Analysis Of Incidence Rates

Description

Incidence rates are counts divided by person-time; mortality rates are a well-known example. Analysis of Incidence Rates offers a detailed discussion of the practical aspects of analyzing incidence rates. Important pitfalls and areas of controversy are discussed. The text is aimed at graduate students, researchers, and analysts in the disciplines of epidemiology, biostatistics, social sciences, economics, and psychology. Features: - Compares and contrasts incidence rates with risks, odds, and hazards. - Shows stratified methods, including standardization, inverse-variance weighting, and Mantel-Haenszel methods - Describes Poisson regression methods for adjusted rate ratios and rate differences. - Examines linear regression for rate differences with an emphasis on common problems. - Gives methods for correcting confidence intervals. - Illustrates problems related to collapsibility. - Explores extensions of count models for rates, including negative binomial regression, methods for clustered data, and the analysis of longitudinal data. Also, reviews controversies and limitations. - Presents matched cohort methods in detail. - Gives marginal methods for converting adjusted rate ratios to rate differences, and vice versa. - Demonstrates instrumental variable methods. - Compares Poisson regression with the Cox proportional hazards model. Also, introduces Royston-Parmar models. - All data and analyses are in online Stata files which readers can download. Cover......Page 1 Half Title......Page 2 Series Page ......Page 3 Title Page......Page 4 Copyright Page......Page 5 Table of Contents......Page 6 Preface......Page 16 Author ......Page 18 1.1 Karl Pearson and Spurious Correlation......Page 20 1.2 Jerzy Neyman, Storks, and Babies......Page 22 1.3 Is Poisson Regression the Solution to the Stork Problem?......Page 25 1.4 Further Reading......Page 26 2.2 Closed and Open Populations......Page 28 2.4 Numerators for Rates: Counts......Page 29 2.6 Prevalence Proportions......Page 31 2.7 Denominators for Rates: Count Denominators for Incidence Proportions (Risks)......Page 32 2.8 Denominators for Rates: Person-Time for Incidence Rates......Page 34 2.10 Rate Denominators Other than Person-Time......Page 36 2.11 Different Incidence Rates Tell Different Stories......Page 37 2.12 Potential Advantages of Incidence Rates Compared With Incidence Proportions (Risks)......Page 38 2.13 Potential Advantages of Incidence Proportions (Risks) Compared with Incidence Rates......Page 41 2.15 Radioactive Decay: An Example of Exponential Decline......Page 42 2.17 Relationships Between Rates, Risks, and Hazards......Page 45 2.18 Further Reading......Page 48 3.1 Estimated Associations and Causal Effects......Page 50 3.2 Sources of Bias in Estimates of Causal Effect......Page 51 3.4 Ratios and Differences for Risks and Rates......Page 52 3.5 Relationships between Measures of Association in a Closed Population......Page 53 3.6 The Hypothetical TEXCO Study......Page 54 3.7 Breaking the Rules: Army Data for Companies A and B......Page 58 3.8 Relationships between Odds Ratios, Risk Ratios, and Rate Ratios in Case-Control Studies......Page 60 3.10 Convergence Problems for Estimating Associations......Page 64 3.11 Some History Regarding the Choice between Ratios and Differences......Page 65 3.12 Other Influences on the Choice between Use of Ratios or Differences......Page 67 3.13 The Data May Sometimes Be Used to Choose between a Ratio or a Difference......Page 68 4.1 Alpha Particle Radiation......Page 72 4.3 Prussian Soldiers Kicked to Death by Horses......Page 75 4.4 Variances, Standard Deviations, and Standard Errors for Counts and Rates......Page 76 4.5 An Example: Mortality from Alzheimer’s Disease......Page 80 4.6 Large Sample P-values for Counts, Rates, and Their Differences using the Wald Statistic......Page 81 4.8 Large Sample P-values for Counts, Rates, and Their Differences using the Score Statistic......Page 83 4.9 Large Sample Confidence Intervals for Counts, Rates, and Their Differences......Page 84 4.10 Large Sample P-values for Counts, Rates, and Their Ratios......Page 86 4.12 A Constant Rate Based on More Person-Time Is More Precise......Page 88 4.13 Exact Methods......Page 90 4.14 What Is a Poisson Process?......Page 93 4.15 Simulated Examples......Page 95 4.16 What If the Data Are Not from a Poisson Process? Part 1, Overdispersion......Page 96 4.18 Must Anything Be Rare?......Page 98 4.19 Bicyclist Deaths in 2010 and 2011......Page 100 5.1 Florence Nightingale, William Farr, and Hospital Mortality Rates. Debate in 1864......Page 102 5.2 Florence Nightingale, William Farr, and Hospital Mortality Rates. Debate in 1996–1997......Page 106 5.3 Criticism of Rates in the British Medical Journal in 1995......Page 107 5.4 Criticism of Incidence Rates in 2009......Page 109 6. Stratified Analysis: Standardized Rates ......Page 112 6.2 External Weights from a Standard Population: Direct Standardization ......Page 113 6.3 Comparing Directly Standardized Rates......Page 118 6.4 Choice of the Standard Influences the Comparison of Standardized Rates......Page 121 6.5 Standardized Comparisons versus Adjusted Comparisons from Variance-Minimizing Methods......Page 122 6.7 Variations on Directly Standardized Rates......Page 124 6.8 Internal Weights from a Population: Indirect Standardization......Page 125 6.9 The Standardized Mortality Ratio (SMR)......Page 126 6.10 Advantages of SMRs Compared with SRRs (Ratios of Directly Standardized Rates)......Page 127 6.11 Disadvantages of SMRs Compared with SRRs (Ratios of Directly Standardized Rates)......Page 128 6.13 P-values for Directly Standardized Rates......Page 130 6.14 Confidence Intervals for Directly Standardized Rates......Page 131 6.15 P-values and CIs for SRRs (Ratios of Directly Standardized Rates)......Page 132 6.17 Small Sample P-values and CIs for SMRs......Page 133 6.19 Standardization Is Not Always the Best Choice......Page 134 7.1 Inverse-variance Methods......Page 136 7.2 Inverse-Variance Analysis of Rate Ratios......Page 137 7.3 Inverse-Variance Analysis of Rate Differences......Page 140 7.5 Mantel-Haenszel Methods......Page 141 7.6 Mantel-Haenszel Analysis of Rate Ratios......Page 142 7.7 Mantel-Haenszel Analysis of Rate Differences......Page 143 7.8 P-values for Stratified Rate Ratios or Differences......Page 144 7.9 Analysis of Sparse Data......Page 145 7.11 Stratified Methods versus Regression......Page 146 8.1 What Is Collapsibility?......Page 148 8.2 The British X-Trial: Introducing Variation in Risk......Page 149 8.3 Rate Ratios and Differences Are Noncollapsible because Exposure Influences Person-Time......Page 150 8.4 Which Estimate of the Rate Ratio Should We Prefer?......Page 151 8.5 Behavior of Risk Ratios and Differences......Page 152 8.7 Comparing Risks with Other Outcome Measures......Page 153 8.9 The American X-Cohort Study: 3-Levels of Risk in a Cohort Study......Page 154 8.10 The Swedish X-Cohort Study: A Collapsible Risk Ratio in Confounded Data......Page 158 8.11 A Summary of Findings......Page 160 8.13 Practical Implications: Avoid Common Outcomes......Page 161 8.16 Practical Implications: Uniform Risk......Page 162 8.17 Practical Implications: Use All Events......Page 163 9.1 The Poisson Regression Model for Rate Ratios......Page 164 9.2 A Short Comparison with Ordinary Linear Regression......Page 167 9.3 A Poisson Model without Variables......Page 168 9.4 A Poisson Regression Model with One Explanatory Variable......Page 171 9.6 The Header Information above the Table of Estimates......Page 175 9.7 Using a Generalized Linear Model to Estimate Rate Ratios......Page 177 9.8 A Regression Example: Studying Rates over Time......Page 180 9.9 An Alternative Parameterization for Poisson Models: A Regression Trick......Page 185 9.11 A Short Summary......Page 190 10.1 A Regression Model for Rate Differences......Page 192 10.2 Florida and Alaska Cancer Mortality: Regression Models that Fail......Page 193 10.3 Florida and Alaska Cancer Mortality: Regression Models that Succeed......Page 194 10.5 A Caution......Page 198 11.1 Limitations of Ordinary Least Squares Linear Regression......Page 200 11.2 Florida and Alaska Cancer Mortality Rates......Page 201 11.3 Weighted Least Squares Linear Regression......Page 202 11.4 Importance Weights for Weighted Least Squares Linear Regression......Page 204 11.5 Comparison of Poisson, Weighted Least Squares, and Ordinary Least Squares Regression......Page 205 11.6 Exposure to aCarcinogen: Ordinary Linear Regression Ignores the Precision of Each Rate......Page 210 11.7 Differences in Homicide Rates: Simple Averages versus Population-Weighted Averages......Page 212 11.9 Variance Weighted Least Squares Regression......Page 215 11.10 Cautions regarding Inverse-Variance Weights......Page 218 11.11 Why Use Variance Weighted Least Squares?......Page 219 11.12 A Short Comparison of Weighted Poisson Regression, Variance Weighted Least Squares, and Weighted Linear Regression......Page 220 11.14 Ratios and Spurious Correlation......Page 223 11.15 Linear Regression with ln (Rate) as the Outcome......Page 226 11.17 Summary......Page 227 12.1 Tabular and Graphic Displays......Page 230 12.2 Goodness of Fit Tests: Deviance and Pearson Statistics......Page 231 12.3 A Conditional Moment Chi-Squared Test of Fit......Page 234 12.4 Limitations of Goodness-of-Fit Statistics......Page 235 12.6 Robust Variance Estimator as a Test of Fit ......Page 236 12.8 Comparing Models using Akaike and Bayesian Information Criterion......Page 237 12.9 Example 1: Using Stata’s Generalized Linear Model Command to Decide between a Rate Ratio or a Rate Difference Model for the Randomized Controlled Trial of Exercise and Falls......Page 239 12.10 Example 2: A Rate Ratio or a Rate Difference Model for Hypothetical Data Regarding the Association between Fall Rates and Age......Page 243 12.11 A Test of the Model Link......Page 246 12.14 A Caution......Page 249 12.15 Further Reading......Page 250 13.1 Estimating the Variance without Regression......Page 252 13.2 Poisson Regression......Page 253 13.3 Rescaling the Variance using the Pearson Dispersion Statistic......Page 254 13.4 Robust Variance......Page 257 13.6 Using the Robust Variance to Study Length of Hospital Stay......Page 258 13.8 The Bootstrap Idea......Page 261 13.10 The Bootstrap Percentile Method......Page 262 13.13 The Bootstrap-T Method ......Page 263 13.15 Permutation and Randomization......Page 264 13.17 Better Randomization Using the Randomized Block Design of the Original Study......Page 265 13.18 A Summary......Page 266 14.1 Neyman’s Approach to His Data......Page 268 14.2 Using Methods for Incidence Rates......Page 269 14.3 A Model That uses the Stork/Women Ratio......Page 271 15.2 Quadratic Splines......Page 274 15.3 Fractional Polynomials......Page 276 15.5 Which Method Is Best?......Page 278 16.1 An Example: Shoes and Falls......Page 280 16.2 Problem 1: Using Subgroup P-values for Interpretation......Page 282 16.3 Problem 2: Failure to Include Main Effect Terms When Interaction Terms Are Used......Page 283 16.5 Problem 4: Interaction May Be Present on a Ratio Scale but Not on a Difference Scale, and Vice Versa......Page 284 16.6 Problem 5: Failure to Report All Subgroup Estimates in an Evenhanded Manner......Page 285 17. Negative Binomial Regression ......Page 290 17.2 An Example: Accidents among Workers in a Munitions Factory......Page 292 17.3 Introducing Equal Person-Time in the Homicide Data......Page 296 17.4 Letting Person-Time Vary in the Homicide Data......Page 298 17.5 Estimating a Rate Ratio for the Homicide Data......Page 300 17.6 Another Example using Hypothetical Data for Five Regions......Page 302 17.7 Unobserved Heterogeneity......Page 305 17.8 Observing Heterogeneity in the Shoe Data......Page 306 17.10 A Rate Difference Negative Binomial Regression Model......Page 307 17.11 Conclusion......Page 310 18. Clustered Data ......Page 312 18.1 Data from 24 Fictitious Nursing Homes......Page 313 18.3 A Single Random Set of Data for the Nursing Homes......Page 314 18.4 Variance Adjustment Methods......Page 318 18.5 Generalized Estimating Equations (GEE)......Page 320 18.6 Mixed Model Methods......Page 322 18.7 What Do Mixed Models Estimate?......Page 323 18.9 Simulation Results for Some Mixed Models......Page 324 18.11 Which Should We Prefer for Clustered Data, Variance-Adjusted or Mixed Models?......Page 327 18.13 Further Reading......Page 328 19.2 Using Rates to Evaluate Governmental Policies......Page 330 19.3 Study Designs for Governmental Policies......Page 331 19.4 A Fictitious Water Treatment and U.S. Mortality 1999–2013......Page 332 19.5 Poisson Regression......Page 333 19.6 Population-Averaged Estimates (GEE)......Page 335 19.7 Conditional Poisson Regression, a Fixed-Effects Approach......Page 336 19.8 Negative Binomial Regression......Page 339 19.10 Water Treatment in Only 10 States......Page 340 19.12 A Published Study......Page 343 20.2 Matching in Randomized Controlled Trials......Page 346 20.3 Matching in Cohort Studies......Page 347 20.4 Matching to Control Confounding in Some Randomized Trials and Cohort Studies......Page 348 20.5 A Benefit of Matching; Only Matched Sets with at Least One Outcome Are Needed......Page 349 20.6 Studies Designs that Match a Person to Themselves......Page 351 20.7 A Matched Analysis Can Account for Matching Ratios that Are Not Constant......Page 352 20.9 Stratified Methods for Estimating Risk Ratios for Matched Data......Page 353 20.10 Odds Ratios, Risk Ratios, Cell A, and Matched Data......Page 358 20.11 Regression Analysis of Matched Data for the Odds Ratio......Page 360 20.12 Regression Analysis of Matched Data for the Risk Ratio......Page 363 20.13 Matched Analysis of Rates with One Outcome Event......Page 368 20.14 Matched Analysis of Rates for Recurrent Events......Page 374 20.15 The Randomized Trial of Exercise and Falls; Additional Analyses......Page 378 20.16 Final Words......Page 379 21.1 What Are Margins?......Page 380 21.3 Estimating a Rate Difference from a Rate Ratio Model......Page 381 21.4 Death by Age and Sex: A Short Example......Page 382 21.5 Skunk Bite Data: A Long Example......Page 385 21.6 Obtaining the Rate Difference: Crude Rates......Page 386 21.7 Using the Robust Variance......Page 389 21.8 Adjusting for Age......Page 390 21.9 Full Adjustment for Age and Sex......Page 395 21.10 Marginal Commands for Interactions......Page 397 21.11 Marginal Methods for a Continuous Variable......Page 401 21.12 Using a Rate Difference Model to Estimate a Rate Ratio: Use the ln Scale......Page 403 22.1 Cancer Mortality Rate in Alaska......Page 406 22.2 The Rate Ratio for Falling in a Trial of Exercise......Page 410 23.1 A Simple Example......Page 414 23.2 A Perfectly Predicted Outcome......Page 416 23.3 Memory Problems......Page 417 23.4 A Caveat......Page 420 24.2 Analysis by Treatment Received May Yield Biased Estimates of Treatment Effect......Page 422 24.3 Using an Instrumental Variable......Page 424 24.4 Two-Stage Linear Regression for Instrumental Variables......Page 427 24.5 Generalized Method of Moments......Page 428 24.6 Generalized Method of Moments for Rates......Page 430 24.8 There Is No Free Lunch......Page 433 24.9 Final Comments......Page 434 25.2 Poisson Regression and Exponential Proportional Hazards Regression......Page 436 25.3 Poisson and Cox Proportional Hazards Regression......Page 440 25.4 Hypothetical Data for a Rate that Changes over Time......Page 442 25.5 A Piecewise Poisson Model......Page 445 25.6 A More Flexible Poisson Model: Quadratic Splines......Page 446 25.7 Another Flexible Poisson Model: Restricted Cubic Splines......Page 449 25.8 Flexibility with Fractional Polynomials......Page 450 25.9 When Should a Poisson Model Be Used? Randomized Trial of a Terrible Treatment......Page 452 25.10 A Real Randomized Trial, the PLCO Screening Trial......Page 453 25.11 What If Events Are Common?......Page 455 25.13 Collapsibility and Survival Functions......Page 456 25.14 Relaxing the Assumption of Proportional Hazards in the Cox Model......Page 457 25.15 Relaxing the Assumption of Proportional Hazards for the Poisson Model......Page 459 25.16 Relaxing Proportional Hazards for the Royston-Parmar Model......Page 461 25.17 The Life Expectancy Difference or Ratio......Page 462 25.19 A Short Summary ......Page 465 Bibliography......Page 468 Index......Page 486