Basic Statistics: A Primer for the Biomedical Sciences Book

Preface to the Fourth Edition xiii

1 Initial Steps 1

1.1 Reasons for Studying Biostatistics 1

1.2 Initial Steps in Designing a Biomedical Study 2

1.2.1 Setting Objectives 2

1.2.2 Making a Conceptual Model of the Disease Process 3

1.2.3 Estimating the Number of Persons with the Risk Factor or Disease 4

1.3 Common Types of Biomedical Studies 5

1.3.1 Surveys 6

1.3.2 Experiments 7

1.3.3 Clinical Trials 7

1.3.4 Field Trials 9

1.3.5 Prospective Studies 9

1.3.6 Case/Control Studies 10

1.3.7 Other Types of Studies 10

1.3.8 Rating Studies by the Level of Evidence 11

1.3.9 Consort 11

Problems 12

References 12

2 Populations and Samples 13

2.1 Basic Concepts 13

2.2 Definitions of Types of Samples 15

2.2.1 Simple Random Samples 15

2.2.2 Other Types of Random Samples 15

2.2.3 Reasons for Using Simple Random Samples 17

2.3 Methods of Selecting Simple Random Samples 17

2.3.1 Selection of a Small Simple Random Sample 17

2.3.2 Tables of Random Numbers 17

2.3.3 Sampling With and Without Replacement 19

2.4 Application of Sampling Methods in Biomedical Studies 19

2.4.1 Characteristics of a Good Sampling Plan 19

2.4.2 Samples for Surveys 20

2.4.3 Samples for Experiments 21

2.4.4 Samples for Prospective Studies 23

2.4.5 Samples for Case/Control Studies 23

Problems 25

References 26

3 Collecting and Entering Data 27

3.1 Initial Steps 27

3.1.1 Decide What Data You Need 28

3.1.2 Deciding How to Collect the Data 29

3.1.3 Testing the Collection Process 30

3.2 Data Entry 31

3.3 Screening the Data 33

3.4 Code Book 33

Problems 34

References 34

4 Frequency Tables and Their Graphs 35

4.1 Numerical Methods of Organizing Data36

4.1.1 An Ordered Array 36

4.1.2 Stem and Leaf Tables 36

4.1.3 The Frequency Table 38

4.1.4 Relative Frequency Tables 40

4.2 Graphs 40

4.2.1 The Histogram: Equal Class Intervals 41

4.2.2 The Histogram: Unequal Class Intervals 41

4.2.3 Areas Under the Histogram 43

4.2.4 The Frequency Polygon 44

4.2.5 Histograms with Small Class Intervals 45

4.2.6 Distribution Curves 45

Problems 47

References 47

5 Measures of Location and Variability 49

5.1 Measures of Location 50

5.1.1 The Arithmetic Mean 50

5.1.2 The Median 51

5.1.3 Other Measures of Location 52

5.2 Measures of Variability 52

5.2.1 The Variance and the Standard Deviation 52

5.2.2 Other Measures of Variability 54

5.3 Sampling Properties of the Mean and Variance 55

5.4 Considerations in Selecting Appropriate Statistics 57

5.4.1 Relating Statistics and Study Objectives 57

5.4.2 Relating Statistics and Data Quality 58

5.4.3 Relating Statistics to the Type of Data 58

5.5 A Common Graphical Method for Displaying Statistics 60

Problems 61

References 62

6 The Normal Distribution 63

6.1 Properties of the Normal Distribution 64

6.2 Areas Under the Normal Curve 65

6.2.1 Computing the Area Under a Normal Curve 66

6.2.2 Linear Interpolation 68

6.2.3 Interpreting Areas as Probabilities 70

6.3 Importance of the Normal Distribution 70

6.4 Examining Data for Normality 72

6.4.1 Using Histograms and Box Plots 72

6.4.2 Using Normal Probability Plots or Quantile-Quantile Plots 72

6.5 Transformations 75

6.5.1 Finding a Suitable Transformation 76

6.5.2 Assessing the Need for a Transformation 77

Problems 77

References 78

7 Estimation of Population Means: Confidence Intervals 79

7.1 Confidence Intervals 80

7.1.1 An Example 80

7.1.2 Definition of Confidence Interval 81

7.1.3 Choice of Confidence Level 82

7.2 Sample Size Needed for a Desired Confidence Interval 83

7.3 The t Distribution 83

7.4 Confidence Interval for the Mean Using the t Distribution 85

7.5 Estimating the Difference Between Two Means: Unpaired Data 86

7.5.1 The Distribution of &Xbar;1 - &Xbar;2 86

7.5.2 Confidence Intervals for μ1 - μ2: Known Variance 87

7.5.3 Confidence Intervals for μ1 - μ2: UnKnown Variance 88

7.6 Estimating the Difference Between Two Means: Paired Comparison 89

Problems 91

References 93

8 Tests of Hypotheses on Population Means 95

8.1 Tests of Hypotheses for a Single Mean 96

8.1.1 Test for a Single Mean When σ Is Known 96

8.1.2 One-Sided Tests When σ Is Known 99

8.1.3 Summary of Procedures for Test of Hypotheses 100

8.1.4 Test for a Single Mean When σ Is Unknown 101

8.2 Tests for Equality of two Means: Unpaired Data 103

8.2.1 Testing for Equality of Means When σ Is Known 103

8.2.2 Testing for Equality of Means When σ Is Unknown 104

8.3 Testing for Equality of Means: Paired Data 107

8.4 Concepts Used in Statistical Testing 108

8.4.1 Decision to Accept or Reject 108

8.4.2 Two Kinds of Error 109

8.4.3 An Illustration of β 110

8.5 Sample Size 111

8.6 Confidence Intervals Versus Tests 113

8.7 Correcting for Multiple Testing 114

8.8 Reporting the Results 115

Problems 115

References 116

9 Variances: Estimation and Tests 117

9.1 Point Estimates for Variances and Standard Deviations 118

9.2 Testing Whether Two Variances Are Equal: F Test 118

9.3 Approximate t Test 121

9.4 Other Tests 122

Problems 123

References 123

10 Categorical Data: Proportions 125

10.1 Single Population Proportion 126

10.1.1 Graphical Displays of Proportions 126

10.2 Samples from Categorical Data 128

10.3 The Normal Approximation to the Binomial 129

10.3.1 Use of the Normal Approximation to the Binomial 129

10.3.2 Continuity Correction 130

10.4 Confidence Intervals for a Single Population Proportion 130

10.5 Confidence Intervals for the Difference in Two Proportions 131

10.6 Tests of Hypothesis for Population Proportions 133

10.6.1 Tests of Hypothesis for a Single Population Proportion 133

10.6.2 Testing the Equality of Two Population Proportions 134

10.7 Sample Size for Testing Two Proportions 136

10.8 Data Entry and Analysis Using Statistical Programs 137

Problems 138

References 139

11 Categorical Data: Analysis of Two-Way Frequency Tables 141

11.1 Different Types of Tables 142

11.1.1 Tables Based on a Single Sample 142

11.1.2 Tables Based on Two Samples 143

11.1.3 Tables Based on Matched or Paired Samples 144

11.1.4 Relationship Between Type of Study Design and Type of Table 145

11.2 Relative Risk and Odds Ratio 146

11.2.1 Relative Risk 146

11.2.2 Odds Ratios 147

11.3 Chi-Square Tests for Frequency Tables: Two-by-Two Tables 150

11.3.1 Chi-Square Test for a Single Sample: Two-by-Two Tables 150

11.3.2 Chi-Square Test for Two Samples: Two-by-Two Tables 154

11.3.3 Chi-Square Test for Matched Samples: Two-by-Two Tables 155

11.3.4 Assumptions for the Chi-Square Test 156

11.3.5 Necessary Sample Size: Two-by-Two Tables 156

11.3.6 The Continuity Correction: Two-by-Two Tables 157

11.4 Chi-Square Tests for Larger Tables 158

11.4.1 Chi-Square for Larger Tables: Single Sample 158

11.4.2 Interpreting a Significant Test 159

11.4.3 Chi-Square Test for Larger Tables; More Than Two Samples or Outcomes 161

11.4.4 Necessary Sample Size for Large Tables 161

11.5 Remarks 162

Problems 162

References 164

12 Regression and Correlation 165

12.1 The Scatter Diagram: Single Sample 166

12.2 Linear Regression: Single Sample 168

12.2.1 Least-Squares Regression Line 168

12.2.2 Interpreting the Regression Coefficients 170

12.2.3 Plotting the Regression Line 170

12.2.4 The Meaning of the Least-Squares Line 170

12.2.5 The Variance of the Residuals 171

12.2.6 Model Underlying Single-Sample Linear Regression 172

12.2.7 Confidence Intervals in Single-Sample Linear Regression 174

12.2.8 Tests of Hypotheses for Regression Line from a Single Sample 176

12.3 The Correlation Coefficient for Two Variables From a Single Sample 177

12.3.1 Calculation of the Correlation Coefficient 177

12.3.2 The Meaning of the Correlation Coefficient 177

12.3.3 The Population Correlation Coefficient 179

12.3.4 Confidence Intervals for the Correlation Coefficient 179

12.3.5 Test of Hypothesis that ρ = 0 179

12.3.6 Interpreting the Correlation Coefficient 180

12.4 Linear Regression Assuming the Fixed-X Model 180

12.4.1 Model Underlying the Fixed-X Linear Regression 181

12.4.2 Linear Regression Using the Fixed-X Model 181

12.5 Other Topics in Linear Regression 183

12.5.1 Use of Transformations in Linear Regression 183

12.5.2 Effect of Outliers from the Regression Line 184

12.5.3 Multiple Regression 184

Problems 184

References 187

13 Nonparametric Statistics 189

13.1 The Sign Test 190

13.1.1 Sign Test for Large Samples 190

13.1.2 Sign Test When the Sample Size Is Small 191

13.2 The Wilcoxon Signed Ranks Test 192

13.2.1 Wilcoxon Signed Ranks Test for Large Samples 192

13.2.2 Wilcoxon Signed Ranks Test for Small Samples 194

13.3 The Wilcoxon-Mann-Whitney Test 195

13.3.1 Wilcoxon Rank Sum Test for Large Samples 195

13.3.2 Wilcoxon Rank Sum Test for Small Samples 197

13.4 Spearman's Rank Correlation 198

Problems 199

References 199

14 Introduction to Survival Analysis 201

14.1 Survival Analysis Data 202

14.1.1 Describing Time to an Event 202

14.1.2 Example of Measuring Time to an Event 202

14.2 Survival Functions 204

14.2.1 The Death Density Function 204

14.2.2 The Cumulative Death Distribution Function 205

14.2.3 The Survival Function 206

14.2.4 The Hazard Function 207

14.3 Computing Estimates of f(t), S(t), and h(t) 208

14.3.1 Clinical Life Tables 209

14.3.2 Kaplan-Meier Estimate 212

14.4 Comparison of Clinical Life Tables and the Kaplan-Meier Method 214

14.5 Additional Analyses Using Survival Data 215

14.5.1 Comparing the Equality of Survival Functions 215

14.5.2 Regression Analysis of Survival Data 216

Problems 216

References 216

Appendix A Statistical Tables 219

Appendix B Answers to Selected Problems 235

Appendix C Computer Statistical Program Resources 243

C.1 Computer Systems for Biomedical Education and Research 243

C.2 A Brief Indication of Statistics Computer Program Advances and Some Relevant Publications Since 2000 244

C.3 Choices of Computer Statistical Software 248

Bibliography 249

Index 253