Biostatistics (eBook)

Front cover 1
Title page 4
Copyright page 5
Supporting Website 7
Table of Contents 8
Preface 16
Acknowledgments 17
1 Introduction 20
1.1 What Is Biostatistics? 20
1.2 Data — The Key Component of a Study 21
1.3 Design — The Road to Relevant Data 23
1.4 Replication — Part of the Scientific Method 25
1.5 Applying Statistical Methods 26
EXERCISES 26
REFERENCES 27
2 Data and Numbers 28
2.1 Data: Numerical Representation 28
2.2 Observations and Variables 29
2.3 Scales Used with Variables 29
2.4 Reliability and Validity 30
2.5 Randomized Response Technique 32
2.6 Common Data Problems 33
Conclusion 36
EXERCISES 37
REFERENCES 39
3 Descriptive Methods 40
3.1 Introduction to Descriptive Methods 40
3.2 Tabular and Graphical Presentation of Data 41
3.2.1 Frequency Tables 42
3.2.2 Line Graphs 43
3.2.3 Bar Charts 46
3.2.4 Histograms 49
3.2.5 Stem-and-Leaf Plots 54
3.2.6 Dot Plots 56
3.2.7 Scatter Plots 57
3.3 Measures of Central Tendency 58
3.3.1 Mean, Median, and Mode 59
3.3.2 Use of the Measures of Central Tendency 61
3.3.3 The Geometric Mean 61
3.4 Measures of Variability 64
3.4.1 Range and Percentiles 64
3.4.2 Box Plots 66
3.4.3 Variance and Standard Deviation 67
3.5 Rates and Ratios 70
3.5.1 Crude and Specific Rates 71
3.5.2 Adjusted Rates 72
3.6 Measures of Change over Time 74
3.6.1 Linear Growth 74
3.6.2 Geometric Growth 76
3.6.3 Exponential Growth 77
3.7 Correlation Coefficients 79
3.7.1 Pearson Correlation Coefficient 79
3.7.2 Spearman Rank Correlation Coefficient 82
Conclusion 83
EXERCISES 83
REFERENCES 87
4 Probability and Life Tables 90
4.1 A Definition of Probability 90
4.2 Rules for Calculating Probabilities 92
4.2.1 Addition Rule for Probabilities 92
4.2.2 Conditional Probabilities 94
4.2.3 Independent Events 96
4.3 Definitions from Epidemiology 99
4.3.1 Rates and Probabilities 99
4.3.2 Sensitivity, Specificity, and Predicted Value Positive and Negative 100
4.3.3 Receiver Operating Characteristic Plot 102
4.4 Bayes' Theorem 103
4.5 Probability in Sampling 106
4.5.1 Sampling with Replacement 106
4.5.2 Sampling without Replacement 107
4.6 Estimating Probabilities by Simulation 108
4.7 Probability and the Life Table 110
4.7.1 The First Four Columns in the Life Table 112
4.7.2 Some Uses of the Life Table 114
4.7.3 Expected Values in the Life Table 115
4.7.4 Other Expected Values in the Life Table 117
Conclusion 118
EXERCISES 118
REFERENCES 121
5 Probability Distributions 122
5.1 The Binomial Distribution 122
5.1.1 Binomial Probabilities 122
5.1.2 Mean and Variance of the Binomial Distribution 127
5.1.3 Shapes of the Binomial Distribution 129
5.2 The Poisson Distribution 130
5.2.1 Poisson Probabilities 130
5.2.2 Mean and Variance of the Poisson Distribution 132
5.2.3 Finding Poisson Probabilities 133
5.3 The Normal Distribution 135
5.3.1 Normal Probabilities 135
5.3.2 Transforming to the Standard Normal Distribution 137
5.3.3 Calculation of Normal Probabilities 138
5.3.4 The Normal Probability Plot 141
5.4 The Central Limit Theorem 143
5.5 Approximations to the Binomial and Poisson Distributions 145
5.5.1 Normal Approximation to the Binomial Distribution 145
5.5.2 Normal Approximation to the Poisson Distribution 148
Conclusion 150
EXERCISES 150
REFERENCES 152
6 Study Designs 154
6.1 Design: Putting Chance to Work 154
6.2 Sample Surveys and Experiments 156
6.3 Sampling and Sample Designs 157
6.3.1 Sampling Frame 158
6.3.2 Importance of Probability Sampling 159
6.3.3 Simple Random Sampling 160
6.3.4 Systematic Sampling 161
6.3.5 Stratified Random Sampling 163
6.3.6 Cluster Sampling 163
6.3.7 Problems Due to Unintended Sampling 164
6.4 Designed Experiments 167
6.4.1 Comparison Groups and Randomization 168
6.4.2 Random Assignment 169
6.4.3 Sample Size 171
6.4.4 Single- and Double-Blind Experiments 173
6.4.5 Blocking and Extraneous Variables 174
6.4.6 Limitations of Experiments 175
6.5 Variations in Study Designs 177
6.5.1 The Crossover Design 177
6.5.2 The Case-Control Design 178
6.5.3 The Cohort Study Design 179
Conclusion 179
EXERCISES 180
REFERENCES 185
7 Interval Estimation 188
7.1 Prediction, Confidence, and Tolerance Intervals 188
7.2 Distribution-Free Intervals 189
7.2.1 Prediction Interval 189
7.2.2 Confidence Interval 190
7.2.3 Tolerance Interval 194
7.3 Confidence Intervals Based on the Normal Distribution 195
7.3.1 Confidence Interval for the Mean 196
7.3.2 Confidence Interval for a Proportion 201
7.3.3 Confidence Intervals for Crude and Adjusted Rates 204
7.4 Confidence Interval for the Difference of Two Means and Proportions 207
7.4.1 Difference of Two Independent Means 207
7.4.2 Difference of Two Dependent Means 213
7.4.3 Difference of Two Independent Proportions 215
7.4.4 Difference of Two Dependent Proportions 216
7.5 Confidence Interval and Sample Size 217
7.6 Confidence Intervals for Other Measures 219
7.6.1 Confidence Interval for the Variance 220
7.6.2 Confidence Interval for the Pearson Correlation Coefficient 222
7.7 Prediction and Tolerance Intervals Based on the Normal Distribution 224
7.7.1 Prediction Interval 224
7.7.2 Tolerance Interval 225
Conclusion 225
EXERCISES 226
REFERENCES 230
8 Tests of Hypotheses 232
8.1 Preliminaries in Tests of Hypotheses 232
8.1.1 Terms Used in Hypothesis Testing 234
8.1.2 Determination of the Decision Rule 235
8.1.3 Relationship of the Decision Rule, a and ß 237
8.1.4 Conducting the Test 240
8.2 Testing Hypotheses about the Mean 242
8.2.1 Known Variance 242
8.2.2 Unknown Variance 247
8.3 Testing Hypotheses about the Proportion and Rates 248
8.4 Testing Hypotheses about the Variance 250
8.5 Testing Hypotheses about the Pearson Correlation Coefficient 251
8.6 Testing Hypotheses about the Difference of Two Means 253
8.6.1 Difference of Two Independent Means 253
8.6.2 Difference of Two Dependent Means 256
8.7 Testing Hypotheses about the Difference of Two Proportions 257
8.7.1 Difference of Two Independent Proportions 257
8.7.2 Difference of Two Dependent Proportions 258
8.8 Tests of Hypotheses and Sample Size 259
8.9 Statistical and Practical Significance 262
Conclusion 262
EXERCISES 263
REFERENCES 267
9 Nonparametric Tests 268
9.1 Why Nonparametric Tests? 268
9.2 The Sign Test 268
9.3 The Wilcoxon Signed Rank Test 272
9.4 The Wilcoxon Rank Sum Test 276
9.5 The Kruskal-Wallis Test 280
9.6 The Friedman Test 281
Conclusion 283
EXERCISES 283
REFERENCES 287
10 Analysis of Categorical Data 288
10.1 The Goodness-of-Fit Test 288
10.2 The 2 by 2 Contingency Table 292
10.2.1 Comparing Two Independent Binomial Proportions 293
10.2.2 Expected Cell Counts Assuming No Association: Chi-Square Test 293
10.2.3 The Odds Ratio — a Measure of Association 296
10.2.4 Fisher’s Exact Test 298
10.2.5 The Analysis of Matched-Pairs Studies 299
10.3 The r by c Contingency Table 301
10.3.1 Testing Hypothesis of No Association 301
10.3.2 Testing Hypothesis of No Trend 303
10.4 Multiple 2 by 2 Contingency Tables 305
10.4.1 Analyzing the Tables Separately 306
10.4.2 The Cochran-Mantel-Haenszel Test 307
10.4.3 The Mantel-Haenszel Common Odds Ratio 309
Conclusion 310
EXERCISES 310
REFERENCES 314
11 Analysis of Survival Data 316
11.1 Data Collection in Follow-up Studies 316
11.2 The Life-Table Method 318
11.3 The Product-Limit Method 325
11.4 Comparison of Two Survival Distributions 329
11.4.1 The CMH Test 329
11.4.2 The Normal Distribution Approach 332
11.4.3 The Log-Rank Test 332
11.4.4 Use of the CMH Approach with Small Data Sets 333
Conclusion 335
EXERCISES 335
REFERENCES 339
12 Analysis of Variance 342
12.1 Assumptions for Use of the ANOVA 342
12.2 One-Way ANOVA 343
12.2.1 Sums of Squares and Mean Squares 344
12.2.2 The F Statistic 345
12.2.3 The ANOVA Table 346
12.3 Multiple Comparisons 348
12.3.1 Error Rates: Individual and Family 348
12.3.2 The Tukey-Kramer Method 349
12.3.3 Fisher’s Least Significant Difference Method 349
12.3.4 Dunnett’s Method 350
12.4 Two-Way ANOVA for the Randomized Block Design with m Replicates 351
12.5 Two-Way ANOVA with Interaction 354
12.6 Linear Model Representation of the ANOVA 358
12.6.1 The Completely Randomized Design 358
12.6.2 The Randomized Block Design with m Replicates 360
12.6.3 Two- Way ANOVA with Interaction 360
12.7 ANOVA with Unequal Numbers of Observations in Subgroups 361
Conclusion 364
EXERCISES 364
REFERENCES 366
13 Linear Regression 368
13.1 Simple Linear Regression 368
13.1.1 Estimation of the Coefficients 370
13.1.2 The Variance of Y|X 372
13.1.3 The Coefficient of Determination (R2) 374
13.2 Inference about the Coefficients 376
13.2.1 Assumptions for Inference in Linear Regression 376
13.2.2 Regression Diagnostics 377
13.2.3 The Slope Coefficient 380
13.2.4 The Y-intercept 381
13.2.5 An ANOVA Table Summary 382
13.3 Interval Estimation for µY|X and Y|X 383
13.3.1 Confidence Interval for µY|X 383
13.3.2 Prediction Interval for Y|X 385
13.4 Multiple Linear Regression 387
13.4.1 The Multiple Linear Regression Model 387
13.4.2 Specification of a Multiple Regression Model 388
13.4.3 Parameter Estimates, ANOVA, and Diagnostics 393
13.4.4 Multicollinearity Problems 395
13.4.5 Extending the Regression Model: Dummy Variables 397
Conclusion 399
EXERCISES 399
REFERENCES 404
14 Logistic and Proportional Hazards Regression 406
14.1 Simple Logistic Regression 406
14.1.1 Proportion, Odds, and Logit 408
14.1.2 Estimation of Parameters 409
14.1.3 Computer Output 410
14.1.4 Statistical Inference 411
14.2 Multiple Logistic Regression 413
14.2.1 Model and Assumptions 413
14.2.2 Residuals 417
14.2.3 Goodness-of-Fit Statistics 418
14.2.4 The ROC Curve 420
14.3 Ordered Logistic Regression 422
14.4 Conditional Logistic Regression 426
14.5 Introduction to Proportional Hazard Regression 428
Conclusion 434
EXERCISES 435
REFERENCES 438
15 Analysis of Survey Data 440
15.1 Introduction to Design-Based Inference 440
15.2 Components of Design-Based Analysis 441
15.2.1 Sample Weights 441
15.2.2 Poststratification 442
15.2.3 The Design Effect 443
15.3 Strategies for Variance Estimation 445
15.3.1 Replicated Sampling: A General Method 445
15.3.2 Balanced Repeated Replication 446
15.3.3 Jackknife Repeated Replication 447
15.3.4 Linearization Method 449
15.4 Strategies for Analysis 450
15.4.1 Preliminary Analysis 451
15.4.2 Subpopulation Analysis 452
15.5 Some Analytic Examples 453
15.5.1 Descriptive Analysis 453
15.5.2 Contingency Table Analysis 454
15.5.3 Linear and Logistic Regression Analysis 456
Conclusion 459
EXERCISES 459
REFERENCES 461
Appendix A: Review of Basic Mathematical Concepts 464
CHAPTER 3 464
3.1 The Logic of Logarithms 464
3.2 Properties of Logarithms 464
3.3 Natural Logarithms 465
3.4 Conversion between Bases 465
3.5 Exponential Function 465
CHAPTER 4 467
4.1 Factorials 467
4.2 Permutations 467
4.3 Combinations 467
CHAPTER 15 467
15.1 Taylor Series Expansion 467
REFERENCE 469
Appendix B: Statistical Tables 470
Appendix C: Selected Governmental Sources of Biostatistical Data 500
I. Population Census Data 500
II. Vital Statistics 501
III. Sample Surveys 502
IV. Life Tables 503
REFERENCES 504
Appendix D: Solutions to Selected Exercises 506
Chapter 1 506
Chapter 2 506
Chapter 3 506
Chapter 4 507
Chapter 5 507
Chapter 6 507
Chapter 7 508
Chapter 8 509
Chapter 9 509
Chapter 10 509
Chapter 11 510
Chapter 12 510
Chapter 13 510
Chapter 14 511
Chapter 15 511
Index 512