UI-LMP RANK TESTS AGAINST RESTRICTED ALTERNATIVES
by
Michael N, B'oyd
Department of Biostatistics
University of North Carolina, at Chapel Hill
Institute of Statistics Mimeo Series No. 1406
J'u1y 1982
UI~LMP
RANK TESTS
AGAINST RESTRICTED ALTERNATIVES
by
Michael N. Boyd
--
A Dissertation submitted to the faculty of The
University of North Carolina at Chapel Hill in
partial fulfillment of the requirements for the
degree of Doctor of Philosophy in the Department
of Biostatistics
Chapel Hill
1982
Approved by:
Advisor
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V,
MICHAEL N. BOYD. UI-LMP Rank Tests Against Restricted Alternatives
(Under the direction of PRANAB KUMAR SEN.)
v
In their book, Theory of Rank Tests, Hajek and Sidak (1967) formulate a general theorem on locally most powerful (LMP) rank tests for
the one and two-sample location problems as well as the problem of simple
regression.
However, when testing against ordered alternatives one
encounters a family of mUlti-parameter probability densities and no
general LMP rank test exists.
It is desirable to extend this local
optimality property to the multi-parameter setup.
Following Sen (1981), the union-intersection (UI) principle
of
Roy (1953) is used to extend the theory of LMP rank tests to a broad
class of mUlti-parameter alternatives.
The test against ordered alter-
natives in the one-way layout is derived as a special case. The results
of this test correspond to those of Chacko (1963) when logistic scores
are used.
This implies that Chacko's test is optimal when the under-
lying distribution of the data is the logistic distribution.
The problem of extending the UI-LMP rank test method to tests
against ordered alternatives in the complete blocks design is considered.
Three procedures are examined.
tests which use within block rankings.
The first procedure concerns
The second and third procedures
concern tests which use weighted and aligned rankings, respectively. In
addition, methods for the analysis of covariance, and analysis of the
two factor experiment are presented.
ACKNOWLEDGEMENTS
Having Dr. P.K. Sen as my advisor has been the most rewarding
experience of my academic career.
I also wish to thank the other mem-
bers of my committee, Dr. Dana Quade, Dr. Craig Turnbull, Dr. Bert
Kaplan, and Dr. Roger Grimson.
They have indeed been very helpful and
supportive.
During my time as a graduate student in the Department of
Biostatistics, I have been supported by grant T32
~lli
15131 from the
Center for Epidemiologic Studies, National Institute of Mental Health.
Special thanks are due to my wife Beth for making these past four
years in Chapel Hill the happiest years of my life.
Thanks also to our
son Kevin who arrived just in time to help me finish my computer work.
Finally, I wish to express sincere thanks to Jackie O'Neal who
did an excellent job of typing this manuscript.
iii
CONTENTS
ACKNOWLEDGEMENTS
1.
INTRODUCTION AND LITERATURE REVIEW
1.1
Introduction.
·1
1.2
Tests Against Ordered Alternatives for the One-Way
Layout . . . . . . . . . . . . . . . . . . . .
·1
Tests Against Ordered Alternatives for the Two~Way
Layout . . . . .
.... . . . . . . .
.6
1.4
Nonparametric Methods for Higher Order Layouts.
.11
1.5
Relevant Methods.
· 14
1.6
Proposed Research
.16
1.3
2.
OPTI~~L
RANK TESTS FOR ORDERED ALTERNATIVES IN A GENERAL LINEAR
MODEL
--
2.1
Introduction.
.18
2.2
Theory.
.19
2.3
Example
.24
2.4
Asymptotic Nonnull Distrubtion of TN'
.29
2.5
Local Asymptotic Power and Efficiency
.34
2.6
An Application to Mental Health Data.
2.7
An Example Involving Two Factors . .
·44
.47
2.8
Testing for Order in an Analysis of Covariance.
.51
') 0
....
Example
.53
-'
3.
2.10 Testing for Order in a 2 Factor Experiment.
.56
2.11 Summary
.57
OPTI~~L
. . . .
. ....
RANK TESTS FOR ORDERED ALTERNATIVES IN A COMPLETE BLOCKS
DESIGN
.59
3.1
Introduction.
3.2
Within Block Rankings
59
3.3
Example . . . . . . .
.66
3.4
Asymptotic Power and Effeciency Considerations.
69
3.5
Weighted Rankings
70
3.6
Example . . . . .
77
3.7
Aligned Rankings.
.80
iv
..
.
Example
3.9
Testing for Order in an Analysis of Covariance.
3.10 The 2 Factor Experiment Replicated in
Complete Blocks.
3.11 Summary
4.
.85
n
.
.88
.89
MONTE-CARLO RESULTS
4.1
Introduction.
.90
4.2
Simulated Experiments for the One-Way Layout.
.90
4.3
Simulated Experiments for the Complete Blocks
Design
97
Summary
103
4.4
5.
.82
3.8
RECOMMENDATIONS FOR FUTURE RESEARCH
105
REFERENCES
v
CHAPTER 1
Introduction and Literature Review
Introduction
1.1
When data are collected for the purpose of comparing different
treatment effects, a priori the investigator may hypothesize a specific ordering of these effects.
In this case a procedure which tests
the hypothesis of equal treatment effects against the alternative that
not all treatment effects are equal is not appropriate.
For instance,
consider a clinical trial in which different groups of patients receive
'e
varying amounts of a drug.
The alternative hypothesis may be that the
effect of the drug increases with the amount.
A standard analysis of
variance F-test will not be sensitive to the ordered alternative.
A
more powerful test which takes this ordering into account is needed.
Specific procedures depend on the design of the analysis.
cally the problem is to test the null hypothesis,
were
h
H = ]Jl
a
]J.
1
~
...
.
IS
Basi-
HO=]Jl = ... =]Jk
.
.
t h e 1. th treatment mean, agaInst
t h e or d ere d a 1ternatIve
~]Jk
where the subscripts
1
to
k
are arranged according
to the prespecified alternative, and at least one inequality is strict.
1.2
Tests Against Ordered Alternatives for the One-Way Layout
An
early attempt to solve the problem of testing against ordered
alternatives in the one-way layout was made by Jonckheere (1954a). Let
k
be the number of samples.
The test statistic,
J =
L
i <j
u.. ,
1J
is
2
the sum
of the
k(k-l)/2
comparing treatments
i
Mann-Whitney statistics
and
j.
If
J
for
U .. , i<j,
I)
is too large, the hypothesis
of equal treatment effects is rejected in favor of the ordered alternative.
Tables exist for evaluating the significance of
(1971)).
J
(See Odeh
However, these tables are limited and may not exist for a
given number of treatments and sample sizes.
For large samples.
J
is approximately normally distributed, the approximation being better
for large values of
the
i
th
min(nl, ... ,n )
k
where
n.
1
is the sample size for
sample (see Hollander and Wolfe (1973)).
The exact test is
powerful when the treatment means are monotonically increasing.
This
test has been generalized by Patel and Hoel (1973) to the case where
observations are subject to arbitrary right censorship.
Another approach to solving this problem was taken by Bartholomew
(1961) who makes use of parametric theory.
derived under the assumption of normality to
ordered means.
One such test, the
-2
X
Likelihood ratio tests are
te~t
the equality of
test, is derived in section
3.2 of Barlow, Bartholomew, Bremner, and Brunk (1972) .
Let
x.
1
be the
. th
1
sample mean.
Population variances,
associated with each sample are assumed to be known.
lihood estimates of
k
k
under
J.J.
1
I w.x./ I w., and
i=l 1 1 i=l 1
~ ~, i = 1, ... ,k where j.1~
~ =
1
1
A
2
o. ,
1
The maximum like-
•
H are ~. = J.J, for all i, where
O
1
-2
w. = n.o.
Under H the m.l.e. 's are
III
a
is the isutonic regression of
x ..
1
Under
2
Ho all populations have the same variance a. The likelihood ratio
-2
k
2 2
test rejects HO for large values of X = I n. (0~-0) /0. A
k
i=l 1 1
method for calculating the 0~ involves amalgamating some of the ori1
ginal
k
groups into subsets and is given in Barlow et al. (1972).
e-
3
Under
k
L pC-Q"k)pr(X;_l ~ c), c > O. and
1=2
where P(t,k) is the probability that the isotonic
-2
~
H Pr (X
O
k
-2
Pr (X = 0) = P (l, k),
k
c) =
~*
lJ.
regression function
takes exactly
1
is a chi-squared random variable with
P(1,k)
t
t - 1
degrees of freedom.
multivariate normal population with mean vector
known variance matrix
i
Values
are tabulated in Barlow et al. (1972).
Now consider a multivariate approach to this problem.
HI: 8
2
X_
t l
distinct values, and
~
A,
0 (i=l •... ,k)
Given a
8' = (8 , ... ,8 )
and
k
l
H : 8 = 0 vs
O i
on the basis of the sample x(l), ... ,X(n).
it is desired to test
l
Kudo (1963) proposes the test statistic
(X-8)}
;(2 = n(X'A-lX- min (X_8)'A8.20
2
1
X is the sample mean vector, Calculation of X
where
involves finding the minimum of the quadratic form
subject to the condition
8.
1
~
0
for all
1.
l
(X-8)'A- (X-8)
-2
The distribution of
X
is discussed in Kudo (1963).
The results of Kudo may be considered as a generalization of those
of Bartholomew.
H
Under
a
should
the differences
Y
be positive and can be estimated by the successive differences
X2~Xl'"
Y- )
k l
"Y -
k l
=
xk-x _l ,
k
The variance-covariance matrix of
can be calculated when the common variance of the
known.
-2
X
duced by Chacko (1963), is calculated as follows.
observation in the
r..
1)
=
(Y ",·,
l
samples is
Then the procedure of Kudo may be applied.
A nonparametric version of Bartholomew's
for all
k
I
i.
i
th
sample.
test statistic, introLet
x ..
1)
i = I , ... ,k, .i = I •... , n 1.
. th
be the
where
)
n.1 = n
The observations are ranked from smallest to largest where
is the rank of
x..
1)
among all the
are now cal culated from the ranks.
N observations.
The
The form of the statistic is
O*'s
4
-2
Xrank = (12/N(N+l))In.1 (P~1 _ (N+l)/2)2.
Note that in the parametric case the estimates
pooling of adjacent within sample means.
o~
1
depend on some
The estimates for the nonpara-
metric case are different in that they are calculated from ranks
obtained in an overall
1
to
N ranking.
Hence, these ranks depend
on all the observations, not just separate ranks within each sample.
Under
-2
Xrank
H the limiting distribution of
O
-2
X
the null hypothesis distribution of Bartholomew's
et al. (1972)).
-2
The
is the same as
(see Barlow
test is appropriate when the alternative
X
states that the treatment means are monotonically increasing.
-2
X
rank
test does not depend on the assumption of normality.
-2
Xrank
treatment means do not increase linearly, the
The
When
test will be
more powerful than Jonckheere I s test (Chacko (1963)).
The above methods have included specific tests against ordered
alternatives.
Let
tests.
combined
t.;(i,j) = 1
v
i th and J.th
~(i,j) = O.
v
Let
the combined
wise
Puri (1965) has described a family of
. th
1
if the
v
n . T ~~)
J 1J
1J
.th
1J
I
I
i=1 j=i+l
n.n.h .. ,
1 J 1J
X.
1
v
th
observation and otherwise
smallest observation from
x.
observation and other-
J
n.+n.
where
n.+n.
= 1 I J E (i ,j) n (i, j )
v=l
v
v
k
rank
smallest observation from the
samp l
es '1S an
J
satisfying certain restrictions.
k-I
th
samp l
es '1S an
n(i,j) = -1
and
v
n(i,j) = o.
v
Denote he. = T~~) + T~~)
1J
and
if the
k-sample
n.l~~):::
1
I
1 1J
v-I
where the
E(i,j)
v
J
are constants
Then the test statistic is
V=
5
When
test.
E(i,j) = v/(n.+n.)
v
. 1 )
the test is equivalent to Jonckheere's
Asymptotic normality and the asymptotic distribution of this
family of statistics are discussed in Puri (1965).
Tryon and Hettmansperger (1973) follow the approach of Hogg (1965)
to extend Puri's family of statistics. Weighting coefficients are
k
k-l
included to form T = I
I a .. T .. where T..1J is any Puri staN i=l j=i+l 1) 1J
tistic and a .. ;:O: O.
1)
Consider the subclass of statistics
(TlZ,TZ3, ... ,Tk_lk)'
The
authors prove that for each statistic TN there exists an equivalent
v
k
k-l
statistic T* where T* = \ a T
and
a
= I
I a .. ,
N
L
v vv+l
N
v
i=1 j=v+l 1J
v=1
v = 1,Z, ... ,k-l. Thus computation for k - 1 statistics rather than
(k )
Z
is required.
TN - TN ~> 0
of
T*
N
The statistics are equivalent in the sense that
and the Pitman efficienty
when standardized under
with respect to
is one.
In another paper, Shorack (1967) has derived the results of
Bartholomew and Chacko as special cases of a theorem which represents
a generalization of Bartholomew's result.
Chacko's test is also
extended to the case of unequal sample sizes.
Finally, three more methods will be briefly mentioned. Hogg (1965)
considers a parametric procedure similar to that of Jonckheere (1954a).
The test statistic is
I
i<j
(x.-x.),
J
and the differences may be
1
weighted.
Another method is similar to that of Sen (1968).
Here the test
statistic is based on correlating treatment means with predicted treatment rankings.
Lastly, Wallenstein (1980) suggests calculating a test
statistic based on summing over
k - 1 adjacent pairs of two-sample
one-sided Smirnoff-type statistics.
6
1.3
Tests Against Ordered Alternatives for the Two-Way Layout
This section is concerned with four types of procedures for the
two-way layout.
The procedures include within block rankings, aligned
ranks, the tests of Hollander (1967) and Doksum (1967), and weighted
rankings.
The earliest paper was again written by Jonckheere (1954b).
sider a complete blocks design with
k
treatments and
n
Con-
blocks.
Observations are ranked within blocks, and the test statistic is the
sum of the Kendall coefficients of correlation between the predicted
ordering of the treatments and the ranking of each of the
n
blocks.
Jonckheere derives the sampling distribution of the test statistic
and shows its large sampl e distribution to be normal.
Page (1963) uses Spearman rank correlation to calculate a test
statistic for the completely randomized block design with
k
treatments.
Let
L
=
I
treatment in the
.th
1
ordered alternative.
large samples
L
R..
1J
block.
(P.
I
Pl"",Pk
are
is the within block ranking of the
jth
j=l
the predicted ranks and
blocks
n
k
and
n
R.. )
J i=1
\vhere
1J
A large value of
L
reflects the
Exact tables are included in Page (1963).
For
is asymptotically normal.
Jonckheere (1954b) also considers the case when there are one or
more observations per cell and the number of observations per cell is
the same within each treatment group.
Here the tratments are ordered
and assigned appropriate tied rankings based on the number of observations per cell, e.g. if
th
2
r
is the number of observations per cell
th
treatment group then the r
treatment will receive
r-l
1
.
predicted rank "2 (2 + 1) + I 2.. Observat ions are ranked wi thin
r
i=l 1
for the
r
e'
7
each block and the test statistic is again the sum
of the Kendall
coefficients of correlation between the predicted rankings and the
.
observed rankings within each block.
Page's test is extended by Hettmansperger (1975) who considers
the case of more than one observation per cell.
k
L
n
L
n
is the rank
In I)
. . , where R..
L L j R..
1J.
I).
i=l j=l
is the number of observations in the (i,j)th cell.
R..
In ..
j=l
i=l
cell sum and n ..
IJ
T:::
j
The statistic
k
1).
I)
=
The null hypothesis is rejected for large values of
T
which is also
shown to be asymptotically normal.
A competitor of Page's test has been proposed by Pirie and
Hollander (1972).
o_
define
Let
R..
I)
be the within block ranking of
D~ to be the expected value of the
)
a random sample of size
jth
X.. ,
I)
and
order statistic of
from the standard normal distribution.
n
k
k
The test statistic is W = L I j DR ..• For k~4 the normal
n
i=l j:::l
I)
scores technique is shown to be more efficient than Page's test for
k
many alternatives, but not in general.
Pirie and Hollander (1972)
include tables for the null distribution of
approximation is given where
k
{nk(k+I)/12}
EO
=
(W ) ::: 0,
n
W,
n
and the normal
var {W } :::
and
0
n
L (D~)2.
j=l
)
More literature for the two-way layout is provided by Shorack (1967)
who extends the procedures of both Bartholomew and Chacko.
In the para-
metric case, likelihood ratio tests are derived by minimizing the sum of
•
amalgamated squares for the desired effect, subject to the constraint
specified by the ordered alternative.
observation
R. =
J
Ij
r .. /1
I)
x ..
I)
For the nonparametric case, each
is replaced by its rank in the
where
I
.th
1
row.
Let
equals the number of levels of the first factor.
8
R. 's to obtain m dis-
An amalgamation process is applied to the
tinct quantities
m
RIt J, ... ,R1t J.
(12Ij(J(J+l))) i~l t i
(R
The test statistic is
m
1
J
2
] - (J+l)j2) ,
Iti
tion is discussed by Shorack (1967).
denoting the
i
th
.th
block and
X..
based on a sum
Y
be a random variable
IJ
treatment combination in a random-
J
ized complete blocks design with
Let
.
and its asymptotic distribu-
HOllander (1967) introduces a test statistic
of Wilcoxon signed-rank statistics.
-2
X =
r
n
blocks and
k
Let
treatments.
Y (i)
R(i) is the rank of y(i) in the ranking of
= Ix. - x. I and
uv
IU
IV
uv
uv
n
I (i)
y(i), ... ,y(n). Also let T
= I R \jJ (i) where 1/!uv
= I if
uv
uv
uv
uv uv
i=]
X. < X.
and 0 otherwise. Then Y = I T
Hollander (1967)
lU
IV
u<v UV
discusses the asymptotic normality of Y and notes that Y is neither
distribution-free nor asymptotically distribution-free.
A test similar to that of Hollander is proposed by Doksum (1967) .
Using Hollander's notation define random variables
U
uv
= T
uv
I
1jJ (i) .
uv
. I
1=
(U
- U )
where U. =
u.
v.
J
u<v
This test statistic is asymptotically normal and asymp-
Doksum considers the statistic
n
-I
n
I
U ...
i=l IJ
totically distribution-free.
Puri and Sen (1968) generalize the results of HOllander to a
Chernoff-Savage class of tests.
i = I, ... ,n.
zen) = I
uv,o.
Let
tion
V
n
a
or
th
\jJ*(x)
I
u<v
IU
Consider the random variables
0
as the
o.
th
T(n)
uv
X* ,
and
T(n)
uv
- 1jJ(-x), x
~
IV
= n- 1 IE no. Zuv,o.
(n)
0,
and
0
no.
are asymptotically normal.
if
where
Ix*
I
i,uv
is the expected value
E
order statistic of a sample of size
= 1jJ(x)
- X. , u < v = I, ... , k,
smallest observation among
comes from positive or negative
of the
= X.
X~
1,UV
n
x < o.
from a distribuThe test statistic
•
9
Pirie (1974) has made a comparison of the above tests, which he
refers to as tests based on among block rankings or
tests based on within block rankings or
as a measure, Pirie concludes that
W-tests.
W-tests
A-tests,
and
Using their A.R.E.
would more often be pre-
ferred to A-tests.
Another procedure for the two-way layout involves using aligned
ranks.
Sen (1968) considers such a class of aligned rank order tests.
Define the aligned observations as
k
Y.. = X.. - X.
IJ
1J
1
where
X. =
1
I X.. /k, i = 1, ... , n; j = 1, •.. , k . Let R.. be the rank of Y..
j=11 J
IJ
IJ
among all nk = N observations. For every N, a sequence of rank
functions,
·e
£N = (E N1 ,·· .,E NN ) is defined.
Define z(j) = 1 if the a th smallest observation among the N
Na
.th
values of the Y.. 's is from the J
treatment, o otherwise. A
1J
where
class of statistics is defined by
N
n
-1
\'
!...
~ k
(j-~(k+I))TN ./{o2(p )k(k2_1)}~
T* = (12n) 2
N
,J
n
k
j=l
1 n
o2(p ) = {n(k-l)})1 {ENR .. - ENR . }2, and ENR . =
n
I
I
where
k
. =
n, J
Th e statistic for testing against
E Z(j)
. - 1
k
Na Na ' J - , .... , .
a=l
ordered alternatives is
-1
T
k
L
j=l
ENR .. '
IJ
Note that
1.
i=l J=
IJ
1.
T* is asymptotically standard normal.
N
De (1976) extends the methods of Sen by employing the unionintersection principle to formulate a test using aligned ranks.
Fol-
lowing the notation of Sen (1968), De defines a reference class of test
statistics
k
St>.,N =
I b. = 0,
j=l J
and
I
b. (TN"-EN)/{var p
j =1 J
.-
.J
n
II
II
b.
=1 J
(TN"-EN)l}~
J
J
where
Sb N .:. , N(0,1) .
~,
A test against the ordering of treatment effects is derived using
the
VI
principle where the test statistic
and
B is a
10
U
8b where 8 = {~: 8 ~ ". ~ 8 ,
i
k
bEB
.th
with at least one inequality being strict}, e. being the 1
treatk-l
1
ment effect. The lim PH {Q> c} = I P{X9- > c}P(9-,k) and the weights
a
9-=1
P(9-,k) are tabulated in Barlow et al. (1972) and Chacko (1963). Note
set of values of
that
b
such that
Q is related to the
8
c
~esults
of Bartholomew, Chacko, and Shorack.
Quade (1979) proposes a method of weighted within block rankings.
Assuming blocks to have equal underlying variability, those blocks in
which the observed variability is greater are more likely to reflect
true ordering of treatment effects.
Such blocks are termed more
credible and are assigned greater weight.
Using this technique infor-
ation is gained by comparing observations among blocks as well as
within blocks.
Using the above weighted rankings procedure, Silva (1977)
generates a family of distribution-free tests by combining different
measures of variability, different sets of block-scores, and different
sets of treatment-scores.
Test comparisons are made using the expected
significance level method (see Silva and Quade (1980)).
Salama and Quade (1981) extend the procedure of Quade (1979) to
tests against ordered alternatives. The test statistic is w =
n
n
i~l bQiCi/i~l b i where Ci is the correlation between the predicted
th
and observed ranking within the i
block, Q. is the rank of the
1
i
th
that
block with respect to credibility, and the
0::; b ::; ... ::; b .
l
n
Kendall correlation.
bls
are weights such
Distributions are obtained for both Spearman and
Comparison of tests for small experiments is
undertaken using the expected significance level method of Silva and
Quade (1980).
W is shown to be asymptotically normal when
11
as
n
+
and
00
0< V(C) <
where
00
V(C)
is the vari-
ance of the correlation statistic.
Skillings and Wolfe (1977) present a class of tests based on
r
weighted sums of block statistics.
block.
Let
T.
be a statistic on the
I
the
b i '",.>
are nonnegative weighting constants.
the
b
are based on the maximum A. R. E. of
I
S
th
n
Then the test statistics are of the form
i
i
T:
I
b.T. where
i:l 1 I
Criteria for selecting
T.
Different scoring
schemes may be used in different blocks to form the weighted sum in an
optimal way.
This might be done when distributional forms are thought
to be different in different blocks.
In a related paper, Skillings (1978) considers an application of
Jonckheere's test statistic in block designs with unequal scale paran
meters for the blocks.
Let
Jonckheere's statistic be
JK*:
Skillings proposes the statistic
nonnegative weighting constants.
the
b. Ii
1
I
JK ..
i:l
b. 's
n
I
where the
are
I b.JK.
1 1 1
i=l
The main concern is the selection of
JK
=
in the unequal scale situation, but assuming distributional
forms to be the same in all blocks.
The
to
IS
JK*
b. 's
1
are selected to maximize the A.R.E. of
under translation alternatives.
shown.
with respect
Asymptotic normality of
JK
To adaptively select the weighting constants Skillings
suggests estimating the
1.4
JK
~onparametric
o. 's and using them to
I
est ima te the
b.
I
is.
Methods for Higher Order Layouts
First consider a procedure for testing for main effects and interaction proposed by Reinach (1965).
m and
p
Given two factors
levels respectively, let the
k
=
mp
A and
B at
treatment combinations
12
be replicated
times, and let
c
. .) th
( 1,
J
replication of the
j=l, ... ,p;
and
ranked from
I
k, r
observations of the
h
th
ijh
be the observation in the
ijh
treatment combination where
h= l, ... ,c.
to
x
h
th
i=l, ... ,m;
Within each replication, values are
being the rank of
replication.
x.1J"h
among the
k
Reinach defines test statistics
for the main effects and their interaction which are asymptotically disstributed as chi-squared random variables.
A second method using
orthogonal contrasts is also described.
Crouse (1968) proposes a method for testing any effect indepenm factorial experiment with an
dently of the other effects in a
2
equal number of observations per cell.
The strategy is to reduce the
problem of testing for any effect to a two-sample problem and then
apply existing nonparametric two-sample tests.
are similar.
Tests for interaction
Assuming the usual side conditions, independent random
variables are created by summing in such a way as to make all but the
desired term drop out of the model.
Then a two-sample tests of loca-
tion may be used.
This
techniq~e
is used by Mehra and Sen (1969) who describe a
class of conditionally distribution-free tests for interactions in
factorial experiments.
Consider a replicated two factor experiment
with one observation per cell.
factors are at levels
p
tions results in a model
and
Z. =
~l
There are
q.
r
~
+
Let
among all
replicates and the two
The use of intra-block transformaE., i = 1, ... ,n,
~l
interaction parameters and an error matrix.
dimension
n
containing only
These matrices are of
p x q.
r ijk
denote the rank of
N = pqn
Zijk' j=l, ... ,p, k=l, ... ,q,
aligned observations.
A sequence of real numbers
13
UN 1"" ,I N N}
,
'
where
TN , J'k
n, . + n.
IJ.
L
N
=
is defined. Let J
= n
, and IN = (TN 'k)
ijk
,J
N,r,
'k
n
n
. IJ
-1
-1
n
Also, TN 'k = n
n
(n, 'k - n, kijk = n .jk·
,J
'1 IJ
1.
1=
i=l
L
L
).
The test statistic for testing for interactions is
P
q
2
\'
\'
2
2
-1
[n/o (PN)].L
L {T * 'k}
where 0 (P ) = (n(p-l) (q-l))
N
N
'J
P
q
J= 1 k =l
1..
=
\'
L
\'
L (n"k- n , k-· n .,
j=l k=l IJ
1.
I).
+
n.
1 ..
2
).
The cOnditional distribution of
LN under the hypothesis of no interaction converges in probability to
2
X(p-l(q_l)'
Mehra and Smith (1970) also propose tests for the absence of
interaction in a factorial experiment.
metric estimation of contrasts.
Their work involves nonpara-
They claim their procedure to possess
more robust A.R.E. than the method of Mehra and Sen (1969).
In anotjher paper, Sen (1970) considers a class of nonparametric
procedures for testing the various main effects and interactions in a
m
factorial experiement replicated in
2
n
blocks.
Intra-block trans-
formations are used to obtain aligned observations.
This procedure
incorporates the use of some well known rank statistics such as the
sign test and general scores test.
Extensions to confounded designs
are considered.
MacDonald (1971) also considers using ranks in the analysis of a
m
2
.
experIment.
The anlysis is based on the use of orthogonal con-
trasts of ranks where observations are ranked from
each block.
1
to
m
2
within
It should be noted that the rankings are interchangeable
only'when all main effects and interactions are equal to zero, the
rankings being sensitive to all parameters in the model.
Hence, this
14
test is not efficient for sub-hypotheses, only the full hypothesis
that all main effects and interactions are zero.
Relevant Methods
1.5
Rank tests are valid for a broad class of distributions.
a general class of tests involving different rank scores.
question is what score should be used.
blem is to look at local optimality.
:!,est~,
Consider
The basic
One way of solving this proIn their book, Theory of Rank
v
Hajek and Sidak (1967) formulate a general theorem on locally
most powerful (LMP) rank tests.
tests is rarely possible.
Exact evaluation of power for rank
For this reason LMP rank tests have been
developed.
Consider an indexed set of densities
qo
E
H.
A test is LMP for
is uniformlly most powerful at level
{Q6' 0 < ~ < s}
~
H against
for some
s >
a
{q~}, ~ 2 0,
and assume
> 0
at some level
for
H against
a
if it
K
s
o.
v
Hajek and Sidak discuss the one and two-sample location problems
as well as the problem of simple regression.
However, when testing
against ordered alternatives one encounters a family of mUlti-parameter
probability densities and no general LMP rank test exists.
It is
desirable to extend this local optimal ity property to the mu 1ti -parameter
setup.
A method for doing this is proposed by Sen (1981) who uses the
union-intersection
of
LMP
(UI)
principle of Roy (1953) to extend the theory
rank tests to a broad class of multi-parameter alternatives.
As an example consider the
f. (x) = fee
1
-6 ' c.
2
~
~l
(x-~'c.))
~l~l
'
k-sample
i=1, ... ,n,
location-scale model where
are the p.d.f.'s of
n
e·
15
independent random variables.
The vector
of unknown parameters, and the
thesis,
H :
O
Q,
~ =
are known vectors.
c.
~l
is a vector
t:.' = (t:.'
t:.')
~l'~2
The null hypo-
is tested against appropriate alternatives.
To formulate the alternative hypotheses consider sets
rEL:'
and let
f=tir,
Ht:.y:
I,
and
U Ht:.y represent the alteryEf
principle is used to construct the test
t:.;:>:O, Ht:.f =
~
native hypotheses.
statistic
T*
n
=
UI
The
sup{T*(y): Y E flo
n ~
Further detail will be presented in
the next section.
The union-intersection principle of test construction devised
by Roy (1953) is covered in Morrison's book (1976).
Primarily intended
for multivariate tests, it may be extended to other setups.
x~
. . . ."
Np (~,~),
f'Oo..)
elements.
a'
is any nonnull
p-component
row vector of real
""
~'~ ~ N(~'~,~'E~),
Then
tance region
a
and
Let
t-critical
2
t2(~) ~ t a/2,N-l
and a univariate test with accep-
may be performed where
t
a/2,N-l
is
value.
The original multivariate hypothesis of
for all nonnull
only i f
~.
]J
~
]J
~O
is true if and
Acceptance is equivalent to
accepting all univariate hypotheses for varying
acceptance region is the intersection,
=
3.
The multivariate
2
2
n
[t C.~) ~ta/2 N-l]'
a
'
which
2
is the same as specifying t h at max t 2(),,<
~ - ta/ 2
N-1 . Th"'
1 S I S the
a
'
Hotelling
test which is also the likelihood ratio test. In
general, the
VI
test and likelihood ratio test are not the same.
When using the
function.
VI
principle it is necessary to maximize some
For tests against ordered alternatives the function must
be maximized subject to inequality constraints.
of nonlinear programming.
This involves the use
16
Consider maximizing
subject to
(KT) necessary conditions for
the above problem are:
i,
and
f,(:) sQ.
~ ?
~
and
g(x) :::; O.
~
to be a stationary point of
- ""AVg(x)
::: ,....,
0, A.g.(X):::O,
0, V'f(x)
,....
""
1 1
Note that
~
V'f(~)
:::
is convex for all
1.
for all
~
[df(~)
'
oX
l
necessary conditions are also sufficient if
gi (2S)
The Kuhn-Tucker
~
... ,
f(2S)
of(~) )
ox
n
.
The above
is concave and
This material may be found in Taha (1976) .
The purpose of this work is to develop new statistical theory in
the area of nonparametric tests against ordered alternatives.
Exist-
ing literature has been reviewed which concerns testing against ordered
alternatives in both one-way and two-way layouts, the use of weighted
rankings in block designs, and rank tests for higher order layouts.
Chapter 2 is concerned witb constructing tests against ordered
alternatives in the one-way layout.
This problem is attacked using
the union-intersection-Iocally most powerful rank test methodology of
Sen (1981), and literature relating to this topic is also reviewed in
this chapter.
A procedure for testing for order in the analysis of
covariance is described in section 2.8.
This method involves applying
the union-intersection principle to a procedure presented in Puri and
Sen (1971).
The results of Chapter 2 are extended to the complete blocks
design in Chapter 3.
Extensions to tests which use within block rank-
ings, weighted rankings, and aligned rankings are considered.
method for testing against ordered alternatives in a two factor
Also, a
e-
17
experiment with no interaction is described in section 3.9.
Finally,
the last chapters deal with some numerical work concerning asymptotic
power and efficiency, and recommendations for future research.
-e
CHAPTER 2
Optimal Rank Tests for Ordered
Alternatives in a General Linear Model
2.1
Introduction
This chapter uses the UI-LMP rank test methodology to construct
tests against ordered alternatives in the one-way layout.
Applying
this method tn extend the results of De (1976) to the one-way case,
the solution obtained is the same as that of Chacko (1963).
used the UI principle
De has
to construct tests for the two-way layout
while Chacko has used an ad hoc procedure to extend the results of
Bartholomew (1961) to the nonparametric case.
use any optimality criteria.
Neither Chacko nor De
Following the theory of Sen (1981) the
UI principle is used to introduce some local optimality properties.
Hence, this work may be viewed as a justification of the results of
Chacko on the grounds of local optimality as well as an extension of
De's results to the one-way layout.
The optimality properties of this family of tests are derived
from the choice of scores for the rank statistic.
Basically, a prob-
ability density is chosen from which the scores are determined.
a rank test is constructed.
Then
While the scores depend on a certain den-
sity, the test is valid for a broad class of alternatives.
To be more
general, a linear model which includes the one-way layout as a particular case will be considered.
e·
19
2.2
Theory
be N independent random variables with continuous
N
distribution functions F , ... ,F
and continuous probability density
I
n
Let
Xl' ... ,X
f , ... ,f .
functions
I
Consider the multiple regression model
N
f, (x) = f(x-a-c~l::,)
(1)
1
where
~l~
,
1
= I, ... ,N
is a vector of unknown parameters, and the
of known constants.
H : k = Q,
O
c.
~l
are vectors
It is desired to test the null hypothesis,
against appropriate alternatives.
problems the vectors
and
I::,
c.
~l
In a majority of
may be chosen in more than one way.
However, this non-uniqueness of choice should not be a concern as the
ultimate test procedure remains invariant under such a choice.
·e
general, the alternative hypotheses are of the form
and
Hl::,f
=
U
yEf
where the sets
H6y
r
and
H
f::,X
In
: 8 = f::,y. f::,
~
A-'
~
0
depend on the speci-
fic alternative.
Following Sen (1981), the calculation of scores to construct
locally optimal rank tests is the same as in H§jek and Sid§k (1967).
f(x;~)
For a given density
(2)
£(x)
=
let
{f(x;Q)} -1 Cd / a8)f(x;Q)
_ -f' (x)
• 1 = g(x)l
f (x)
-
where
(a/3Q)f(x;Q)
is the vector of partial derivatives taken with
respect to each of the parameters and evaluated at zero,
f(x;Q)
the density function with all parameters set equal to zero, and
g(x) = -f(x)/f(x).
Define
,
is
20
¢(u)
(3)
where
F
f(x;Q).
g(F-l(u)) ,
Then the scores are defined by
::: E¢ (lJ ")
N'1
where
O<u<l
is the distribution function corresponding to the pdf
(4)
n
=
U
NI
< .•. < U
NN
from the uniform
tistics let
are the ordered random variables of a sample of size
(0,1)
distribution.
X.
be the rank of
~i
i=l, ... ,N
,
Ties are disregarded since the
tor of rank statistics
IN
1
To construct the rank sta-
among
Xl' ... ,X
N
for
are assumed continuous.
F.
1
i = 1, ... ,N.
The vec-
consists of elements corresponding to the
elements of the unknown parameter vector
and is defined by
~
N
TNm
(5)
where
p
and hence,
T
=
I
m= l, ... ,p
i=l
is the dimension of
C .•
1J
~.
Note that
can be replaced with
c .. 1J
aN
C.
J
=
° by definition,
in the definition of
Nm·
~~en
the scores are derived from the true underlying distribution
of the data the resulting rank tests will be optimal.
For instance,
if the data come from a normal distribution, and normal scores are
chosen, then the optimal rank test will be constructed.
Using the
above method, logistic, normal, and exponential scores have been determined.
Logistic or Wilcoxon scores are of the form
Chacko (1963) uses these scores.
aN ( 1. )
of
=
E( Z
Ni)'
h
were
ZNi
aN(i) = ~;l - l.
The normal scores are of the form
.
. . f rom a samp I e
IS
t h e 1. th ord er statIstIc
N standard normal random variabl es.
Teichroew (1956) has tabled
21
these values for
are
0
N::; 20:
Finally, the exponential or log rank scores
i
~
f th e torm
a ( 1. ) N
-
I
1
- 1.
N-.i+l
j=l
Constructing the UI-LMP rank test involves forming the statistic
r'IN
on
r r
and maximizing it over all
1:'
rEI>
Note that for every fixed
ylO
~
Y where
~N~
o
E{XI!NIHo}
=
0
is defined as follows.
~N
where
subject to certain restrictions
E
-C
.
-1
Nm:; N
and
Let
N
I
i=l
c·1m
m=l, ... ,p.
Then
N
(7)
I
C
N
i=l
~l
(c. -eN) (c. -eN)
~1
~
~1~,
I
and
·e
N
=
(N_I)-l) [i:;N Ci ))
[~N(i)J I
=
~ ill
where
1=1
(8)
Finally,
(9)
Before considering the UI-LMP rank test against ordered alternatives, tests against orthant alternatives will be examined.
The fol-
lowing restrictions are placed on the alternative hypothesis:
(10)
r =
{r: r ~> ~
with
r' r > O}
The problem now becomes one of maximizing
y'T
~
~N
subject to the
22
constraints
y
~
'"
and
0
y'D y :; constant.
f""o.J
f"o.,I
This problem has been
f"otJN,....,
solved byChinchil1i and Sen (1981 a,b) who make use of the KuhnTucker theorem.
Following the notation of Sen (1981), let
P
=
-
{1, ... ,p} and
(0::. a ::.. P) .
a
a
be the complementary subset.
2P
For each of the
sets
a, IN
be any subset of
Note that
and . .QN
are partitioned
as
,
rT'
T')
IN = '~N(a)' ~N(a) ,
(11 )
and
QN =
(12)
~
~N(aa)
D
-
~N(aa)
D-J
D--
~N(aa) ~N(aa)
Let
k (a)
be the number of elements in the set
dimension of
k(a)
x
D
~N
k(a), etc.
For each
T
(13)
k (a)
is
(aa)
~N(a)
T
~N(a)
For
and
a = 0
D*
~N(a)
define
=
QN'
~N(aa)
v
T
~N(a)
k (a) ,
0
a,
0* (a) = D
~N
x
c
so that the
the dimension of
a
c
-1
- D
a
- D
D
-
~N(aa)
is
define
P,
-- T
-
~N(aa)~N(aa)~N(a)
- D
= 0,
- D-1 - - D
-
~N(aa)~N(aa)~N(aa)
and
for
a=P
define
v
_.
T
~N(a)
IN
The or -LMP statistic is given by
(14 )
where
I(A)
stands for the indicator function of the set A.
both indicator functions are nonzero for only one of the
above expression tells us which
T
~N(a)
to use.
Note that
T
~N(a)
so the
e
23
The asymptotic distribution of
a: 0 cae P,
every
The
rea)
T*
N
is as follows.
First, for
define
are the products of certain normal orthant probabilities
and they sum to one.
Then the
(16)
where
is a chi- squared random variable having
freedom.
Note that for the null set
probability
1
k(a) = 0
and
k (a)
2
Xk(a)
degrees of
~
x
. h
W1t
by definition.
To construct UI-LMP rank tests against ordered alternatives set
-e
(I 7)
where at least one inequality is strict.
i; = Ey
and
g
Consider the transformation
where
1S
a nonsingular
p
x
p
matrix of the form
1
0 0
-1
1 0
0 -1 1
(19)
:1
1 0
0 -1 1
Thus,
for every
r
E
£.
Now
I
r!N
may be written as
24
f
I~
!N'
and the solution for the orthant alternative pro-
blem may be used to solve the ordered alternative problem after transformation.
y'T
use
~ ~N
Instead of working with
~
( E,)-l =- E* ,
~
~
2.3
Example
Setting
D =- ~E*D~N~
E*'.
and its variance
~N
An example will serve to illustrate the preceding methods.
Con-
sider a one-way layout with 3 treatments and 7 observations per treatment.
The observations are ranked from smallest to largest, and the
overall ranking appears below.
Treatment
Ranks
1
1
10
2
4
16
8
3
2
13
19
6
15
7
18
9
3
5
21
17
12
11
20
14
The model of interest is
F.(x) =- F(x-8.)
J
J
(1)
where
8.
J
j=-1,2,3
is the effect of the jth treatment.
(2)
Now the model may be expressed as
where
,
j=-1,2,3.
Define
25
£1
£8
H
a
:
o:s; 11
21
:s; 6
31
,
(0,0) ,
:::
:=
:::
£14 ::: (1,0)'
:::
£21
:=
(0,1)
6
=
0
£15 :::
It is desired to test
:::
:::
H : 6
O 21
:=
£7
31
I
against the ordered al ternati ve
Alternatively, the model may be defined as
where
£1
_.
:::
:::
£8
(0,0)'
£7 =
£14=(1,0)'
£15 = ... = £21
and
·e
H :
O
lin = 6 32 = 0
:=
(1,1)'
is tested against
H
a : 112l ?:
1132?:
0,
results will be the same for both of these models.
O.
The
Note that a model
containing 3 parameters, one for each treatment, will result in a sinN
gular variance matrix
QN
L aN(i) = 0 implies
i=l
(T , T , T ) are not
N2
N3
N1
since the fact that
that the set of 3 treatment statistics
linearly independent.
Logistic scores have been chosen so that the end result may be
compared with Chacko's test.
~lN(i) = [~N+l
,-'
1
~-
'N+l
1)'
The scores are
=
[i-ll
11'
i-11] ,
11
'
i=1, ... ,2l.
Calculate
10
IT'
and
21
23
aN(R.) = 11
i=lSNl
L
26
To compute the variance matrix
QN'
first calculate
and
.'iN
TIlen
1 -11
2
66l:1
49
o
~N
~ ,
and
IN
10
= [IT'
23)
IT
t
.
The next step is to apply the transformation which changes this
problem from one of testing against the ordered alternative to one of
testing against an orthant alternative.
H : O:s; L1 :s; L1 ,
a
3l
21
L1
31
- .6
21
~
O.
which is the same as testing that
The needed transformation matrix is
E =
since
and
The alternative hypothesis is
C~
TI
L1
21
~
0
and
27
Now
IN
and
£N
may be calculated as
and
~,N =
P
.Now for each
that for
a = 0,
E*D~N~E*'
~
a~{I,2}
T
~N(a)
=
r
calculate
and for
0,
al ID .
49
= -66
~N(a)
and
a = {I,2},
remembering
0*
~N(a)
v
~
IN(a)
=
IN·
The
fol1mving tables contain the necessary components for calculation.
-
.
e
-
a
-
a
~N(a)
T'
I'
~N(~
~
0
~N(aal
o ~N (aa)
o o -- ~-1
o -~N(aa)
~N(aa)
~N(aa)
~
O}
{2}
33/11
23/11
49 (2)
66
~(1)
66
49
66 (1)
~(2)
66
66
49 (2)
{2}
O}
23/11
33/11
4~(2)
66
49
66 (1)
~(1)
49 (2)
66 .
66
49(2)
a
T
~N(a)
=
~
~
T
- D
~N(a)
~-1
- 0
~
-- T
-
~N(aa)~N(aa)~N(a)
0*
66
~N(a)
=0
~N(aa)
-D
- 0 -- D ~N(aa)~N(aa)~N(aa)
0
0
{I}
43/22
147/132
{2}
13/22
147/132
{1,2}
[33/11~
23/UJ
49 [ 2 1J
66 1 2
The UI-LMP rank test statistic is
T* =
N
I {cIN(a)£~(~)tN(a))IcIN(a) 2 Q)I(Q~(i)IN(a) ~ o}
0~a~P
: U~i, in ;;[-i -~J [~;jinJ
= 6.3748 .
28
Note that applying Chacko's test to these data yields the same results,
-2
Xrank
i.e.
= 6.3748.
T*
To evaluate the significance of
P{T~ ~ xiHO}
lim
N-roo
=
N
use the expression
r(a)p{x~(a) ~ x}
I
09~P
where
The
rea)
may be calculated as follows.
For
P
= {1,2},
r(0)
is the
probability that both variates from a bivariate normal distribution
are negative.
Gupta (1963) gives the expression for calculating this
probability as
f(x,y;p)dxdy
where
1
1
+
4
2n
arc sin
= -
~
Therefore,
r({1,2})
r (0)
1
1
= -4 +
~ as
p =
arc sin (!z) =
2n
1..
4
1(49/66)(2)(49/66)(2)
+
_1 (~)
2n 6
= 1/3.
Now
is the probability that both variates from a bivariate normal
distribution are positive, and by sYmmetry
r({I})
This correla(49/66) 0)
is the correlation between the two variates.
p
tion may be derived from the matrix
= 1:2.
p
r({1,2}) =
r(~).
Finally,
is the probability that of two bivariate normal random vari-
abIes the first is positive and the second is negative.
This proba-
bility equals the probability of a univariate standard normal random
variable being positive minus
and by sywnetry
1 -
~
(1) +
r({l,2}),
r({l}) = r({2}).
~
(.988425)
i.e.
r({l}) = !z-r({l,2}),
Hence, the p-value is
+~(. 988425) +}(. 958721)J
= . 018 .
29
2.4
T*
Asymptotic Nonnull Distribution of
N
In order to investigate asymptotic power and efficiency properties of
T*
N
in (2.2.14), its nonnull distribution must be determined.
{H }
N
To do this, first consider a sequence
of alternative hypotheses
where
(1)
Under the assumptions needed to construct the tests of section 2.2 (see
Sen (1981) assumptions (I)-(V)), Chinchilliand Sen (1981a) have shown
the sequence of probability measures under
that under
{H }
N
to be contiguous to
H '
O
A thorough discussion of continguity is contained in Chapter VI
of
H~jek
and
Sid~k
(1967).
Let
{P } and
N
{QN}
be two sequences of
absolutely continuous probability measures on measure space
Then the sequence of measures
{QN}
if for any sequence of events
~(c ~),
(~'~'~N)'
is said to be contiguous to
{P }
N
(2)
Chinchilli and Sen (1981a) use this contiguity to establish the
limiting distribution of
IN
under
{H
lim N-ID~N
=
}
N
as
(3)
where
and
N"tOO
D both exist and are of full
~
30
rank.
Given the asymptotic
of
nOl~ality
under
the
in (2.2.14) is now determined.
distribution of the statistic
The problem is to find
T
Note that
under
may be expressed as
~N(a)
T
~N(a)
(5)
~lfhere
mea)
(6)
and
c
~a
and
c-
are matrices such that
~a
(7)
Thus
c T
~a~N
N
-kzv
l'
~N(a)
= T~N(a)'
.s:.a-IN
._.-
and
= .-IN(a-) .
have asymptotically normal distri-
butions, and if their covariance is a null vector they are asymptotically independent.
To show this note that
Cov(T
~N(a)
-0
- 0
-1
-- l'
-
T
-
~N(aa)~N(aa)~N(a)'~N(a)
,
-1
Cov(IN(a) ,IN (a) ) - Cov(QN(aa)!2N(aa)IN(a) ,IN(a))
o
_ ..
~N(aa)
0
- 0
-1
-- 0
--
~N(ua)~N(aa)~N(aa)
=0
Now, ignoring the summation sign, (4) may be written as
31
(9)
P{I(QN(~)IN(a)~Q)}p{(r~(a)Q~(~)fN(a))I(IN(a)~Q»~}
=
p{ ICQN (~) IN (a) ~Q) }p{ I
cIN(a) ~~) }p{ cI~ (a)QN(~)IN (a/xl IN (a) ~Q}
=
r*(a)p{(r~(a)QN(~)rN(a))>xIIN(a)~Q}
where
(10)
It is now desired to show that the expression in (9) is equal to
(11)
·e
This is accomplished by applying lemma 3.1 of Bartholomew (1961). First
note that there exists a matrix
D*
(12)
~N(a)
C such that
~
=~
CC
I
Then
(13)
Let
(14 )
and express
D*-l
¥
v
TI
(15 )
Since
i"
~N(a)~N(a)~N(a)
~N(a)
V
T
~N
is normal, and
(C I)
~
as
-1 C -IVT
~
~N(a)
=
ZIZ
~
~
32
Var (z)
(16)
= ~k(a)
I
the elements of
z
= ( z1 ' ... , zk ( a) )
variables with means
lJ-:'
1
i
are independent normal random
say, and unit variance.
Now make a transformation to the polar coordinates
(r,8 , . .. ,
1
Let
(17) Z. = r cos 8 - ... cos 8k ( ) . 2 sin
a
1
1
defining
sin
cos 8
0
-1-
:: 1.
8 kCa )-i-l '
i=l, ... ,k(a)
Now
becomes
(18) P{r
Since
r
2
2
> x Irestrictions depending on
is independent of
8 "" ,8 (a) -1
1
k
8 " .. ,8 (a)_1'
1
k
only} .
the restrictions on the
angles can be ignored which implies that the probability in (9) is
independent of the restriction
INCa) ~ 0,
and may be expressed as (11).
Furthermore, since
(19)
the asymptotic nonnull distribution of
lim P{T*
. N~x} =
(20)
N+=
where
2
X [k(a),~(a)]
(21)
r*(a)P{X
2
can be written as
[k(a),~(a)]~x}
is a random variable having the noncentral chi-
squared distribution with
parameter
I
Q'cacP
T*N
k(a)
degrees of freedom, and noncentrality
e·
33
The method used to calculate the
section 2.2 that the
rea)
r*(a)
follows.
Recall from
are the products of certain normal orthant
probabilities for the null mean case and sum to
For the nonnull
1.
case the mean is shifted from the origin, and some adjustment must be
made.
For a given set
a,
define
R(a)
to be
+1
or
-1
times the
probability of the subspace or region corresponding to the shift of
the mean from the origin, the sign depending on the direction of the
shift.
Then
(22)
r* (a) ::: r (a) + R(a)
where
rea)
also sum to
·e
to the
r*(a)
is calculated as in the null case.
1
Note that the
I
R(a) ::: O. More weight is given
a
associated with a positive shift in the mean.
which implies that
For illustrative purposes consider calculating
p ::: 2.
Assume that
to calculate
(23)
IN
r*({1,2})
note that
r*({},2}) ::: P{T
>O}
~N
~
""""N
-jJ
(24)
>
-jJ}
rooJ
("'oJ
::: r({l,2})
r*({l})
r*(a)
is distributed normally with mean
p{ (T
To calculate
r*(a)
+
R({l,2}) .
use the relation
r*({l}) ::: P{T
::: P{T
NI
N1
>
0, T
N2
>
O} - r*({l,2})
>
O}
when
~.
Then
34
where the marginal distribution of
calculation of
r*({1,2}).
r*({2})
T
NI
is univariate nonnal.
is similar, and
1'*(0)
=
The
l-r*({l}) - r*({2})
Numerical methods or approximation fonnulas may be needed
to calculate these probabilities.
Some comments may now be made regarding the asymptotic power and
efficiency of the UI-LMP rank test.
value
x,
For a given
a-level critical
the asymptotic power function may be expressed as
a
(25)
In general, both the asymptotic power and efficiency are functions of
all possible
~(a)'s
where
a
is an element of
Here the power function depends on
~(a)
2P
possible sets.
in a very complicated way,
and Pitman efficiency is not generally applicable.
Hence, a numerical
study is needed to further investigate the asymptotic power and efficiency of this class of tests, and this will be dealt with in Chapter
4.
This study will compare the UI-LMP rank test to other tests for
both the cases of equally and unequally spaced ordered alternatives.
Tests such as Jonckheere'Stest attain their maximum power under equal
spacing.
The UI-LMP rank test is valid for any spacing of the alter-
native, and its power is expected to be greater than that of either
tests when there is unequal spacing of means under the alternative.
2.5
Local Asymptotic Power and Efficiency
When the mean under the alternative is very close to the origin,
local asymptotic power and efficiency may be investigated.
Writing the
e·
35
o~
alternative as
(1)
where
13(0:) =
I
~
is fixed, the power function
2
r*(a)p{x [k(a),L'l(a)]
a
can be expressed as a function of
~Xo:l{HN}}
and examined as
0,
0
Compar-
O.
+
ing the slope of this power function \vith a slope from another test
produces a measure of local efficiency.
This method will now be
described in general terms, followed by an example for the bivariate
case.
As in section 2.4, express
r* (a) = r (a)
(2)
where
·e
r*(a)
R(a)
parameter
as
+
R(a)
is now expanded as a function of
L'l(a)
O.
The noncentrality
can also be written as a function of
0,
namely
(3)
where
);!,(a)
=
EcIN(a) I{H }).
N
Now for a given set
a,
the noncentral
chi-squared random variable in (1) can be expressed as an average over
central chi-squared random variables having different Poisson weights
and expanded to a power series in
o.
The product
of the two components in (1) is also a power series in
O.
Summing
depending on
L'l(a),
over all possible
a,
the slope of the power function at the origin
can be calculated and compared to that of another test to yield a measure of local efficiency.
Consider the following bivariate example.
distributed normally with mean
first step is to calculate
o~
R({1,2})
Assuming that
and finite variance
so that
r*({1,2})
QN'
=
IN
is
the
r({1,2})
+
36
R({1,2})
may be obtained.
Making the transformation
(4 )
where
is a diagonal matrix such that
d
don matrix of
IN'
the probability
dOd' ::: p
~~~
r*({1,2})
is the correla-
is written as
(5)
Now
z
is distributed nonnally with mean
r*({I,2})
is equal to
of the shaded
r({l,2})
0
and variance
£,
and
plus the bivariate normal probability
region below.
I
I
I
I
I
: (hI' h 2 )
I
J----------------
(6)
::: r({I,2}) + R({l,2}) .
e·
37
R({1,2}),
To calculate
the bivariate normal density is first
expressed as the product of marginal and conditional densities, and
the area of one strip is calculated,
R({1,2})
other strip, and
Then the same is done for the
is the sum of the probabilities of the two
strips plus the probability of the rectangle defined by the coordinates,
h ,h ,
l 2
and the origin.
!
The bivariate normal density of
is written
as follows:
(7) g (2
1
, Z )
2
1
=:
C2
2n l-p
1
12-;~
=:
·e
where
~(zl)
cx{
~J
-Z
exp
"'l
(Z2 _ 2 Z 2 + Z2)
2
1
1 2
2
2 (l-p )
I
1
1
J
-f I 2n(l-p 2 )
ex{
1
~
2 (l-p 2)J
(2
_
2
Z )2
1
is the standard normal density function, and
is the conditional density of
strip along the
axis is
(8)
Making the transformation,
(9)
(8) is now written as
2
2
given
zl'
g(z2 1zl)
The probability of the
38
_ z2
1/2
e
---
_ rh.1
- J
where
¢(zl)
a
e-Z~/2I[
PZ1]~
I¢ --lldZ
I2TI L ~_p2~
I
is the standard normal distribution function.
P zl
Taylor's theorem to expand ¢l( - ] around the point
a
Using
yields
Q
e·
(10)
If:~(hl)
2
~
- ¢(O)J
+
l
n; 1
1 p
where the fact that
t
zl¢(zl)dz
a
expanding
¢(h )
l
E_ 2
l
~
~(O)
= ¢(a) - ¢(b)
in a Taylor series around
order terms involving
hI
(since
of the first strip is written as
hI
+
o(h~)
is used.
0,
Finally,
and ignoring higher
is very small), the probability
j
~-ili
1
(11 )
~
- ¢(hl)J
2
_h
e 1/2
39
=
I
<P
1
2
(0)2
{ hI - 2 hI
2.-
l}
r 2 /2;
l-p
Similarly, the probability of the strip along the
zl-axis
is calcu-
lated as
(12)
and that of the rectangle as
(13 )
for
hI
and
-e
h
2
=b
where
o
a1
a = {1,2},
and
both small.
0 + b
and
b
0
al
R({1,2})
is written as
2
are the above coefficients of
2
0 , respectively.
1
After calculating
(IS)
a2
Thus
r*({1,2}), r*({l})
can be expressed as
40
A Taylor series expansion of
~(oTl)
around
0
yields
(16)
and (15) can be written as
where higher order tenns involving 0 are ignored.
r*({2})
is similar, and the respective
R(a)
The expression for
may be obtained by sub-
traction using the relation
R(a) = r*(a) - rea) .
(18)
As in (14) denote
R(a)
(19)
R(a)
where the
a.
as
b 's
are the coefficients of the 0 terms for a given set
a
Note that while
r*(0)
may be obtained by subtraction, it is not
a factor as
To express the power function in (1) as a function of
consider writing
as
6
(20) P{/[k(a),6(a)] 2:x } = e- (a)
a
e
-6 (a)
0,
r~ {2
>}
~ Xk(a) - x a
00
I
(6(a))
r!
r=O
{ 2
r
P{ 2
>}
Xk (a)+2r- x a
>}
+ 6(a)P Xk (a)+2 - x a
J
1 6 2 (a)P{X 2 (a)+42: x } + ...
+ "2
k
a
first
41
1 3
2
- ;[ 1'1 (a)P{xk(a)+42: \x} + •..
~
= P{ X ( a) 2: Xa} + 0
2
(~ ~' (a) Q~ ( ~ ) ~ (a) J (P { X~ (a) +2 2:
-P{X~(a)
where
= 21
c aO
Now, using (2),
~
'( )0*-1
('
a ~N(a)~ aj,
Xa
}
4
2: x }) + 0(0 )
a
and
(14), and (20), (1) is written as
(21) Sea) = Ir*(a)P{/[k(a),l'1(a)] 2: x }
a
a
=
I(r(a)p{X~(n)
2: \.x}) + 82Ir(a)caOCa1 +
a
2\
a
2
+ 8 L b a 2 p { Xk (a) 2:
a
a,oLbalP{X~(a)
> \}
a
X
a} +
oIba/{X~(a)
a
a}
3
0 (
0 )
+ ozGr(a)CaoC al +
~
2:X
Lba2P(X~(a»x";lJ
a
3
+ 0(0 ).
42
Having expressed the power function as a function of
6
when
6
0,
-+
a measure of local asymptotic efficiency may be obtained by comparing
the slope at the origin of the above function to that of a competing
test.
This slope is
dS(a~)
(22)
do
where
b
al
I
0=0
is the coefficient of
0
In the expansion of
R(a).
Consider Jonckheere's test, and the normal scores test as two
competitors of the
UI-LMP
rank test.
Let the alternative be defined
as in (2.4.1), and assume that each of the
number of observations
n
such that
of Puri (1965), and letting
V
N
k
N = nk.
= N- 3/ 2v,
samples has the same
Following theorem 5.2
the appropriate limiting
distributions may be written as
(23)
and
(24)
where
J,
and
NS
stand for Jonckheere's test statistic, and the nor-
mal scores test statistic, respectively.
The asymptotic power of these
tests is generally written as
(25)
where
l
a
is the upper
100a
percentage point of
For the case of loca 1 al ternatives write
This amounts to writing
A.
1
as
~(x).
)1* = 0 (y*)
where
0
-+
O.
43
(26)
),
0
:::
1
oy 1
0
for
,
i::: 1, ... ,k .
Expanding (25) in a Taylor series around
T
a
yields
2
S(V )::: <P(T )+<jJ(T )[Y: 0-2T )+0(0 ) ,
N
a
a 0
a
(27)
and taking the deri vati ve of this power function at the point
gives
(28)
Now for Jonckheere's test
flj
(29)
and (28)
--* :::
oj
1S
{
I
18
}
2
(Y.-Yo) 0 ,
lIT(k -1) i<j
J 1
expressed as
(30)
For the normal scores test
{+-
(31)
I
o. (Yo-Yo)}o,
J 1
k - 1 l<J
and (28) is written as
(32)
mNS ::: ep(T
a
Finally, writing (22) as
(33)
){+
k -1
I
i<j
(y,-y,)}.
J 1
0::: 0
44
the local asymptotic efficiency of the
to Jonckheere's test
1S
m*
-y,
rank test with respect
and with respect to the normal scores
m
test its asymptotic local efficiency is
2.6
UI-LMP
m*/~S.
An Application to Mental Health Data
Consider the following hypothetical example which was suggested
by Turnbull (1982) regarding an ongoing study of depressed inpatients.
The study objective was to test the efficacy of isocarboxazid, an antidepressant also known as Marplan.
The impetus for this hypothetical
example was an extension of the work of Davidson et al. (1981). Suppose a double blind clinical trial was conducted and 38 patients were
observed for a period of up to 6 weeks.
During this period 12 patients
received a placebo, 13 patients received 30 mg of Marplan, and 13
patients received 50 mg of Marplan.
Each patient was assessed for
depression at baseline, and at the end of the fourth week using the
first 17 items of the Hamilton Depression Scale (see Hamilton (1967)).
The observation on each patient was the sum of the Hamilton scores at
week
a
minus the sum of that patient's scores at week 4.
This obser-
vation is termed the change score, and theoretically could range from
-52
to
52.
Suppose the investigator is interested in testing for an ordering
of the three treatment groups.
A priori one could hypothesize that at
the end of the 4 weeks, the patients receiving 50 mg of Marplan would
have greater change scores than the 30 mg group, which in turn would
have greater change scores than the placebo group.
observations appear below.
The rants of the
45
Treatment
Ranks
Placebo
•
8 14
2
7 23
3
1
5 13 17 11 12
Marplan (30 mg)
32 16 15 10 26 25 30 27 28
Marp1an (50 mg)
38 24 22 20 34 33 37 35 36 19 29 31 6
9 18 21 4
Using the same model as in section 2.3, define
£1
=
=
£12
=
£13
=
...
=:
£25
=
£26
=
...
=
£38
=
(0,0) ,
(l, 0) ,
(0,1) , ,
data are not derived from a continuous distribution, because the
observations are obtained by summing the scores of the first 17 questions the data may be asymptotically normally distributed.
tational simplicity logistic scores of the form
are used.
aNCi)
For compu-
+\- 1 = 2i;939
= N2
These scores coorespond to the locally optimal choice of
scores when the underlying distribution is logistic.
A very similar
case holds for the normal scores.
The vector
T
~N
is calculated as
(T
25
T
=
NI
I
,T
and
38
I
i=26
To calculate the variance of
aN CR.)
IN'
I
1
N2
15/39 ,
aNCR i )
i=13
N1
=
221
39
calculate
),
where
46
N
I
C
~N
(£i-SN)(£i-SN),
i=l
1
1963
L:481
= 1444
-48n
963J'
and
•
Then
.-PN
2
= AN~·N
.-
1 1963
4446 L:48l
=
To test against the ordered alternative,
-48ll
96~
.
and
IN
QN
are transformed
to
=
IN
8(
[1
L9
1
15/39J
221/39
=
(236/39) ,
221/39
and
,~
QN
=
II
[1
L9
respectively.
G TI1
e-
1
~64
= 4446
482
48TI
963 '
The components needed to calculate the
UI-LMP
1
r963
Ij 4446 L:481
-48Il
96~ 1
statistic
are given below.
a
-
a
~
~
T'
Tf
-
~
o ~N(aa)
0
o ~N(aa)
~
o -~N(aa)
~J-l
o --
--
~N (a).
{l} {2}
236/39 221/39 964/4446 482/4446 482/4446 963/4446 4446/963
{2}
-
~N(aa)
--
v
~
~
~-1
~
T
= T
-0
- 0 -- T "N (a)
"N (a) ""1\J (aa) ~ (aa) ~ (a)
~
a
{2}
,2}
~N(aa)
{l} 221/39 236/39 963/4446 482/4446 482/4446 964/4446 4446/964
a
{l}
{l
~N(a)
0*
~(a)
t"'V
!"oJ
::: 0
-0
-
~(aa) ~N(aa)~N(aa)~N(aa)
3.22
.163
2.64
.164
(236
221)'
l39 ' 39
f'oI-I
""
- 0 -- 0
1
4446
~64
482
48TI
963
47
The UI-LMP rank statistic is
.
r
221) 117
963
39
18316 l-482
[236/39)~
-4821
96~ 221/39 ~
::: 211.83 ,
and the asymptotic p-va1ue is
O.
Remembering that this is a ficti-
tious example, the investigator's hypothesis is overwhelmingly supported.
An Example Involving Two Factors
2.7
Tests against ordered alternatives in randomized blocks with more
-e
than one observation per cell have been considered by Hettmansperger
(1975), and Skillings and Wolfe (1977, 1978).
for
X.1)'k
i
Consider the model
= 1, ... , p
j=l, ... ,q,
and
where
B.
k
=
1 , ... ,n ..
1J
is the random effect of one factor,
. th
the J
treatment, and
J
CI,).) th
cell.
Let
is the number of observations in the
n ..
I)
n.
)
=
is the effect of
T.
1
q
P
I
i=l
it may be reasonable to assume
n .. ,
1J
that
and
B.
1
I
N =
=
is no point in testing for order among the
n ..
j =1
a
)
for all
B.
1
In many cases
i,
and there
if the investigator
is only concerned with the ordering of the treatment effects.
For
this situation, the results of Chapter 2 may be applied directly.
An example is now presented using a set of hypothetical cat data.
Nine fictitious cats, all new mothers of the same breed, were selected
48
for the study.
Their litter sizes vary.
Three different types of
ration were randomly allocated to three treatment groups.
consist of three randomly assigned Ii tters each.
there is no mother effect, i.e., no litter effect.
These groups
It is assumed that
The weights of the
kittens were recorded when they reached the age of 3 months.
The
investigator hypothesized before the study that the kittens who were
fed ration 3 would be the heaviest, and that ration 2 kittens would be
heavier than ration 1 kittens.
As it happened, the hirthweights of
all kittens were virtually identical.
The data appear below.
!2ypothetical Cat
Cat
Treatment 1
Treatment 2
Treatment 3
l- ~
Data
Kitten weight in grams at 3 months
822
737
741
935
938
1128
547
3
958
598
1601
4
5
6
867
1033
1404
666
731
714
921
979
1417
533
595
897
7
8
9
1088
1360
1312
765
870
1091
962
1298
377
802
877
e
836
1407
Ignoring the cat subscript, define the model
for
(2)
where
T.
J
is the effect of the J·th ration,
£1
=
= £11 = (0,0) ,
£12
= £24 = (1,0)'
£25 =
= £35 = (0,1) ,
tl j
.Q,=l, ... ,N
1 = 'j-T 1 ,
and
-
49
To test
~21
H :
O
-
~31 =
tions are ranked from
1
0
against
to
N,
Ha : 0
~ ~21 ~ ~31
the observa-
and the logistic scores
£-18
18
(3)
are used to calculate the LMP rank statistics.
Cat
Treatment 1
Ranks of kitten weights at 3 months
3
22
5
35
9
10
20
21
28
4
5
6
15
25
32
6
8
7
19
24
34
4
18
7
8
9
26
31
30
11
27
1
12
16
17
23
29
1
2
Treatment 2
Treatment 3
·e
To calculate the vector
IN'
13
=
I
£=]
3
14
2
33
first calculate
35
T
NI
The ranks appear below.
24
c n aN (R.Q,)
=
I
9.,=]2
aN(R£)
7
=- 8 ,
and
35
I
T
N2
£=1
35
I
c9,2 a N(R9,)
£=25
25
aN(R£) = 18
To calculate the variance of
IN =
[- 18 ,
7
25)
18
note that
N
fN
=
1 r286
(£9,-SN)(££-SN)' = 35 L:143
£=1
I
-14~ ,
264
50
and
35
108
Then
1 r 286
108 L:143
-1431
26tU
.
The transformation matrix needed to construct the test against
ordered alternatives is
and the transformed vector, and its variance are
e·
and
P
.-N
respectively.
= E*D E*-l =
~ ~N~
1 ~64
108 U2l
12n
264J
The components needed for the calculation of the UI-LMP
statistic are given below.
a
a
{l}
{2}
{2}
{l}
II
~N(a)
25
18
Ti_
~N(a)
o
~
D
-
o -
~-1
o --
D (aa)
--
~N(aa)
121/108
264/108
108/264
121/108
264/108
108/264
~N(aa)
~N(aa)
~N(aa)
25
18
264/108
121/108
1
264/108
121/108
~N
51
v
T
a
~
~N(a)
~
=T
~N(a)
~~1
-0
~
- D -- T
,. . ."
D*
-
~N(a)
~N(aa)~N(aa)~N(a)
"'"
=D
""-1
-D
~N(aa)
91
o
{I}
157/432
5005/2592
{2}
67/72
5005/2592
{I, 2}
[1 ,
[264
1
25) ,
18
""
- D -- D -
~N(aa)~N(aa)~N(aa)
108
1211
264j
U2l
The VI-LMP rank statistic is
I fl et'·
D*-l 1
)I (1
~N(a)~N(a)~N(a)
~N(a)
T*
N
a
r.
=
II(~ 1
,
25)
108
>
-
I
0) (D*-!
~
T -
~N(a)~N(a)
<
-
O)}
~
1 24
18 5005 l-ll
::: 0.8575 .
·e
To evaluate the significance of
T*
N
=
I
use the expression
r (a)
a
where
r(0) = r({l,2}) =
=
p-value
21 -
.326 ::: .174.
1- [.326(1)
+
J+ 21
Tf
arc sin
p{ ~
X ( a)
s x}
l21'J = .326,
[ 264
and
reO}) =
Then the p-values is calculated as
.174(.645561)
+
.174(.645561)
+
.326(.348677)]
.336
which does not support the investigator's hypothesis.
2.8
Testing for Order in an Analysis of Covariance
To construct
vr -LMP
rank tests against ordered alternatives in
the analysis of covariance, one must consider the conditional
52
distribution of the principal variate given the covariates.
While
this is a difficult problem, DI rank tests can be constructed quite
easily by combining the method in section 5.9 of Puri and Sen (1971)
with the procedure already described in this chapter.
The basic problem is that all permutations of the rankings are
not equally likely under the covariance model.
There is a need to
eliminate or adjust for the effect of the covariates.
Following Puri
where
and Sen (1971), consider a stochastic vector
Xl
is the primary variate, and
stochastic vector.
~
= (X , ... ,X )
2
p
is a concomitant
Interest lies in testing for order among the
locations of the first variate while utilizing the information contained in the covariates.
Let
N be the number of observation vec-
tors, and define the matrix of ranks as
(1)
~
(R (1)
R(l)
11
ln
R(1)
pI
R (1)
(c)l
ln
c
R
l
=
pn
l
R(c) J
pn
c
th
is the number of observations in the k
treatment group,
c
k = 1, ... ,c, and
I n k = N. Each row of ~ is a permutation of
k=l
1, ... ,N, i. e. for each variate, the observations are ranked from I
where
to
n
N.
k
Define the rank statistics
I
(2)
where
nk
c
ak
is
Further define
N
I
a=l
o
(l)
c kaN(R. )
a
la
or
as
k=l,'H,c,
i=l, ... ,p
is calculated as in (2.2.4).
1,
where
V(E~)
1
c
- I
N k=l
53
To utilize the information contained in the covariates, Puri and
Sen (1971) suggest fitting a linear regression of
T~~).
on
(k)
T N2 " ' "
The adjusted test statistics are defined as
(4)
k=l, ... ,c
where
is the cofactor of
V..
1J
T*
Nq
and
for
v .. (!t*)
1J
;;.,,~
k, q=l, .. .',c
in
V(B:-*).
""N
The covariance
is defined as
(5 )
okq
where
is the Kronecker delta, and
Denote this covariance by
d
kq
,
Ivl
and let
the covariance matrix of the test statistics.
-e
statistics
T*
- (T*Nl"'" T*)'
~N Nc '
been defined, the
Since
IN
(T
=
inverse of
tistics
2.9
(T
N1
, ... ,T
D*
~(3)
Nl
VI
statistic
Nc
)'
is the determinant of
D = ((d q ))
k
be
~
Now that the vector of
and its covariance matrix
D have
may be calculated as in section 2.2.
is of rank
c-l,
either the generalized
must be used, or alternatively the vector of sta-
,··· ,T
_ )
Nc l
may be calculated as in example 2.3.
Example
A swine research unit was interested in comparing three different
feeds.
Thirty young pigs were randomly assigned to three groups of
size ten each, and fed the appropriate feed for an equal number of
days.
The researchers were interested in finding out if feed 1
resulted in lower weight gain than feed 2, and if feed 3 resulted in
the largest weight
gain
(Xl)'
~ain.
In addition to the primary variate, weight
the researchers wished to include the covariates age
S4
(X ),
2
and birthweight
in the study.
(X )
3
The matrix of ranks
appears below.
Xl
X
2
X
3
26
12
1
14
16
30
20
9
28
26
14
3
2
27
28
26
13
24
6
16
9
15
2
13
4
10
20
11
5
6
27
19
29
3
22
EN
17
23
18
9
24
22
30
23
25
11
17
16
24
15
11
8
20
7
12
8
3
4
18
30
28
21
15
25
27
21
19
29
10
14
8
29
19
13
21
23
2S
7
12
1
5
22
17
18
10
5
1
2
4
7
6
As in previous examp 1 C'~;, let
and define
1\1
tested against
:=:
8
Ha :
k
- 8
for
1
o:s: lI
2l
:s: lI
two covariates. To calculate
as
k
31
Feed 1
8
:=:
Feed 2
e·
Feed 3
be the effect of k
k
2,3.
Then
HO =
lin
:=:
th
lI
31
treatment,
:=:
0
is
whi 1 e adjust ing for the effect of the
(k)
T
' k
Ni
:=:
1,2, i
:=:
1,2,3,
first define
55
..
=
£1
=
=
and use the logistic scores
=
£30
aN (cd
T(k)
Ni
Then using (2.8.2) the
(0,0) ,
= £20 = (l, 0) ,
£11
£21
=
£10
(0,1)'
2a
2a-31
-1 =
N+l
31
=
are calculated as
19
,
T (2)
N1
T(l) = -12 ,
N2
155
T(2)
-12
= 155 ,
~(2)
T(l)
N1
Tel)
N3
155
cev .. (R *)))
1J ~ N
a=1, ... ,30.
N2
lN3
=
24
,
155
9
= 155 ,
=
1
155
is cal culated using (2.8.3), and its value
is
·e
r
1
14415
The cofactors of
v 1J
..
(RN~)
~l·
4495
-289
1269
for
V
ll
-289
4495
2991
l269~
2991
4495
i = 1,2,3,
=
.
J' = 1
are
.054
V21 - .025
V31 =-.032
and the determinant of
V(B~)
Ivi
IS
=
.014 .
Finally, using (2.8.4) and (2.8.5)' the vector
are calculated as
T* = (.133 ,
~N
.178)' ,
IN'
and its variance
56
and
-. 0091
0= r'.018
~
009
L:-.
.011U
Now the UI statistic is calculated just as in previous examples.
Remember that the vector
(.331, .178)
IN
must be transformed to
which has variance
I,
test the ordered alternative.
Ul test statistic is
N* =
T
0 =
Q*Q£*' =
T*
= ~
E*T*
=
~N
~N
[018
.009
,00TI
.018 '
to
Applying (2.2.14), and (2.2.16), the
5.41,
and the asymptotic p-value is
.029.
2.10 Testing for Order in a 2 Factor Experiment
In a
2
factor experiment, an investigator may wish to test for
order among the levels of both factors.
2
factors
easily.
is assumed null, such tests can be constructed quite
Consider the following model.
(1)
X"
IJ k
,
IS
where
factor
If the interaction between the
1
= ]J+S.1 +T.J
Let
+E"
IJ'k
~ k th 0b
'
' th 1 eve 1 of
tIle
servat10n
correspon d'1ng to th
el
and the jth level of factor
j = 1, ... ,p, k
= 1, . .. ,n.
Is.
E'S
.
1
are
1J
1
J
To
construct tests
=
1
n
i = 1, ... ,m,
observations in each
IT.J
.
= 0,
are in effect.
J
i.i.d.
continuous distribution function F(x. 'k),
f(x, 'k)'
and
There are exactly
cell, and the usual side conditions,
It is assumed that the
2,
random variables with
and continuous
p.d.f
against ordered alternatives, first
note that an overall ranking of the observations is inappropriate since
the rankings are interchangeable only when both main effects are equal
to zero.
Therefore, a transformation must be made to eliminate the
e-
57
effects of one of the factors from the model.
Then the ordering of
the effects of the remaining factor may be tested.
Following the technique introduced by Crouse (1968), the transformed variables
ill
(2)
Y' k
.I
=
L
X. 'k
1.1
i=1
can be regarded as
tion with
c.d.f.
the sum of
the
B.1
p
for
k = 1, ... ,n ,
random and independent samples from a populawhere
m i.i.d.
j = 1 , ... ,p,
G is the distribution function of
random variables each with
F.
c.d.f.
Now
effects have been eliminated from the model, and the problem
reduces to one of testing against ordered alternatives in the one-way
layout.
·e
The results of section 2.2 may be used for this problem, and
the optimal scores are now calculated
LMP
using the
c.d.f.
G(Yjk). The
rank test against the hypothesis of an ordering of the
S.
IS
1
is
constructed in the same way.
If the
E"
I) k
have mean zero and finite variance . no matter if
normal i ty holds or not,
the sum
IE,I)'k
i
will be approximately normal
if the number of elements is not too small.
For this reason, a non-
parametric procedure which is good for nearly normal distributions is
appealing.
2.11 Summary
In summary, Sen (1981) has proposed a method of test construction for multi-parameter problems which results in certain optimality
properties.
This method uses the UI principl e to extend the Htij ek-
Sidtik LMP property from the two-sample case to the multi-parameter
58
case.
Applying this theory to the multiple regression model, the
test against ordered alternatives has been derived as a special case.
The results of this test correspond to the test of Chacko which implies
that Chacko's test is optimal when the underlying distribution of the
data is the logistic distribution.
The nonnulldistribution of the VI statistic has been established
in order that asymptotic power and efficiency might be investigated.
In general, expressions for power and efficiency are very complicated,
but for the case of local alternatives such expressions may be obtained,
and this has been discussed.
Finally, an application to mental health
data, a simple extension to the two factor case when the effects of
one factor can be ignored, a test for the analysis of covariance problem,
and
a brief discussion on testing for order in a two factor
experiment have been presented.
The next chapter will be concerned
with extending this theory to the analysis of randomized blocks.
e·
CHAPTER 3
Optimal Rank Tests for Ordered
Alternatives in a Complete Blocks Design
3.1
Introduction
Now consider the problem of extending the
VI-LMP
rank test
method to tests against ordered alternatives in block designs.
methods will be examined.
within block rankings.
The first method concerns tests which use
The second and third methods concern tests
which use aligned and weighted rankings, respectively.
three methods will yield a different result.
depends on the method used.
Each of these
The optimality property
In general terms, each method will involve
applying the Hajek-~idak theory to a different type of density.
for each result, the
Three
Then
principle is applied to construct three dif-
VI
ferent types of tests.
Also, comments are made regarding the analysis
of covariance, and the
2
plete blocks.
3.2
factor experiment replicated in
n
com-
The within block rankings case is considered first.
Within Block Rankings
X.. , i == 1 , ... ,n, j := 1, ... ,k be a random vari ab I e corre1J
.th
} '1 th a f
·
span d 1ng
to t1e
n blocks and the J
of k treatments
Let
in a complete blocks design.
The
X..
1J
independent with distribution functions
for the
F.(X-8.)
1
f (x-8 ).
i
j
i
J
th
block are assumed
and continuous pdf's
Alternatively, following the notation of Chapter 2, the pdf
60
X..
for the random variable
may be defined as
IJ
f .. (x) :: f .. (x-a-c!lI)
(1)
1J
where
c.
and
-J
II
IJ
-J-
are defined as before.
Blocks are assumed indepen-
dent, and block effects are assumed additive.
HO:
again of the form
;:.,
::
The null hypothesis is
O.
Within each block the observations are ranked from
Let
Then
the
R..
1]
R.
x
n
.th
observation in the
J
.th
is the k-vector of ranks for the
-1
k
.th
be the rank of the
1
1
to
k.
block .
1
block, and
R
is
Before deriving the
matrix of within block rankings.
locally optimal scores, the following assumptions are made.
Assume
that for every
1
where the sets
(2)
Then
f·e(x;S)
-1
-
f~
IX
f~
IX
IX
For almost all
x
(4)
f~
-
-
X E I,
1
-
1
yf r
-
the limit
lI~O
1
-
-
j
XE r
rXJlf~IX (x;lIy)
Idx
-
=:
foo If:<
IX
_00
(x; 0) Idx
-
Let
(3/3l1)f. (x;lIv)
(x;O):: lim ;:.,-l{f. (X;lIy)-f(x;O) I
exists, and for every
_00
=:
-
(x;lIy) - y'f,s(X;lIy)
and
IX
(X;lIy)
-e:: lIy,-
and for
a.e.
f~
lim
and
'-'0.1
exists
1I~0
-
are as previously defined.
f
Cd/(6)f.
(x;6)
......
1
(3)
(5)
-y -r
rand
is absolutely continuous
I, f. (X;lIy)
<
00
.
~
61
These assumptions follow those stated in Sen (1981) and allow the theory
v
of Hajek and Sidak (1967) to be applied to the multi-parameter model.
They are general in nature and cover the more specific case defined by
the model in (1).
~r,
Now, instead of using the vector
tive hypothesis is formed
using the scalar quantities
quantities shall be denoted by
the alternac!y~.
~J~
These
j=l, ... ,k.
~c.,
J
To derive the optimal statistics, first note that the' probability
of observing a given set of within block rankings is
n
=
(6)
JR
•••
k
n.k
f IT IT
f. (x .. -6
J i=l j=l
1
: fr {J .. .f n
j=l
R.
1=1
~1
=
fr
{f···J{
i=l
R.
n
j=l
1J
.)!T TT dx ..
i=l j=l
J
f. (x .. -6.)
1
f.
1
1J
ex<
J
n
j=l
1J
dx .. }
1J
.;.6c.)dx· 1 · .. dX· k} .
1J
J
1
1
~1
Under
H'
O
the probability of observing any set of within block rank-
ings is
(7)
Following the Neyman-Pearson Lemma, form the ratio
(8)
g (8) / go (8) = (k!)
nfr!f·
.. J-IT f. ex . .;~c J<) dx. 1 ... dx. k}
i=l l R. j
1
1J
1
1
~1
{rJ."
TT
. 1
R
= (kl) n n
1:::
~i
Jr
k
1
TTf. (x .. ;OJdX· l <<<dx' k
. 1 1J
1
1
J
r
;~c .)-ITf. (x. <;o)JdX<l<
. (x.1J<
frf
.
1
J
J
<
J
1
1J
1
0
odX·1 k}
62
= (k!)nfr{k\ + /::,
i=l
.
I J... Jrq..(f.
~
R.
j=l
1
(x .. ;/::'c.)-f. (x .. ;0))
IJ
J
1
IJ
~1
k
L I J···Iq..(£.
L4
1
(k!)nfT{k ,1r+/::'k!
i=l .
j=l
;o~.(x.
;/::'Cn)~dX'l"
.dX· }
nIl
N
NIlk
n
f. (x.
· 111m
m=J+
x
:=
'-1
R.
1
n
1
N=
(x .. ;/::'c.)-f. (x .. ;0))
IJ
J
1
IJ
~1
k
x nf.(x.
· 111m
m=J+
n (
n~l+tJk!
i==l
l
I
. -1
;O)n£.(X.n;/::'Cn)~dX·l···dX·k}
nIl
IN
NIl
N=
I ."JrG1-(f.(x .. ;tJc.)-f.(x .. ;0))
k
j=1
R.
1
Ll
IJ
J
1
IJ
~1
k
rTf.(x.
· 111m
m=J+
x
l+~k! i I== 1
I I···I~-(f.(X
. = 1 R.
~l
J
L:.
1
. -1
;O)nf.(X.n;/::'Cn)~dX·l·
.. dX·k}
nIl
IN
NIl
N=
.. ;6C.)-f.(X .. ;0))
]
1
1J
1J
k
x
TI
·
m=J+
+
'-1
f.(x. ;0)nf.(X.n;6Cn)QdX·1···dX·k
111m
n 1 1
IN
NIl
0(6)
N=
when
6
is sufficiently small .
Following the proof of Theorem 4.8 of Hajek and
~idak
(1967, pp. 71-73)
which applies here with minimal differences, it can be shown that
I f···fL~·i.
n
= 1+6k!
I
J
. 1
i=l P'
R.
1
(x 1..J ;O)nf.
(x.1m
..J.' 1
m, J
;O)~:lTrdX"
J
J
.
1
~l
n
:=
1-6k!
I c·f···J
I
i=l j=l J
n
= 1-6
I
R.
~1
tI }
f.(x .. ;O) k
~k
/ ( 1J 0)
(x. ;0) ITdx ..
. x..;
111m
.
1J
1
IJ
m=
J
.. ;0)
L c.E {-f.1 (x IJ
k
i=l j=l J
R.
fi(xij;O) ~1
TI£.
63
= 1-~
k
n
L L c.E
'=1 .
J=
1
where
z..
is the
1J
jth
1)
{-f. (z .. ,O)}
.
1
1)
1
1J
f.(z .. ;0)
order statistic of a random variable having
f.. Then for a given
the pdf
the LMP intrablock rank statistic
1
is given by
k
n
I
(0)
I
c.E
{-f.
i~l j=l J
where
c.
1J
fk(zij;O)
i s give n by
J
Let
(z .. ;O)}
1
c ! y, j::: 1 , . . . , k .
. ~J~
-f.(Z ..
(0) _ E
1
1J
.
1
k
J
J = , •.• , ,
{ f.(z .. ;0)"
i
;O)}
o
3
1
denote the optimal scores.
1J
Then (10) may be rewritten as
n
(11)
TW(Y)
~
The scores
a.(j)
=
k
I I c.a. (R .. )
i=l j=l J 1 1J
may be calculated as
1
. _ [k-1) foo f.(x)[F.(x)]
.
j -1 [l-F.(x)J k
-j dx,
(12)a.(J)- .. k '
1
J- 1
1
1
.
1
_00
i=l, ... ,n; j=l, ... ,k,
and are the same scores as in Chapter 2 except that these are calculated
from samples of size
The
a. (j)
1
k
rather than
nk.
are also the same as the optimal scores givin in
Puri and Sen (1971, p. 271).
integrated by parts.
For
To see this, the above score function is
i = 1, ... ,n
and
j = l, ... ,k
64
a.(j)
1
= E{
-f.(Z .. ;O)}
1
1J
f.(z .. ;O)
1
1J
k-IJJoofi(Z)
f. (z)
= -k ( j-l
=
_00
.
k-j
[1\ (z)] j-l [l-F
f i (z)dz
i (z)]
1
-k(~-llJfOOrF.
(z)]j-I[I_F. (z)]k-jdf. (z)
lJ1
1
1
(00
- J f.(z)[F.(z)]
1
1
j _I
(k-j)[l-F.(z)]
1
k _ j -1
I
f.(z)dd
1}
+k(k-l) [ k-2Jfoo
j-2
f 21 (z) [F i (z) ]j-2 [l-FCz)] k-j dz
which is the form of the optimal score function obtained in Chapter 7
of Puri and Sen (1971).
The advantage in using this form is that the
scores may be easier to calculate.
Also note that if the assumption of constant error variance is
a (j) = a(j)
i
is assumed,
made, the scores are the same for each block, i.e.
for
every
i.
If
a
the
scores
may
be
derived
heteroscedastic
using
the
model
model
65
I fC-SiJ
f. (x) -- -O.
1
where
O.
1
1
1
.th
common variance of the
1
a (j)
f.
.th
is the
S.
block.
2
o.
block effect, and
1
Then the score
1
is
a. (j) :: :. a (j) , and
1
1
is the same with
1
replaced by
(10) becomes a weighted average.
f.
If the
If the o.
1
o. 's
are known,
'5
are unknown they may
]
be estimated using the block range or some other estimate of the block
variance.
Some function of this estimate may be used, e.g. the rank
of the range.
More attention will be paid to this point in the sec-
tion on weighted rankings. The
are introduced only if it is
O. '5
1
believed that they differ among blocks.
Recall that the statistic
I w::
where
:::
Wm
where
p
ing
the model.
I I
i=1 j:::l
is defined as
c. a.(R .. ),
Jm 1
1J
is the dimension of
k,
m=l, ... ,p
and the
c.
Jm
Once the vector of optimal rank statistics,
is formed, the
Tank test.
ylT,v
~
~I
UI
principle
r
Y
f
f
1
depend-
T = (T '" .,T ), ,
Wl
Wp
W
UI-LMP
subj ect to certain restric-
"1
r, E{r'~W IHO} = a and
is defined as follows.
Let
. th
(2.2.9), only calculated for the
i.
l
Ow-1
~
block .
var{r'~wIHo} = ~y'O~W~y
be defined as
QN
in
Then for the hama-
are all equal, and
°"WI. =
°
For the heteroscedastic model where the scores are of the
scedastic model the
form
or
1'.
QW
for all
0
As in Chapter 2, this involves forming the statistic
For every fixed
where
are
is applied to construct the
and maximizing it over all
tions on
y'T~W
~
k
n
T
011
and
(TWl,···,TWp )',
(13)
in (10) is given by
Ok
~
~i
1
n
a. (j) = 0." a (j), Q = )
1
w
1
1
2" QWi .
1=1 o.
1
66
The results of Chapter 2 may be applied to construct the UI-LMP
rank test.
A transformation is used to change the problem from one
of testing against the ordered alternative to one of testing against
an orthant alternative.
xiIw
The Kuhn-Tucker solution for maximizing
r~
subject to the constraints
same as in (2.2.14).
Q and
X'Q,w X :: constant is the
The exact small sample distribution may be
obtained by considering the
(k 1)n
possible equally likely within
block ranking permutations, whereas the asymptotic distribution of
the
UI
statistic
is as in (2.2.16).
An example follows.
Consider a compl ete blocks design with
n:: 7
blocks.
k
=
3
treatments and
The within block rankings appear below.
Block
Treatment
1
2
3
4
5
6
7
I
I
1
1
1
3
I
I
2
3
2
2
3
1
2
2
3
2
3
3
2
2
3
3
The model of interest is
(1)
where
F .. ex) :: F.(x-8.) ,
1J
1
J
8.
J
is the effect of the
i:::l, ... ,7; j=1,2,3
. th
]
treatment.
define
(2)
6.
= 8.-8 1 '
i1
J
Now the model may be experessed as
j=1,2,3.
As in example 2.3,
67
i=1, ... ,7; j=1,2,3
where
£1
=
(0,0) ,
£2
= (l, 0) ,
c
=
~3
(0,1) ,
Assume that block effects are additive and error variance is constant,
and test
H : (1,21
O
a : a ~ 6 21
H
~
6
:=
==
31
°
against the ordered alternative
(1,31 '
For convenience, use the logistic scores
a. (j)
I
-e
aU)
=
2L_
:=
k+l
j -2
1 . - -2-
j==1,2,3; i==1, ... ,7 .
Now calculate
3
7
T
W1
=
7
L I
c '1 a (R, ,) ==
i:=l j =1
J
I
J
I
a (R
i==l
i2
) = !z
and
T
W2
:=
7
3
I
I
i=l j =1
7
I
a(R· )=2
c '2a(R .. ) ==
I 3
J
I J
i=l
To compute the variance matrix
£w
=
n~,
first calculate
k
C
~k
-11
?J'
\ (c.-ck)(c.-c ),
j~l ~J ~
~J ~k
and
:£k -
A~
III
Then
-11
?J'
68
and
Iw
(1/2 , 2)' .
0:
The transformation matrix of example 2.3 is used here to calculate
~
I w = £*Iw = (5/2 , 2) ,
and
~
QW - '"E*D""w'"E*'
t
Calculate
-
a
a
~W(a)
T'
T' -
~W(a)
~(a)
---
and
=
0*
7/l2IT
as before.
~W(a)
o
0
D
-
o -
~-l
o --
o --
~W(aa)
~W(aa)
~W(aa)
~W(aa)
~W(aa)
O}
{2}
5/2
2
7/12(2)
7/12(1)
7/12(1)
7/12(2)
12/14
{2}
O}
2
5/2
7/12(2)
7/12(1)
7/12(1)
7/12(2)
12/14
ea
~
~
1
~W(a)
~
o:T
~W(a)
~-l
-0
- 0
~
-- T
-
~W(aa)~W(aa)~W(a)
0*
~W(a)
~
=0
~W(aa)
~
-D
~-l
- D
The
-1/2
7/8
7/4
7/8
7 \.1r2 1)2
IT
UI-LMP
=
rank test statistic
6.0
It
is
-
~W(aa)~W(aa)~W(aa)
o
{l,2}
~
-- 0
69
To evaluate the significance of
T*
W
I
.
use the expression
p{ ~
r (a)
X ( a)
0~a~P
where the
D*
~N(a)
rea)
differ
:;; x}
are the same as in example 2.3 since
D*
and
~W(a)
only by the multiplication of a scalar quantity.
This
yields a p-value of
1-
3.4
G
(1)
+~
(.985694)
+i; (.985694) +~
(.950213)J = .021 .
Asymptotic Power and Efficiency Considerations
The results of sections 2.4 and 2.5 may be applied to the test of
section 3.2 \vith only slight modification.
{H }
N
Of alternative hypotheses where
Again consider a sequence
N =
nk,
and
e
(1)
where
Under the assumptions of section 3.2 ((2)-(5)), and following
Chinchilli and Sen (198la), it can be shown that the sequence of probability measures under
{H }
N
is contiguous to that under
result is used to establish the limiting distribution of
Iw
as
(2)
where
k
'\ (c.-ck)(c.-c )',
j~l ~J ~
~J ~ k
and
H .
O
is the variance of
This
under
70
It is assumed that the
lim N-1D
~W
exists and is of full rank.
N-+ro
The asymptotic nonnull distribution of the
VI
T*
statistic
may be obtained directly from the results of section 2.4.
W
Define .
(3 )
where
mea)
is defined similarly to (2.4.6). Then the asymptotic non-
null distribution of
(4 )
lim
N-+ro
where
ing
W
is written as
P{fTW~X}= I
r*(a)p{/[k(a),i:I(a)]
0'=-a ,::Y
x2 [k(a),i:I(a)]
k(a)
T*
~x}
is a noncentral chi-squared random variable hav-
degrees of freedom, and noncentrality parameter
(S)
The
r*(a)
are defined as in section 2.4.
The asymptotic power function for the within block rankings test
may be written as in (2.4.22).
General expressions for power and
efficiency are complicated, and this test will be included in the
numerical study of chapter 4.
For local alternatives, the methods of
section 2.5 may be applied to look at local asymptotic power and
efficiency.
3.5
Weighted Rankings
The weighted rankings test of Salama and Quade (1981) was reviewed
in Chapter I.
This concept of utilizing the information contained in
a comparison of the blocks is intuitively appealing, and it is desired
to incorporate this concept into the LMP rank test framework.
To do
..
71
this, the notation and assumptions of section 3.2 are adopted, and the
additional assumption that
f.(x .. ,lIy.) = f(X .. ,lIy.)
(1)
1
1]]
1]
J
for
i = 1, ... , n
is made.
To calculate the optimal scores for the weighted rankings proceclure, the joint density of the within block rankings and the variables
used for weighting must be considered.
Some measure of variability,
e.g. the block range, is used to weight the rankings within each block.
Let
Then
v.
be the chosen measure of variability for the
1
9
i th
block.
is defined as
(2)
where
Q.
1
is the rank of
block rnakings
The matrix of within
v.
1
R is defined as before.
In the following derivation, the block effects are taken to be
euqal to zero for all
i
since both the ranks, and the
invariant under tanslation.
v.
1
are
Under the null hypothesis the probability
of observing a given set of within block rankings and block variability
rankings can be written as
n
(3)
po(E,9) =
k
k
n
~ f(x .. ;0) r-r r-l- dx ..
fE,9 f r-r
i=l j=l
1J
. i=] j=1
1]
•••
Under the alternative this probability is written as
(4 )
n
f.. .f fr
i=l j=l
~,g
fr
n
f( x .. ; 1I Y .)
dx ..
1J
J
i=] j=l
1J
72
To derive the locally optimal scores, first expand the function
f(x ..
;~y.)
)
g(x)
=
1)
in a Taylor series around the point
-fl (x)/f(x)
= O.
Now letting
as in Chapter 2, (4) may be written as
(R,Q)=J···ffr
-rrf(X 1J.. ;0) LIr-~y.g(x
.. )+o(~)lfr TrdX.
R n .
J
1J
~.
1J
(5) P A
y ,LI ~ ~
~
~y.
)
0
0
~';:S
f
0
J
1
1
)
n k
)~ n k
~n k
IT ITf (X .. ; 0) 1 - L I ~y. g (x. 0) +o(~) IT TTdx ..
r(
R, Qj i
j
1J
L i=l j =1 J 1J
i
j
1J
= ...
1
h.· f... f
nk
n k
n k
g(x .. )IT ITf(x .. ;O)IT ITdx. o+o(~)
.. J R Q
1J .
.
1J
.
.
1J
1 J
N' ~
. 1 J
1
J
=PO(R,Q)-~L
~
~
Following the Neyman--Pearson lemma, form the ratio
nk
~ Lh·
ij J
P y L\(R,Q)
(6)
f ... Jrg (x. .) ITn ITk f (x. . ; 0) orrn ITdx.
k
.+
~---:::::.....::-'-=: 1 -
PO (~,
R,g
~
g)
0
j
1J
i
j
1J
-----------(n k
n k
••• JIT ITf( x. . ; 0)
dx.
R Q
1J
1J
~'~ 1
J
1
J
1J
f
IT IT
0
=:
Zi S
v.
1
0
Io J[n.LE ( g (x.1J.) IR, Q) ) +
o(~)
+
o(~)
(~)
0
•
1
are the order statistics from a sample of size
corresponding to the
that the
0
1 - ~ k y.
J
where the
i
pod.f.
f (x .. ), i
1J
= 1, ... ,n,
j
=:
1, ... ,k .
are functions of these order statistics.
mal scores are defined by
k
Note
Then the opti-
73
(7)
a Q.£ :::: E(g(zi£)
19)
1
I: [[:g(Zi2)m(Zi2Ivi)dZi~n[~:_~][rr(Vi)JQi-l
o
n-Q.
1 dIT (V.)
x [1-IT(v.)]
1
where
is the conditional p.d.f. of the
m(z',e,lv.)
1
1
given the measure of variabi Ii ty for the
marginal distribution function of
LMP
In order for a
. th
£th
1
order statistic
block, and
1
is the
IT (v.)
1
v ..
1
weighted rankings test to exist, the quantity
say
must be expressible as the product of two score functions,
aQ . .Q,
1
c.Q,
This will be the case if
and
(")
,) g(z·n)m(z.
n!v.)dz'
n
Lx,
Ix,
1
Ix,
(8)
::::
c x,n h(v .)
o
1
_co
where
h(v.)
some function of
IS
1
v..
1
not depends on the conditional density
~~ether
this is the case or
m(zi.Q,lv ).
i
If
a .£
Q1
is not
LY ·La
the product of two scores then the statistics are of the form
which is not a linear combination of the weighted rankings.
. J. Q.R ..
J 1 1 1J
Thus, while
it is possible to construct such tests, the weighted rankings procedure
is not generally justifiable from the LMP criteria.
Consider briefly the case when the
k
2
and
v. : : (k-l)s.
1
J=
-
1
(x .. -x i )
2
for
i:::: 1, ... ,n.
k
i
::
Xl"
and
v.
1
=:
_
L (z 1].. -z.)
. 1
1
J'=
Denote the
. th
J
1J
normal order statistic from a sample of size
2
are distributed normally,
1J
\'
L
=:
1.
X..
.
written (see Durbin (1961)) as
2
k
by
z ...
Note that
1J
Now the order statistics may be
74
(9)
Z ..
:::
1J
where the
u..
1J
X.
1
+ u .. IV:
1J
for
1
j:::l, ... ,k
are symmetrically dependent random variables marginally
distributed as the order statistics from a uniform
k
(-1,1)
2
distribu-
k
I u .. ::: 1, and I u .. ::: O. Geoj:::l 1J
j:::l 1J
randomly distributed points which lie on
tion, and satisfying the conditions
metrically, the
u.. are k
1J
the surface of a unit sphere of dimension
independently of
k - I,
and are distributed
v ..
1
Remembering that
g(ziZ):::
side of (8) becomes E(Z. nlv.),
1 x,
(10)
E (Z , n
1
Iv '.)
the left hand
1J
and using (9) this can be written as
Z. n
Ix,
for normal
::: IV:
E (u .. ) .
1
1J
Ix,]
In relation to the expression in (8),
expected value of the
jth
afore mentioned constraints.
X.. ,
h(v.) :::
1
IV:,
1
and
is the
uniform order statistic subject to the
Thus, while the locally optimal weighted
rankings scores exist in this case, their calculation is quite complicated.
The following ad hoc statistics are suggested on the grounds that
they are intuitively related to the
cases, depending on the form of
LMP
a .£,
Q1
statistics, and in some
may satisfy the
LNP
Define the vector of weighted rank statistics as
(11 )
where
k
n
(12)
TQm :::
I
i:::l
d (Q.)
]
I
j:::l
c. a(R .. )
Jm
1J
for
m:::l, ... ,p.
criteria.
75
Note that the calculation of
involves weighting the
within block ranking statistic
The
d(Q.).
1
in (3.2.12)
1
that
f(x .. ) .
The density
1J
q.
1
be the
n(v.)
1
. th
1
.th
p.d.f.
will of course depend on
n(v.).
1
d(Q.) _ E{-ll¥(V i ;O)
]
n(v.;O)
1
I Q}
=
~
n(v.),
the
1
block, rather
1
order statistic from a sample of
1
variables all having the
(13)
the differ-
is based on
d (Q.)
of the chosen variability measure for the
Let
f(x .. ) .
1J
random
n
Then
E{-n'(qi;O)}
n(q.;O)
1
While these scores are intuitively appealing,
-e
a (R .. ) ,
1J
ence being that the calculation of the
p.d.f.
with the block score
are defined similarly to the
d(Q. )
LMP
their calculation
depends on obtaining a closed form for the density
n (v. ),
will in most cases be quite complicated.
is the range of
the
.th
1
block, the distribution function of
II(v.) =
(14 )
When
1
v.
1
V.
1
1
and this
is expressed as
kjrOO f(x .. )[F(v.+x .. )-F(x .. )] k-l dx ..
1J
1
1J
1J
1J
Quat ing from Sarhan and Greenberg (1962) \\lho give this expression along
with the
c.d.f.
of the midsum'
"These beautiful formulas are not of
much use since, in general, the probability functions
F(w-x)
c;mnot be expressed by
F(x)."
F(x+v)
and
This is the case when the
x..
1J
are distributed as logistic random variables.
Calculation of the
d(Q.)
1
IS
possible for the case when the
are distributed as standard normal random variables, and v.
1
k
_ 2
2
taken to be (k-l)s.
For this situation the
L (x .. -x.)
1
distributed as
2
X _
k l
. I
J=
1J
1
x..
1J
is
v.
1
random variables, and the block scores are
are
76
calculated as
(15)
E{
-nl(qi;O)}
k-3 {I}
1
- - - - E - +n(q. ;0)
2
q.
2
1
where
1
. the l.th or d er statIstIc
. . f rom a samp 1e o
f n
IS
q.
1
abIes having the chi-squared distribution with
k-l
random vari-
degrees of freedom.
Using the approximation
E{-
TI I
(16)
'IT
(q i ; ()) }
(q. ; 0)
1
(see Cox and Hinkley (1979) p. 189) where
function of a
is now the distribution
random variable, (15) may be written as
l
1
n'(v.)/n(v.)
1
e-
l'
k-3 rIT -1 - Qi )) -1 + -1
n+l
2 .
2
d (Q. )
(17)
Here, since
IT
1
is a drecreasing function in
weight is given to the less variable blocks.
v. ,
Of course when
(15) is a constant, and the blocks are weighted uniformly.
can b:.C:lf"iat::.s,:,:~~y
and
1
~=l
IJ
1
when the
Xij
1
more
k
The
==
3
d(Q.)
1
are normal random variables,
J
This result is in conflict with the numerical results of Silva
(1977), and Salama and Quade (1981).
Their work indicates that more
weight should be given to the more variable blocks when the data are
normal and the sample variance is used to measure within block variability.
While the ad hoc statistics of (12) are intuitively related
to the LMP statistics, it must be remembered that the resulting scores
77
depend on the choice of density, and the choice of within block variability measure.
appropriate.
For certain cases, this ad hoc statistic may not be
In Chapter 4 the above case will be examined numeri-
cally, and some conclusion will be sought as to the weighting of the
within block rankings.
3.6
Exampl c::.
For purposes of illustrating the weighted rankings test the LMP
within block ranking scores are used as in example 3.3, and the block
weighting are selected arbitrarily.
While the resulting test will
not be the LMP weighted rankings test as defined in section 3.5, the
UI principle is used to produce a valid test against ordered alternatives.
The data of example 3.3 appear below.
of interest are the same as in section 3.3.
ro'h
.th bl OC k.• and
tel
Q.1
Or
The model and hypotheses
Let
v.
1
be the range
In their paper, Salama and
its rank.
Quade (1981) suggest using linear weights.
These weights are adopted
here, and denoted by
(1)
b(Q.)
1
==
for
Q.
1
i
==
1, ... ,7 .
Then the weighted rankings statistics are defined by
3
7
I
(2)
i==l
where the
a (R .. )
1
in (3.3.3).
J
b (Q. )
1
I
j=l
c. a(R .. )
Jm
1J
m = 1.2
are the logistic scores, and the
c.
Jm
are defined
7P>
Block
Treatment
1
2
3
4
5
6
7
1
38
51
39
43
57
49
40
2
54
63
46
56
47
62
50
3
45
79
SR
53
52
72
55
Range
16
28
19
13
10
23
15
4
7
5
2
1
6
3
Q.1
Calculations yield
IQ
: : [~ QJ'
2' 2
which has variance
QWk
being defined as in section 3.3.
The transformation matrix needed
to test for order is again
and the transformed vector, and its variance are
E*T~Q
~
~
QQ
The
v
T
~Q(a)'
and
D* (a)
~~Q
:::
E*D~Q~E*'
~~
:::
35 [2
3
lJ
U
for the various sets
a
are presented below.
e
79
'"
~
1
a
-
=T
~-1
-D
- [)
'"
-- T
-
-Q(a) -Q(a) -Q(aa)-Q(aa)-Q(a)
D*
~
=D
-Q(a) :q(aa)
-0
~-l
- 0
~
-- 0
-
~(aa)~(aa)~(aa)
o
{l,2}
The
31/4
35/2
4
35/2
35
(13,21/2)'
UI
3
r2 n
II
~
statistic is given by
\'{(T!
0*-1 T
)I(TNQ(a)
"'Q(a)~Q(a)-Q(a)
T* =
Q
L
a
=
8.1571 .
To evaluate the significance of
where the
l'
<O)}
>0)1(0* - T -Q(a)~Q(a) --
- '"
(a)
TO
use the expression
are again the same as in example 2.3, and this yields
a p-value of
1 -
G
(1) +
~
(.995711) + ;
(.983068)J = .007 .
For the sequence of alternatives in (3.4.1), the limiting distribm:ion of
(3)
where
(4)
IQ
is
80
where
2
8
Q
lim N-1D
NQ
N-+=
and
=
k
~k
=
),
.. \L (c.-ck)(c.-c
~J ~
~J ~k
J =1
exists and is of full rank.
If
~(a)
assuming that the
is a matrix such that
~
v
(5)
IQ(a)
=
m(a)IQ '
then the asymptot ic nonnull di stribution of
T* is written as
Q
L r*(a)p{x 2 [k(a) ,li(a)] ~ x}
(6)
a
where
(7)
and the
3.7
r*(a)
are defined as in section 2.4.
Aligned Rankings
The aligned rankings procedure (see Puri and Sen (1971)) makes
use of a transformation to subtract off the block effects in the randomized complete blocks model.
ranked from
1
to
N,
Then the aligned observations are
and statistics are calculated in an effort to
compare observations among blocks as well as among treatments.
As in
the weighted rankings procedure, it is hoped that some of the information lost as a result of considering the within block rankings alone
will be recovered.
Define the aligned observations as
(1)
Y.. = X.. - X.
1J
1
J
1
i=l, ... ,n, j=l, ... ,k.
.
81
To derive the LMP aligned rank test, the joint density of the
must first be considered.
densities as the
Then the density
rro
ri
TT
i=1
j=l
k
n
(2)
f(v)
:0:
/..,
n
IS
k-variate
are correlated within each block, and indepen-
Y..
IJ
dent among blocks.
This density is the product of
Y ..
IJ
may be written as
f (x. +y .. ; 6 y. ) dx .
J
1
IJ
J
1
_00
=
ro
f
.. •
_00
Let
peE)
"
J
IT f ex. +y .. ; 6y J.) TTdx.
.
1
J
1
_00
n
k
IT
as in (3.5.5).
f(x.+y., ,.t,y.)
J]
1
Then peE)
.
in a Taylor series around
]
IT
_00
~
=J···f
R
f"V
= 0
J
'
_00
1
L:oo
_00
r
oo
6y.
is written as
=f· R.. f IItfOOj ... JooIT. ,IT f CX. +y .. ;
1
1)
)
co~n
k
1
J
J
J
0) (l-6y . g (x. +y .. ) +0 (6)) Trdx dy
J
1
1)
.
1
~
1
l ~
n k
n
-,
1
-.l
Ii ... )
r .
""IT.IT f(x.+y
.. ;0) I 1-2 L6yog(x +y 0)+0(6) ITdx.ld;r
1
I)
JON
r
rN
.
11
._00
g(x +y )
r rt
_00
= -f'
(x +y )/f(x +y )
r r£
r r£
Proceeding as before, form the ratio
(4)
1
1
ooJr IIlfoo ... foo i ]IT
fCX. +y .. ; 6y.) frdXJd l
'
1
1)
J i
1
R
f"<o..J
where
1J
be the probability of observing a given set of aligned rank-
ings, and expand
(3) peE) =Jr.
roon
rN
for
r=l, ... ,n, t=l, ... k.
82
and the locally optimal scores are of the form
(5)
These scores are not easily expressed in a closed form since their
actual
the
computation depends on the joint distribution of
Y..
1J
X.
1
and all
However, the following approximation may be used.
'So
Let
(6)
where
F
is the c.d.f. of the
X..
1J
'So
While the scores of (6) are
not strictly LMP, they provide an adequate approximation since the
block effects are taken to be null.
3.8
Example
The data of sections 3.3 and 3.6 are again used to calculate an
example.
The observations are first aligned by subtracting off their
block averages as in (3.7.1).
ranks appear below.
The aligned observations, and their
Note that there is one tie, between
Yll
and
Block
2
Treatment
1
3
-13.3(1)
- 8.7 (3)
8.3(18) - 1.3(9)
- 1.7(8)
-7.7(5)
2
-
3
.7(10)
14.7(21)
10.3(19)
4
-7.7(6)
5
5 (1 5) -12(2)
5.3(16) -5 (7)
2.3(14)
6
o(11)
7
-8'\4)
1 (12)
10 7 (13)
11 (20)
6.7(17)
but since both the observations are in the same treatment group it does
not matter how the tie is broken.
The model and hypotheses of section
83
3.3 are retained, and tested here.
Define
(1)
where
£1
£2
If
Dj 1 =
F
ej - e1 '
an c1
t
est
(0,0)
i
:::
(l, 0)
I
= (0,1)'
£3
and
:::
H0: '" 2 1 = '" 31 = 0
vs
the scores in
is taken to be the standard normal c.d.f.
(3.7.6) become
(2)
-e
i=1, ... ,7" j=1,2,3,
and the aligned rankings statistics are defined by
7
T
( 3)
The vector
Am
=
I
3
L
i=l j=1
c
cjJ-
jm
1[R-2l.
.. )
m=1,2.
N+1
IA
IA
and its variance
Q
A
::: (.8,5.1)
I
,
is calculated as
r2
= 2.66,
'
L:'l
where
(4)
is the same as in section 2.3, and
is defined as
~
nk
2
0A'" _. (n(k-l))-1 \\ (aeR .. ) - aeRo ))
LL
ij
I)
I.
84
where
aCR. )
1.
1 k
k
L aCR I).. )
j
To test against the ordered alternative, make the following transformation:
I A = E*T ,-,- ~l
~
~A
and
E*D~A~E*'
~
fACa )'
a
and
Q;Ca)
~
':i.
1
~A(a)
= T
= 2. 66
for the various sets
'v-l
~
- 0 -- T
~
-0
-
~A(a) ~ACaa)~A(aa)~ACa)
o
o
{l}
3.4
{2}
2.2
{l, 2}
~
a
are presented below.
~
0*
~ACa)
~
=0
~ACaa)
-0
~-l
- D
~
-- 0
-
~ACaa)~ACaa)~ACaa)
3.99
(5.9,5.1)'
The UI-aligned ranking statistic is given by
T* = \'
A
L
a
{CT'
D*-l T
)I(T~A(a)
~ACa)~A(a)~ACa)
~s 9,5.1)(·125{i
> 0)1(0* -
- ~
T -
< OJ}
~A(a)~A(a) - ~
'fl [~: ;1]
::: 7.6825 ,
and its approximate p-value, calculated as in the previous examples, is
e-
85
1 -
~
(1)
+-}
(.994424)
+~
(.978533)J = .009 .
For the alternatives of (3.4.1), the limiting distribution of
T
~A
is
(5)
where the
lim N- 1QA
exists and is of full rank.
Let
mea)
be the
N-7=
matrix such that
bution of
rAta) = m(a)I .
A
Then the asymptotic nonnull distri-
is written as
(6)
where
(7)
and the
3.9
r*(a)
are defined as in section 2.4.
Testing for Order in an Analysis of Covariance
As in section 2.8, ur rank tests for the analysis of covariance
can be constructed quite easily by combining the ur principle
existing methods.
with
For the within block rankings case, Gerig (1975) has
proposed a multivariate test which can be readily adapted to form the
ur test.
Let
X..
1J
jth treatment,
be the primary variate in the i
i
= 1, .. ,n.
Y.. = (Y~., ... ,Y~.) '.
~lJ
1.1
1)
j
= 1, ... ,k.
th
block receiving the
and define the covariates as
Observations are ranked for each variate within
blocks, and this ranking is denoted by
R~ ., s
1J
= I, ... , p where
p
= q+l.
86
1
The ranks
S ::
R..
1)
2, ... ,p
s
R.. ,
correspond to the primary variate, and the ranks
correspond to the covariates.
1 nk
II
n..
T :: -
(1)
m
1)
Define the statistics
1
c. a(R .. )
1)
Jm
1)
and
1 nk
s
== c. a(R .. )
m n.. Jill
1J
t
U
(2)
n
s==t+l, t==l, .... q.
1J
Let
and
Then
~
== (I' .g')',
~
is a
-n
(4 )
~
( 1 \J
identity matrix,
(k-l) x (k-l)
CCa .. B))), V :: ((V)),
rs
Var{~}:: ~
~k-l - lk-lj~k-l
matrix of ones, the symbol
1J~
1
1
q
q
(U 1 • ... ,Uk _1 ' ... , U1 ' •.. , Uk _1) , •
and has variance
6 .- 1 [1
(3)
!2 ::
)
@~
where
'
J
~k-l
is a
(k-l)
@ denotes the Kronecker product
x
(k-l)
CA
@ ~ ::
and
nk
\',
r
s
V
rs :: n(k-I) LL a(Rij)a(R ij )
ij
1
r,s::l, ... ,p.
Writing
(5)
the vector of statistics adjusted for the covariates is defined as
(6)
and has variance
(7)
T
~a
'T'
1 -
87
Now the UI statistic can be calculated as before.
For the case
of weighted rankings, the same method may be used after weighting the
within block rankings with the appropriate block weights.
The same
p variates within a block.
block weights are used for all
A similar method for aligned rankings is presented by Puri and
Sen (1971).
cell be
Again let the p-variate response vector for the
Z .. = (X .. , Y .. )
1J '~lJ
observations are
(i,j)
th
vvhere
~lJ
I
q
Y .. = (Y .. , .•. ,Y . . ).
k ~lJ
1J
1J
1
Z~. ;:; ~lJ
Z.. - -k \ Z. "
~ ~lJ
J
aligned observation for the
s
th
vat ions for each variate being ranked from
statistics
T ,
Nm
1
q
~Nm = CSNm",·,SNm)
and
(8)
T
R~.
and
~lJ
1J
(i,j)th
The aligned
is the rank of the
variate, the aligned obser1
to
nk
= N.
Define the
where
I nk
1
= c. a (R .. )
Nm
n .. Jm
1J
1J
n
and
1 nk
(9)
= --
II
c. a (R: . )
Jm
1J
n .,
1J
Al so, define the
p
x
p
permutation covariance matrix as
1
nk
--C'k----l)
II
a(R:.)aCR~.)
n
. .
1J
1J
(10)
1J
(11)
define
(12)
s=t+l, t=l, ... ,q .
T*
Nm
V (EN) =
N
r,s=l, ... ,p.
88
IN = (TNI ,·· .,TNk _1)',
Then the vector of adjusted statistics is
and
its variance is
D 1[I _[_1)
k-I
(13)
~
=
n
IVN,11.2
J
~k-l
~k-lJ
where
(14)
Once
V
N,11.2
T*
~N
Q
and
in the usual
==V
N,ll
-V
-1
V
V
~N,12~N,22~N,21
have been obtained, the
UI
statistic is calculated
manner.
3.10 The 2 Factor Experiment Replicated in
n
Complete Blocks
Consider the situation described in section 2.10, only now there
is one observation per cell for each of
n 2
factor experiments.
To
test for order among the levels of one of the factors, the observations
are first summed over the levels of the other factor, and then one of
the 3 methods described in this chapter can be applied.
Let
i==l, .... n
(1)
j=I, ... ,p,
k = 1 .... ,q ,
where
S.
1
level of factor
tor
T. I S
J
(2)
2.
. th
is the
1
1,
and
block effect,
Yk
T.
J
is the effect of the
is the effect of the
The usual side conditions hold.
define
q
I
y .. =
X"k
1J
k=l 1J
k
th
. th
J
level of fac-
To test for order among the
e-
89
The
Y.. 's
can be regarded as
1)
function
where
G(y .. )
1J
dom variables.
nk
random variables with distribution
G is the c.d.f. of the sum of
q
Li.d. ran-
Now one of the previously described complete blocks
methods may be used to test for order among the
Similarly, the levels of factor
2
p
levels of factor
1.
can be tested for order.
3.11 Summary_
In this chapter, the
UI-LMP
rank test method has been extended
to tests against ordered alternatives in the complete blocks design.
For the within block rankings case, the
to the test of Chapter 2.
UI-LMP
rank test is similar
For both the weighted rankings, and aligned
rankings procedures, no general closed forms have been obtained for the
LMP
scores.
suggested.
In both cases approximations to the
LMP scores have been
Some computations suggest care should be exercised when
arbitrarily selecting block weighting scores for the weighted rankings
test.
Some further numerical work will be presented in chapter 4.
Finally, in sections 3.9 and 3.10, methods have been suggested for testing for order in the analysis of covariance, and the
ment replicated in
n
complete blocks.
2
factor experi-
CHAPTER 4
Monte-Carlo Results
4.1
Introduction
Chinchilli and Sen (1981 a,b) report via large sample theory that
the UI statistic follows a weighted chi-squared distribution which is
refered to in the literature as the
-2
X
distribution.
The purpose
of this chapter is to investigate the power of the UI-LMP rank tests
discussed in Chapters 2 and 3 for samples ranging in size from 30 to
125 using the weighted
X
2
two ad hoc FORTRAN programs.
approximation.
Simulations were run using
The first program compares the UI-LMP
rank test of Chapter 2 with Jonckheere's test, and the Normal Scores
test.
The second program compares the tests described in Chapter 3 for
the analysis of complete blocks.
Page's test.
These tests are also compared with
A more detailed description of the simulations and their
results is presented in sections 4.2, and 4.3.
4.2
Simulated Experiments for the One-Way Layout
For each of the following 32 cases, sao independent experiments
\liere simulated hy generating normal random samples and calculating the
UI-LMP rank statistic, Jonckheere's statistic, and the Normal Scores
statistic for each sample.
imations,
Using the appropriate large sample approx-
the proportion of rejections at the
recorded for each test.
Note that for
k>3
a= .05
level was
the approximations of
91
Gupta (1963) are used to calculate the UI critical values.
Simula-
tions were TIln for different comhinations of the number of treatments,
number of observations per treatment (equal sample sizes were used),
and type of alternative.
Let
k
be the number of treatments, and
observations per treatment.
m be the number of
Three cases are considered:
the case when
all means are equal, the case when the means are equally spaced, and
the case when the means are unequally spaced.
m considered are
k
= 3,4,5,
and
The values of
m = 10,15,20,25.
When
k
k
and
= 3,
unequal spacing is a possibility, but since the UI statistic is concerned with
k-l
parameters it is not very interesting.
Hence, 32
situations are examined.
Table 1 gives the proportion of rejections for all combinations
of
k
and
m when the vector of treatment means is null.
Tables
2a and 2b give the results when the treatment means are equally spaced.
To generate Table 2a, mean vectors of the form
used where
2N
-~
k
=
(1,2, ... ,k)
3,4,5.
-~
N
were used.
(0,1,2,5)'
-~
(1,2 •... ,k)
were
For Tab Ie 2b, mean vectors of the form
The unequally spaced means case is repre-
sented by Tables 3a and 3b.
were
N
and
tive mean vectors were
For
-1
N
k::: 4, the respective mean vectors
(0,1,4,10), and for
N-~ (0,1,2,2,5),
and
k:::5,
the respec-
N-~ (0,1,4,4,10).
92
TABLE 1
Proportion of Rejections
(a = .05) for Case of Equal Means
93
TABLE 2a
(0.
:=
•
05)
Proportion of Rejections
for Case of Equally Spaced Means
94
TABLE 2b
(ex
=
.05)
Proportion of Rejections
for Case of Equally Spaced Means
k
m
Jonckheere
Normal Scores
UI -LMP
3
10
.440
.356
.390
3
15
.478
.408
.402
3
20
.440
.400
.382
3
25
.464
.456
.404
4
10
.670
.594
.576
4
15
.656
.598
.552
4
20
.680
.652
.558
4
25
.712
.678
.578
5
10
.848
.800
.730
5
15
.854
.824
.722
5
20
.866
.832
.754
5
25
.838
.812
.744
e
,
95
TABLE 3a
(ex = .05)
Proportion of Rejections
for Case of Unequally Spaced Means
96
TABLE 3b
(a. = .05)
Proportion of Rejections
for Case of Unequally Spaced Means
97
The rejection proportions shown in Table 1 should all be around
.05.
..
However, for the Normal Scores test all twelve proportions are
less than .05 and for the UI-LMP test all twelve are greater than .05.
This implies the UI-LMP test has an unfair advantage over the other
tests in the power comparisons, and the Normal Scores test is handicapped.
This also indicates that those two large sample approximations
are not good for the sample sizes tested.
Looking at Tables 2a-3b, the UI-LMP test has not done very well
since its rejection proportions are the smallest.
A priori it was
hypothesized that the UI-LMP test would be most powerful for the case
of unequally spaced means, and at least as powerful for the case of
equally spaced means.
It is possible that the weighted
mation would be better for larger sample sizes.
bons for
rea)
2
X
approxi-
Also, the approxima-
might have added further variation to the asymptotic
values.
4.3
Simulated Experiments for the Complete Blocks Design
The methods of section 4.2 were also used to simulate experiments
for the complete blocks design.
each sample:
Five statistics were calculated for
and
Page's statistic,
is the statistic of section 3.2,
T*
T*
A
T*
A
where
T*
W
is the statistic of section 3.7,
T*
T
OL
is based on a linear weighting as in section 3.6, and the statis-
tic
T
OC
Now
m corresponds to the number of blocks.
QL
and
QC
are weighted rankings statistics.
The statistic
and
is the same as in (3.5.12) with block weights as in (3.5.17).
98
TABLE 4
(a
=
Proportion of Rejections
.05) for Case of Equal Means
k
ill
Page
T*
W
T
QL
T
QC
T*
3
10
.046
.078
.082
.078
.090
3
15
.042
.084
.078
.084
.082
3
20
.056
.078
.070
.072
.058
3
25
.054
.060
.048
.058
.048
4
10
.060
.090
.080
.056
.080
4
15
.046
.064
.060
.066
.054
4
20
.058
.058
.056
.082
.060
4
25
.054
.086
.080
.048
.070
5
10
.052
.078
.080
.066
. 082
5
15
.058
.064
.066
.060
.060
5
20
.060
.068
.072
.052
.070
5
25
.056
.086
.040
.088
.088
A
e
.
99
TABLE Sa
(a= .05)
r
e
Proportion of Rejections
for Case of Equally Spaced Means
k
m
Page
T*
W
T
QL
T
QC
T*
3
10
.172
.150
.192
.] 48
.196
3
15
.156
.184
.186
.184
.204
3
20
.176
. ] 54
.178
.150
.184
3
25
.182
.164
.154
.164
.168
4
10
.278
.220
.206
.160
.218
4
IS
.274
.250
.240
.166
.262
4
20
.200
.174
.198
.150
.210
4
25
.244
.194
.218
.142
.238
5
10
.328
.228
.252
.166
.274
5
15
.354
.246
.254
.146
.298
5
20
.344
.286
.286
.150
.300
5
25
.312
.244
.250
.122
.274
A
100
TABLE Sb
(0.::= .05)
Proportion of Rejections
for Case of Equally Spaced Means
101
TABLE 6a
(ex = .05)
r
e
Proportion of Rejections
for Case of Unequally Spaced Means
k
ill
Page
T*
w
T*QL
Toe
T*
4
10
.504
.394
.396
.314
.454
4
15
.462
.394
.410
.270
.454
4
20
.454
.404
.414
.218
.456
4
25
.444
.356
.390
.202
.424
5
10
.388
.322
.324
.188
.376
5
15
.368
.314
.306
.136
.340
5
20
.352
.304
.336
.178
.348
5
25
.380
.326
.316
.134
.352
A
102
TABLE 6b
(0,::= .05)
Proportion of Rejections
for Case of Unequally Spaced Means
103
Table 4 shows rejection proportions greater than .05 for all of
the VI statistics.
In Tables Sa-6b,
T
A
compares favorably with
Page's test, but the difference in their null rejection proportions
must be considered.
Page's test
has higher rejection proportions
than the other three VI tests.
Comparing the VI tests with each other yields a clearer interpretation.
Here the aligned ranking statistic
T*
A
has consistently
=3
higher rej ecdon propoTtions than the other VI statistics.
When
k
T*
QL
k=4
and
5.
is better than
T*
IV'
and perhaps slightly better when
Also some light has been shed on the question raised in section
3.5 as to whether the blocks should be weighted inversely for the
weighted rakings statistic when the data are normal and the block variances are used to determine the block rankings.
yields far lower rejection proportions that
The statistic
T5L'
These results sup-
port the findings of Silva (1977), and Salama and Quade (1981).
poor performance of
T5c
T5c
The
may be due to the fact that some blocks are
weighted negatively, some blocks are weighted positively, and some of
the blocks may be given a weighting of
O.
Thus, rather than adding
strength to the analysis, these block weightings cancel out the information available from considering the blocks.
Adding a constant so
that these weights are all positive should be tried in the future.
4.4
Summary
Simulations have been run in an effort to investigate the asymptot-
ic
2
X
power of the VI tests described in Chapters 2 and 3.
approximations used for the
VI
The wSighted
statistics do not appear to be
104
very stable for the sample sizes selected.
However, the D1 tests for
the complete blocks design may be compared with some degree of confidence.
Here the UI aligned rankings statistic appears to be doing the
best job.
CHAPTER 5
Recommendations for Future Research
The UI-LMP rank test method presented in Chapter 2 has been
extended to tests against ordered alternatives in the analysis of complete blocks when within block rankings are used.
However, for both
the weighted rankings, and aligned rankings procedures, no general
closed forms have been obtained for the LMP scores.
cases have been discussed.
Some special
One area of research may involve further
investigation of such special cases.
Another problem which needs more
work is that of constructing UI-LM7J rank tests for the analysis of
covariance.
In both Chapters 2 and 3, ad hoc procedures have been sug-
gested which do not have the LMP property.
To construct UI-LMP tests,
the conditional distribution of the principal variate given the covariates must be considered.
In sections 2.10 and 3.10, analysis of the two factor experiment
has been discussed.
An interesting area of research deals with test-
ing for order in higher order layouts.
Factorial experiments are effi-
cient in the sense that they allow for simultaneous investigation of two
or more factors.
It is conceivable that an investigator may hypothesize
some ordering among the levels of one or more factors.
Currently there
are no tests against ordered alternatives in higher order layouts, and
research could be conducted toward this end.
106
Initially the problem of how to define the ordered alternatives
must be solved.
The ordering of interactions should be considered.
Once this problem is solved efforts should be made to apply the UI-LMP
rank test procedure.
Perhaps weighted rankings can be incorporated, the
weightings relating in some way to the other factors.
Consider as an example the two factor model with main effects
a.
1
that
and
La.
. 1
1
S.
and interaction
J
= LS'
=
.')
J
L(aS) ..
.
1J
=
1
(as) ...
I)
L(aS) ..
.
1J
J
The usual side conditions are
= O.
To define the ordering for
the levelsof a given factor, the interaction of the factors should be
taken into account.
Because of the above conditions, the usual con-
cept of ordering is not meaningful.
Perhaps a solution to this problem may be obtained using a technique simi 1ar to profi I e analys is.
Now, instead of mu ltivariate response
vectors for various groups, the levels of another factor are of interest.
Standard plots of means for the levels of one factor at the vari-
ous levels of another factor to show interaction in a two factor ANOVA
are quite similar to the plots associated with profile analysis.
A discussion of profile analysis is found in Norrison (1976). The
questions of interest concern parallelism of group means, differences
among groups, and differences among means within groups.
testing these hypotheses are well known.
Methods for
These same questions are
applicable to the problem at hand. Thus, the problem of testing against
order in higher order layouts can be investigated in light of the
methods of profile analysis.
Most recently, Chinchilli and Sen (1982) have considered multivariate linear rank statistics for profile analysis. They make use of the
107
ur principle while assuming that the response-sample interactions are
null, and also advocate the use of a preliminary test for these interactions.
In addition, more numerical work should be done to further
..
investigate the asymptotic power of the UI-LMP rank test.
While some
useful resuJ 1:5 have beE'n presented in Chapter 4, it wou] d be helpful
to run the simulations for some much larger sample sizes.
Also, simulations should be done to investigate the power of the
analysis of covariance methods presented in Chapters 2 and 3.
ent,
At pres-
not much has been written in this important area.
Finally, in the parametric case beta approximations are usually
E approxima-
available for small sample sizes.
These are known as
tions (see Barlow et al. (1972)).
In view of the simulation work of
Chapter 4 it may be worthwhile to investigate the adequacy of the
approximations for the various rank statistics discussed.
~
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