The Biostatistics of Aging From Gompertzian Mortality to an Index of Aging-Relatedness

A practical and clarifying approach to aging and aging-related diseases

Providing a thorough and extensive theoretical framework, The Biostatistics of Aging: From Gompertzian Mortality to an Index of Aging-Relatedness addresses the surprisingly subtlenotion—with consequential biomedical and public health relevance—of what it means for acondition to be related to aging. In this pursuit, the book presents a new quantitative methodto examine the relative contributions of genetic and environmental factors to mortality anddisease incidence in a population.

With input from evolutionary biology, population genetics, demography, and epidemiology, this medically motivated book describes an index of aging-relatedness and also features:

• Original results on the asymptotic behavior of the minimum of time-to-event random variables, which extends those of the classical statistical theory of extreme values
• A comprehensive and satisfactory explanation based on biological principles of the Gompertz pattern of mortality in human populations
• The development of an evolution-based model of causation relevant to mortality and aging-related diseases of complex etiology
• An explanation of how and why the description of human mortality by the Gompertz distribution can be improved upon from first principles
• The amply illustrated analysis of real-world data, including a program for conducting the analysis written in the freely available R statistical software
• Technical appendices including mathematical material as well as an extensive and multidisciplinary bibliography on aging and aging-related diseases

The Biostatistics of Aging: From Gompertzian Mortality to an Index of Aging-Relatedness is an excellent resource for practitioners and researchers with an interest in aging and aging-related diseases from the fields of medicine, biology, gerontology, biostatistics, epidemiology, demography, and public health.

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

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