Exercises and Solutions in Biostatistical Theory

Drawn from nearly four decades of Lawrence L. Kupper’s teaching experiences as a distinguished professor in the Department of Biostatistics at the University of North Carolina, Exercises and Solutions in Biostatistical Theory presents theoretical statistical concepts, numerous exercises, and detailed solutions that span topics from basic probability to statistical inference. The text links theoretical biostatistical principles to real-world situations, including some of the authors’ own biostatistical work that has addressed complicated design and analysis issues in the health sciences.

This classroom-tested material is arranged sequentially starting with a chapter on basic probability theory, followed by chapters on univariate distribution theory and multivariate distribution theory. The last two chapters on statistical inference cover estimation theory and hypothesis testing theory. Each chapter begins with an in-depth introduction that summarizes the biostatistical principles needed to help solve the exercises. Exercises range in level of difficulty from fairly basic to more challenging (identified with asterisks).

By working through the exercises and detailed solutions in this book, students will develop a deep understanding of the principles of biostatistical theory. The text shows how the biostatistical theory is effectively used to address important biostatistical issues in a variety of real-world settings. Mastering the theoretical biostatistical principles described in the book will prepare students for successful study of higher-level statistical theory and will help them become better biostatisticians.

Lawrence L. Kupper, Ph.D., is an emeritus alumni distinguished professor of biostatistics at the University of North Carolina at Chapel Hill. Dr. Kupper has received several teaching and research awards during his career and has been involved with many research areas in the health sciences, including epidemiology, environmental and occupational health, maternal and child health, and medicine. His research interests concern the development of innovative biostatistical methods for the design and analysis of public health research studies.
Brian H. Neelon, Ph.D., is a research statistician with the Children’s Environmental Health Initiative (CEHI) in the Nicholas School of the Environment at Duke University. Before working at Duke University, Dr. Neelon was a Postdoctoral Research Fellow in the Department of Health Care Policy at Harvard University. He earned his Ph.D. from the University of North Carolina. His research interests include Bayesian methods, longitudinal data analysis, finite mixture models, and health policy statistics.
Sean M. O’Brien, Ph.D., is an assistant professor of biostatistics and bioinformatics at the Duke University School of Medicine. Dr. O’Brien earned his Ph.D. from the University of North Carolina. His research interests include statistical methods for healthcare provider profiling, observational studies, and Bayesian data analysis.

Basic Probability TheoryCounting Formulas (N-tuples, permutations, combinations, Pascal’s identity, Vandermonde’s identity)Probability Formulas (union, intersection, complement, mutually exclusive events, conditional probability, independence, partitions, Bayes’ theorem)

Univariate Distribution TheoryDiscrete and Continuous Random Variables Cumulative Distribution Functions Median and Mode Expectation Theory Some Important Expectations (mean, variance, moments, moment generating function, probability generating function)Inequalities Involving ExpectationsSome Important Probability Distributions for Discrete Random VariablesSome Important Distributions (i.e., Density Functions) for Continuous Random Variables

Multivariate Distribution TheoryDiscrete and Continuous Multivariate DistributionsMultivariate Cumulative Distribution Functions Expectation Theory (covariance, correlation, moment generating function)Marginal Distributions Conditional Distributions and Expectations Mutual Independence among a Set of Random Variables Random Sample Some Important Multivariate Discrete and Continuous Probability DistributionsSpecial Topics of Interest (mean and variance of a linear function, convergence in distribution and the Central Limit Theorem, order statistics, transformations)

Estimation TheoryPoint Estimation of Population Parameters (method of moments, unweighted and weighted least squares, maximum likelihood)Data Reduction and Joint Sufficiency (Factorization Theorem)Methods for Evaluating the Properties of a Point Estimator (mean-squared error, Cramér–Rao lower bound, efficiency, completeness, Rao–Blackwell theorem)Interval Estimation of Population Parameters (normal distribution-based exact intervals, Slutsky’s theorem, consistency, maximum-likelihood-based approximate intervals)

Hypothesis Testing TheoryBasic Principles (simple and composite hypotheses, null and alternative hypotheses, Type I and Type II errors, power, P-value)Most Powerful (MP) and Uniformly Most Powerful (UMP) Tests (Neyman–Pearson Lemma)Large-Sample ML-Based Methods for Testing a Simple Null Hypothesis versus a Composite Alternative Hypothesis (likelihood ratio, Wald, and score tests)Large-Sample ML-Based Methods for Testing a Composite Null Hypothesis versus a Composite Alternative Hypothesis (likelihood ratio, Wald, and score tests)

Appendix: Useful Mathematical Results

References

Index

Exercises and Solutions appear at the end of each chapter.