Discrete distributions: applications in the health sciences

296 Pages·2004·1.214 MB·English

by  Daniel Zelterman

Discrete
Distributions
Applications in the Health Sciences
Daniel Zelterman
Division of Biostatistics,
Epidemiology and Public Health,
Yale University, USA
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ISBN0-470-86888-0(alk.paper)
1.Medicalsciences—Mathematics.2.Medicalsciences—Mathematicalmodels.3.
Medicine—Mathematics.4.Distribution(Probabilitytheory)5.Probabilities.I.Title.II.
Series.
RA853.M3Z4562004
(cid:1) (cid:1)
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2004041199
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To the memory of my father, M.Z.
D.Z.
Contents
Preface xi
Acknowledgements xvii
About the Author xix
1 Introduction 1
1.1 Discrete Distributions in General . . . . . . . . . . . . . . . . . . 1
1.2 Multivariate Discrete Distributions. . . . . . . . . . . . . . . . . . 4
1.3 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 The Multinomial Distribution . . . . . . . . . . . . . . . . . . . . 8
1.5 Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Negative Binomial Distribution . . . . . . . . . . . . . . . . . . . 13
1.7 Hypergeometric Distribution . . . . . . . . . . . . . . . . . . . . . 17
1.7.1 Negative hypergeometric distribution . . . . . . . . . . . . 19
1.7.2 Extended hypergeometric distribution . . . . . . . . . . . . 20
1.8 Stirling’s Approximation . . . . . . . . . . . . . . . . . . . . . . . 23
2 Maximum Negative Binomial Distribution 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.1 Outfitting the ark . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.2 Medical screening application . . . . . . . . . . . . . . . . 30
2.2 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.1 Shapes of the distribution . . . . . . . . . . . . . . . . . . 34
2.2.2 Moments of the distribution . . . . . . . . . . . . . . . . . 36
2.2.3 Modes of the distribution . . . . . . . . . . . . . . . . . . 38
2.3 Asymptotic Approximations . . . . . . . . . . . . . . . . . . . . . 39
2.3.1 Large values of c and p (cid:1)=1/2 . . . . . . . . . . . . . . 40
2.3.2 Large values of c and p =1/2 . . . . . . . . . . . . . . 42
2.3.3 Extreme values of p . . . . . . . . . . . . . . . . . . . . . 43
2.4 Estimation of p . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.4.1 The likelihood function . . . . . . . . . . . . . . . . . . . 45
2.4.2 The EM estimate . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.3 A Bayesian estimate of p . . . . . . . . . . . . . . . . . . 52
vii
viii CONTENTS
2.5 Programs and Numerical Results. . . . . . . . . . . . . . . . . . . 53
2.6 Appendix: The Likelihood Kernel . . . . . . . . . . . . . . . . . . 55
3 The Maximum Negative Hypergeometric Distribution 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 The Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 Properties and Approximations. . . . . . . . . . . . . . . . . . . . 61
3.3.1 Modes of the distribution . . . . . . . . . . . . . . . . . . 65
3.3.2 A gamma approximation . . . . . . . . . . . . . . . . . . . 66
3.3.3 A half-normal approximation . . . . . . . . . . . . . . . . 67
3.3.4 A normal approximation . . . . . . . . . . . . . . . . . . . 68
3.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.1 The half-normal approximation . . . . . . . . . . . . . . . 72
3.5.2 The normal approximate distribution . . . . . . . . . . . . 73
4 Univariate Discrete Distributions for Use with Twins 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 The Univariate Twins Distribution . . . . . . . . . . . . . . . . . . 80
4.3 Measures of Association in Twins . . . . . . . . . . . . . . . . . . 84
4.4 The Danish Twin Registry . . . . . . . . . . . . . . . . . . . . . . 88
4.4.1 Estimate of the effect . . . . . . . . . . . . . . . . . . . . 90
4.4.2 Approximations . . . . . . . . . . . . . . . . . . . . . . . 92
4.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5.1 The univariate twins distribution . . . . . . . . . . . . . . 93
4.5.2 Approximating distributions . . . . . . . . . . . . . . . . . 94
4.6 Programs for the Univariate Twins Distribution . . . . . . . . . . . 97
5 Multivariate Distributions for Twins 103
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Conditional Distributions . . . . . . . . . . . . . . . . . . . . . . . 105
5.2.1 Univariate conditional distribution. . . . . . . . . . . . . . 105
5.2.2 Conditional association measure . . . . . . . . . . . . . . . 109
5.3 Conditional inference for the Danish twins . . . . . . . . . . . . . 111
5.4 Simultaneous Multivariate Distributions . . . . . . . . . . . . . . . 115
5.5 Multivariate Examination of the Twins . . . . . . . . . . . . . . . 118
5.6 Infinitesimal Multivariate Methods. . . . . . . . . . . . . . . . . . 119
5.6.1 Models with no dependence . . . . . . . . . . . . . . . . . 120
5.6.2 Models for dependence . . . . . . . . . . . . . . . . . . . 121
5.6.3 The infinitesimal data . . . . . . . . . . . . . . . . . . . . 124
5.7 Computer Programs. . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.7.1 Conditional distribution and association models in SAS . . 125
5.7.2 Fortran program for multivariate inference . . . . . . . . . 132
CONTENTS ix
6 Frequency Models for Family Disease Clusters 141
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.1.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.1.2 Sampling methods employed . . . . . . . . . . . . . . . . 146
6.1.3 Incidence and clustering . . . . . . . . . . . . . . . . . . . 147
6.2 Exact Inference Under Homogeneous Risk . . . . . . . . . . . . . 148
6.2.1 Enumeration algorithm . . . . . . . . . . . . . . . . . . . . 151
6.2.2 Ascertainment sampling . . . . . . . . . . . . . . . . . . . 152
6.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.3.1 IPF in COPD families . . . . . . . . . . . . . . . . . . . . 153
6.3.2 Childhood cancer syndrome . . . . . . . . . . . . . . . . . 154
6.3.3 Childhood mortality in Brazil . . . . . . . . . . . . . . . . 155
6.3.4 Household T.cruzi infections . . . . . . . . . . . . . . . . 156
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.5 Appendix: Mathematical Details . . . . . . . . . . . . . . . . . . . 158
6.5.1 The distribution of family frequencies . . . . . . . . . . . 158
6.5.2 A model for covariates . . . . . . . . . . . . . . . . . . . . 160
6.5.3 Ascertainment sampling . . . . . . . . . . . . . . . . . . . 161
6.6 Program for Exact Test of Homogeneity . . . . . . . . . . . . . . 161
7 Sums of Dependent Bernoulli’s and Disease Clusters 173
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.2 Conditional Models . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.2.1 General results for conditional models . . . . . . . . . . . 176
7.2.2 Family history model . . . . . . . . . . . . . . . . . . . . 179
7.2.3 Incremental risk model . . . . . . . . . . . . . . . . . . . . 183
7.2.4 The exchangeable, beta-binomial distribution . . . . . . . . 186
7.2.5 Application to IPF example . . . . . . . . . . . . . . . . . 188
7.3 Exchangeable Models. . . . . . . . . . . . . . . . . . . . . . . . . 189
7.3.1 Exchangeable family history . . . . . . . . . . . . . . . . . 195
7.3.2 Exchangeable incremental risk model . . . . . . . . . . . . 200
7.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.5 Appendix: Proof of Exchangeable Distribution . . . . . . . . . . . 206
8 Weighted Binomial Distributions and Disease Clusters 209
8.1 Weighted Models and Clustering. . . . . . . . . . . . . . . . . . . 209
8.2 The Altham Distribution . . . . . . . . . . . . . . . . . . . . . . . 212
8.3 Application to Childhood Mortality Data . . . . . . . . . . . . . . 218
8.4 A Log-linear Weighted Distribution . . . . . . . . . . . . . . . . . 224
8.5 Quadratic Weighted Distributions . . . . . . . . . . . . . . . . . . 227
8.6 Weighted Distributions in General . . . . . . . . . . . . . . . . . . 230
8.7 Family History Log-linear Model . . . . . . . . . . . . . . . . . . 233
8.8 Summary Measures and IPF Example . . . . . . . . . . . . . . . . 234
8.9 SAS Program for Clustered Family Data . . . . . . . . . . . . . . 235
x CONTENTS
9 Applications to Teratology Experiments 243
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
9.2 Dominant Lethal Assay. . . . . . . . . . . . . . . . . . . . . . . . 246
9.3 Shell Toxicology Experiment. . . . . . . . . . . . . . . . . . . . . 250
9.4 Toxicology of 2,4,5 T . . . . . . . . . . . . . . . . . . . . . . . . 255
Complements 265
References 267
Index 273
Preface
Thisbookdescribesanumberofnewdiscretedistributionsthatariseinthestatisti-
calexaminationofrealexamples.Theseexamplesaredrawnfrommyinteractions
with researchers studying cancer, epidemiology, demography, and other health-
related disciplines. In every case, an understanding of the issues surrounding the
data provides the chief motivation for the subsequent development of the mathe-
matical models. Without the data and the helpful discussions with others in these
different disciplines, this book would not have been possible.
Theworkhererepresentsseveralyears’experienceoflookingatseveralexam-
ples of real data. In each of these examples, there was an important question to
be answered through the use of statistical modeling. The principal examples that
gave rise to these models are as follows:
• A design of a medical screening protocol for a genetic marker (Chapters 2
and 3)
• A measure of the association in lifespans of twins (Chapters 4 and 5)
• Multiple cases of a disease occurring in the same family or animal litter
(Chapters 6 through 9).
The distributions that came about in the examination of these three examples
haveanalogiestosamplingtwodifferentcoloredballsfromanurn.Samplingballs
from an urn is a useful teaching tool but, unfortunately, tends to lack practical
importance. Similarly, the urn models will be briefly mentioned when they are
applicable but will not be emphasized. The distributions developed here describe
sampling procedures from the urn
• until c of each color are observed (Chapter 3),
• two balls at a time until the urn is emptied (Chapter 4),
• a handful of balls at a time until the urn is emptied (Chapter 6),
respectively,correspondingtothethreemotivatingsituations.Table 1isanoutline
of the distributions discussed in this book and their specific references to the text
that follows.
xi
xii PREFACE
Theoriginalsourcematerialforalloftheseexamplesanddistributionshasbeen
published by myself and with coauthors, in some cases. The published material
has appeared in various journals and book chapters. My first aim is to unify these
results in one place.
Asecondaimistoprovideadditionaldetailforthevariousmodels.Onedraw-
back to journal publication is the terse nature required of the authors of such
articles. I anticipate a more leisurely style that includes more discussion of the
motivation of the methods, rather than the headlong rush to obtain mathematical
results.
Software will allow the reader to repeat the data analyses presented here and

extendthesetootherdatasetsaswell.S-PlusorR provideahigher-levelplatform,
allowingtheusertoomitmanyofthetediousbookkeepingchoresthatFORTRAN
or C++ ask of the programmer. There are several settings here where extensive
iterationsareneededandtheseprogramsarewritteninFORTRAN.Thesesituations
appearinChapters 5and 6,whereexactsignificancelevelsareobtainedonlyafter
completely enumerating huge numbers of all possible outcomes. There are also
settingsinChapters 4and 8wherethedatatakesatabularformandtheregression
diagnosticsinSASmakethatlanguagetheidealtoolforthedataanalysis.Iexpect
that the software that is included in this text will be posted on a web site.
TheaudienceIamaimingatarethosedoinganadvancedgraduatelevelcourse
and those doing research in the field. There are a number of useful techniques
identified here that reappear in several places. These methods will prove useful to
others doing work in the study of discrete distributions.
The work of Johnson, Kotz, and Kemp (Univariate Discrete Distributions,
1992, Wiley) is a popular book, now in its second edition. I would expect that
individuals who bought that book would be interested in this book as well. The
Johnson text is an outstanding reference encyclopedia of methods and references.
ThecompanionvolumecoveringmultivariatedistributionsisthatbyJohnson,Kotz,
and Balakrishnan (1997).
TherecentbookbyBalakrishnanandKoutras(2002)alsodescribesalargenum-
berofdiscretedistributions.Twoout-of-printbooks,UrnModels (1977)byJohnson
and Kotz and Random Allocations, the translation of Sluqai˘nye razmeweni(cid:1)
by Kolchin, Sevast’yanov and Chistyakov(1978), are specific to the mathematical
developmentofurnmodelsanddonotputanyspecialemphasisonapplications.In
contrast,Iwanttocoverlessmaterial,includenewmaterial,andmotivatetheseideas
byseveralintriguingnumericalexamples.
Brief overview of contents
AbriefoverviewofthemodelsdescribedinthisvolumeisgiveninTable 1.These
distributions can be divided into the two categories as to sampling from either an
infiniteorfromafinitesizedpopulation.Finite-sizedpopulationsareassumedtobe
sampledwithoutreplacement.Inafinitepopulation,theprobabilityoffutureevents
PREFACE xiii
Table 1 Comparison of related sampling distributions
Infinite population Finite population
Sampling scheme or with replacement without replacement
Predetermined Binomial Hypergeometric
number of items distribution (1.6) distribution (1.17)
Until c Negative binomial Negative hypergeometric
successes distribution (1.13) distribution (1.21)
Until either Riff shuffle Minimum negative
c successes or the minimum hypergeometric
or c failures negative binomial distribution (3.5)
distribution (2.1)
Until at least Maximum negative Maximum negative
c successes and binomial distribution hypergeometric
c failures (2.4) distribution (3.6), (3.7)
Two items The twins
at a time distribution (4.2)
A handful of The family frequency
items at a time distribution (6.3)
dependsonthecompositionofindividualsalreadysampled.Ininfinitepopulations,
the distribution of future events is independent of those of the past.
The binomial distribution describes the number of successes when a predeter-
minednumberofitemsareindependentlysampledfromaninfinitelylargeBernoulli
population. The hypergeometric distribution is the corresponding model when the
population has a finite size. The negative binomial and negative hypergeometric
distributions describe the number of trials necessary in order to obtain a specified
number of ‘successes.’
The minimum and maximum distributions of Table 1 refer to the number of
trials necessaryin order to observe at least a specified number of successesand/or
failures.Themaximumnegativebinomialisthedistributionofthenumberoftrials
neededinordertoobtainatleast c successesand c failures.Oneexampleofthisis
theproblemfacingNoahwhenstockingtheark.Howmanywildanimalsneedtobe
capturedinordertoobtainone(orseven)ofbothsexes?Theoriginalmotivationfor
thedevelopmentofthesedistributionsaroseinconnectionwithageneticscreening
clinical trial amongcolon-cancer patients.The maximum negative hypergeometric
distribution is the finite population analogy to this model. These distributions are
described in Chapters 2 and 3.
The twins distribution describes the behavior of samples from a finite popu-
lation (or ‘urn’) taken two at a time. This distribution is described in Chapter 4.
Generalizations of this distribution that examine dependent observations are given