permutation of the rows and columns,
where
~
and
~
are square matrices and
2
consists
of zeros only, then, V * is non-singular. 11
b
'rha condition can also be proved to be sufficient.
frequ~ncies
N
ij
are the same for each block, then
.
.-< (71 ijj ,»
and
=
«71 jj ,»
If the
84
(3.3.11)
If the rank of Yb is c(p-1), the test statrstic is taken to be
(3.3.12)
where C is a (p-1)c x pc matrix of the form
1'01
.
C
= diag
1'01
(D, •.• , D)
"'"
(c) "'"
and D is a [(p-1)x p]matrix of linearly independent contrasts.
1'01
.
ia4efined as
£ in
If D
"'"
(2.2.2:14), Wb takes the form
and when the number of non.. com~ared sets is S > 1,
a
.
~ S~l[!b(S)
Wb
.. E(!b(S) IPb)J '£~El )C£(s)Zo(s)£ts )J'"'l,g(s )C~b(S)
(,.,.14)
a£t~~ ~~al~~b~~,
(that
i.m'l;tli~~ th~t :Nj
.. ~
Wb -
it o~n b~ found th~t, if th~ r&nk of !b is (p-1)c,
c!
-kk l
~.L ~l N ~
> ,,) J
P
j~l
tT~;)
,
..
E(T~~)IPb)J[T~r)
Nj
..
.
E(T~1')IPb~J
»
(:5.,.1')
where
~
-kk.' is the general element of the inverse of the matrix
«~ ~~kk')).
i
J.
It has to be recalled that, in this case as well as in the other
two mUltivariate cases, if either the design is connected and the rank
of!b is r < (p~l)c or if there are S (> 1) non-compared sets of
treatments and the rank is less than (p-S)c, the value of the statistic
Wb is no longer invariant, under different choices of £(or D(s) 's).
The next theorem states the distribution of the test statistic
Under the hypothesis H and assumptions (3.12), (3.13) and A , if
o
3
i)
kk.'
«P
.. ,»
b JJ
b... co
ii)
The Ni'S are uniformly bounded
iii)
Ifb goes to infinity with b,
.
. 1J.m
=
)
«pJJ(kk.'
.. ,
»
has rank c(p-l)
j
then the ;permut;.ation distribution of W
b as defined in (2.3.3 ..12) is
asympotically the distribution ofax2 variable with c(pel) d.f.
The proof is given in theorems 7.1. 3 and 7.1. 4 of Chapt.er 7.
The theorem extends to the case in which the rank of «p~:» is
JJ
r.
From the previous theorem, follows that the asymptotic
permutation test is given by the rule:
Reject Ho if W ~
b
2
xr,a'
86
where r is the rank of
!b~ld x~,~
is such that
(O<~<l).
3.4
Case MUIV
Let us define the aligned (vector) yields and errors by
1
where
Y.
"'~
== -N
.
~
.
""
,
1,
" " "
,
== 1,
" ""
,
== 1,
:p Nij
·f;1 ~f;l Y.. ~
J -. XI- "'~J XI
==
and
......
with
1
N..
P
~J
e. == -N ·L:1 ~L:1 e .. ~
"'~
i J==. XI=:· "'~JXI
The cummulative distribution function of
€.
N~
== (€. 11'
"'~.
... ,
.
€.
"!l..
IN
il
' ... , ~pl' ... ,
.-....
€.
N
"!l..p ip
)
remains the same under permutation of its N columns, because of
i
assumption Ay
It can be written that
"
#I
•
,
... ,
". e,
SO that the hY]?othesis H
o
reduces to testing the interchangeability
of
~'11'
... ,
~. .
under shift alternatives.
J'iN
,. .... , "'~p
Z. N .
1.
. 1
~.
1p
)
Let
. .., ,
.. . ,
. .- ,
,,,,
i = 1,
j = .1,
1,
~
::
k
= I,
b
p
~
C
~nd
(1)
rill'
.r. ==
N.I.
(2)
rill'
·.- .,
•
t
• ,
(.1)
(1)
r 11N ' r i21' ,.
il
(2)
(2)
l'ilNil,ri21'
(c)
• • e..-"
... ,
(1)
(1)
r'~ 2Ni2 "."r.lP l' .(2)
(2)
ri2Ni2,,··,ripl'
(c)
r ilNi,/r121'
.- .. ,
= 1, , .. ,
For each variable, the ranks in tpe blocks
th~
r
(1)
ipN.
lP
(2)
r.lpN.
lp
b),
ar~
r
(c)
'
ipN.
. lp
(3,4,5)
a subset of N.1
firs 1;; ;N natl.+ral numbers,
The test is bast::d on intra.. . block
!i'
... ,
... ,
(1
numbers out of
.. ,
. ,. .
., .
· .-.
(c)
rill'
... ,
:Perm~tation
ThenUJ:r).ber qf configurations der;i.ved :from a
of the columns of
giv~n
SE;!t ofb blocks
h
b
.11 (N.).',
1=1
:)...
each configuration
b,av~ng
Let us denote th:l,s
(conditionally) the same ;prQbab1li ty,
un~form
probability distribution over the
b
~Ill N.,'
.,L i".
J,
88
equSj,lly likely
~eltii2ations
by Pb •
:p
Under Ph'
Nit
(k)
(k)tt:1 ) :: ttl h~l .l'ith _ .. (k)
E· ( r14tU-b
.
N.~
- ri
·tJ
(;.. 4.6)
t
if
i = i'
j = j'
t = AI
for any i, i
l
= 1,
2, .•• , b; j,
A' :: 1, ... , Ni/j/ ; k, k'
= 11
.1
'
= 1,
2, ""
.•• , Pi t :: 1, ..• , Nij ,
c.
The proofs are similar to the ones of the expressions (3.3.2) a..p.d
(3.';.3) in the case MUIl, since in both models 5\ is originated py the
invarianae of the matrix of ranks under permutation of its columns.
But in this case,
is not equal to
oppositelyth~
other,
89
Slince the set of ranks in each roW' of the i.:!ilib.1oCk ts not any-longer
t4e set of the first Ni
n~tura1
numbers •
1,: . ~ ., :p
.(.. j- '"
1
(3.4.8)
k "'.1, .•• , c .•
Condit~ona.l1y,
(;3.4.10)
where
(3.4.11 )
For each k,
t
r,r(k) ",N(N+1)
j=.l
bj
(3.4.12 )
2
and
j
\lk '.k"',l'
\
2,
'" 1,
as a consequence, the rank of the covariance matrix of
is a.t most
(~"T.l)Q
there are s(> .1)
p).,
,
, c.
Zb'
(3.4.13)
.
(1. 2. 4.1),
if the desig;n b connected and is at most (p-S)c if
non~compared
sets of treatments.
90
(3.4.14)
with «11 ..
,,»
, J.JJ
defined as in (3.3.9)·
Similarly to (3.3.10), the covariance matrix;' of !b* can be
writte,n as
The sufficient condition (Section 3.3, page 83) based on
Tawski's theorem for the
non~singularity of ~b*
applies to this case
as well.
(3.4.16)
If the ra,nk of
!b
is c(p-.1), the ,test statistic is
where ,..,C is defined as in (3.2.13).
For~as
(3.3.~3),
(3.3.14) and (3.3.15) apply to this case after
,.obviDUs"adt:\.ptations.Also, comments on the selection of ,..,C when the
r~
of Ybis r < (p-S)c (8
:It
1) apply.
The next theorem gives the distribution
of W.
.
b
Under the hypothesis H and assumptions (3.1.2),
o
if
91
i) .
(U'
.
kk'
lim ((P'h'j')) = «P.~;)) has rank c(p.. l)
0"'00
"'J
JJ
ii)
The
iii)
N~'S
lim
b ... CO
are uniformly bounded
~ ~ h f 0 then,
' as defined in (3.:4.17) is
the permut;ation distribution of
B,symptot;I.c~lly the d:Lstri1;lution ofaX2 with c(p.. l) d.!.
The proof i~ given in theorems 7. .1.5 and 7. .l.6·of Chapter 7.
The theot'em can be extended to the case of
$e~$ of treatments and rank: of
Yb
sf> 1)
non-compared
is '(p-S)c,and to the situatio,q in
which there are S(__ 1) non-c.ompared sets and rarik
l"..1Ii:i
.(p-B)c;· Of
co\U'se, thede:f'tnition of W has to be cbanged accordingly.,
b
The
a~ymptotic
permutation test fOF Ho
~sgiven
by ·the rule,
Accep't' F! if w., .X~· .
o
'b< r,a
wh~re r is the rank of Y:q and X2
.-
r,a.
is defined by the equation·
(0 < a < 1j.
92
4.
MULTIVARIATE MIXED MODEL WITH SEVERAL (EVENTUALLY
ONE OR ZERO) RESULTS PER TREATMENT AND
INCOMPLETE RESPONSES - CASE lMUII
4.l
In90mplete
Introduction
mult~response
designs are characterized by the
fact that not all the variates are measured on each experimental unit.
Such a situation arises when it is eit0er physically impossible or
otqerwise inadv!sable or inconvenient to measure each response on each
\Ulit.
Even in the pare,m.etric theory, the problem of handling of multi-
variate data with observl:j.tions missing onSQmeor all the variables
under
~tudy
In the
pre~ented
has only been solved in a very limited number of situations.
non-par~etric
framework almost no results have been
on the subject.
I~cluded
in this chapter is a discussion of a permutation-rank
test for t'he hypothesis of no treatment effects in an incomplete mu.Ltiresponse mixed model.
discusse~ ~~
Th~
~he
experiment is of the kind that was
Chapter 3, Section 3.3.
set of coorqinates is divided in two groups:
interest and
th~
c.lasSificat~qn,
one of primary
other of secondary importance, and aocording to this
a variate-wise design is superimposed pn the
complete layout.
The
de~pmination
"primary interest" is used in a technical sense,
and it il;l applied to that group of variables that is measured in all
ex~erimental ~its.
The reason for doing it can be either that these
variables are actually more :i,mportant than the other variables, or,
93
that other pX'act~ca1 situations (cost, facilities at hand) are taken
into consideratiop.
Withqut; .l.os·s of genera.li ty, it can be ass1,1med that the variables
of primary interest are the ones identified by the set of superscripts
(1, 2, .•. ~ q}.
The model and tpe test are given in the next section.
4.2
'rha Model and
The Test
i
Af3 in the usual Iilituat;ion of several j;ria.l.s per treatment, it is
as/iiurned that
thei~ s1,1bject is submitted to
N.
4
t~:i.als.
n~
(fi;
:I
r:
j
N .
iJ
Thetria.is afe divided intq two groups:
one consisting of
N:j.) tria.l.l:! and tqe other consisting of the remaining (N
i
~ n ~
i
ones,
For the;::
c
t#,a~s
in the first grQ)lp, the responses are vectors ()f
OOm;po4~ntsw:p.1*e~h<;:lres:ponses
of the tr;La.ls in the second grol.lp
have only q cqord;i.nates (thE) ones that correspond to the var;i.ab.les of
.less impor:t;a,nce) .
.The n:umber
Uj:
:l,.s fixed, but whicb n
i
tri~ls
(out of the N ) are
i
gotng tp be com.plete ano. whioh are not, i;3 determip.ed at ra.p.dom.
'the seJ,eotion of the Ai trials is ma.de iJ;l. a way that gives to
EUl.fs.et of n
i
trials O'J.lt of the N , eqwa.l ehance of having a
i
OOIll:plete respQJ;l.se.
volved intben
i
The seJection disregards which treatments are in-
OOIP.I1lete trials.
For a given treatment, all the N
ij
trials can 'be oom:\?l,.et;e, or only part of them, or none.
The symbol n .'! rep,resentfJ the number of complete trials that
i
cOrrespond to
treat~ent
varies from zero to N ..
iJ
f1~ed.
The S,ymbol
i~ ~sed
to
1ndic~te
j in block i.
n .. is a random v&riab1e
l,J
~hat
On the other hand, N..
is cpnsiqered to be
l,J
the set of indexes of the tria1@ with complete
:re~poAses;
s
different
~orms.
n:i
The symbol
b
(k)
J
~nd~Ga~es tne nwmber qf blockS in which the k~ variable of t~e ~
tf~at~en~
:is
measureq~
Th~
outcome of the experiment can be
represen~ed
by a set of i.p.coP'lple1:ie matrices of the form .
21 = (1i' ,1l~
... ,
, r
Y'11\1
,....~.J.,J.lIil
'Y
""i 2..l'
... ,
... , ""'J.p
Y. 1"'" Y. N ]
,."",l,Pip
(i = .:1., ... , b)
~Uch
that for each j, nij out of the N..
l,.:J
vec~or~
95
h~vec coprdina,t:es whi.le the other (N .. - n .. ) vectors have only
J,J
+~
CJl ~<
O}.
qnangEl
~
'I'h!plilet of q coqrdinates of primary importance does not
un
Given
C~n
be
first;
ei thaI' j or i,
~y
thaUt~ht
'block, .its entries
\Df as
st~ge consist~
a,r~sing;
f;roraa two-stage sampling process.
The
in tne random Selection of a set of n. indexes,
l
s'n. ,
;L
aut of the N inde:>\!e$ that ident,i:fy· the trials of the block.
i
The selec~ion is assumed to pe IIlt:locJ;e
qy
g~v;i.ng eq;1ialpropaoilities
'Fhe' procedwe associates wi th e;ach set
a p!U'ticu,],ar [0
(f'AAc~:llOp,.
¥:.
X1i + q ;x. {N - n 1 )]-variate9'Pll);u..lative dis'tribu.tion
i
96
';['he
pQs~~ble
cumulat~ve
dist~i~ution
functio~s
a~sociated
with the sets
{Sn
i}
are the marginals obta:\,.ned from a certain fixed (c x Ni) ..variate
c~u~ative
distribution function F. (y), y being defined as in (3.1.1),
~""
""
by int~grating on the last (c-g) va~iables of the columns whose
indexes do not belong to the corresponding sets
SIS.
n;j,
It is ~s~~ed that Fi(l) f~1fills the conditions ().1.2) and
(3.1.3) of SectiQn 3.1,
F(y)
1-,..,
1'~"
= G (y
i.,.,
-
m.) = Gi(e)
""+
~"""
(4.2.1)
wnl1re Ill. is a (c x N.)
ma,tri~ with g~neral entry
J.
,..,~
(k)
mij,e
:;::
(k) + b~k)
'1'j
~
Jt is also assumed that the N
i
2.1 :;: Xi ..
~i
i
=·1,
j
;::;;
1,
R.
;::;;
1,
· ... ,
·.. ,
·. ,
k
:c:
.1,
·.. ,
co~umns
b
p
~
c.
of the (c x Ni ) matrix
are interchangeable random va,riables.
97
(~)
FS
n
obtained
Ot' any
by
inti~gra,ting'
with reflpect t.o the las,t (c-q)-variables
group of co.).:urnns of..!i are invariant under the grot+pof trans-
formatic?lls: G such that g
gl
i
belongi~g
to
IS
th~ gro~p
G is q.efined as a pr.oduct, g
= glg2'
with
of permutations of the complete columns and
g2 be.longing to the group of :permutations of the incomp.lete ones.
The'
hy.poth~si$.
tnat i$, wl:\!lted to be tested is:
'1"
(k.);
(k)
.~
2
'", '1'
=
= '1'P(k)
The teEilt is 'bal:\ed on intrablock ra.rtking.
=
0
The ranking is made
The ranks run from 1 to N. for the
J.
first (l,.'. va.rjtates and fr.om .1 to 4
i
for t14e last (c-q).
Le1:; ",,:1
1'L be the incomplet.e ma.trix of ranks that corresponds to the
~±ncomp.lete~.m~trix ~i.'
fQ>fma
tlh~t
Given a,. matrix ~i' let. us co.q.sider all the
it Gould have taken if its entries would have been the
result of other samples ofindex:es and all the treatments would haye
bee~ ~q~ally
effective.
BecaUse- of the l;prelevance of the treatments and the random
proces~
o:.t" the firs,t. stage, the com1"llete
columns of "",J.
R. cou.ld belong
'l"
to any set of tr:Llq;ls as well as the ones:
~ctua.lly
chosen..
Also,
under H and hecause of statement .A' , above, given any set of trials,
a
5
98
~.;Ll
tqe tnat;rices that cQUld be obtained from R. by permuting the
"'~
, complete columns
the incomplete ones
~d
amon~ themselv~s
would be
equally pr(}bable.
So, given the actual value of R., there are
"'~
N.:
~
=
[ni,:(N.n.)1]
~
~
, (NiJ
n.
~
assoc~ated inoomplet~
ties.
~
As
matrices that have, under H , equal probabilio
conSequence, the
b
b
_ n. )1 ( Ni )
.rrlN.:
=',rrln.:(N.
~=
~
~~
~
~
~
n.~
potentiaJ,.
ou~comes
assooiated with the set of incomplete matrices
( .Bl , }32' ... ,
ar~,
unq,~r
H ,
o
equ~lly
~}
probable as well.
Ip this cell of the space of possible results, a uniform
p!!o'bability law ' \ can be defined (under
c~n
H
o
)' and a conditional test
beconstfuoted.
Th~
test is based on the statistics:
b
Nij
• .E ' /J.E
~:;=
l
JfI=
l
(k)
R.. /J
~JJfI
if .1
~
k :s:: q
if (q+l) :s:: k :s:: c.
j = 1, ... , p.
99
S
n ..
~J
is the set of indexes that identify the complete trials of treatment j
i,q, ttei~ block.
that b
j
i
It is considered that the N. 's and N.
,IS
::I,.J
~
are such
0 for any trea.tmElnt and that the salllple is la.rge enoug).1 so
that a.:Uthe variables of every treatment are present in at least one
'bl0o.k, of
the~xperiment, ~.~.
, it is assumed that
T~~) >
0 "IT j,k.
Unde.rP , the following results are obtained:
b
b
i~l
N.+l
(+)
= 1,
k
•.. , q
(4.2.4)
b
n. n.+l
. £1 N. . N~ (..2:..- )
~::::.
~J
.
.
~
2
k :::: q+I, ••• , c.
I
(k) .. (k l )
Cov(Tb .· , Tb · , rb )
. J
J
equals to
(4.2.5)
k :::: q+1, ••. , c; k' :::: q+1, ••• , c.
N.+1
2
(+)],
(4.2.6)
100
if
't •.• ,
'1; k'
= 1,
... , q.
if'
k
= 1,
2, ••• , '1; k'
= '1+1, ••. ,
c.
The treatment totals fulfill the relationships:
b
i~l
N (N +1)
i
i
2
I
if
1 :10 k s;
'1
(4.2.8)
b
n. (n. +1)
i~l
2
~
~
if' '1+1
.:10:
k
:10:
c.
Also,
for any j and any pair (k, k').
It follows, the,n, that:
ha$i~t
~
most,
r~k c(p~l).
If the number of non-compared sets is
(>1), the~ the rank is, at most, c(p-S).
(4.;;.10)
.101
(4.2.12)
The~atrix
!b can be written as
(4.2.13 )
wh~re
with
(CTl ijj /)) is defined as in (2.3.3.9) and the matrices
genera.lertries~s defined
by (4.2.10), (4.2.11) and (4.2..12)
respectively, have dimensions q x q, q x (c-q) and (c ... q) c (c ...
gJ.
A,lf'lo, as in (2.3.3.10), the covariance matrix of !b* ca,n be
written:
iW,O
b
Vb *
:;:
i~l(('l'lijj/*»
I~.)
--- - .
I
"",;Lqq . : ""'.;Lqc
®
. -- -
..
(
~!
"",;LqC
-
...
I~.
I "",lCC
In the same way that in the complete response case, it can he
said tl::\at if
102
I,
I
I
I
I
: §!icc
are non-singular, so is the matrix !b'
Also, a necessary and ~ufficient condition for ((~ .. "*)) to be
~JJ
non-singular is that it cannot be transformed in a matrix of the
form
by the same permutation of the rows and colunms, where! and
.s are
0 has ~ll its entries equal to zero.
""
::;: N foJ;' any block i,
sq~~+e matric~s ~d
IfN
ij
j
Z,R'
.
~
iqq
I Z~:
I
I
....~ ""~qc
- - - - - -1- - - - ..,..
I
I
I
Z~.
i
""~cc
(4.2.15)
.
If all theN.. = 1, thev.b * is positive definite if the second
~J
factor of thela'onecker product is non-singular.
Wl:1en the J;'ank; of v. is c(p-l), the test statistic for· testing H
.
0
b
;is
(4.2.16)
where C is a (p-l) c 4 pc matrix of the form
""
103
C = diag(D, • '0' D),
""'
""'
(c) ""'
-
a,nd D is a (p-1):x: l' matrix of contrasts.
With D defined as C in
""'
""'
(2.2.2·14 )
(4.2.17)
If the number of non-compared sets is S> 1, and r(V (s»=c(p -1)
b
s
S
Wb =
-1
S~l[Tb(S) - E(Tb(s)ltPb)]/C(S)[C(s)Vb(s)C(s)] £(s,)[Tb(s)
(4.2..18 )
The rem,qrks about the situations in which the rank. of
.Yb
is
r«p",S)c ,(whereS (::=: 1) is the number of non-compared sets) made for
the analogous complete case are also valid in this case of incomplete
res;ponses.
Thefp11owi:ng theorem states the distribution of the test
statistic Who
Under the hypothesis Hand the assumption (3.2..1), (3.2. 2) and
o
i)
(kk )
" I») has rank.
lim « Pb " 1' » = «p ~~' ») and «p.j(kk
JJ
J
b... oo
JJ
c(p-l)
ii)
the N. I S are uniformly bounded
iii)
b. goes to infini ty wi th b,
:L
J
the aSVffiptot;ic (permutational) distribution of W as defined in
b
2
(4.2.17) is the distribution ofaX with c(p-1) d.f.
104
The proof is given in Chapter 7; theorems 7.1.11 and. 7.1.12.
If there are S non-compared sets, W (defined in (4.2.18)) has a
b
X2-distribution with c(p-S) d.f.
If there are
S(~
1) non-compared sets of treatments and the rank
2
of Vb is less than c(p-S), say r, still Wb is distributed as a X
variable with r degrees of freedom.
The asymptotic permutation test for H is given by the rule
o
Reject H if W ~X2
b
0
r,OI
2
Accept H if W < X ,
0
b
r,OI
2
where r is the rank of ,Y.b and Xr,OI is such that
4.3 A Numerical Example
Some records of the memorizing of sonnets are presented in a
paper by Gordon (1933).
The data are the scores obtained for
thirteen subjects submitted to the following trial:
Ten Shakespeare's sonnets were used as material to be memorized.
th
Tbeex}?er.imenter read sonnet "A" (Shakespeare IS 5r) aloud once.
SUbject then wrote down all that he could recall.
The
The experimenter
read the same sonnet through again, and the subject, without referring
to his firS!t trial, wrote all that he cou.ld recall.
This procedure
continued until tbe sonnet had been heard five times and five
reproductions had been written.
These results were scored' and the
105
~ra,des
(of only five out of the ten sonnets) are shown on the following
page,
'Xhe multivariate model MUIl can be superimposed on these data,
under which each score is represented by the equation
y~~)
=
~J
where
b~k)
~
the sonnet,
(k)
b ~k) + 'T (k) + e ..
j
~
~J
(
= 1,
•
fJ
•
,
= 1,
•
•
til
,
= 1,
· .. ,
is the subject effect associated with the
'T~k)
J
is the sonnet effect (in the
is tb,e error term.
k~
U)
5
,
5
~
recall of
~~~)
recall) and
~J
The test statistic given by (3.3.12) was computed
£or the example and the result obtained was W
b
= 1092,56
which
indicates that the effects of the sonnets are significantly distinct.
The same data have been modified to illustrate the test
proposed in this chapter.
Some treatments (sonnets) were omitted in some blocks according
to a fixed treatment-wise design.
Afterwards, random experiments were
conducted in order to select the incomplete trials.
The blanks obtained by this procedure are shown in the table by
the shadowed areas.
The total scores for each sonnet and trial are shown in the
table on page .107.
The number
~n
parenthesis express the number of blocks that has
been ad<:J,ed to give the tota.l in each cell.
The test statistic (4.2.16)
was cO.\'llputed for the incomplete data, and the result was Wb
=
937.59,
a highly significant result, as it was the case with the. complete design.
"Treat-
A
ment
SUbject
Hey
. TRIAIS
I II III IV
I
c
B
V
I
TRIAIS
I II III IV
'·1
V·
D
TRIAIS.
I II III IV
V
I
TRIAIS
I II III IV
E
V
I
TRLAIS
I 11111 IV.
V
I 5 15 10 27 50 I 1520 34 37
DUI16 14
Ma
Gr
Wy
Sc
Wa
I 826
Mc
I
I)
26
25 4(3 70
r /o/.Y5IIW /lJf5/ NYJ
6 25
Sm
46 61 72
25 36
Ke
Ed
I 8 19 13 19 21112 10 13 23 31 11}15 27 :50 36 120 32 37 44 53 114 24 37 44 53
Go
III
Mi'
I 8 9 22 39 60 117 4149 77 79 150 "81 103 108 108 '25 43
21 20 25 44 1 6 19 24 31 41 122 37
26
45
56 127 31 46 50 62 130 33'47 51 63
60 73
87 III 4053 76
82
f-'
o
0\
., '
, "
..
,
.
107
Sonnet
Trial
".
A
B
C
D
E
I
11. 0(9)
18.0(9)
22.0(9)
29·0(9)
25.0(9)
II
.14.0(9)
15.5(9)
23.5(9)
25.0(9) .
27.0(9)
III
11. 5(9)
15·5(9)
21. 0(9)
28.5(9)
28.5(9)
IV
9.5(5)
8.5(5)
5.0(4)
6.0(3)
4.0 (3 )
V
8.0(5)
6.0(5)
5.0(4)
6.0(3 )
5·0(3 )
108
5.
ON FURTHER APPLICATIONS OF MIXED MODELS, SUMMARY AND
SUGGESTIONS FOR FUTURE RESEARCH
5..1 Introdu.ction
The first section of the present chapter discusses the
application of mixed models to the analysis of experimental situations
other than the random block design type.
The following section
presents a summary of the principal points of this research and some
·suggestions for further research are made at the end.
5.2 Some Further Applications of Mixed Models
In the previous chapters, new mixed models have been proposed
that are useful for studying the outcomes of several types of
experiments.
This section is devoted to some comments about how the existing
models can be adapted to make them suitable for the study of some
experimental situations other than random blocks.
In applying rank tests, the ranking can be app.lied to the
original observations as well as to functions of them.
If the
original model inc.1udes nuisance parameters, it may be possible "to
clean" the model by a transformation of the basic variables.
If this
is the case, Care has to be put in the interpretation of the error
terms.
th~
Even if tQe original errors were assumed to be independent,
new ones will not, in general, be so.
deal of help in this kind of application.
Mixed models offer a great
Their feasibility comes
from tQe fact that they do not necessarily assume independence of the
error terms associated with the same block.
109
As an illustration of these principles, two experimental
situations and their models will be discussed in the next paragraphs.
5.2.1 Test of the Interaction Terms in a Factorial Design Laid in
Corrw1ete Blocks with Unegual Number of Observations in the Cells
Let us consider the case of a factor A with P levels and another
B with q levels.
Let Yij.R,h represent the yield in the h
th
trial of the combination
Qf the levels (j,.t),(j being a level of A and i- a level of B), in the
;i.th block.
The following model is assumed to describe the data:
Yij.R,h = b.J.
where b.,
J.
+
and
'r.
J
'r
j + 0 i- + Yji- + eiji-h
i
= 1,
j
= 1,
i-
I,
.... ,
., .. ,
... ,
h
= 1,
... ,
b
P
q
Niji-
stand for the effects of the block, factor A and B
o~
JCI
respectivelyj and Y.
,
to be tested.
~
JJCI
,represent the interaction effects that are wanted
The e's are the error terms and it is assumed that the
vectors
=
e!
.....J.
(e· l1,
J. .
e' llN
, ... , eJ.·pnl' ... , e. N
)
J . . . . 11
':L
J.pq •
J.pq
J.. .
are independently distributed with a continuous cumulative distribution
function G.•
J.
One restriction is imposed on the cell frequencies: Niji- = Niith
for each block and level of B. That means that the i
block has a
pattern that can be represented as follows:
110
K
1
2
2
Yill .
Yi21
(Nil)
(Nil)
Yi 2.1
Yi22
(N )
i2
...
.. .
. ..
.. .
...
Y'1. l q
Y':L 2q
(N. )
:Lq
...
Y.:Lpq
(N
q
i2
)
(N1q )
where
Y.lJ..t,I
'A
...
1
P
Y.:Lp1
...
(Nil)
YiP2
(N
i2
)
(N. )
:Lq
stands for the mean of the observations of the (j~) cell
and Nit is the frequency i.n all the cells of the same row IIt ll .'
The hypothesis to be tested is
H'
o'
~gainst alternatives H : ,...,
Y
l
! ,...,o.
Let us consider the transformed variables, Z.
'A'
:LJ..t,I
Zij.t
h
Yi'
were
J'
::::
Y. .t
:LJ
-.
and Y
A.
1. • ..t,I
Y.l.j.
Y.:L' t + Y.:L'
. ht e d means
are th e unwe:Lg
row and tth co.1umn respectively; Y.
1.'
vallles of the matrix (Y
ijt
i
::::
1,
j
::::
1,
t
::::
.1,
0
·.. ,
·.. ,
·.. ,
b
P
CJ.
f the·Y-.' . A I S:Ln
•
th e J
.th
l.J..t,I
is the unweighted mean of the pq
)), associated with the i th block.
111
The Zij1,'S are free from the nuisance parameters b,
~f
the
hOld,
'l"
and (), and
restr~ctions
th~y
can be expressed as:
where
Eij1,
=
-
-
eij1,' - e ij . - ei.1, + e i ··
t;Uld thee's are unweighted mean!;!.
The model assumes that the joint distribution of
{f:lij1,h: j :;:: l~ ... , p, 1, = 1, .•. , q, h = 1, ... , Nij1,}
is invariant
un~er
the group of permutation!;! of the
su,'bscripts of the e's.
At the same time, the joint distribution of the E'S is invariant
under the group of permutations generated by interchanging. the "j"
suoscripts of the E'S, for each block i.
As the suoscripts "j" interchange, so do the vectors
E!. = (E.l.J.'1' E.,
2' ... , E.,.) .
l.J
l.Jq
.....;LJ
.112
The model can be restated in terms of the
ZI S
as follows:
Let each block to be formed by p vectors
(Z'l'
Z, 2' •.. , ""1.p
Z, )
""1.. ",,:I,
where
Z', , = (Z.1.J.'1' Z~J'2·'
..• , Z"1.Jq )
""1.J
~
and the Zls are such that
with
y',
J
Let the
E. IS
",,1.
= (y '1'
J
... , y, ) •
Jq
have joint cumulative distribution function that is
symmetric in its p arguments (~'1'
",,1..
... ,
The hypothesis H can be written
o
y, = 0
""J
""
for
j = 1, ... , p.
The test statistic to be used for testing H depends on further
o
assumpti.ons on the distribution of the errors and the pattern of
frequencies.
i)
If additivity of the blocks is assumed to hold
and the same pattern o£ frequencies is used in
each block, the mixed model MUIV can be used
to describe the situation and the corresponding
.113
statistic can be used for the test of the
nUllit~
of the interactions.
i~)
If additivity of the blocks does not hold and/or
the pattern of frequencies varies with the blocks,
mixed model MUIl is the appropriate one and so is
the related test statistic for the test of H •
o
J
•
One Analysis of Covanance-Type of Problem with Unegual Number
of Ob@ervationsPer Cell
A fictitious
Suppo~e
ex~ple
will help to illustrate the problem.
that a study is performed about the effect on the
success in learning produced by a change in some condition related to
an
the metl;lod of teaching and,f"or this purpose,
ducted on twelve individuals.
experiment is con-
Each individual is submitted to a
series of trials and the number of right answers in each trial is
recorded.
is changed.
until
In the midqle of the experiment, the condition under study
The e:ICperiment continues under the modified condition
~other
series of tests have been performed by the same
..
following model is considered to be appropriate for the situation:
Yij..e :::: b.~ + T.J + (3i(X i i.
-
x) + eiji.
(:
::::
1,
::::
1,
::::
1,
• •• J
· .. ,
·., ,
b
::::
12
2
P
N..
~J
114
where b. represents the effect of the individual,
l
the condition,
X
jt
~.
J
is the effect of
represents the serial number of the
the average increment in score per trial due to
tria~
le~rning
and
~i
is
under either
conditions.
For the purpose of the study, b. and
l
~.x.n
l JXi
are nuisance
para~
meters that are wanted to be eliminated.
Let Zijt be the "adjusted" value of Yijt •
Z.. n
lJXi
= Y.. n ~ ~. (x. n
lJXi
l
JXi
-x),
with
p N..
lJ
(X
·l:l
jt
J=. t~l
=
~i
p Nij
j~l t~l (X jt
"
and
~
x.J )Y.lJ't
- )2
- x.
J
N..
lJ
t~l X jt
=
x.J
whose location parameter is
N ..
lJ
(~.
J
'
+ b.), so that
l
The new error term Eij is a linear combination of the previous
e's.
If the Y's are assumed to be diagonally symmetric, ai test of
~j
=0
can be performed by computing the test statistic (2.2.2.13) on
the adjusted values "Z".
115
5.3
Summary
Most of the current tests based on ranks are only applicable to
simple experimental designs.
Looking for a wider application of these
procedures, this work has been addressed to the search for tests that
are suitable for the analysis of experiments having incomplete and/or
un;palanced
desi~ns"
with univariate as well as multivariate responses.
In relation to multivariate data, a test has been developed that
allows for
mi~sing
observations in some variables of the responses.
7.3.1 Underlf!ng Designs and Models
I i .
_
i
The legitimate useo! rank tests requires that a null hypothesis
b~
specified and a model for certain functions of the observations be
established such that the related null distribution of the test
statistic to be used be app.lica,ble.
~he
models that underlie the tests proposed in this thesis can be
oonsidered as extensions of the non-parametric mixed models I, II and
IV defined by Koch and Sen
(1968).
The designs to which they are
applicable can be assimilated to the designs known by the names of
"
i)
generalized randomized block design
ii)
randomized block design with various result!? per ce.l.l
iii)
incomplete randomized block design.
For mixed model experiments, each of b randomly selected sUbjects
res~onds
to each of p treatments exactly once.
The outcome of the
experiment constitutes an observation matrix composed of b vectors,
eachofl<;l,ngth p.
treatment effects.
The hypothesis of interest is the equa.lity of
·u6
In Chapter 2 (Section 2.2) non-parametric models have been extended to the situ.ation in which several (or eventually only one or more)
results are associated with each treatment, so that the vector of
observations of a given subject, call it "i", can be symbolized as
follows:
=
(Y· 11 , . ., Y' lN ' Yi21 , ... , Y. 2N '
J.. .
J.. i1
J. i2'
i
where Y.. n (j =.1, ... , p, JJ.J~
:=
:=
... ,
Y. l' ... , Y. N
J.p.
1, ... , b
J.p ip
)
(5.}.1.1)
th
1, ... , N.. ) represents the J-- obserJ.J
vation of the jth treatment; N.. is the number of observations
J.J
cQrresponding to the j~ treatment in the block and
P
N. = .l:1 N..
J.
is the
J =.
J.J
1eng~h of the i th block.
The pbrase "results per treatment" has been deliberately chosen
to be ambiguous.
It may indicate either "results of replicated trials
of a given treatment" or "replicated results of a unique trail of the
trElatment".
In the first case, the design in the background is a type
of generalized block experiment; in the second case, it is a kind of
randomized block design with several observations per cell.
In the
partiCUlar situation in which the numbers (Nij } are only ones or zeros,
the design is called incomplete b1oc~ design.
Assu.mptions analogous to the ones that originate models I and IV
have been used to define tWQ models, called mixed model DI and UIV,
117"
that can be looked at as their extensions.
(The extension of case II
is in the work by Benard and van Elteren (1953)).
Case ill can be applied to any type
of design
mentioned in the
previous paragraph, while case UIV (as well as the model underlying
the test of Benard and van E1teren) is suitable only for either
generaliz'ed randomized block designs or incomplete block designs.
Also, a new model is introduced (Section 2.3) which is especially
useful for the description of experiments with several observations
per cell, that can be thought of as another type of extension of case
II.
It has been identified as case UII .
o
The distinctive characteristics of models illI and UII
o
become
apparent writing down the covariance matrix of the vector Y. (under
",J.
H ) that is implicit in both models.
Let N.. be constant, with
o
J.J
respect to treatments and blocks, and let ......J.
Z. symbolize the covariance
matrix of ......J.
Y..
Then, for model UII;
0'"
Z.
......J.
=
pO'"
pO'"
while for model illI ,
o
2
2
2
2
pO'" ,
0'"
(JO'"
••
,
pO'"
o
,.
0
,
pO'"
•
a
$
,
0'"
2
,
2
It
2
2
2
118
0"1'
·.. ,
2
Pl 0"1'
2
P.10"1'
·... ,
0"1'
2
I:. '"
",,;I.
p<Y
pO"
2
2
,
II
••
,
· .. ,
,
In Chapter 3
2
pO"
2
2
pO"
•
•
0'
2
pO"' ,
,
·... ,
,
·.. ,
,
• a. • ,
2
pO"
th
•
,
... ,
,
2
pO"
pO"
2
2
2
0"1'
2
Pl 0"1'
II
•
III
,
(I
••
,
P10"1
2
0"1
a further extension of cases UI, UII and UIV had
been made to include multiresponses.
to the i
11 .. -
The observations corresponding
subject can be symbolized as in (5.3.1.1), but each Y
ijt
has to be interpreted as a vector of "ct! coordinates instead of as a
number.
Three models have been defined and identified by the symbols MUI,
MUII, MUIV, respectively, a denomination that makes clear the type of
univariate model to which they are related.
In Chapter 4 a model has been created in order to cope with the
'problem of incomplete responses in a multivariate general incomplete
block design.
The model has been identified by IMUIIand is an extension of
case MUIr.
In relation to model IMUII, a vector response is said to
be incomplete if it includes only a subset of the t!ct! variables that
the research takes into account.
i
th
block, has n
i
It is assumed in the model that the
complete responses and (N
i
- n ) incomplete ones.
i
The set of variables that form the incomplete responses varies with
neither. the block in the experiment nor the trial in the block.
Models UIIo and IMUII have a common characteristic that has to be
remarked.
In both cases, the actual design of the experiment has been
119
conceived
a~
a realized sample from a larger population of designs.
In case UII , the allocations of the frequencies to the treatments of
o
each block ar~'the result of random experiments and this is also the
case with the selection of the complete trials of each block in the
mixed model IMUII.
5.3.2
The Tests
The test statistics that are proposed in this work are extensions
of the ones u.3ed by Wilcoxon, Kruskal and Friedman.
The underlying
theory for these test procedures is discussed in Chatterjee and Sen
(1966), Puri and Sen (1966), Koch and Sen (1968), Sen (1968), Gerig
The tests are based on some function of the observations.
After
the experiment has been performed, the actual values taken by these
fUnctions are ranked.
Cases I and II use intrablock ranking and,in
case IV, a general ranking on the whole set of blocks is performed.
The space of all possible sets of ranks that can be obtained by
the indicated procedures is considered.
A partition of this space is
chosen in such a way that,under the null
hypothesi~a11
lying in the same cell are equally probab.le.
the points
Given an actual result,
say R , the test of the hypothesis is performed in the cell, S(R ),
o
0
to which the point belongs.
All the (conditional) tests presented in this thesis are based on
criterion statistics that are quadratic forms that can be symbolically
written as:
120
where T is a vector whose components are the sums (over the whole
'"
experiment) of the ranks that correspond to each treatment and
variable separately.
Only for one of the tests, related to
diffe~ent
model UII , there is a
o
definition for T.
In that case, T
/'OJ
'"
is the unweighted sum of the p averages of the ranks for treatments 1,
E(T) and V = V(T) are the (conditional) expectation vector
'"
'"
'"
and covariance matrix of T, C is a convenient matrix of linearly in'" '"
dependent vectors that breaks the singul.ari ty of V.
'"
Restricted to S(R ), the conditional distribution function of W
2, .•. , p.
o
can be found.
Nevertheless,' since in the case of unbalanced designs,
many parameters are involved, only asymptotic distribution of W has
been considered.
In all the cases, the resulting asymptotic
2
(conditional) laws are of the X -type, and are independent of the
p~obabilistic
structure of the cell, so they can be used also as an
unconditional approximation.
In cases UII
o
and IMUII, because of the random device that has
been used for selecting the design, the space of possible results is
enlarged.
Accordingly, given the actual value R , the cell S(R ) is
o
enlarged, associating with R
o
0
the results that would have been
produced if other patterns had been selected in the first stage.
Under H, a uniform (condi tiona.i) law is defined in the cel.i and the
o
test is carried out as in the fixed design situation.
5· 3· 3 Examples
Two numerical examples have been presented; the first is designed
as an incomplete block experiment, with several observations per cell.
121
Three mixed models (cases I, ill, illI ) have been superimposed on the
o
data and the corresponding tests statistics computed.
The most
significant results were obtained for the mixed model illI , but as the
o
number of blocks do not justify a large sample approximation, the
result has to be considered with caution.
The data of the second example refers to a psychological test
performed on thirteen subjects.
Each subject was submitted to five
different treatments once, and five variables were measured on each
trial.
j).
The design was balanced and complete,
(N ..
l.J
=1
for any i and
A mixed model MUll was assumed to be appropriate for describing
the data and the corresponding test statistic was computed.
Later on,
some treatments and variables were deleted to make the data conform
to an unequal frequency multivariate block design of the type that
can be described by a mixed model lMUII.
The new set of data was
analyzed by the corresponding method and a result highly coincidental
with that given for the complete data was found.
Two other (non-numerical) examples are given at the beginning of
this chapter to illustrate the feasibility of mixed models in the
description of different practical situations.
5.4 Suggestions for Further Research
i)
The models presented in this work can be classified
according to two criterions,
a)
!.~.,
Assumption on the distribution of the errors
in
~h€
same
block (diagonal symmetry, compound
122
symmetry, additivety, fragmented compound
symmetry).
b)
Nature of the responses (univariate, complete
and incomplete mUltivariate).
The crossed classification of the models originate
twelve cases, but only seven out of them have been
considered.
The remaining five cases need to be
developed.
ii)
Other patterns of incomplete responses can be
ceived of
an~
con~
accordingly, new mixed models intro-
duced.
iii)
The idea, of choosing incomplete designs at random
can be used in relation to other non-parametric
tests.
iv)
The power properties of the new tests has to be
studied.
Power comparisons of tests with complete
and incomplete responses has to be done.
v)
New models can be developed for experiments with a
complex design in the blocks.
123
6.
LIST OF REFERENCBS
Abbreviations of Journals
AMS
=
Annals of Mathematical Statistics
JASA
=
Journal of the Americal Statistical Association
JRSS
=
Journal of the Royal Statistics gociety
Benard, A. and Ph. Van Elteren (1953). A generalization of the method
of rn r.ankings. Indagationes, Mathematicae, Vol. 15, pp. 358-369.
Bennet, B. M.
(1962).
On multivariate sign tests.
JRSSB, 24:195-261.
Bickel, P. J. (1965). On some asymptotically non-parametric
competi tors of Hotelling I s T2. AMS 36: 160-.173.
Blumen, T.
(1958).
A new bivariate sign test.
JASA 53:448-156.
Bradley, R. A. and M. E. Terry (1952). Rank analysis of i ncomp.lete
block designs I. The method of paired comparisons. Biometrika
49:205-214.
Chatterjee, S. K.
(1966).
A bivariate sign test for location.
AMS
37 :.1771-1782.
Chatterjee, S. K. and P. K. Sen (1964). Nonparametric tests for the
bivariate two sample location problem. Calcutta Statistical
Association, Bulletin 13, pp. 18-58.
Chatterjee, S. K. and P. K. Sen (1965). Nonparametric tests for the
multi-sample multivariate .location prob.lem. Essays in Probability
and Statistical in Memory of S. N. Roy. (Ed. by Bose, et. al.),
University of North Carolina Press: 197-228.
Chernoff, H. and J. R.Savage (1958). Asymptotic normality and
efficiency of certain non-parametric test statistics. AMS 29:
972-994.
David, H. (1963).
New York.
The Method of Paired Comparisons.
Hafner, Inc.,
Davidson, R. R. and R. A. Bradley (1969). Multivariate paired comparisons: The extension of a.univariate model and associated
estimation and test procedures. Biometrika 56:81-95.
Dixon, W. J. and A. M. Mood
JASA 41:557-566.
(1946).
The statistical sign test.
124
Durbin, J. (1951). Incomplete blocks in ranking experiments.
Journal of Psychology (Statistical Section) 4:85-90.
British
Eisenhart, C. (1947). The assumptions underlying the analysis of
variance. Biometrics 3:1-24.
Fisher, R. A. (1925). Statistical Methods for Research Workers.
Oliver and Boyd, Ltd., Edimburgh.
Fisher, R. A. (1935).
Ltd., Edimburgh.
The Design of Experiments. Oliver and Boyd,
Fraser, D. A. s. (1957). Nonparametric Methods in Statistics.
Wiley and Sons, Inc., New York.
John
Friedman, M. (1937). The use of ranks to avoid the assumption of
normality implicit in the analysis of variance. JASA32:675-701.
2
Gerig, T. M. (1969). A multivariate extension of Friedman's X test.
JASA 64:.1595-.1608.
r
Gordon, K. (1933). Some records of the memorizing of sonnets.
Experimental Psychology 16:701-708.
H~jek, J.
(1969). A Course in Nonparametric Statistics.
Publ. Co., San Francisco, Calif.
I
v
Hajek, J. and F. Sidak (1967).
New York and London.
Hodges, J. L.
(1955).
Theory of Rank Tests.
A bivariate sign test.
Holden Day
Academic Press,
AMS 26:523-527.
Kempthorne, O. (1952). The Design and Analysis of Experiments.
Wiley and Sons, Inc., New York.
John
Kempthorne, o. (1955). The randomization theory of experimental
inference. JASA 50:946-967.
Kendall, M. G. (1948).
Publ. Co., London.
Rank Correlation Methods.
Charles Griffin
Koch, G.
(1969). Some aspects of the statistical analysis of "'Split
experiments in comp.1ete.1y randomized layouts. JASA 64:
485-505·
p~ot"
Koch, G. (1970). The use of non-parametric methods in the statistical
analysis of a complex split plot experiment. Biometrics 23: .
.1 05-128.
Koch, G. and P. K. Sen (1968). Some aspects of the statistical
analysis of the mixed model. Biometrics 24:27-48.
125
Kruskal, W. H. (1952). A nonparametric test for the several-sample
problem. AMS 23 :525-540.
Lehmann, E. L.
(1953).
The power of rank tests.
Lehmann, E. L. and C. Stein
parametric hypothesis.
AMS 24:23-43.
(1949). On the theory of some nonAMS 20:28-45.
Mann, H. B and D. R. Whitney (1947). On a test of whether one of
two random variables is stochastically larger than the other.
AMS 18:50-60.
Mehri:t, K. L. and P. K. Sen (1969). On a class of condi tiona1.1y
distribution free tests for interactions in factorial experiments.
AMS 40:658-664.
Neyman, J., K. Iwaszkiewicz and S. T. Kolodziejczyk (1935).
Statistical problems in agricultural experimentation. Supp.L
JRSS 2:107-154.
Pi tman, E. J. G. (1937). Significance tests which may be applied to
samples from any populations I, Supl. JRSS4:ll9-l30.
Puri, M. L. (.1970). Nonparametric techniques in statistical inference.
Cambridge University Press, Cambridge, Mass.
P~ri,
M. L. and P. K. Sen (1966). On a class of multi-sample rankorder tests. Sankhya Series A-28:353-376.
Puri, M. L. and P. K. Sen (1969). On the asymptotic theory of rank
test for experiments involving paired comparisons. AMS 21:163-173.
Puri, M. L. and P. K. Sen (1971). Nonparametric Methods in Multivariate Analysis. John Wiley and Sons, Inc., New York.
Puri, M. L. and H. D. Shane (1970). Statistical inference in incomplete block designs. Proceedings of the First International
Symposium on Nonparametric Techniques in Statistical Inference.
Indiana University, Bloomington, Indiana.
Roy, N. S. and N. S. Gnanadesikan (1959). Some contributions to /iliOVA
in bne or more dimensions: II. AMS 30:318-340.
Roy, N. S. and Cobb Whitfield (1960). Mixed models with normal error
and possibly non-normal other random effects: Part II: The
multivariate case. AMS 31:958-968.
Savage, T.~. (1962). Bibliography of Nonparametric Statistics.
Harvard University Press, Cambridge, Mass.
/
Scheffe, H.
(1952).
JASA 47:381-500.
An analysis of variance for paired comparisons.
126
,
Scheffe, H. (1959).
New York.
The Analysis of Variance.
John Wiley and Sons,
Sen, P. K. (1968). On a class of aligned rank tests in two-way
layout. AMS 39:1115-1124.
Sen, P. K. (1969). Non-parametric tests for multivariate interchangeability, Part II: The problem of MANOVA in two-way
layouts. Sankhya Series A, 31: Part 2, pp. 145-156.
Sen, P. K. and H. A. David (1968). Paired comparisons for paired
characteristics. AMS 39:200-208.
Sen, P. K. and M. L. Puri (1967). On the theory of rank order tests
for location in the multivariate one sample problem. AMS 38:
.12.16-.1228.
Shane, H. D. and M. L. Puri (1969). Rank order tests for multivariate paired comparisons. AMS 40:2101-2117.
Siegel, S. (1956). Non-Parametric Statistics for the Behavioral
Sciences. McGraw-Hill Co.~, Inc., New York.
Sverdrup, E. (1952). The limit distribution of a continuous
function of random variables. Skandinavisk Acturarietidskrift,
pp. 1-10.
Tausski, O. (1949). A recurring theorem of determinants.
Mathematical Monthly 56:672-676.
American
Terry, M. E. (1952). Some rank order tests which are most powerful
against specific parametric alternatives. AMS 23:346-366.
Wilcoxon, F. (1945). Individual comparisons by ranking methods.
Biometrics, pp. 80-83.
Wilk, M. B. (1955). The randomization analysis of a generalized
randomized block design. Biometrika 42:70-79.
Wi.lk, M. B.
models.
and O. Kempthorne (1955).
JASA 50:1144-1167.
Fixed, mixed and random
Wilks, S. S. (1952). Mathematical Statistics.
Sons, Inc., New York.
John Wiley and
127
7.
A P PEN D I X
128
7.
.APPENDIX
In many of the proofs of the previous chapter, the following
tneorem, due to Tausski, has been mentioned.
IILet ((a )) be an n x n matrix with complex
ik
elements such that:
n
Ian.. I ~ kfi'
.I: Ia.J.k I
(i
=:
1, ... , n)
with equality in at most (n-l) cases.
As~ume
further that the matrix cannot be transformed
into a matrix of the form
by the same
permu~ation
of the rows and
columns, where P and Q are square matrices
'"
'"
and 0 consists of zero only.
It follows that
. Dennit;i.on
A random vector l' = (11 " 1 ' ..• , 1 ) is said to be
2n
-n
n
qn
asymptotically q-variate normal with mean f.l n and dispe,rsion ,.",n
I: , if
a' (Y ,.",,.,,,n
a' I:
,.",
is asymptotically N(O, 1) for
f.l
n)
a
,.",n·,.",
every~'
=
(aI' a , ..• , a q ) in the
2
129
vector space X.
q
If X can have at most rank r, then the rank of the
q
asymptotic distribution is r.
7.1
Limit Theorems
Several theorems that refer to the asymptotic null distribution
of the test criteria proposed in this thesis are given in this section
of the Appendix.
Before considering each of them separately, some
remarks will be made that apply to all theorems.
Since the conditional distribution of W is constructed from the
b
actual data of the experiment, it can be symbolized by Fb(w,~) where
~
stands for the set of matrices
(R , R , ... , R} .
"".1 ",,2
;:;"0
The expression asymptotic distribution of W actually means
limit of Fb(w,~) when b goes to infinitY#
In all models, this
convergence is studied for selected types of sequences (~} that fulfill specific: conditions (for instance, the one quoted in the theorems
under
a)). As
will be seen later on, the asymptotic results do not
depend on which of the selected sequences
~
has been taken in con.
sideration, so they are also valid unconditionally.
In fact, the un-
conditional cumulative distribution function of Wb is
and the proof that Fb(w) goes to F 2(w) can be done in the same way
X
that Corollary 3 to Theorem 3.2.2 in the paper by Chatterjee and Sen
(1964 ).
130
7.1..1
Theorem
If the hypothesis H and the assumptions that correspond to case
o
MUI hold, and
lim (( Pkk'
.. , ) )
b JJ
b-+oo
i)
c (p-l)
ii)
the N.
iii)
for anyj andk,
are uniformly bounded
IS
l
limVar(Tb(~)IPb) =00,
b-+oo
then
J
the joint asymptotic permutation distribution of
(t~~):
= 1, ... , p; k = 1, ... ,
j
c}
with
(k)
t
=
bj
and
T~~~ as
T~~)
J
-
E(T~~) I~\)
~ i
()
[Var(Tb~
(7·1..1. 1)
IPb )]
defined in (3.2.8), is a pc-variate normal of rank c(p-l)
with mean vector zero and dispersion matrix
or
Proof
Let
(a ~k) :
J
j
= 1, .... , p; k
== 1,
... , c}
.131
be arbitrary constants such that
~
.L.
J=l
(k)
a"
J
=0
for each k, and bto be large enough so that
for any j andk.
Let
Setting
(k) (k' ) (kk' )
E E E E a"
a ",
PbJ· ", = b.b '
J
J
(k)
a.
b
N
ij
k k' j j '
tn
b
J
can be written as
c
:Q
k~+ j~l ~ i~l .Q,~.1
~
~bJ'
N. +1
(Rij.Q, -
-1-)
b
::; ----...;;-""'---..-------- = i~l
l~
b
with
(k)
cpa.
k~l j~l ~
Cf
~i
bj
::;
The random variables
(~i: i
are independent and
= 1,
..• , b}
~i'
132
N. .1
J.J
X
(kl )
N +1
.
[#-1 (Ri!.t - -1-)],
'
so that
(7 •.1.1.2)
Since N (and so forth, the ranks in the i th block) are bounded,
i
as well as the a~k),s, a constant M can be found such that for any
J
configuration of ranks and any i, j and k,
Then, for any i,
Igbi I ~
c
P
k~l j~l
M
1
O"bj
Ab
:JkY -:i :; Kb'
Let us call ~ the Lyapunoff quotient.
(7.1.1.:3), . it/follows' that
froPl (7.1.1. 3 ),
(7·1.1.3)
From (7·1.1.2) and
133
It is known that
lim
b.... oo
CI~
> 0
since, in the limit, Cl b is a positive definite quadratic form by i).
Also, by iii),
goes to 00 wi th b; then
lim ~
b.... oo
=0
,
what implies that the Lyapunoff condition for the asymptotic normality
of
lib
7.1.2
is fulfilled.
From this result, theorem 7.1. 1 is proved.
Theorem
Under thecondi tions stated in theorem 7.1.1, the asymptotic
permutation distribution of Wb defined in (3.2.13) is the distributio4
2
ofaX variab.le with c(p-l) d.f.
Proof
Since W can be written as a continuous function of the vector
b
~b
-_
(1)
(1). (2)
(2)
(c)
(c)
(t b1 ' •.. , t bp , t bl , .•• , t bp , ... , tbl ' ... , t
),
bP
theorem 7.1.2 follows by applying the results of Sverdrup (1952) to
the one stated in theorem 7.1.1.
7.1. 3'l'heorem
l.f the hypothesis Ho and the assumptions
that correspond to case.
.
MUIl hold} and the fo.llovring condi tions are satisfied}
i)
/
(kk )
«p
»
"
b
b... co
. JJ
lim
where
I
has
be
rank c (p-l)}
ii)
iii)
the N, 's are uniformly bounded}
J.
b'
·for'
.
lim b J :.:: h 1= 0 for some constant h}
b.-tco
then the joint asymptotic. (permutational) distribution of
j
:.:: .l}. 2} "'} p; k :.::
l} 2}
••
00
J
c}}
Dei'
with
t(k)
bj
andT~~)
T(~) - E('ll(~) Ip
bJ·
bJ
b
)
= '[Var(T~~) lo\)]ft
defined by (3,3.4)} is the one of a pc-variate normal variable,·
of rank c(p-l)} with vector zero mean and dispersion matrix
,·ritl
(kk / )
«Pbjj
»
I
or
cone
Proof.
2
FJ.' rst } not e
th ta
J.' n th1'S
.
(f.b(k, )
. caBe}
J
probability one.
> 0 for anY'J' arlcl k .,r1'ttl
•.
135
Let
ta~k\
J
j :::;1, ... , p;k:::;l, ... ,
o}
be arbitrary constants s"Uch that
for each k.
Let
Defining
~
can be written as
b
.1::
J.:::;.
1
f!.. •
-bJ.
w:i,th
,
i
:::; 1, .... , b.
The gbi's are independent random variables and under the
conditional law P ,
b
136
1
== '"7;;""
b
c
P
c
(k)
p
(k / )
a.
a'l
CJ"bj
(fbj'
k~l k/~1 j~1 j 1~1 ~ ~
Nij IX
As a consequence of condition ii), there exists a constant M such
that for any configuration of ranks and any i, j and k,
Then, for any i,
Igbi I :s;
c
k
M
.t~1 j~1 ~
CJ"bj
If
S,
1
J
8b
==
~.
is the Lyapunoff quotient, we have, because of (7.1.3.1) and
Also,
a:
(k) 2
bj
==
~.
P Nit
Ni +1 2
b.~ N..
(N.-N.
·)t~l
h~l(R'th
2)
1J
1
1J _ .
.
1
.L.
1==1
N. (N.-1)
1
1;
1€
S
b.
J
~ b. K for some K
J
f o.
1
137
can attain occurs when the Nij'S are either ones or zeros.
The second
follows from the result:
because the R's are different integers.
Even if ties exist, it can
be prpven that
N.+l 2
_J._)
2
.... 1
'" '4'
From (7.1. 3.4), it follows that if b. goes to infinity with b,
(J'~~)
2
J
goes to infinity as well (no matter the values of j and k); then
l goes to zero and Lyapunoff condition for the normality of ~ if
fulfilled.
7..1. 4 Theorem
Under the conditions stated in theorem 7.1.2, the asymptotic
permutation distribution of W defined in (3.3.10) is the distribution
b
2
ofaX -variable with c(p-l) d.f ..
Proof
As given in theorem 7.1.2.
138
7· 1. 5 Theorem
If the hypothesis H and the assumptions that correspond to case
o
MUIV hold, and
i)
(kk / )
.lim (( P ' . I
b JJ
b... oo
(kk / )
' )) where ((p
= ((Pkk
jj/
jjl )) has rank
))
c(p-l)
the N. 's are uniformly bounded
ii)
~
b,
=h f
lim ~b
iii)
b... oo
0 for some constant h,
the joint asymptotic (permutational) distribution of
(t~~):
j
= 1,
... , p, k
= 1,
... , s},
where
=
t
bj
'th T(k)
bj defined by
W~
T~~)
J
-
E(T~~) I~\)
()
J
i
[Var (Tb~ I~\) ]
(3.4.81
.
(7.1.5.1)
is the one of a pc-variate normal
variable of rank c(p-l), with mean vector zero and dispersion matrix
(kk / )
((Pbjjl ))
or
((p
(kk l
jjl
)).
Proof
Let
(a~k):
J
j
=
1, ... , p; k
=
.1, ... , c}
139
be arbitrary constants such that
for each k.
Let
If we call
~
can be written
with
~i
i
::;
:::: 1, ... , b.
The gbi's are independent random variables and conditionally to r ,
b
E(gbi1rb) :::: 0
(k)
.1
c
c
12
p
: :; b:" k~l k/~l j~l j/~l
b
(k / )
I
(kk/)
(k) ~Nij/(Ojj/Ni-Nij)6ti
'
(Jbj (Jbj
a.
a'
so that
b
i~l Var(gbi1rb) :::: 1.
.140
The Lyapunoff condition for the asymptotic normality of
lim
b-+CO
S,
t1b
is
= 0,
where
Let us consider the inequality
Because of relation (7 ..1. 5.4 )., it is seen that the proof of
(7.1.5.3) can be made in two parts, by showing that in the set of all
possible configuration of ranks
for some positive constant
~
(for any i), and
b)
for some positive constant 5 (for any j and k).
Statement a) follows almost immediately since the maximum value
'of'
141
can be taken to be less than N.
a consequence}
As
where m is a fixed constant depending on the
a~k),s
and the N..
~J
IS.
Besides this} by assumption ii) there exists a constant} say n}
such that N.
~
~
nV i} what implies that N ~ nb.
gether with (7.1.5.5) proves statement
This inequality to-
a).
According to (3.4.10)}
(k)2
Since the ranks are all different numbers} ~bj
is Positi~e (with
probability one).
For a fixed N.}
the two minimum values that
~
N.. (N. -N .. )
~~
(~ 0)
~
Ni -1)
~J
can attain are O. (that happens when either
N..
= 0 or N..
.
~J~J
and 1 (that occurs if eitherN~J'
...
=1
or N..
~J
= N.
~
- 1).
=
N.)
~
Let us put 1
in place of N.. in the expression of
~J
if N.. > O.
;LJ
Then} accordingly}
and
J
142
N' t
l
p
(k)2
min O"b j
b
:i::
min i ~l
(k)
t~l h~h (ri th
On the other hand, because the aligment is made around the mean, in
each block at least one deviation lies to each side of the origin.
Let r(k) be the rank of the origin if the origin is ,in the set
a
y~k)):
l
i
= 1, ... , b; j
= .1, ... ,
p; i-
=
1, ... , N.. }.
lJ
If it is not so, let r(k) be the average of the ranks of the two
a
inbetween which the origin stands.
P
Nit
Then, for any variable,
(k) _ r-(k))2
t ~.l h~.l(rl·th i
:i::'1
---~N~-----N
i
i
()
()
• (k
ffiln
r. th - r k )2
t,h
a
l
The first inequality follows
in each block at
least one
imp.lies that at
least one
The second in-
equality follows from assumption ii).
It also follows from the definition of r(k) that since in each
a
block at least one rank. .lies to the right and one .lies to the left of
z· min(;t~(k)
ieS pS t,h itO.
and
'" r(ki»2 ~ 2(l2 + 22 + ••• +
0
I
th~s S~ 1~ of the order qf b;'
The other
thr~e
mayor may ,not b€llo.p.g
or odd) ,lead to
(tna~ ar~se
oases
tha set::; of
~o
l?!tm~lar
from the
ra~s
and b j can be e;1.ther E)ven
I3tatements.
It has been assumed (a~sv.m:ptiQn· iii) that ~ J 8iges .lin~arly to '
inffp,ity witp. b.
~7,.l.5.8), ;it
lJ;Ipen, oombining
follows
.
(k)2, .E'"',,
rn:lr,n fJbj
~ 1€Sb'~n
,.,
f'pr som~ oonst~t
do~s
not
1t
f4>X!
(7..),.• 5.6), (7.1.5.7) and
th~t
"
.
reSl,l~ts
N.
P
:1. t· ($.).
"'(k) 2
ttl htl (r~th .. 1'1 ) "
,I
T
N~i"
J.
1
I
I
~ 0
6, and statement'Q) :1,:;: prove<l,
Note that
de~end
on whioh
~
fpllow~
from the
~rOQf'bf stateme.n~s· a)-and
apd
j
,~e
3
b
involved in
some cqnstlplt K,and this s/;lows tha,t
lim l:;; O.
b'" c:x;>
~~~).
p) that
t~e
:pr90f
144
Then ~ is asymptotically normally distributed and theorem
7.1. 5' is proved.
7. 1. 6 Theorem
Under the assumptions stated" in' theorem' 7~.l.5,· theasymptbtic
permutation distribution of W defined in
b
(3.4.16) is the distribution
2
ofaX variation with (p-l) d.f.
Proof
As in theorem 7.1.2.
7·1. 7 Thyorem
Under the model described in Section 2.3, and the hypothesis H,
o
and the assumptions
i)
lim
b-+ co
«Pb J..J ,))
=
«p J..J ,))
where
«PJ..J ,)).
has rank
(p-l)
ii)
iii)
the N. 's are uniformly bounded
J.
2
lim O':b.
b-+co
=
COfor any j (j
=
1, •.• , p),
J
the joint asymptotic (permutational) distribution of
(t
bj
:
j
=
1, 2, •.• ,
p} ,
with
and T has been defined in (2.3.7), is a p-variate normal distribution
bj
of rank (p-l) with mean vector zero and dispersion matrix
or
Proof
Let b be large enough so that
Cf
2
>
bj
0
for any j and let
{ a. : j = 1, 2, •.. , :p}
J
be arbitrary constants such that
:P
.2::
J=.1
Define
Letting
lib
can be wri tten as
a.
J
= O.
146
with
N.
pa.
L.: ....J.
l
P
= j=l O"bj
The
~i I S
N. +1
(_l_))
2
are independent random variables and
N. N. +1
N. N.+l
a. a. ,
(R .. - 2.(-l2--))(R."
- pl(_l2-))
L: L: .....J. ....J...:. E _J.::.lJ....._ _p~._~--,-_..;;J.~J~'_-=-_-=-_
Ab
j j ' O"j O"jl
:f\dd;1..ngover the blocks,
Let us consider the Lyapunoff
Since the N.
J.
'-8
quo~ient
(and so the ranks in the blocks) are bounded, as
Wella.s are the a.'s, a constant M can be found such that for any conJ
figuration of ranks and any block and treatment
This illlP.lies that
Then,
By assumption i)
is finite; and this result, together with assumptions iii), implies
that
lim
b-.c:o
I1,
=
O.
The-theorem is provep..
7~ 1. 8 Theorem
,
IJ
Under the conditions sta.ted
~n
theorem 7.1.7, the asymptotic
:,perm.utation distribution of W
b defined in (2.).15) is
thedistri1;l~'\iJi,oA
ofaX2.vari-a'ble with (p..l) d. f.
Proof
Same as in theorem 7.1.2.
7.1.2
Theorem
Under the model described in Section
and the
2.4 and the hypqthests Ho'
as~umptions
i)
lim «Pb'j'»
J
b-+co
=
«P.,,»
JJ
- where «p",»
- JJ
(p"'l)
\ .
has
. ra,nk-
148
ii)
the N.
iii)
Var(TbRj I~\) >
~
IS
are uniformly bounded
€
> 0 for any
j .(j, .. 1, •••:,p).
the joint asymptotic (permutational) distribution of
with
,
where T
has been defined in (2.3.8),is a p-variate normal
bRj
distribution with mean vector zero and dispersion matrix
or
((POOl))'
JJ
7. 1. 10 Theorem
Under the conditions stated in theorem 7.1. 8, the asymptotic,
permutation distribution of W defined in {2.3.16) is the,distribution
bR
2
ofaX -variable with (p-l) d.f.
'Proof
The proofs of the last theorems follow the pattern of those
. theorems 7.1.6 and 7.1.7, and are omitted.
of
7.1.11 Theorem
When the hypothesis H is true, under the model described in
o
Chapter 4 (with conditions 4.2.1,4.2.2 and A holding), and if
5
(kk / )
.
i)
.lJ.m (( P .. I
b JJ
b.... ro
»
. (kk
= ((p
jj /
/ ).
(kk
mth ((p jjl '
» of rank
c(p-l)
the N. 's are uniformly bounded
ii)
l
b.
iii)
lim ~b = h ~ 0 for some constant h,
b.... ro
the joint asymptotic (permutational) distribution of
(t~~):
j
= 1,
... , p; k
= 1,
2, ... , c} ,
with
T~~)
defined in (4.2.8), is a pc-variate normal of rank c(p-l) with
mean vector zero and dispersion matrix
or
Proof
Let
r1. a(.k) .
J
J' =
1 , 2,
••• ,
p., k
="1
• •• ,
c'}
.
.150
be arbitrary constants such that
for each k.
Let
P
·~l
~1
(k)
(k)
t~ \
a.J
-bJ.
c
12
(k)(k)
[Var(E:~l j~l a j tbj)J
_ J=
ITht -
c
k·
=.
i
Calling
~
can be written as the sum of b variables:
with
(k)
n. n.+l
1
+ .~
~ ~ [ ~ R(k) - N. . N~ (...2:- ) ] t:=.,.. •
j=l k=q+l:TkJ hESn .. ij~
2
~J.
J/j'
(J' •
J
~J
It can be shown that under the two stage law
b
i~.l var(~i IPb ) = 1.
~
1\\
·b
151
Because of assumption ii), there exists a constant M such that for
any configuration of ranks and any i, j and k, satisfy the two inequali ties:
and
la~k)ln
S~
J
.KJE n..
1.J
R"
1.J.KJn
Then, since the variances
- N..
1.J
~~~)
are positive w.p.l., it can be
written that
Let us designate the Lyapunoff quotient with!.
From (7.1.11.2)
and(7.1.11.3), it follows that
The statement about the positiveness
If k
=q
+ 1, ..• , c
(N.-N .. ) P
(k)2 c n1.' n1..+1 2
(k) 2 __ b~ .N..
1.J 1. 1.J
~
[R
() ]
a:
4J.
~
4J
-bj
i=l
N.1. (N.1. -1) t=1 hESn'1. . i th
N.
2
. 1.
t
b
N.. (N.-N .. )
1.J 1. 1.J
1.=1 N (N -1)
i i
~.~
152
b~k)
for some constant m f O.
J
was defined on page
93 •
For
k ::; 1, 2, ••. , q,
cr~~)
2
coincides with the variance defined in (3.3.6) with j ::; j' for
the complete mu1tiresponse design.
If
k ::; (q+1), ..• ,
cr~~)
0,
2
goes to infinity with b, since by the random selection of the
comI.>lete colwnns of the design b ~k) goes to infini ty wi th b., in
J
J
probabili ty, and b. goes to infinity with b by assumption iii).
J
(k)2
The proof that crbj
goes to infinity with b for the variables
k ::; 1, 2, .•• , q
follows by the same arguments given when proving the same statement
about the variance of the complete multiresponse case.
Then for any j
and k,
::;00.
As the limit of 8 is finite (see ,assumption i)),
b
lim S- ::; 0
b-+oo
and the theorem is proven.
153
7.1.12
Theorem
Under the conditions stated in theorem 7.1.11, the asymptotic
permutation distribution of W (given in 4.2.17) is the distribution
b
2
ofaX -variate with c(p-l) d.f.
The theorems of this appendix could have been stated in a more
general form considering that the rank of
(kk /)
«p jjl
)),
,w:a.s anumher r< c(p-l).
In such cases, the arbitrary constants should have been chosen in
the subspace X spanned by
r
where C is an r x pc matrix such that evC/. is ndn-singular.
~
7.2
Computation of Terms that Enter in the Formulas for the
Expected Values, Variances and Covariances
of Section 2.3
In this part of the Appendix, some conditional expectations that
are key terms in the development of the expressions 2.3.10, 2.3.11,
2.3.13, and 2.3.14 will be calculated.
It has to be noticed that according to the model of Section 2.3,
for any block i, N.. is a positive random variable with values in the
~J
set
"
.. ,
154
and such that the event
(N..
l,J
= ~)
u
occurs if and only if the fixed jth treatment belongs to the random
i
set Su of n., i.e.,
~
--
i
= ~)
U
P(N..
l.J
Therefore
Nij
Ui
:Pu
=-
:P
Nij
i
.
EL~lR4J'
n) = u~
.E1 EL.E1 R.. n IN.. =M )P(N.. ;::~)
~
~ ~
~=
~J~
~J
u
~J
u
and
E(R .. n IN.. =~)
~J~
J:.. ~. ,
~J
u
it is the average of all the ranks associated with the set Si of
u
treatments.
Let us call
then,
N. .
U.
~J
E( n.E1 R. . n)
~=
~J~
=
R
it·
~ .~
.E 1VJ. (
tlu
.E . -.-.) -.
U=.1 u t-ES l :pl~:P
u
u u
u.
=
~
.E
U=.1
Since
is a partition of the p treatments involved in the experiment,
so,
155
Furthermore,
N..
lJ
E(~~lRij~)
2
N..
lJ 2
= E(~§lRij~)
U.
=
N..
l
lJ 2
.
i
~lE(,,~lR
.. n IN.. =W)P(N.. =M )
u=.
~=.
lJ~
lJ u
lJ U
~ 2
h~lRith P~
=
irv?
Pu u
=
U
i
N
K.
.j
= U=.
~lE( ~=
J~l
lJ
'.
ii.
n/~lR
. . nR . . n, IN.. =W)P(N.. =M. )
= lJ~
lJ~
lJ U
lJ U
~.
·~I=~I
~
U.
ii
.,
iPu
= ~lM (M -l)E(R . . nR. 'nl N.. =M )pu- u u
. lJ~ lJ~
lJ u
l
Ui
=
.
l
.
. l
~lM (M-l) ~ i
u u
. t€S
U-.
U
•
=
P
Adding the results obtained for the two expectations on the right side
of equation
(7.2.1)
,......-----------.,.
Also, ifij
+
ui
j/,
U·
i:
w=.:L
u'
!
i Ni · Nijl
EE( r;J . ~
.
u~l:e=l £/=1 lJ£'
!
u
IN =M
ij I£,I ij u'
l
R .. R
i
~
[ r; l. h~lR.thJ
teS
. 1
u
N.
,,=~)P(N
.. =M1.,
U.
lJ U
lJ
N.
'I=~~
u:
lJ
i
2
-
~
2
r;. (h~lR·th)
teSl.
i
i
pu Pu- l
1
u
---+
P
p
Then,
N. (N. +1) 2
[
11]
2
p
2
- t~l Rit ·
p(p-l)
7.3 Computation of Terms
tha~
•
Enter in the Formulas for
the Expected Values, Variances and Covariances
4.2
of Section
Most of these terms involve expectations of sums of products of
ranks such that one factor is related to one variable, say k, and the
other factor to a second variable k
'
(k and k l
can be equal).
The
157
procedures that have to be used depend on which group of variables (the
group of primary mterest or that of secondary importance) k and k'
belong to.
Three cases have to be distinguished.
Case A:
k andk' belong to the set
{q + 1, ... , c}
of secondary interest.
Case B:
k and k' belong to the set
(1, ... , q}
of primary interest.
Case C:
k belongs to the set
{l, ... , q}
and k' to
(q+l, ... , c}.
In finding the expectations, the variables n .. play an important
lJ
role.
From the sampling scheme used for selecting
the n.1 complete
.
t.ria.ls, it follows that the n .. 's have the hypergeometric distribution,
lJ
i·~· ,
N..
Pen .. =n? .)
lJ lJ
Ni ·
= f'f\r J
~".,n.
1
1
(n?lJ.)
:r
N.-N..
( oJ)( 1 iSJ)
n .. n.-n ..
lJ
l lJ
N.
( 1)
n.
1
fi /,
If i
n ~J
.. and n'~ l J
. , are independent but ii" i' = i
(and j
.
.' .
f
j/),
. the joint probability distribution is given as follows:
P(n .. =n o.. ,n.. =n 0
.. )
, :::;
~J
~J
~J
~J
.. The symbols
S
n ..
~J
and
S
N.. -n ..
~J
wi.l.l represent the set of
1,
~J
indexes in block "i" and treatment "j" for
which the trials are complete and incomplete respectively,
Also,
and
S
N.. -n ..
~J
are disjoint
set~
~J
such that
sus
n ij
Nij-nij
=
S ,
Nij
the set of all the trials for the combination (i,j).
The expectations are computed under the law P ,
b
159
7.3.1 Case A
=
Nij
(k)
Nij
Z-o ECZS
R.
'Aln
..
)f
.. )
n .. XlE
lJXI
lJ
N . n. (nlJ
lJ
n. .
l l
lJ
=
n. +1
Nij
N..
Z-o n .. (2- )f
nlJ
.. -
lJ
2
lJ
N .,n.
l
l
(n .. ).
lJ
After rearrangement of factors and a change of variables, the
last expression can be written as:
Since for any parameters
it fo.l.lows that
By similar ar uments, i t can be proved that:
p
t~l
(k) (k / )
ECx.:
XlE S
R .. AR ..
n ..
lJ
;
Let j 1= j/, then
A
lJXI lJXI
)
=
160
::::;
Nij
Nijl ['
(k))(
(k/)
N.1J. ,N·· I
L=O . L =OE \OLS
R .. o ol'i:.
R.. ';o,)ln .. ,n .. ,]f
1J (n .. ,n.
)
n. .
n .. I
XlE
1JXI XI E S
,1J"XI
1J 1J
N., n . 1 J 1 j/
1J
1J
n. .
n .. I
1 1
1J
1J
Nij
Nijl
= n ..L_- O n .. L_OELL
ES
, -.
1J
XI
1J
Once that n
ij
and n
n ij
ij
o,L
XI E S
(k) (k / )
Nij,Nijl
R. 'oR. ',olln .. ,n .. ,]fN
(n .. ,n. "),
1JXI·1J XI
1J 1J'
.,n.
l.J 1J
n ij ,
1 1
, are fixed, the expectation can be obtained by
using the same arguments used in the case of the analogous unconditional
expectation for the complete model MUll.
Nij
Nij I n ..n .. ,
n. (n. +1) 2
.... _
1J 1J .[ (1 1
)
[ ....._
n.":'-O n,":'/-O n.(n.-l)
2
1J;
1J
1 1
Then,
(k) (k')
P
t~lh~S
X f
nit
X
1
N..N.. ,
=1~
1J
n. (n. +1) 2
[( 1 1
N.1 N.-.l)
1
Then,
]
Nij,Nijl
(n .. ,n.. ,)
N .,n.
1J 1J
1
The double sum adds to one.
RithRith
2
)_
.
161
•
Also
n..
Rij.e)(t~S
n ..
1.J
Nij .
= n .•I:-oE[(nI:
XlE s
1.J
=
+
Nij
I:_OECI:
n •• XI€ S
1.J.
n
··
€s
n.
nI:nlI:
XI, XI
(k/)
(k)
1.JXI
Rijt )] =
1.J
XlE
N..
R .. n )In .. ]fN1. J
R.. n)CI:
n1.' '
J
(k / ) . .
(k)
E[ (.e~s
sn.
.
1.J
1.JXI
1.J
•
1.
... )
,n.1. (n1.J
(k) (k / )
R.1.JJCI
'nR"n
+
1.JXI
ij
(k) (k / )
Nij
R.. nR .. n1In .. )f
(n .. ).
N
1.JJCI 1.JXI
1.J
.• , n.
1.J
.
1.
1.J
1.
For fixed n . . , it can be shown, as in the cornp.lete case, that,
1.J
(k) .(k / ) t
E(R1.JXI
.. nR1.JXI
.. ) =
. . nn. 1.J
p
t·
~-.·l ·h.I:
·. ·
ES
nit
R (k)R. (kl )
ith ith
n.1.
and
(k),(k/)
E(R. 'nR"nlln .. )
1.JJCI 1.JJCI
1.J
1 . n. (n. +1) 2
= n.(n.-1)[(
1.
1.
1.
21.
)
162
So, the expectation above is equal to:
NiJ· n.. p
(k) (k i )
n .. (n .. -1) n. (n. +1) 2
~ r~ ~
~
R R
+ . ~~ lJ
[( l
l
)
n .. =01. n . t=1 h€S
ith ith
n. n. -1)
2
lJ
l
nit
l
~
P
N'
- t~.l hT~
R.(k)R.(k/)]}f l j
(
)
~
th th
N n n ..
nit l
l
i' i lJ
n.l
p
= N••
lJ -N
. t~l
-. h~S
€
nit
/
R(k) (k ) N -1
.thR.th
ij
N.. -1
l
l
~ f lJ
(n .. )
n
n .. =0 N.-l,n.-l lJ
i
lJ
l
l
l
+
N.. (N .. -1) n. (n.+l) '2
lJ. lJ
[( l 1 )
N. (N. -1)
2
l
l
(k) (k / )
P
- t~1 hts
RithR i th ] X
nit
N.. -2 N
-2
ij
X n ..~-Of
- n.- 2 ,n.-"2(n..
lJ ).
lJ
l
l
"
lJ
Then,
(k)R(k / )
itn ith
R
n
+
N.. (N.. -1) n. (n.+l) 2
lJ lJ . [( l l
)
N (N
i
i
-1)
2
i
•
163
7.3.2 Case B
~
(k)
~_
'nln.. )
E(R.lJk
if the trial is in
8
(N.. -n .. )
lJ
lJ
and
if the trial is in
•
80,
By
R~~)
t=l hEO(N. _
) lth
.
it nit
N - n. .
l =J .
l
l
similar arguments, it can be proved that,
164
Let j
Fjf,
theh
.
Let us call I, II, III and IV the four terms on the right hand of
the previous equality
I
==
Nij
(k) (k f )
Nij,Nij f
Z=OECZS.
/J,ZS
R. 'JlR. '~Jlfln .. n.. ,)fN
(n.. ,n. 'f)'
n.~J.
""E
~J ~J
."ni
~J
~J
nij "" E nij , ~J"" ~J ""
~
E(R. 'JlR. 'Jl,ln.. ,n. j ,) ==
~J""J.J""~J
~
( t~l
=
,
""'
h€S
' (k) ) (~""'
Rith t~l h€S
~t
~
(k I ))
~ ""'
Rith - t~l h€S
~t
~t
R(k )R (k f )
ith ith
(The argument by which this inequality is obtained is similar to the
one used in the complete case, except that in that ca::;e ni
=
Ni , so that
165
x
N..
Nij N .. ,
. N'J'
J.
J.J ,
X
I:I: -0 n ..n .. ,f
nJ. J (n .. ,n .. ,) ::::
n ij - O n ij ,J.J J.J Ni' i
J.J J.J
Then,
For fixed values of n ij , n ij , (j, j' :::: 1, •.• , P),Riji, and Rij,.t, with
and
iU'e independent variables since complete and incomplete trials permute
each other independently.
Then,
166
N ·
Ni j l
i
(k)(k / )
II ==
L:~
L: _ E( nL:
n I 'E._
R. . nR • ., n I n.. -l'. n. . I) X
n .. -0 n . . ,-0 JfJ Es
.It.l. lJ
lJ
lJ
n .. JfJ €S (N.. I -no . I) lJ JfJ lJ JfJ
lJ
lJ
lJ
I
with
==
So,
II ==
N..N.. ,
l~ . l J
P
L:
L:
N. N.-l) t==l hES
l
l
(k) P
L:
L:
ith t==l hES(N
R
nit
(k/)
) ith
R
it-nit
X
or,
III
c~
be solved as term II, while IV is the
case in which both trials (jiJ)
complete columns.
~d
It follows that
~alogou$
of I for the
(j/iJI) belong to the set of in-
167
•
Adding the
E
If j
~our
terms,
NiJ"R(k) Ni. J"/ (k/)
. )(
)J
£,=1 ij£, £,~=lRij / R-'
[( E
= j/,
N.. N.",
= N.1.~~J.
[(
N. -.1)
1.
1.
N. (N.+l) 2
1. 1.
)
2
it follows by similar arguments than the ones used
for the analogous expectation in case A that
+
N.. (N.. -.1)
.1. J .. 1.~
N" (N" -1)
1.
7.3.3
1.
N. (Ni+l) 2
[( 1.
.•..•
)
2
Case C
Let j 1= j', then
.
1
168
Let us call I and II the two terms on the right hand of the equality
above,
x
N.. ,N.. ,
N..N.. , n. (n.+1) P
'~J
~J (
) _ ~~. ~J [~ ~
f N. ,no
' nij,nij , - N.N. -1)
2
t~l htS
~
~
~
'N . .
II
= n,.~~o
~J
_
N.
.,
P-oECCZ
""E S (N
n ~J
.. ,-
N..N,.,
~~ ~J
- N.N,-l) ni
~
~
~
n.;+l
(oJ.)
.. -n~J
,. )
~J
R~~~)C,Zs
~J""
"" E
nit
Ri th
R~~:~,)ln';iJn
.. ,Jx
~J ""
~J
..10,1,
,(k)
P
~ t~l iJ~S(N
n';J"
oJ.
(k)
it-nit
)Rith
Then, from I and II,
NiJ· .(k)
,(k')
N..N."
n.+1
N.+1
"R
)( "
)J - ~J ~J C' (~ )N (~ )E[ (iJ;t1
ijiJ iJ'€'Sn." Rij' t'
- Ni (Ni -1) ni -2-' i ~
~J
I'
•
The first term on the right hand is:
_
I -
N. .
~J
E_OE[CES
n..
.!:IE
~J
n
(k)
R.. nHnL
s
~J.!:I.!:IE
ij
I
(k )
n ij
N
ij
_
R.. n )In.. ]f
(n.. )N• ,no ~J
~JXI
~J
~
~
and the second term is,
_
II -
Nij
(k)
.(k/).
Nij
L_OE[(.E S
R.. n)C,L
R.. n,)!n .. ]f
(n .)
s
Ni,n4 i J
n ~J
.. i.e (N .. -n .. ) ~JXI XI E n. . ~J""
~J
...
~J
~J
~J
.,•
Then,
cd
NORTH CAROLINA STATE UNIVERSITY
INSTITUTE OF STATISTICS
(Mimeo Series available at 1¢ per page)
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mixture problem. Ph.D. Thesis - 1971
BURROWS, PETER. Consequences of selection in finite populations. Ph.D. Thesis July 1971
CAMBANIAS, STAMATIS. The equivalence at singularity of stochastic processes and
other measures they induce on L.
KRIER, NICOLAS. Linear representation of derived Shear's planes.
LINDGREN, GEORG. Discrete wave analysis of continuous stochastic processes.
DUTTA, KALYAN. On the asymtotic properties of some robust estimators in certain
multivariate stationary autoregressive processes.
KOCH, GARY G., H. DENNIS TOLLEY and WILLIAM D. JOHNSON. A linear
models approach to the analysis of survival and multi-dimensional contingency tables.
LEADBETTER, M. R. Point processes generated by level crossings.
CHAKRAVARTI, M. Confidence set for the ratio of means of two normal distributions
when the ratio of variances is unknown.
CHAKRAVARTI, 1. l\1. On generalized inverses in a linear associative algebra and
their applications in the analysis of a class of designs.
LACHENBRUCH, PETER A and JAN PERRY. Testing equality of proportion of
success of several correlated binomial variates.
SEHGAL, JHE MOHAN. Indices of fertility derived from data on length of birth intervals using different ascertainment plans. ph.D. Thesis 1971
BROGAN, D. R. and J. SEDRANSK. Pooling correlated observations; Bayesian and
preliminary test of significance approaches.
KOCH, GARY, WILLIAM D. JOHNSON, and H. DENNIS TOLLEY. An application
of linear models to analyze categorical data pertaining to the relationship betWeen survival and extent of disease.
OPPENHEIMER, LEONARD. Estimation of a mixture of exponentials for complete
and censored samples. (Thesis)
DALEY, D. J. Some problems in the theory of Jl<?int processes.
FABER, JOOP A J. Precision of sampling by dots for proportions of land use classes.
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ARROW, R. J., F. J. GOULD and S. M. HOWE. A general saddle-point result for constrained optimizations. 1971
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EVANS, J. T. and F. J. GOULD. On using equality constrained algorithms for inequality constrained problems.
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selection of variables problem in general linear models.
HOWE, STEPHEN. New conditions for exactness of a simple penalty function.
DOWLING, T. A A q-analog of the partition lattice. November, 1971.
ORDOUKHANI, NASSER. A stochastic model for plant spacing. 1971. Ph.D. thesis
MASON, ROBERT E. Non-linear estimation by parabolic approximation. Ph.D. thesis
RAJPUT, BALRAM S. Gaussian measures on L,. spaces.
CAMBANIS, STAMATIS and B. S. RAJPUT. Some zero-one laws for Gaussian
processes.
JOHNSON, N. L. Inferences on sample size: Sequences of samples.
BAKER, CHARLES R. Zero-one laws for Gaussian measures on Banach space.
KOCH, GARY G. and B. G. GREENBERG. The growth curve model approach to
the statistical analysis of large data files.
KOCH, GARY G. The use of non-parametric methods in the statistical analysis of the
the two-period change-over design.
O'REILLY, FEDERICO JORGE. On goodness-of-fit tests based on Rao-Blackwell distribution function estimators. Ph.D. Thesis
GOULD, S. J. Proximate linear programming: An experimental study of a modified
simplex algorithm for solving linear programs with inexact data.
KOCH, GARY G., PETER B. IMREY and DONALD W. REINFURT. Linear model
analysis of categorical data with incomplete response vectors.
BASU, SUJIT KUMAR. Improved density version of the central limit theorem.
CLEVELAND, WILLIAM S. Estimation of parameters in distributed lag econometric
models.