Introduction to Biostatistics
Introduction to Biostatistics
Larry Winner Department of Statistics
University of Florida
July 8, 2004
2
Contents
1 Introduction
7
1.1 Populations and Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Types of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.1 Quantitative vs Qualitative Variables . . . . . . . . . . . . . . . . . . . . . . 8
1.2.2 Dependent vs Independent Variables . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Parameters and Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4 Graphical Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Basic Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.1 Diagnostic Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Basic Study Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6.1 Observational Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6.2 Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.6.3 Other Study Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.7 Reliability and Validity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Random Variables and Probability Distributions
35
2.1 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.1 Statistical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Sampling Distributions and the Central Limit Theorem . . . . . . . . . . . . . . . . 39
2.2.1 Distribution of Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3 Statistical Inference ? Hypothesis Testing
45
3.1 Introduction to Hypothesis Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Large?Sample Tests Concerning ?1 - ?2 (Parallel Groups) . . . . . . . . . . . 47 3.2 Elements of Hypothesis Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.1 Significance Level of a Test (Size of a Test) . . . . . . . . . . . . . . . . . . . 48
3.2.2 Power of a Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Sample Size Calculations to Obtain a Test With Fixed Power . . . . . . . . . . . . . 51
3.4 Small?Sample Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.1 Parallel Groups Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.4.2 Crossover Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3
4
CONTENTS
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4 Statistical Inference ? Interval Estimation
65
4.1 Large?Sample Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 Small?Sample Confidence Intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.1 Parallel Groups Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.2 Crossover Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Categorical Data Analysis
71
5.1 2 ? 2 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.1 Relative Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.1.2 Odds Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.1.3 Extension to r ? 2 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.1.4 Difference Between 2 Proportions (Absolute Risk) . . . . . . . . . . . . . . . 76
5.1.5 Small?Sample Inference -- Fisher's Exact Test . . . . . . . . . . . . . . . . . 78
5.1.6 McNemar's Test for Crossover Designs . . . . . . . . . . . . . . . . . . . . . . 79
5.1.7 Mantel?Haenszel Estimate for Stratified Samples . . . . . . . . . . . . . . . . 81
5.2 Nominal Explanatory and Response Variables . . . . . . . . . . . . . . . . . . . . . . 83
5.3 Ordinal Explanatory and Response Variables . . . . . . . . . . . . . . . . . . . . . . 85
5.4 Nominal Explanatory and Ordinal Response Variable . . . . . . . . . . . . . . . . . . 87
5.5 Assessing Agreement Among Raters . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
6 Experimental Design and the Analysis of Variance
97
6.1 Completely Randomized Design (CRD) For Parallel Groups Studies . . . . . . . . . 97
6.1.1 Test Based on Normally Distributed Data . . . . . . . . . . . . . . . . . . . . 98
6.1.2 Test Based on Non?Normal Data . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.2 Randomized Block Design (RBD) For Crossover Studies . . . . . . . . . . . . . . . . 104
6.2.1 Test Based on Normally Distributed Data . . . . . . . . . . . . . . . . . . . . 104
6.2.2 Friedman's Test for the Randomized Block Design . . . . . . . . . . . . . . . 108
6.3 Other Frequently Encountered Experimental Designs . . . . . . . . . . . . . . . . . . 109
6.3.1 Factorial Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3.2 Crossover Designs With Sequence and Period Effects . . . . . . . . . . . . . . 114
6.3.3 Repeated Measures Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7 Linear Regression and Correlation
129
7.1 Least Squares Estimation of 0 and 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 130 7.1.1 Inferences Concerning 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
7.2 Correlation Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.3 The Analysis of Variance Approach to Regression . . . . . . . . . . . . . . . . . . . . 137
7.4 Multiple Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
CONTENTS
5
7.4.1 Testing for Association Between the Response and the Set of Explanatory Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4.2 Testing for Association Between the Response and an Individual Explanatory Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
8 Logistic and Nonlinear Regression
151
8.1 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2 Nonlinear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
9 Survival Analysis
163
9.1 Estimating a Survival Function -- Kaplan?Meier Estimates . . . . . . . . . . . . . . 163
9.2 Log?Rank Test to Compare 2 Population Survival Functions . . . . . . . . . . . . . 166
9.3 Relative Risk Regression (Proportional Hazards Model) . . . . . . . . . . . . . . . . 168
9.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
10 Special Topics in Pharmaceutics
179
10.1 Assessment of Pharmaceutical Bioequivalence . . . . . . . . . . . . . . . . . . . . . . 179
10.2 Dose?Response Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
10.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
A Statistical Tables
187
B Bibliography
193
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