Discrete distributions Applications in the Health Sciences.pdf
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Daniel Zelterman
John Wiley & Sons, 2005-10-18 - 306 页

There have been many advances in the theory and applications of discrete distributions in recent years. They can be applied to a wide range of problems, particularly in the health sciences, although a good understanding of their properties is very important. Discrete Distributions: Applications in the Health Sciences describes a number of new discrete distributions that arise in the statistical examination of real examples. For each example, an understanding of the issues surrounding the data provides the motivation for the subsequent development of the statistical models.

• Provides an overview of discrete distributions and their applications in the health sciences.
• Focuses on real examples, giving readers an insight into the utility of the models.
• Describes the properties of each distribution, and the methods that led to their development.
• Presents a range of examples from the health sciences, including cancer, epidemiology, and demography.
• Features discussion of software implementation – in SAS, Fortran and R – enabling readers to apply the methods to their own problems.
• Written in an accessible style, suitable for applied statisticians and numerate health scientists.
• Software and data sets are made available on the Web.
  

Discrete Distributions: Applications in the Health Sciences provides a practical introduction to these powerful statistical tools and their applications, suitable for researchers and graduate students from statistics and biostatistics. The focus on applications, and the accessible style of the book, make it an excellent practical reference source for practitioners from the health sciences.



Contents
Preface xi
Acknowledgements xvii
About the Author xix
1 Introduction 1
1.1 DiscreteDistributionsinGeneral .................. 1
1.2 Multivariate Discrete Distributions . . . . . . . . . . . . . . . . . . 4
1.3 BinomialDistribution......................... 5
1.4 TheMultinomialDistribution .................... 8
1.5 PoissonDistribution ......................... 10
1.6 NegativeBinomialDistribution ................... 13
1.7 HypergeometricDistribution..................... 17
1.7.1 Negative hypergeometric distribution . . . . . . . . . . . . 19
1.7.2 Extended hypergeometric distribution . . . . . . . . . . . . 20
1.8 Stirling’sApproximation....................... 23
2 Maximum Negative Binomial Distribution 25
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.1.1 Outfitting the ark . . . . . . . . . . . . . . . . . . . . . . . 29
2.1.2 Medicalscreeningapplication................ 30
2.2 ElementaryProperties ........................ 33
2.2.1 Shapesofthedistribution .................. 34
2.2.2 Momentsofthedistribution................. 36
2.2.3 Modesofthedistribution .................. 38
2.3 AsymptoticApproximations ..................... 39
2.3.1 Large values of c and p
= 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 TheEMestimate....................... 49
2.4.3 A Bayesian estimate of p .................. 52
2.5 ProgramsandNumericalResults................... 53
2.6 Appendix: The Likelihood Kernel . . . . . . . . . . . . . . . . . . 55
3 The Maximum Negative Hypergeometric Distribution 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 TheDistribution ........................... 60
3.3 PropertiesandApproximations.................... 61
3.3.1 Modesofthedistribution .................. 65
3.3.2 Agammaapproximation................... 66
3.3.3 A half-normal approximation . . . . . . . . . . . . . . . . 67
3.3.4 Anormalapproximation................... 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 MeasuresofAssociationinTwins.................. 84
4.4 TheDanishTwinRegistry ...................... 88
4.4.1 Estimateoftheeffect .................... 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 Modelsfordependence ................... 121
5.6.3 Theinfinitesimaldata .................... 124
5.7 ComputerPrograms.......................... 125
5.7.1 Conditional distribution and association models in SAS . . 125
5.7.2 Fortran program for multivariate inference . . . . . . . . . 132
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 Incidenceandclustering................... 147
6.2 Exact Inference Under Homogeneous Risk . . . . . . . . . . . . . 148
6.2.1 Enumerationalgorithm.................... 151
6.2.2 Ascertainmentsampling ................... 152
6.3 NumericalExamples ......................... 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. cruziinfections ................ 156
6.4 Conclusions.............................. 157
6.5 Appendix:MathematicalDetails................... 158
6.5.1 The distribution of family frequencies . . . . . . . . . . . 158
6.5.2 Amodelforcovariates.................... 160
6.5.3 Ascertainmentsampling ................... 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 ConditionalModels.......................... 176
7.2.1 General results for conditional models . . . . . . . . . . . 176
7.2.2 Familyhistorymodel .................... 179
7.2.3 Incrementalriskmodel.................... 183
7.2.4 The exchangeable, beta-binomial distribution . . . . . . . . 186
7.2.5 ApplicationtoIPFexample ................. 188
7.3 ExchangeableModels......................... 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 WeightedModelsandClustering................... 209
8.2 TheAlthamDistribution ....................... 212
8.3 Application to Childhood Mortality Data . . . . . . . . . . . . . . 218
8.4 ALog-linearWeightedDistribution ................. 224
8.5 QuadraticWeightedDistributions .................. 227
8.6 WeightedDistributionsinGeneral.................. 230
8.7 FamilyHistoryLog-linearModel .................. 233
8.8 SummaryMeasuresandIPFExample................ 234
8.9 SASProgramforClusteredFamilyData .............. 235
9 Applications to Teratology Experiments 243
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
9.2 DominantLethalAssay........................ 246
9.3 Shell Toxicology Experiment . . . . . . . . . . . . . . . . . . . . . 250
9.4 Toxicologyof2,4,5T ........................ 255
Complements 265
References 267
Index 273