Testing Problems with Linear or Angular Inequality Constraints

302 Pages·1990·10.12 MB·English

by  Johan C. Akkerboom (auth.)

Lecture Notes in
Statistics
Edited by J. Berger, S. Fienberg, J. Gani,
K. Krickeberg, I. Olkin, and B. Singer
62
Johan C. Akkerboom
Testing Problems with Linear or
Angular Inequality Constraints
Spri nger-Verlag
New York Berlin Heidelberg London Paris Tokyo Hong Kong
Author
Johan C. Akkerboom
Department of Statistical Methods, Central Bureau of Statistics
P.O. Box 4481,6401 CZ Heerlen, The Netherlands
Mathematical Subject Classification: 62-02, 62F03, 62F05, 62005, 62P15,
62P10, 65U05, 90C25
ISBN-13: 978-0-387-97232-9 e-ISBN-13: 978-1-4612-3392-3
001 :10.1007/978-1-4612-3392-3
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© Springer-Verlag Berlin Heidelberg 1990
2847/3140-543210 - Printed on acid-free paper
PREFACE
The present Lecture Notes in Statistics volume gives me the oppor
tunity to publish my doctoral thesis, cf. AKKERBOOM (1988). On
October 21, 1988, I received my Ph.D. in Mathematics and Physical
Sciences from the University of Groningen. In the following eight
months, my wife Iris and I were happily awaiting the birth of our
daughter Milena. At that time nothing else would have bothered me,
if it had not been for the following circumstances in professional
life.
First, just before I had finished the thesis, there appeared the
comprehensive book about order restricted statistical inference by
ROBERTSON ET AL. (1988). This book provided me with an extra stimu
lus to try and have the thesis published. The former book offers a
wealth of theory based on isotonic regression and direct applica
tion of the likelihood ratio principle. The approach worked out in
the latter one is about "Circular Likelihood Ratio Tests" that
arise if the order restricted alternative is substituted by a
circular-cone-shaped one and the likelihood ratio principle is
applied to the "auxiliary problem" thus obtained. This Lecture
Notes in Statistics volume may help to let statistical practice -or
maybe future statistical software- decide about the usefulness of
"CLR-test methodology" as a simple way of dealing with order re
stricted testing problems.
Secondly, Prof. Dr. W. Schaafsma, who -together with Dr. T.A.B.
Snijders- acted as my Ph.D. advisor, encouraged me to correct some
flaws and inaccuracies in the thesis. I am confident that enough
errors have remained to prevent his and my worries to be eliminated
altogether, but I trust the added Author Index (see References) and
Subject Index will make it easy for the reader to find his way.
Einighausen, October 19, 1989
ACKNOWLEDGEMENTS
One of the first things that struck me while enjoying the lectures
of Willem Schaafsma was his careful definition of the criteria
"most stringent" and "most stringent somewhere most powerful".
Later on I learned to appreciate the nice and simple geometry
behind the "most stringent linear tests" that he constructed for
many well-known testing problems with a restricted alternative in
the form of a polyhedral cone. (The geometrical results were partly
due to L.J. Smid.) Soon after Prof. R. Pincus published his paper
on the likelihood ratio test against a circular-cone-shaped alter
native, there was an exchange of letters about the practical need
for combining the results of Pincus with those about linear tests
against a polyhedral-cone-shaped alternative. I am very grateful to
Willem that he put me on this track, and, at about the same time,
urged me to follow a career in statistical consulting. I learned a
great deal from his witty and painstaking way of commenting on my
manuscripts, as well as from his human understanding of working
together with researchers from other disciplines.
I am grateful to Tom Snijders because he encouraged Willem to
involve me in the study of testing problems with a restricted
alternative. Tom's work on the relevant asymptotic optimality
theory made me aware of many fascinating mathematical intricacies,
which I still do not feel able to grasp the way he did. lowe much
to Tom's appreciation of the many facets of mathematical statistics
and its applications in society at large. I was amused both by
Tom's unorthodox way of becoming a professional statistician and by
Willem's unorthodox way of teaching the subject.
For their helpful discussions I am grateful to Prof. R. Pincus
(Karl Weierstrag Institut, Berlin, German Democratic Republic) and
to Prof. D.W. Muller and Dr. E. Mammen (Heidelberg University;
Federal German Republic).
VI
My fellow student Ton Steerneman accompanied me on my first
orientation in the subject of the present study. I thank him for
allowing me to use his results and insights after he got engaged in
other research projects.
My first work on the present study, in 1979 at the Department of
Mathematics of Groningen State University, was made possible by a
grant of the Netherlands Organization for the Advancement of Pure
Research (Z.Y.O., nowadays called N.Y.O.).
My work at the Pediatrics Department of Groningen State Univer
sity, from January 1980 until July 1985, made me aware of the
tremendous gap between "theoretical" and "applied statistics".
There were many persons that guided me as a "neophyte" statistical/
medical researcher on a fascinating slalom between mathematical
rigor and the everyday worries of making useful contributions to
the medical sciences, whether by statistical insight or by straight
thinking. In this respect I am grateful to Ben Humphrey and Sijmon
Terpstra, to name only the first two persons that come to my mind.
I thank the Stichting Kinderoncologie Groningen (S.K.O.G.) for the
time I was allowed to spend on this study.
A lecture by Youter Keller about personal computing drew my
attention to the kaleidoscopic world of the Department of Statisti
cal Methods at the Netherlands Central Bureau of Statistics, where
I have been employed since July 1985. I thank my present employer
for the support I enjoyed.
Special thanks are due to my colleagues Marly Odekerken-Smeets
and Pierre Reynders for their computational contributions to the
preparation of Figs. 4.6.1, 4.6.2, 4.8.1-4.8.5, and 5.1.1. Thanks
are due to Frans Bovenlander, Ed van Uden, and their colleagues for
the technical preparation of the illustrations and the appendices.
ABSTRACT
The present monograph gives a self-contained account of a generally
applicable and promising approach to testing problems with an
alternative restricted by linear inequality constraints. In such a
problem, the alternative is essentially a pointed r-dimensional
polyhedral cone (2~r<oo). The monograph focuses on the practical
implementation of Pincus's idea (1) to replace the linear inequal
ities (the polyhedral cone) by one angular inequality (a circular
cone), and (2) then to apply the likelihood ratio (LR-) principle.
Implementation requires that a suitable member be selected from the
class of available circular likelihood ratio (CLR-) tests.
Chapter 1 introduces the general class of testing problems for
which CLR-tests can be constructed. A number of case studies are
introduced -to be worked out later in Chapter 5. Topics include
testing the monotonicity assumptions of Mokken's one-dimensional
scaling model, and one-sided treatment comparisons in ANOVA-layouts
and in a two-period crossover trial.
Chapter 2 summarizes the basic results from the literature about
LR-tests and "linear" tests. Chapter 3 treats the CLR-test associ
ated with an arbitrary "substituting" circular cone. Critical
values are tabulated for r=2, ... ,13. For the "reversed problem",
where the null hypothesis is defined by a polyhedral cone, a CLR
test is derived.
Chapter 4 contains the main results of the book. In the normal
case, the most stringent CLR-test is determined as an improvement
on the most stringent linear test due to Schaafsma and Smid. In
other cases, Snijders's asymptotic optimality criterion "everywhere
asymptotically MS-1lr" applies to the MS-CLR test for the appropriate
limiting problem, where 1lr is the class of CLR-tests for that prob
lem. Other choices for the substituting circular cone, besides the
one associated with the MS-CLR test, are discussed. Overall power
properties of the CLR-test turn out to be most satisfactory if the
opening angle is taken somewhat smaller than in the MS-CLR case,
while the axis is the same (the central direction of the polyhedral
VIII
cone). For many well-known applications, relevant formulas are
given so that CLR-tests can be easily constructed. E.g. for a2
known, the test statistic can be computed from the opening angle of
the substituting circular cone, the linear statistic associated
with the axis, and the x2-statistic that is ordinarily used for
testing against the unrestricted alternative.
CONTENTS1)
1 Testing problems with linear inequality constraints 1
1.0 General introduction and outline of results 1
1.0.1 Various shortcuts through the study 5
1.1* Notations 7
1.2 Testing statistical hypotheses 10
1.2.1 The Neyman-Pearson approach to testing statistical hypotheses 11
1.2.2 The selection of a test for (HO ,H1): restricting principles 16
1.2.3 The selection of a test for (HO ,H1): ordering principles 22
1.3 Cases from statistical practice 24
1.3.1 Test expectancy in educational psychology 25
1.3.2 Predatory behavior of hungry beetles 29
1.3.3 The assumption of double monotony in Mokken's latent trait model 34
1.4 The general problem with the alternative restricted by
linear inequalities 37
1.5 The canonical form: testing against the pointed poly-
hedral cone K 44
1.6 Particular classes of testing problems with the alter-
native restricted by linear inequalities 49
1.6.1 Testing against the positive orthant in IRm;
the combination of m independent one-sample tests 50
1.6.2 Testing homogeneity against upward trend 53
1.6.3 Testing additivity of effects against positive interaction
in a two-way analysis of variance 58
1.6.4 Testing goodness of fit for a multinomial distribution
with the alternative restricted by stochastic inequality 60
1.6.5 Testing homogeneity against stochastic trend
in a doubly-ordered k-by-m contingency table 62
1.6.6 Testing independence against stochastic positive association
in a doubly-ordered k-by-m contingency table 64
1.7 Problems with the null hypothesis restricted by linear
inequalities 65
2 The main problem: testing against the pointed polyhedral
cone K 70
2.0 Introduction and summary 70
1) Sections marked by an asterisk may be skipped at cursory reading.
x
2.1* Linear inequality constraints and the geometry of poly-
hedral cones 73
2.1.1 Extreme half-spaces and extreme rays 74
2.1.2 Partitioning a polyhedral cone by its facets 77
2.1.3 Orthogonal projection onto a closed convex cone 80
2.1.4 Orthogonal projection onto a polyhedral cone 81
2.2 Linear tests 83
2.2.1 Minimizing the maximum shortcoming over K within and with respect to
the class of somewhere most powerful (similar size-a) tests 83
2.2.2 The minimax ray and the minimax angle of K 86
2.3 Likelihood ratio tests 89
2.3.1 The Ef- and E2-statistics for testing against a closed convex cone 90
2.3.2 The null distributions of Ef and E2 94
2.3.3 The LR-test for the combination of tests problem 97
2.3.4 The LR-test against upward trend in a one-way analysis of variance 97
2.4 Testing a polyhedral-cone-shaped null hypothesis 101
2.4.1 The LR-tests 103
2.4.2 The union-intersection test (q2=1) 107
3 A modification of the main problem: testing against a
circular cone 110
3.0 Introduction and summary 110
3.1* An angular inequality constraint and the geometry of
circular cones 112
3.2 Likelihood ratio tests for the modified problem 114
3.2.1 The ~ - and If -statistics lor tes..!-ing against a circular cone 115
Ei
3.2.2 The null distributions of and E2 117
3.3* Computation of critical values of the likelihood ratio
test statistics for the modified problem 122
3.3.1 Tables of critical values 123
3.3.2 The program CRVCLRI (q2=1) 124
3.3.3 Evaluating a mixture of Beta-distributions
with parameters assuming successive half-integer values 125
3.4 A reduction of the modified problem by sufficiency and
invariance 129
3.5 Easy-to-use combination procedures for the reduced
modified problem 132
3.6* Other procedures for the reduced modified problem
(a2=1) 135
3.7 Some theory about the power properties of invariant
tests (a2=1) 137
3.7.1 Admissibility with respect to the class of translation-invariant
tests 139
3.7.2 Some monotonicity properties of the power function of the LR-test 141
3.7.3 Power properties when the test statistic preserves a partial order 144
3.8 Testing a circular-cone-shaped null hypothesis 149
4 Circular likelihood ratio tests for the main problem 155
4.0 Introduction and summary 155
4.1 Replacing the polyhedral cone K by some circular cone 158
4.2* Computation of the power of circular likelihood ratio
(CLR-) tests (a2=1) 162
4.3 Minimization of the maximum shortcoming of CLR-tests
over K (a2=1) 163
4.3.1 The criterion MS-CLR level-a 165
4.3.2 The criteria AMS-CLR level-a and EAMS-CLR level-a 169
4.3.3 The maximum shortcoming on half-lines and the MS-CLR test 172
4.4 The minimax ray and angle of K for some particular
cases 180
4.4.1 The positive orthant 183
4.4.2 Upward trend 183
4.4.3 Positive interaction in a two-way analysis of variance 187
4.4.4 Multinomial distributions stochastically larger
than a specified multinomial distribution 188
4.4.5 Stochastic trend in a k-by-m table 190
4.4.6 Stochastic positive association in a k-by-m table 191
4.5 The maximin ray and angle of K for some particular
cases 193
4.5.1 A duality correspondence 193
4.5.2 Pairs of dual problems 195
4.6 The use of CLR-tests 199
4.6.1 An example: the symmetrical one-sided multiple comparison problem 199
4.6.2 CLR-tests for the general problem or for the reversed general
problem 206
4.7 Power comparisons 208
4.7.1 The symmetrical one-sided multiple comparison problem 209
4.7.2 Testing homogeneity against upward trend 212
4.8 Graphs of the minimax angle and the maximin angle of K
for some particular cases 217
5 Applications 221
5.1 One-sided treatment comparison in the two-period cross-
over trial with binary outcomes 221
5.1.1 Testing exchangeability against a restricted alternative 221
5.1.2 The asymptotic problem of testing against the polyhedral cone K 224
5.1.3 The geometry of K and its circumscribed cone 226
5.1.4 The geometry of K and its inscribed cone 231
5.1.5 Linear tests 233
5.1.6 Replacing K by some circular cone; circular likelihood ratio
tests 235
5.2 Test expectancy in educational psychology 239