The Biostatistics of Aging E-Book

Gilberto Levy

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 515

Veröffentlichungsjahr: 2014

Beschreibung

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.

Leseprobe

CONTENTS

Cover

Title page

Copyright page

Dedication One

Dedication Two

PREFACE AND ACKNOWLEDGMENT

1 INTRODUCTION

2 AN ACCOUNT OF GOMPERTZIAN MORTALITY THROUGH STATISTICAL AND EVOLUTIONARY ARGUMENTS

2.1 THE STATISTICAL THEORY OF EXTREME VALUES

2.2 THE EVOLUTIONARY THEORY OF AGING

3 THE ARGUMENT AGAINST GOMPERTZIAN MORTALITY

3.1 DEPARTURES FROM THE GOMPERTZ MODEL

3.2 AN EVOLUTION-BASED MODEL OF CAUSATION

4 THE INDEX OF AGING-RELATEDNESS

4.1 A SURVIVAL MIXTURE MODEL OF THE GOMPERTZ AND WEIBULL DISTRIBUTIONS

4.2 DEFINITION AND INTERPRETATION OF THE INDEX OF AGING-RELATEDNESS

4.3 THE SURVIVAL MIXTURE MODEL AND COMPETING RISKS

4.4 ESTIMATION OF THE MODEL PARAMETERS

4.5 ILLUSTRATIVE APPLICATION: THE ISRAELI ISCHEMIC HEART DISEASE STUDY

4.6 PRECISION OF ESTIMATION

5 DISCUSSION: IMPLICATIONS

5.1 THE MEANING OF THE GOMPERTZ PARAMETER

5.2 AGE AS A RISK FACTOR FOR DISEASE

5.3 ARE AGING-RELATED DISEASES AN INTEGRAL PART OF AGING?

5.4 BIOLOGICAL VERSUS CHRONOLOGICAL AGING

5.5 THE PUBLIC HEALTH NOTION OF COMPRESSION OF MORBIDITY

5.6 A PICTURE OF AGING FOR THE TWENTY-FIRST CENTURY

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

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

APPENDIX C: SOME PROPERTIES OF THE GOMPERTZ AND WEIBULL DISTRIBUTIONS

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

APPENDIX E: EXPECTATION–CONDITIONAL MAXIMIZATION (ECM) ALGORITHM

APPENDIX F: R PROGRAM

REFERENCES

AUTHOR INDEX

SUBJECT INDEX

Eula

List of Tables

Chapter 3

TABLE 3.1 Combinations of categories of component causes into sufficient causesa,b

TABLE 3.2 Characteristics of two groups of sufficient causes

Chapter 4

TABLE 4.1 Articles presenting a thorough and comprehensive criticism of the use of the heritability index in human studies

TABLE 4.2 Distinction between the CR and CS modelsa

TABLE 4.3 List of the ICD codes used to identify extrinsic causes of death (adapted from Carnes et al., 2006)a

TABLE 4.4 Intrinsic mortality by 5-year age intervals

TABLE 4.5 Results of fitting the mixture model to intrinsic mortality data, overall and stratified by smoking status. The estimated index of aging-relatedness corresponds to . Parameter estimates are followed by 95% CIs in parenthesis

TABLE 4.6 Results of fitting the mixture model to intrinsic mortality data, using the original data and bootstrapped samples 4, 9, and 25 times larger than the original sample size. Parameter estimates are followed by 95% CIs in parenthesis

Guide

Cover

Table of Contents

Begin Reading

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THE BIOSTATISTICS OF AGING

From Gompertzian Mortality to an Index of Aging-Relatedness

GILBERTO LEVY

Rio de Janeiro, Brazil

BRUCE LEVIN

Department of BiostatisticsColumbia UniversityNew York, NY

Copyright © 2014 by John Wiley & Sons, Inc. All rights reserved

Published by John Wiley & Sons, Inc., Hoboken, New JerseyPublished simultaneously in Canada

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Library of Congress Cataloging-in-Publication Data:

Levy, Gilberto, author. The biostatistics of aging : from Gompertzian mortality and to an index of aging-relatedness / Gilberto Levy, Bruce Levin.  pages cm Includes bibliographical references and index.

 ISBN 978-1-118-64585-7 (cloth)1. Aging–Statistical methods. 2. Mortality–Statistical methods. 3. Life spans (Biology) I. Levin, Bruce (Bruce A.), 1948– author. II. Title. QH529.L48 2013 571.8′78–dc23

2013034339

To my parents Menache and Norma LevyG.L.

To my wife BettyB.L

The decay of vitality with age is a biological fact most recognize in themselves and none fail to recognize in others; but the biometry of the subject is a difficult undertaking.

Greenwood, M. and Irwin, J. O. (1939). The biostatistics of senility. Human Biology, 11, 1–23.

PREFACE AND ACKNOWLEDGMENT

The purpose of this book is to describe a new quantitative method to examine the relative contributions of genetic factors and lifetime exposures to rates of mortality and disease incidence in a population. The book is highly multidisciplinary. The theoretical foundations of the work presented here involve the fields of statistics, evolutionary biology, demography, and epidemiology and should be of interest to those in these fields. Moreover, in its applications the work is broadly relevant to medicine, aging, and public health. Researchers and practitioners in these areas are also target audiences. We expect readers to comprise a spectrum from the more mathematically inclined to the more biologically inclined, though of course there will be readers who have expertise in both domains. This made the choice of level of presentation especially difficult. We chose to tilt the balance in favor of reaching a wide audience, at the cost of possibly making some (though hopefully not all) of the material in a field seem basic for an expert in that field. We also endeavored to make the book more widely appealing by keeping the denser mathematical material in a separate section in Chapter 2 and in a few appendices, and by providing summary pictures and statements after a series of mathematical results and at the end of some elaborate arguments. We bring the results and arguments from different areas of knowledge together starting in Section 2.2.1, and we hope the reader’s forbearance will be rewarded with some interesting synergies.

On a personal note, we have enjoyed collaborating on this project immensely and have learned a great deal from each other. From one perspective, the statistical modeling required was of the most precious kind as it derived from careful evolutionary and causal thinking, leading inexorably to consideration of one special model, rather than a plethora of them. From another perspective, the evolutionary arguments benefited from giving them a sound statistical underpinning and the clarity that mathematics can bring to an argument. In the interchange of ideas, we could hardly have had more fun.

The book greatly benefited from an illustrative application of the proposed method using data from the Israeli Ischemic Heart Disease (IIHD) study (Section 4.5). This was possible thanks to a collaboration with Uri Goldbourt, Ph.D., of the Division of Epidemiology and Preventive Medicine at the Sackler Faculty of Medicine of Tel Aviv University. The IIHD study collected mortality information on more than 10,000 subjects over a 43-year follow-up period (1963–2006), and Dr. Goldbourt has been involved with the study for nearly half a century. We deeply appreciate his invaluable help and responsiveness and are grateful for his graciousness in allowing us to report some of our analytic results from these data here.

GILBERTO LEVYRio de Janeiro

BRUCE LEVINNew York

1INTRODUCTION

The so-called aging-related diseases currently constitute a major public health concern, and their importance only tends to increase with the increase in absolute and relative numbers of older people in the population. The qualification “aging-related” is commonly used in the medical literature for diseases or disorders in a wide range of categories (e.g., neurodegeneration, cardiovascular, metabolic, neoplasia) and affecting virtually every organ system. In addition to aging-related, other terms that are often used with the same meaning are age-related, age-dependent, and age-associated. Although usually no formal definition is given, these terms are generally employed as referring to diseases whose age-specific point incidence rates (or, briefly, “incidence”) increase with increasing age.

However, some authors have drawn a fundamental distinction among these terms. While considerations about the relation between disease and aging go back a long way (Blumenthal, 2003), perhaps one of the earliest discussions specifically pointing to that terminological distinction is to be found in Kohn (1963), who noted, “it is useful to make two categories of the bad things that happen to people with increasing age—basic aging processes and age-related diseases, and to consider that the latter may be conditioned by, or dependent on, the former.” He then distinguished between a category of diseases that shows an increasing incidence with increasing age and a category that shows “a less clear-cut, age-related increase in incidence.” Two decades later, Kohn (1982) proposed that age-related diseases could be categorized in three ways: diseases that are normal aging processes themselves, diseases in which the incidence increases with increasing age, and diseases that have more serious consequences the older the affected persons. The more precise distinction that is most relevant to this work was given by Brody and Schneider (1986), who described two classes of “chronic diseases and disorders of old age” as follows: “Age-dependent diseases and disorders are defined as those whose pathogenesis appears to involve the normal aging of the host. Mortality and morbidity from age-dependent diseases and disorders (e.g. coronary artery disease and Alzheimer’s disease) increase exponentially. Age-related diseases and disorders, on the other hand, have a temporal relationship to the host but are not necessarily related to aging processes. They occur at a specific age and then decline in frequency or continue at less than an exponential rate of increase (e.g. multiple sclerosis and amyotrophic lateral sclerosis).”1

Particularly relevant to this work, Brody and Schneider (1986) suggested that the group of diseases related to the aging process is characterized by an exponential increase in age-specific incidence or mortality rates, as opposed to “less than an exponential rate of increase.” However, they did not provide a basis rooted in biological or statistical principles for that notion. Similarly, Kohn (1963) had considered, without justification from first principles, that an exponential increase in cause (disease)-specific (DS) mortality rates with age “is characteristic of deaths due to basic age-related processes and suggests the extent to which a disease is related to such processes.” Brody and Schneider (1986) illustrated such notion by plotting DS mortality rates by age for cardiovascular disease and cancer, representing the groups with and without exponential increase in age-specific rates, respectively (Fig. 1.1).

FIGURE 1.1 Cause (disease)-specific mortality rates by age for cardiovascular disease (left) and all cancers (right), data from the United States, 1978 (reproduced from Journal of Chronic Diseases, 39, Brody, J. A. and Schneider, E. L., Diseases and disorders of aging: An hypothesis, pages 871–876, Copyright 1986, with permission from Elsevier).

In the context of a meta-analysis of dementia prevalence, Ritchie and Kildea (1995) distinguished between an “ageing-related disorder” (“caused by the ageing process itself”) and an “age-related disorder” (“occurring within a specific age range”). Thus, they suggested that one category had a causal relation to the aging process and the other did not (labeled “ageing-related” and “age-related,” respectively), as Brody and Schneider (1986) had done before but instead labeling the first category “age-dependent.” As an example that this distinction continues to provoke and underlie the debate about the relation between diseases and aging in the twenty-first century, even if the causality notion is not always explicitly conveyed, Blumenthal (2003) offered the following “note on terminology” in his article titled “The aging-disease dichotomy: true or false?”: “In this essay I have used the term aging-associated disease rather than age-related disease. This choice is to emphasize that the primary focus here has been on diseases with age at onset in the senescent period of the life span, the oldest old, rather than through progressive periods of the total life span.”2

Yet, in a sense, the relation between diseases and aging has eluded medical thinking. While employing separate terms or categories for qualitatively different relations between diseases and aging seems warranted, it may not be clear under which category a disease falls given how its incidence increases with age. On the other hand, diseases considered to be in the same category may show different rates of increase in incidence rates with age. This may be seen as reflecting the fact that the aforementioned distinction arises from an underlying relation on a continuous scale, which therefore might better be considered using a quantitative approach. Although the authors quoted in the preceding text have attempted to clarify the meaning of aging-relatedness, the quantification of aging-relatedness has not been addressed at all in the medical, biostatistical, epidemiological, or demographic literature. Hence, we aimed to develop an index of aging-relatedness, as a means of quantifying and elucidating the underlying meaning of aging-relatedness.

Since the increase of mortality rates with age is an expression of aging at the population level, the notion of aging-relatedness applies as well, and perhaps even more naturally, to mortality. Medawar (1955) distinguished between a personal measure of aging, which “purports to measure a process that takes place in the life history of an individual animal,” and a statistical or actuarial measure of aging, “which is founded upon the mortality of a population of individuals and which bears only indirectly upon the changes that occur within the lifetime of anyone.” The assumed relevance of the mortality experience of a population to the physiological process of aging of its members is reflected in other authors’ definitions of aging or senescence. For instance, Maynard Smith (1962) stated, “Ageing processes may be defined as those which render individuals more susceptible as they grow older to the various factors, intrinsic or extrinsic, which may cause death.” Similarly, Comfort (1979, p. 21) stated, “Senescence shows itself as an increasing probability of death with increasing chronological age: the study of senescence is the study of the group of processes, different in different organisms, which lead to this increase in vulnerability.” More recent statements include Kirkwood’s (1985): “The pattern of mortality experienced by human populations serves to illustrate what is most commonly understood by the term aging. Following the attainment of sexual maturity and a peak of vitality which occurs early in adulthood, a long period of progressive deterioration takes place during which individuals become increasingly likely to die.” Or Finch’s (1990, p. 5): “Senescence is mainly used to describe age-related changes in an organism that adversely affect its vitality and functions, but most importantly, increase the mortality rate as a function of time.” The implicit assumption in all these authors’ definitions—that the pattern of age at death (or schedule of age-specific mortality rates) in a population parallels functional changes in the organism—is supported by experimental research with a variety of organisms (Austad, 2001).

Consistent with this premise, the index of aging-relatedness that we propose in this book is based on the schedule of age-specific mortality rates at a given point in time, through the use of time-to-event population-based data. Likewise, the index of aging-relatedness as applied to specific diseases is based on age-specific incidence rates or DS and age-specific mortality rates. The terms aging-related and aging-relatedness are used in this work without reference to the terminological distinctions in the preceding text. Indeed, developing a quantitative index of aging-relatedness would turn these distinctions moot. This is also to say that no claim of causality in the sense expressed by Brody and Schneider (1986) and Ritchie and Kildea (1995) is made in connection with the proposed index of aging-relatedness. Rather, in Chapters 2 and 3, we develop an extensive theoretical framework for the proposed index of aging-relatedness involving the statistical theory of extreme values and the evolutionary theory of aging, both of which rest on solid ground. We start by considering the biological basis of the Gompertz survival distribution, which is precisely characterized by the exponentially increasing hazard rate referred to by Kohn (1963) and Brody and Schneider (1986) and has long played a central role in demography for describing the survival time of human populations. The theoretical framework then includes (i) original mathematical results on the asymptotic behavior of the minimum of time-to-event random variables, extending those of the classical statistical theory of extreme values (Section 2.1.2.); (ii) an account of the Gompertz pattern of mortality in human populations, using those results on the statistical theory of extreme values and arguments based on the evolutionary theory of aging (Section 2.2.1.); and (iii) the development of the sufficient and component causes model of causation in epidemiology into an evolution-based model of causation, relevant to mortality and aging-related diseases of complex etiology (Section 3.2). While these are necessary steps toward devising the proposed index of aging-relatedness, each stands on its own as a theoretical contribution.

The index of aging-relatedness is presented in Chapter 4. The evolution-based model of causation provides the motivation for a statistical model for describing mortality and incidence of aging-related diseases of complex etiology involving a mixture of the Gompertz and Weibull distributions. This creates a framework for interpreting the index of aging-relatedness, which is defined as a parameter of this model (Sections 4.1 and 4.2). We describe the estimation procedures for obtaining the index and present an illustrative application to a real set of data (Sections 4.4 and 4.5). Although the overall presentation of this book proceeds from Gompertzian mortality to the index of aging-relatedness, we originally set out to develop an index of aging-relatedness. As we considered the theoretical basis for various proposals and found the mixture model index especially appealing from a theoretical viewpoint, the scope of the work widened considerably—while remaining in essence a medically motivated quantitative/statistical pursuit—to involve other disciplines such as demography, epidemiology, evolutionary biology, and population genetics. With this widening scope came a deeper reach. Even as our motivation was at first sight purely conceptual, a practical biomedical and public health relevance of the index arises from its interpretation, in a special sense, in terms of genetic and environmental contributions to mortality or disease incidence in a population. As a consequence, despite an ostensibly narrow initial aim, there are widespread implications of our theoretical framework and the index of aging-relatedness. These are discussed in Chapter 5. In its implications, the work presented in this book is additionally relevant to the fields of gerontology and geriatrics (Sections 5.3 and 5.4), as well as any medical specialty whose practitioners deal with aging-related diseases, but the most direct and practical implications are for public health (Sections 5.5 and 5.6) now and into the future.

Notes

1

Reproduced from

Journal of Chronic Diseases

,

39

, Brody, J. A. and Schneider, E. L., Diseases and disorders of aging: An hypothesis, pages 871–876, Copyright 1986, with permission from Elsevier.

2

Reproduced from

The Journals of Gerontology Series A: Biological Sciences and Medical Sciences

,

58

, Blumenthal, H. T., The aging–disease dichotomy: True or false? pages M138–M145, Copyright 2003, with permission from Oxford University Press.

2AN ACCOUNT OF GOMPERTZIAN MORTALITY THROUGH STATISTICAL AND EVOLUTIONARY ARGUMENTS

While several statistical models of age-specific mortality rates have been developed, including those based on the Weibull and logistic distributions, the most widely applied to mortality data in demography and gerontology has been the Gompertz model. Benjamin Gompertz was a nineteenth-century British mathematician and actuary who observed that the death rate in humans within a certain range of adult ages increased geometrically as age increased arithmetically (Gompertz, 1825). Thus, the Gompertz equation was developed based on empirical human mortality observations to describe an exponential relation between age-specific mortality rates and age:

(2.1)

where hG(x) is the hazard function (also called hazard rate function, instantaneous death rate, or force of mortality) for the Gompertz model, λ > 0 is a parameter denoting the initial mortality rate (at birth or another arbitrary age), and θ is an exponential rate parameter. The parameter λ has also been called the vulnerability parameter, while θ has been called the rate of aging or Gompertz parameter (Carey, 1999; Sacher, 1977). The derivative of Equation 2.1,

(2.2)

shows that the rate of change of the Gompertz hazard function at a given age is proportional to the value of the hazard function at that age. For θ > 0, hG(x) is a monotonically increasing function of x, and for θ < 0, hG(x) is monotonically decreasing. For θ = 0, hG(x) reduces to the constant hazard function of the exponential distribution. By taking the logarithm of both sides of Equation 2.1, we obtain

(2.3)

which shows that when there is a good fit of the Gompertz distribution to mortality data, a plot of the log of the mortality rates by age (a semilog plot) follows a closely linear relation with slope θ and intercept log λ.

In contrast, the Weibull distribution was developed by the Swedish engineer Waloddi Weibull, in the context of modeling the strength of materials (Weibull, 1939, 1951). Similar to the role played by the Gompertz distribution in demography, the Weibull distribution plays a prominent role in reliability theory, which applies to the failure of mechanical devices (Barlow and Proschan, 1981, p. 73; Rausand and Høyland, 2004, pp. 37–41). While the Gompertz equation describes an exponential relation between age-specific mortality (or failure) rates and age, the Weibull hazard function describes a power relation: