Matrix methods in health demography: a new approach to the stochastic analysis of healthy longevity and DALYs

Abstract

BackgroundIncreases in human longevity have made it critical to distinguish healthy longevity from longevity without regard to health. Current methods focus on expectations of healthy longevity, and are often limited to binary health outcomes (e.g., disabled vs. not disabled). We present a new matrix formulation for the statistics of healthy longevity, based on health prevalence data and Markov chain theory, applicable to any kind of health outcome and which provides variances and higher moments as well as expectations of healthy life.MethodThe model is based on a Markov chain description of the life course coupled with the moments of health outcomes (“rewards”) at each age or stage. As an example, we apply the method to nine European countries using the SHARE survey data on the binary outcome of disability as measured by activities of daily living, and the continuous health outcome of hand grip strength.ResultsWe provide analytical formulas for the mean, variance, coefficient of variation, skewness and other statistical properties of healthy longevity. The analysis is applicable to binary, categorical, ordinal, or interval scale health outcomes. The results are easily evaluated in any matrix-oriented software. The SHARE results reveal familiar patterns for the expectation of life and of healthy life: women live longer than men but spend less time in a healthy condition. New results on the variance shows that the standard deviation of remaining healthy life declines with age, but the coefficient of variation is nearly constant. Remaining grip strength years decrease with age more dramatically than healthy years but their variability pattern is similar to the pattern of healthy years. Patterns are similar across nine European countries.ConclusionsThe method extends, in several directions, current calculations of health expectancy (HE) and disability-adjusted life years (DALYs). It applies to both categorical and continuous health outcomes, to combinations of multiple outcomes (e.g., death and disability in the formulation of DALYs) and to age- or stage-classified models. It reveals previously unreported patterns of variation among individuals in the outcomes of healthy longevity.Electronic supplementary materialThe online version of this article (10.1186/s12963-018-0165-5) contains supplementary material, which is available to authorized users.

Estimating the number and length of episodes in disability using a Markov chain approach

“…To estimate the expected number and length of episodes, we use Markov chains with rewards [ 22 , 23 ]. An excellent introduction to Markov chains with rewards with application to population health was given by Caswell and Zarulli [ 20 ]. Generally, Markov chains with rewards are based on assigning rewards to transitions between states, and then allow for the calculation of the expected value of rewards.…”

“…Using a discrete-time, homogeneous Markov chain with finite state space, we show how a general method for Markov chains with rewards presented by van Daalen and Caswell [ 19 ] and recently discussed in this journal by Caswell and Zarulli [ 20 ] can be used for the calculation of the expected number and length of episodes spent in a state. Modifying their method suffices to arrive at an easily applicable approach.…”

Flexible transition timing in discrete-time multistate life tables using Markov chains with rewards

“…The theory was introduced by Caswell (2011) in the context of lifetime reproduction, and the possibility of analyzing longevity was introduced. The theory has since been applied to the fertility transition (S. van Daalen & Caswell, 2015), to lifetime income and expenditures (Caswell & Kluge, 2015), to healthy longevity and disability adjusted life years (Caswell & Zarulli, 2018), to episodes of disability (Dudel & Myrskylä, 2020), and to periods of malnourishment (Owoeye et al, 2020). The complete theory for the demographic Markov chains with rewards, including simple expressions for all the statistical moments and sensitivity analysis, is given in van Daalen and .…”

Sensitivity Analysis of Longevity and Life Disparity

“…An important extension of Markov chain models for longevity is the incorporation of "rewards" to represent the value, in some sense, of the length of life, extending methods developed for dynamic programming (Howard 1960). The rewards include the production of offspring (Caswell 2011;Caswell 2015, 2017), the accumulation of income and expenditures (Caswell and Kluge 2015) and healthy longevity (Caswell and Zarulli 2018). The sensitivity analysis of these important models is derived in van Daalen and Caswell (2017).…”

“…Calculations of life lost from mortality due to specific causes of death play a central role in the calculations of disability-adjusted life years (DALYs) used in calculations of the burden of diseases (e.g., Devleesschauwer et al 2014;GBD 2016 DALYs andHALE Collaborators 2017). See Caswell and Zarulli (2018) for the relationship between DALY calculations and Markov chain methods, and for a calculation of the variance in life lost.…”