The Biostatistics of Aging

GILBERTO LEVY, MD, DRPH, is a neurologist with a primary research interest in aging and aging-related diseases, particularly dementia. He conducted clinical research within the spheres of epidemiological studies and clinical trials at Columbia University for more than ten years. Dr. Levy is the author of over thirty journal articles and three book chapters. BRUCE LEVIN, PHD, is Professor of Biostatistics and past chair of the Department of Biostatistics in the Mailman School of Public Health at Columbia University. A Fellow of the American Statistical Association, his research interests include sequential selection procedures and their use in adaptive clinical trial designs. He is the coauthor (with J.L. Fleiss and M.C. Paik) of Statistical Methods for Rates and Proportions, Third Edition, published by Wiley.

PREFACE AND ACKNOWLEDGMENT ix

1 Introduction 1

2 An Account of Gompertzian Mortality through Statistical and Evolutionary Arguments 6

2.1 The Statistical Theory of Extreme Values 10

2.2 The Evolutionary Theory of Aging 36

3 The Argument against Gompertzian Mortality 69

3.1 Departures from the Gompertz Model 70

3.2 An Evolution-Based Model of Causation 72

4 The Index of Aging-Relatedness 93

4.1 A Survival Mixture Model of the Gompertz and Weibull Distributions 94

4.2 Definition and Interpretation of the Index of Aging-Relatedness 97

4.3 The Survival Mixture Model and Competing Risks 103

4.4 Estimation of the Model Parameters 107

4.5 Illustrative Application: The Israeli Ischemic Heart Disease Study 109

4.6 Precision of Estimation 122

5 Discussion: Implications 128

5.1 The Meaning of the Gompertz Parameter 128

5.2 Age as a Risk Factor for Disease 132

5.3 Are Aging-Related Diseases an Integral Part of Aging? 134

5.4 Biological versus Chronological Aging 135

5.5 The Public Health Notion of Compression of Morbidity 138

5.6 A Picture of Aging for the Twenty-First Century 143

APPENDIX A: PROOFS OF RESULTS IN SECTION 2.1.2 WITH SOME EXTENSIONS 154

APPENDIX B: DERIVATION OF HAMILTON’S EQUATION FOR THE FORCE OF NATURAL SELECTION ON MORTALITY 170

APPENDIX C: SOME PROPERTIES OF THE GOMPERTZ AND WEIBULL DISTRIBUTIONS 174

APPENDIX D: FIRST AND SECOND PARTIAL DERIVATIVES OF THE MIXTURE LOG-LIKELIHOOD FUNCTION 178

APPENDIX E: EXPECTATION–CONDITIONAL MAXIMIZATION (ECM) ALGORITHM 183

APPENDIX F: R PROGRAM 190

REFERENCES 226

AUTHOR INDEX 245

SUBJECT INDEX 253