Biostatistics A Bayesian Introduction

by Woodworth, George G.

• ISBN13: 9780471468424
• ISBN10: 0471468428
• Edition: 1st
• Format: Hardcover
• Copyright: 2004-09-06
• Publisher: Wiley-Interscience

Summary

An essential introductory text linking traditional biostatistics with bayesian methods In recent years, Bayesian methods have seen an explosion of interest, with applications in fields including biochemistry, ecology, medicine, oncology, pharmacology, and public health. As an interpretive system integrating data with observation, the Bayesian approach provides a nuanced yet mathematically rigorous means of conceptualizing biomedical statistics--from diagnostic tests to DNA evidence. Biostatistics: A Bayesian Introduction offers a pioneering approach by presenting the foundations of biostatistics through the Bayesian lens. Using easily understood, classic Dutch Book thought experiments to derive subjective probability from a simple principle of rationality, the book connects statistical science with scientific reasoning. The author shows how to compute, interpret, and report Bayesian statistical analyses in practice, and illustrates how to reinterpret traditional statistical reporting--such as confidence intervals, margins of error, and one-sided p-values--in Bayesian terms. Topics covered include: * Probability and subjective probability * Distributions and descriptive statistics * Continuous probability distributions * Comparing rates and means * Linear models and statistical adjustment * Logistic regression and adjusted odds ratios * Survival analysis * Hierarchical models and meta-analysis * Decision theory and sample size determination The book includes extensive problem sets and references in each chapter, as well as complete instructions on computer analysis with the versatile SAS and WinBUGS software packages as well as the Excel spreadsheet program. For professionals and students, Biostatistics: A Bayesian Introduction offers an unique, real-world entry point into a remarkable alternative method of interpreting statistical data.

Author Biography

GEORGE G. WOODWORTH, PhD, holds dual professorships in statistics and public health at the University of Iowa. The author of more than 100 individual and collaborative publications, he received his PhD from the University of Minnesota.

Table of Contents

Contents	vii
Preface	xiii
Introduction to Statistical Science	1	(8)
The science of evidence	1	(1)
Degrees of Belief	2	(1)
Knowing is Believing	3	(2)
Other views of statistical science	5	(1)
Preview of Statistical Reasoning	6	(1)
Exercises	7	(1)
References	8	(1)
Probability	9	(16)
Casino Probabilities	9	(2)
Probability	11	(2)
Combining Sentences with And, Or, and Not	11	(1)
Conditional Probability	12	(1)
Laws of Probability	13	(2)
The Product Rule	13	(1)
The Addition Rule	14	(1)
The Complementation Rule	15	(1)
The Sure Thing Rule	15	(1)
Statistical Independence	15	(1)
Fair Bets	16	(1)
Fair Prices	17	(1)
House Odds, Fair Odds, and Statistical Odds	18	(1)
Expected value	19	(4)
Exercises	23	(1)
References	24	(1)
Subjective Probability	25	(22)
Interpretations of Probability	25	(1)
Definitions of Probability	25	(1)
Objectivity and Science	26	(1)
Thought Experiments to Expose Inconsistency	27	(1)
Subjective Probability and Degree of Belief	28	(1)
Books and Dutch Books	29	(1)
Rationality	30	(2)
The Laws of Probability via Dutch Book Arguments	32	(9)
A Subjective Casino	32	(5)
Sure Things, Sure losers, Complements	37	(1)
Fractional and Multiple Bets	37	(1)
House Odds Revisited	38	(1)
Conditional Probability and the Product Rule	38	(3)
Independence Revisited	41	(1)
Expected Value Revisited	41	(2)
Exercises	43	(2)
References and Additional Readings	45	(2)
Distributions and Descriptive Statistics	47	(16)
Introduction	47	(1)
Cases and Variables	47	(2)
Distributions	49	(5)
Stemplots	49	(3)
Frequency Tabulations	52	(1)
Histograms	53	(1)
Descriptive Statistics	54	(5)
Quantiles	55	(2)
Moments	57	(2)
Computing Descriptive Statistics	59	(1)
Exercises	60	(1)
References	61	(2)
Statistical Inference	63	(26)
Models	63	(3)
Updating beliefs	66	(10)
Bayes' Rule	66	(5)
Bayes' Rule in Diagnostic Testing	71	(1)
Bayes' Rule in DNA Evidence	72	(4)
Bayes' Rule in the Analysis of Rates	76	(6)
The Importance of Studying Rates	76	(1)
Box Models	77	(1)
A Toy Research Project: Spinning a Penny	78	(3)
Posterior Distributions	81	(1)
Credible sets	82	(1)
Bernoulli Processes	83	(3)
Exercises	86	(1)
References and Further Reading	87	(2)
Continuous Probability Distributions	89	(20)
Introduction	89	(1)
Continuously Distributed Probability	89	(2)
Probability Density Functions	91	(5)
Probability as Area	91	(1)
Normal Density Functions	92	(1)
Computing Areas under the Normal Density	93	(3)
Posterior Distribution of a Bernoulli Success Rate	96	(2)
Approximate credible intervals	98	(2)
The Beta Family of Distributions	100	(6)
Beta Densities	100	(2)
Beta Posterior Distribution of a Rate	102	(2)
Informative Beta Priors	104	(2)
Exercises	106	(2)
References	108	(1)
Comparing Two Rates	109	(32)
Posterior Distribution of a Difference	109	(1)
Models, Likelihoods, and Bayes' Rule	109	(5)
An approximate Posterior Distribution of Δ	114	(1)
Auxiliary and Hidden Hypotheses	115	(1)
Conventional Statistical Practices	116	(5)
Significance Testing and p Values	116	(2)
Confidence Intervals	118	(1)
Confidence Intervals and Flat-Prior Credible Intervals	119	(1)
Making Sense of p Values and Statistical Significance	119	(2)
Quantification of Comparative Risk	121	(12)
Prospective Studies	121	(1)
Relative Risk	122	(3)
Interpreting Published Confidence Intervals	125	(1)
Retrospective Studies and Odds Ratios	126	(4)
Posterior Distribution of the Log Odds Ratio	130	(3)
Computer Analysis of Rates and Proportions	133	(4)
WinBUGS	133	(1)
The SAS System	133	(4)
Exercises	137	(2)
References	139	(2)
Inference on Means	141	(38)
Models for Measurement Data	141	(2)
The t Family of Distributions	143	(4)
Comparing Two Means	147	(6)
Sampling from Approximately Normal Boxes	147	(2)
Interpreting Conventional Statistical Reports	149	(2)
Interpreting p Values	151	(2)
Making Data Normal by Transformations	153	(11)
A Single Nonnormal Sample	153	(6)
Comparing the Medians of Two Nonnormal Boxes	159	(1)
Box--Cox Power Transformations	159	(2)
Posterior Distribution of the Difference of two Medians	161	(3)
Analyzing Designed Experiments	164	(5)
Randomization, Control, and Blinding	164	(1)
Varieties of Experimental Designs	165	(4)
Comparing Means and Analyzing Experiments in SAS	169	(4)
Comparing Two or More Means	169	(2)
Analyzing Completely Randomized Experiments	171	(2)
Analyzing Crossover Experiments	173	(1)
Supplement: The Box-Cox Procedure	173	(2)
Exercises	175	(3)
References	178	(1)
Linear Models and Statistical Adjustment	179	(30)
Ethical Treatment of Human Subjects	179	(1)
The Need for Statistical Adjustment	179	(1)
Regression	180	(8)
Linear Regression	181	(2)
Computing the Slope and Intercept	183	(2)
Residuals and Residual Variation	185	(3)
Posterior Predictive Distribution	188	(1)
Statistical Adjustment Via Multiple Regression	188	(3)
Design Variables	191	(4)
Combining Equations via Boolean Variables	191	(2)
Reference Categories	193	(2)
Interpreting Published Multiple Regressions	195	(4)
Computing Adjusted Means and Differences in SAS	199	(6)
Analysis of Crossover and Related Designs	205	(2)
Exercises	207	(1)
References	208	(1)
Logistic Regression	209	(14)
Introduction	209	(2)
Interpreting Logistic Regression Coefficients	211	(1)
Adjusted Odds Ratios	212	(1)
A Case Study: Surviving Melanoma	213	(5)
Computing Logistic Regression with SAS Software	218	(3)
Exercises	221	(1)
References	222	(1)
Hierarchical Models	223	(22)
Meta-analysis	223	(2)
Hierarchical Models	225	(1)
Normal Approximation	226	(1)
Exchangeability	227	(1)
The Statistical Model: Prior and Likelihood	228	(3)
The Prior	229	(1)
Likelihood of the Data	229	(2)
Simulation of the Posterior Distribution	231	(5)
Borrowing Strength	236	(4)
The Gamma Family of Distributions	240	(2)
Exercises	242	(2)
References	244	(1)
Time to Event Analysis	245	(34)
Introduction	245	(1)
Life Tables	245	(3)
Survival Analysis	248	(12)
The Kaplan--Meier Survival Function Estimate	249	(3)
Interpretation of Kaplan--Meier Statistics	252	(3)
Comparing Two Survival Curves	255	(1)
Unanticipated Consequences of Complete Ignorance	255	(3)
Interpreting p values	258	(1)
Computing Kaplan--Meier Estimates	258	(2)
Regression Methods	260	(14)
Proportional Hazards Regression	261	(6)
Computing Proportional Hazards Regressions	267	(3)
Weibull Regression	270	(1)
Weibull Regression with WinBUGS	271	(3)
The Human Hazard Function	274	(2)
Exercises	276	(2)
References	278	(1)
Decision Analysis	279	(20)
Evidence and Decisions	279	(1)
Acts and Consequences	280	(2)
Utility of Non-monetary Consequences	282	(1)
Deciding on the Right Sample Size	283	(8)
Noninferiority Studies	283	(4)
The Optimal Terminal Decision	287	(1)
The Preposterior Distribution of Future Observations	288	(1)
Optimal Sample Size	289	(2)
Informal Sample Size Selection	291	(5)
Controlling Credible Interval Width	292	(1)
Estimating a Bernoulli Success Rate	293	(3)
Exercises	296	(2)
References	298	(1)
A Tables	299	(6)
Left tail areas of the standard normal distribution	300	(2)
Quantiles of the standard t distribution family	302	(3)
B Introduction to WinBUGS	305	(28)
Introduction	305	(1)
Specifying the Model --- Prior and Likelihood	306	(1)
Inference about a single proportion	307	(8)
Setting Up the Model	308	(2)
Computing the Posterior Distribution	310	(5)
Two rates -- Difference, Relative Risk, and Odds Ratio	315	(4)
``For'' Loops	319	(2)
Data Entry	321	(2)
Embedding a Data Table in WinBUGS	321	(2)
Loading Data from an Embedded Table	323	(1)
Placing Output in a Fold	323	(1)
Additional Resources	324	(1)
Weibull Proportional Hazards Regression	324	(8)
References	332	(1)
C Introduction to SAS System Software	333	(12)
Data Entry --- The DATA Step	334	(3)
Statistical Analyses --- The PROC Step	337	(3)
Importing files into The SAS System	340	(5)
Permanent SAS System Binary Files	340	(2)
Importing Data Base Files and Tab-Delimited Files	342	(3)
Index	345