About the author.Preface.1 Population Proportion or Prevalence.1.1 Binomial sampling.1.2 Cluster sampling.1.3 Inverse sampling.Exercises.References.2 Risk Difference.2.1 Independent binomial sampling.2.2 A series of independent binomial sampling procedures.2.2.1 Summary interval estimators.2.2.2 Test for the homogeneity of risk difference.2.3 Independent cluster sampling.2.4 Paired-sample data.2.5 Independent negative binomial sampling (inverse sampling).2.6 Independent poisson sampling.2.7 Stratified poisson sampling.Exercises.References.3 Relative Difference.3.1 Independent binomial sampling.3.2 A series of independent binomial sampling procedures.3.2.1 Asymptotic interval estimators.3.2.2 Test for the homogeneity of relative difference.3.3 Independent cluster sampling.3.4 Paired-sample data.3.5 Independent inverse sampling.Exercises.References.4 Relative Risk.4.1 Independent binomial sampling.4.2 A series of independent binomial sampling procedures.4.2.1 Asymptotic interval estimators.4.2.2 Test for the homogeneity of risk ratio.4.3 Independent cluster sampling.4.4 Paired-sample data.4.5 Independent inverse sampling.4.5.1 Uniformly minimum variance unbiased estimator of relative risk.4.5.2 Interval estimators of relative risk.4.6 Independent poisson sampling.4.7 Stratified poisson sampling.Exercises.References.5 Odds Ratio.5.1 Independent binomial sampling.5.1.1 Asymptotic interval estimators.5.1.2 Exact confidence interval.5.2 A series of independent binomial sampling procedures.5.2.1 Asymptotic interval estimators.5.2.2 Exact confidence interval.5.2.3 Test for homogeneity of the odds ratio.5.3 Independent cluster sampling.5.4 One-to-one matched sampling.5.5 Logistic modeling.5.5.1 Estimation under multinomial or independent binomial sampling.5.5.2 Estimation in the case of paired-sample data.5.6 Independent inverse sampling.5.7 Negative multinomial sampling for paired-sample data.Exercises.References.6 Generalized Odds Ratio.6.1 Independent multinomial sampling.6.2 Data with repeated measurements (or under cluster sampling).6.3 Paired-sample data.6.4 Mixed negative multinomial and multinomial sampling.Exercises.References.7 Attributable Risk.7.1 Study designs with no confounders.7.1.1 Cross-sectional sampling.7.1.2 Case-control studies.7.2 Study designs with confounders.7.2.1 Cross-sectional sampling.7.2.2 Case-control studies.7.3 Case-control studies with matched pairs.7.4 Multiple levels of exposure in case-control studies.7.5 Logistic modeling in case-control studies.7.5.1 Logistic model containing only the exposure variables of interest.7.5.2 Logistic regression model containing both exposure and confounding variables.7.6 Case-control studies under inverse sampling.Exercises.References.8 Number Needed to Treat.8.1 Independent binomial sampling.8.2 A series of independent binomial sampling procedures.8.3 Independent cluster sampling.8.4 Paired-sample data.Exercises.References.Appendix Maximum Likelihood Estimator and Large-Sample Theory.A.1: The maximum likelihood estimator, Wald's test, the score test, and the asymptotic likelihood ratio test.A.2: The delta method and its applications.References.Answers to Selected Exercises.Index.
Preface. 1. Probability and Sample Spaces. Why Study Probability? Probability. Sample Spaces. Some Properties of Probabilities. Finding Probabilities of Events. Conclusions. Explorations. 2. Permutations and Combinations: Choosing the Best Candidate Acceptance Sampling. Permutations. Counting Principle. Permutations with Some Objects Alike. Permuting Only Some of the Objects. Combinations. General Addition Theorem and Applications. Conclusions. Explorations. 3. Conditional Probability. Introduction. Some Notation. Bayes' Theorem. Conclusions. Explorations. 4. Geometric Probability. Conclusion. Explorations. 5. Random Variables and Discrete Probability Distributions-Uniform, Binomial, Hypergeometric, and Geometric Distributions. Introduction. Discrete Uniform Distribution. Mean and Variance of a Discrete Random Variable. Intervals, sigma , and German Tanks. Sums. Binomial Probability Distribution. Mean and Variance of the Binomial Distribution. Sums. Hypergeometric Distribution. Other Properties of the Hypergeometric Distribution. Geometric Probability Distribution. Conclusions. Explorations. 6. Seven-Game Series in Sports. Introduction. Seven-Game Series. Winning the First Game. How Long Should the Series Last? Conclusions. Explorations. 7. Waiting Time Problems. Waiting for the First Success. The Mythical Island. Waiting for the Second Success. Waiting for the r th Success. Mean of the Negative Binomial. Collecting Cereal Box Prizes. Heads Before Tails. Waiting for Patterns. Expected Waiting Time for HH. Expected Waiting Time for TH. An Unfair Game with a Fair Coin. Three Tosses. Who Pays for Lunch? Expected Number of Lunches. Negative Hypergeometric Distribution. Mean and Variance of the Negative Hypergeometric. Negative Binomial Approximation. The Meaning of the Mean. First Occurrences. Waiting Time for c Special Items to Occur. Estimating k. Conclusions. Explorations. 8. Continuous Probability Distributions: Sums, the Normal Distribution, and the Central Limit Theorem Bivariate Random Variables. Uniform Random Variable. Sums. A Fact About Means. Normal Probability Distribution. Facts About Normal Curves. Bivariate Random Variables. Variance. Central Limit Theorem: Sums. Central Limit Theorem: Means. Central Limit Theorem. Expected Values and Bivariate Random Variables. Means and Variances of Means. A Note on the Uniform Distribution. Conclusions. Explorations. 9. Statistical Inference I. Estimation. Confidence Intervals. Hypothesis Testing. beta and the Power of a Test. p -Value for a Test. Conclusions. Explorations. 10. Statistical Inference II: Continuous Probability Distributions II-Comparing Two Samples. The Chi-Squared Distribution. Statistical Inference on the Variance. Student t Distribution. Testing the Ratio of Variances: The F Distribution. Tests on Means from Two Samples. Conclusions. Explorations. 11. Statistical Process Control. Control Charts. Estimating sigma Using the Sample Standard Deviations. Estimating sigma Using the Sample Ranges. Control Charts for Attributes. np Control Chart. p Chart. Some Characteristics of Control Charts. Some Additional Tests for Control Charts. Conclusions. Explorations. 12. Nonparametric Methods. Introduction. The Rank Sum Test. Order Statistics. Median. Maximum. Runs. Some Theory of Runs. Conclusions. Explorations. 13. Least Squares, Medians, and the Indy 500. Introduction. Least Squares. Principle of Least Squares. Influential Observations. The Indy 500. A Test for Linearity: The Analysis of Variance. A Caution. Nonlinear Models. The Median-Median Line. When Are the Lines Identical? Determining the Median-Median Line. Analysis for Years 1911-1969. Conclusions. Explorations. 14. Sampling. Simple Random Sampling. Stratification. Proportional Allocation. Optimal Allocation. Some Practical Considerations. Strata. Conclusions. Explorations. 15. Design of Experiments. Yates Algorithm. Randomization and Some Notation. Confounding. Multiple Observations. Design Models and Multiple Regression Models. Testing the Effects for Significance. Conclusions. Explorations. 16. Recursions and Probability. Introduction. Conclusions. Explorations. 17. Generating Functions and the Central Limit Theorem. Means and Variances. A Normal Approximation. Conclusions. Explorations. Bibliography. Where to Learn More. Index.
Preface.Acknowledgements.About the Author.1 Introduction.1.1 Discrete Distributions in General.1.2 Multivariate Discrete Distributions.1.3 Binomial Distribution.1.4 The Multinomial Distribution.1.5 Poisson Distribution.1.6 Negative Binomial Distribution.1.7 Hypergeometric Distribution.1.7.1 Negative hypergeometric distribution.1.7.2 Extended hypergeometric distribution.1.8 Stirling's Approximation.2 Maximum Negative Binomial Distribution.2.1 Introduction.2.1.1 Outfitting the ark.2.1.2 Medical screening application.2.2 Elementary Properties.2.2.1 Shapes of the distribution.2.2.2 Moments of the distribution.2.2.3 Modes of the distribution.2.3 Asymptotic Approximations.2.3.1 Large values of c and p 1/2.2.3.2 Large values of c and p != 1/2.2.3.3 Extreme values of p.2.4 Estimation of p.2.4.1 The likelihood function.2.4.2 The EM estimate.2.4.3 A Bayesian estimate of p.2.5 Programs and Numerical Results.2.6 Appendix: The Likelihood Kernel.3 The Maximum Negative Hypergeometric Distribution.3.1 Introduction.3.2 The Distribution.3.3 Properties and Approximations.3.3.1 Modes of the distribution.3.3.2 A gamma approximation.3.3.3 A half-normal approximation.3.3.4 A normal approximation.3.4 Estimation.3.5 Appendix.3.5.1 The half-normal approximation.3.5.2 The normal approximate distribution.4 Univariate Discrete Distributions for Use with Twins.4.1 Introduction.4.2 The Univariate Twins Distribution.4.3 Measures of Association in Twins.4.4 The Danish Twin Registry.4.4.1 Estimate of the effect.4.4.2 Approximations.4.5 Appendix.4.5.1 The univariate twins distribution.4.5.2 Approximating distributions.4.6 Programs for the Univariate Twins Distribution .5 Multivariate Distributions for Twins.5.1 Introduction.5.2 Conditional Distributions.5.2.1 Univariate conditional distribution.5.2.2 Conditional association measure.5.3 Conditional inference for the Danish twins.5.4 Simultaneous Multivariate Distributions.5.5 Multivariate Examination of the Twins.5.6 Infinitesimal Multivariate Methods.5.6.1 Models with no dependence.5.6.2 Models for dependence.5.6.3 The infinitesimal data.5.7 Computer Programs.5.7.1 Conditional distribution and association models in SAS.5.7.2 Fortran program for multivariate inference.6 Frequency Models for Family Disease Clusters.6.1 Introduction.6.1.1 Examples.6.1.2 Sampling methods employed.6.1.3 Incidence and clustering.6.2 Exact Inference Under Homogeneous Risk.6.2.1 Enumeration algorithm.6.2.2 Ascertainment sampling.6.3 Numerical Examples.6.3.1 IPF in COPD families.6.3.2 Childhood cancer syndrome.6.3.3 Childhood mortality in Brazil.6.3.4 Household T. cruzi infections.6.4 Conclusions.6.5 Appendix: Mathematical Details.6.5.1 The distribution of family frequencies.6.5.2 A model for covariates.6.5.3 Ascertainment sampling.6.6 Program for Exact Test of Homogeneity.7 Sums of Dependent Bernoulli's and Disease Clusters.7.1 Introduction.7.2 Conditional Models.7.2.1 General results for conditional models.7.2.2 Family history model.7.2.3 Incremental risk model.7.2.4 The exchangeable, beta-binomial distribution.7.2.5 Application to IPF example.7.3 Exchangeable Models.7.3.1 Exchangeable family history.7.3.2 Exchangeable incremental risk model.7.4 Applications.7.5 Appendix: Proof of Exchangeable Distribution.8 Weighted Binomial Distributions and Disease Clusters.8.1 Weighted Models and Clustering.8.2 The Altham Distribution.8.3 Application to Childhood Mortality Data.8.4 A Log-linear Weighted Distribution.8.5 Quadratic Weighted Distributions.8.6 Weighted Distributions in General.8.7 Family History Log-linear Model.8.8 Summary Measures and IPF Example.8.9 SAS Program for Clustered Family Data.9 Applications to Teratology Experiments.9.1 Introduction.9.2 Dominant Lethal Assay.9.3 Shell Toxicology Experiment.9.4 Toxicology of 2,4,5 T.Complements.References.Index.
This entry-level text offers clear and concise guidelines on how to select, construct, interpret, and evaluate count data. Written for researchers with little or no background in advanced statistics, the book presents treatments of all major models using numerous tables, insets, and detailed modeling suggestions. It begins by demonstrating the fundamentals of modeling count data, including a thorough presentation of the Poisson model. It then works up to an analysis of the problem of overdispersion and of the negative binomial model, and finally to the many variations that can be made to the base count models. Examples in Stata, R, and SAS code enable readers to adapt models for their own purposes, making the text an ideal resource for researchers working in health, ecology, econometrics, transportation, and other fields.
Preliminaries The Poisson process and distribution Waiting time distributions for a Poisson process Statistical estimation theory Generating a Poisson process Nonhomogeneous Poisson process Binomial, geometric, and negative binomial distributions Statistical Life Length Distributions Stochastic life length models Models based on the hazard rate General remarks on large systems Reliability of Various Arrangements of Units Series and parallel arrangements Series-parallel and parallel-series systems Various arrangements of switches Standby redundancy Reliability of a One-Unit Repairable System Exponential times to failure and repair Generalizations Reliability of a Two-Unit Repairable System Steady-state analysis Time-dependent analysis via Laplace transform On model 2(c) Continuous-Time Markov Chains The general case Reliability of three-unit repairable systems Steady-state results for the n-unit repairable system Pure birth and death processes Some statistical considerations First Passage Time for Systems Reliability Two-unit repairable systems Repairable systems with three (or more) units Repair time follows a general distribution Embedded Markov Chains and Systems Reliability Computations of steady-state probabilities Mean first passage times Integral Equations in Reliability Theory Introduction Example 1: Renewal process with a general distribution Example 2: One-unit repairable system Example 3: Effect of preventive replacements or maintenance Example 4: Two-unit repairable system Example 5: One out of n repairable systems Example 6: Section 7.3 revisited Example 7: First passage time distribution References Index A Problems and Comments section appears at the end of each chapter.
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