ReviewThe quadratic hazard model for analyzing longitudinal data on aging, health, and the life span
Highlights
► An approach to analyses of longitudinal data on aging, health, and longevity. ► Stochastic process models of human aging, health, and longevity is suggested. ► Incorporating state of the art advances in aging research into the model structure. ► Model deals with resistance to stresses, adaptive capacity, and physiological norms.
Introduction
Age patterns of human mortality rates demonstrate remarkable regularities in different populations: they decline in childhood, exponentially increase in adult ages, and tend to decelerate and level off at the oldest-old ages. Demographers have developed a number of parametric descriptions of mortality curves capturing all aspects of their variation with age [1], [2], [3], [4]. Biodemographers and gerontologists have aimed to explain observed features of mortality curves using emerging theoretical concepts [5], [6], [7], [8], [9], [10], [11], [12]. Since the chances of death are affected by internal and external stresses challenging deteriorating defense mechanisms in an aging body, the shape of the age trajectories of mortality curves is likely to reflect the average effects of such deterioration, as well as external influences. In other words, if one wants to make conclusions about the aging process developing in individuals comprising the population under study by investigating the age pattern of the corresponding mortality curve, one has to develop a description of such a curve in terms of parameters characterizing the processes which are likely to contribute to the shape of this curve. Such a description can be done using models of aging-related changes represented in longitudinal data merged with data on health and survival events.
Statistical methods for joint modeling of longitudinal and survival (time-to-event) data have been developed during the past several decades using both frequentist and Bayesian approaches (see recent reviews in [13], [14], [15], [16] and references therein). Commonly, time-to-event data are analyzed using the proportional hazards model and the usual approach to modeling longitudinal data involves a mixed effects model [17] with the assumption that the random effects or individual-specific parameters are normally distributed, see [18], [19], [20], [21] among others). Several papers have developed semiparametric approaches that do not rely on normality of the distribution for the random effects or individual parameters (see, e.g., [22], [23], [24], [25], [26]). Further developments include modeling longitudinal data using a stochastic process, either an (integrated) Ornstein–Uhlenbeck or Wiener process, that allows a more flexible description of individual longitudinal dynamics and provides a better fit compared to the usual random effects model (see, e.g., [27], [28], [29], [30], [31], [32]).
To be useful as a tool for extracting new information, models of longitudinal data have to be based on realistic assumptions and reflect knowledge and evidence accumulated in the field. Epidemiological studies of risks of disease and death show that the conditional hazards of such events considered as functions of given risk variables often have U- or J-shapes [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], so the model of aging-related changes has to take this reality into account. The risk variables as well as their effects on the risks of corresponding events experience aging-related changes (different for distinct individuals) and are measured with certain periodicity in longitudinal studies of aging, health, and longevity. Such data make it possible to evaluate regularities of aging-related changes and their effects on health and survival. An important class of models for joint analyses of longitudinal and time-to-event data incorporating a stochastic process for description of longitudinal measurements is based on this biologically-justified assumption of a quadratic hazard (i.e., U-shaped in general and J-shaped for variables that can take values only on one side of the U-curve) considered as a function of risk factors (i.e., physiological variables). Quadratic hazard models have been developed and intensively applied in studies of longitudinal data [44], [45], [46], [47], [48]. The advantage of this approach is that it allows for incorporation of new insights and ideas appearing in the course of research on aging. The prototype of a model discussed in this paper was suggested in [44]. Yashin [45], [49] investigated conditions for preserving a Gaussian distribution property of the stochastic covariates under the operation of conditional averaging, and found that model [44] satisfies these conditions. An important property of this model is that the age trajectory of the total mortality rate can be explicitly represented in terms of the first two moments of the conditional distribution of the processes involved in the description of the conditional mortality risk. This property, as well as its flexibility in describing age trajectories of factors affecting conditional risk, make this model a valuable tool for studying aging, health, and longevity using longitudinal data.
Despite the availability of efficient statistical methods for analyzing longitudinal data, and considerable progress in understanding various aspects of human aging, health, and longevity, many facts and research findings remain largely disconnected. Researchers working with data on aging usually deal with small portions of information used for addressing specific research question using standard statistical methods. Such methods, however, largely ignore available knowledge, new research findings, and emerging theoretical concepts about aging in the process of data analyses. As a result, the potential of many large longitudinal datasets, as well as the knowledge accumulated in the field, remains underused.
Progress in this area would be substantially facilitated if researchers had a tool for analyzing the wealth of available data and were able to incorporate important facts, research findings, and emerging theoretical concepts in the analyses. Several such concepts capable of capturing fundamental features of aging-related changes are currently under development. They are related to the notion of allostatic load [50], the decline in adaptive capacity (homeostenosis) [51], [52], [53], [54], the decline in resistance to stresses [5], [55], [56], [57], aging-related physiological norms, and heterogeneity in longitudinal data.
In this paper, we describe a mathematical model for analyzing longitudinal data on aging, health, and mortality that incorporates the four concepts of aging described above (i.e., the notions of age-dependent physiological norms, allostatic load, adaptive capacity, and resistance to stress), review applications of this model to analyses of longitudinal data, and investigate its potential for performing more comprehensive analyses of such longitudinal data.
An initial version of such a model was suggested in [58]. Its various extensions have been applied in different contexts to investigate mechanisms of aging-related changes in connection with morbidity/mortality risks. This includes analyses of age trajectories of different physiological indices (such as blood glucose, body mass index, cholesterol, diastolic blood pressure, hematocrit, pulse pressure, and pulse rate) in relation to mortality/morbidity risks [59], [60], [61], [62]; applications to “indices of cumulative deficits,” which have proved to be useful for analyses of a wide spectrum of information in relation to health- and aging-related changes and better characterize the aging phenotype than chronological age [63]; and analyses of trajectories of medical costs in relation to mortality risks [64]. Extended versions of this model also have been used in analyses of dependent competing risks [65], [66], heterogeneity in longitudinal data [67], analyses of genetic effects on age trajectories of physiological indices [68], joint analyses of individual health histories and physiological aging [69], and joint analyses of data collected using different observational plans [70].
Section snippets
General description
To analyze longitudinal data on age-dependent changes in physiological states, we propose a dynamic model describing the trajectories of the individual physiological variables and their influence on mortality risks, which have the J-, or U-shape considered as a function of risk factors [71]. Let (where t is age) be a k-dimensional stochastic process describing a continuously changing vector of risk factors/covariates (e.g., physiological variables), and Z be a vector of time-independent
Discussion
In experimental studies of aging using populations of laboratory animals, the sensitivity of the individual aging process to external disturbances (e.g., medical interventions), or genetic manipulations often is evaluated by comparing empirical survival functions (or mortality rates) constructed for populations in the experimental and control groups. Similarly, the slope of the logarithm of the mortality curve at the adult and old ages often is interpreted as the aging rate [79]. The
Acknowledgements
The work was partially supported by grants R01AG030612, R01AG030198 and R01AG032319 from the National Institute on Aging. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institute on Aging or the National Institutes of Health.
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