Estimating national disability risk

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Abstract

In this article, I provide a method to rebuild the active and disabled life expectancy (ALE and DLE) on the basis of ‘current’ death and disability risks, and to measure disability risk. This method uses national-level data, and is based on two main assumptions. The first is the Gompertz assumption that death rate rises with age exponentially, and the second is the Cox assumption that death rates of active status are proportional to those of disabled status across age. Applying this method to the US data, I find that the disability risk has increased between 1970 and 1990 for both men and women aged 40 and older. Situations in which above assumptions could be removed are also discussed.

Introduction

Disability data have been collected in censuses, for example in the US and Canada (Weeks, 1999, p. 52). These data provide the proportions of disabled population in each age group for both sexes. Combining with mortality data collected from vital statistics, disability proportions are used to address the active life expectancy (ALE), an indication of living how long and how well, that is, free from disability.

Sullivan (1971) provided the first calculation of the ALE, which is now called the prevalence method. In this method, first the death rates are used to construct an ordinary life table; disability proportions are then used to divide the person-years lived in each age group into disabled and active status. The total-person-years lived from each age are then calculated for both disabled and active status. Dividing the active total-person-years by the number of survivors at the corresponding age, the result could be interpreted as the number of years lived in active status (ALE), for an average person in the synthetic cohort, if current death rates and disability proportions prevailed through the cohort's lifetime. The difference between the life expectancy (LE) and ALE is naturally called the disabled life expectancy (DLE).

The prevalence method has generated meaningful studies based on national-level data. For example, Crimmins et al. (1989) found that in the US in 1970s most of increase in LE was DLE, implying the longer life of Americans was more in disabled years. They found later (Crimmins et al., 1997) that this trend had been reversed in 1980s, and Cambois et al. (2001) reported a similar change in France. These findings could provide a quantitative basis to study the conjecture (e.g. Fries, 1980) of whether or not people would be healthier when they live longer.

In the prevalence method, however, there are two obvious problems yet to be solved. Life table may include event other than death, but such event should happen in a certain time interval in order for its measures to be compared between different times or regions. The age-specific disability proportions, however, does not measure the risk of disability in a certain time interval; they are rather records of mixed events, including disability and mortality, accumulated through a long period of history. Comparing the prevalence ALE between different times is hence questionable. A decline in the number of individuals who become disabled in a certain time interval, for example, may not reduce the disability proportions which were formed earlier, and hence may not lower the prevalence ALE. Using the prevalence ALE to compare disability risks between different regions is equally problematic; since it may not be clear to what time interval the comparison refers. The first problem, therefore, is how to make the ALE refer to current disability risk. Further, reducing mortality of active people faster than that of disabled will lead to lower disability proportions and hence higher ALE. In other words, higher ALE does not necessarily reflect lower disability risk. Thus, the second problem is how to measure disability risk.

The double-decrement (e.g., Katz et al., 1983) and the multistate models (e.g., Land et al., 1994) could provide the basis to solve the two problems. In fact, the multistate model can describe the dynamics of transitions between more health status than only active and disabled (e.g., Crimmins et al., 1994; Manton and Land, 2000). The power of these models, however, is built upon panel data that record each individual's health status at two different times. These data are not yet available at the national level. Because the sizes of the sampled data used in these models are too small to calculate age-specific death or transition rates, assumptions about how these rates change with age have to be introduced. These assumptions typically include that the death and transition rates increase with age linearly or exponentially. For almost 200 years, demographers have been using the Gompertz law (see Preston et al., 2001, p. 192) that death rate rises with age exponentially from some starting age around 50 years. Recent modifications of this assumption are only for the oldest ages, usually 90 years and older (e.g., Horiuchi and Coale, 1990). The assumptions that the transition rates (between active and disabled status) increase with age linearly or exponentially, however, remain to be examined.

Using prevalence data, the relationship between mortality and disability can still be disentangled, as in the method I will propose in this paper. This method does not assume how the transition rates change with age, but requires assumptions that can be examined and have been widely used. The first assumption is the way that death rate changes with age, which I choose the Gompertz law. The second assumption is about how the ratio of death rates of active to disabled status changes over age. I choose the Cox proportional hazard model (Cox, 1972), in which the ratio of death rates of active to disabled status is constant over age, as the second assumption. The Cox assumption has been used to model death rates of different health status (Lee, 1992) and has been working well in small sample data (Lee, 1997). This method needs also other two supplementary assumptions that are related with the data whose age group covers n years. The third assumption is about how the number of population in a certain age group changes over a certain time interval. I assume it changes linearly over time for 2n years. The fourth assumption is that death rates and disability proportions remain constant for 2n years. The third and fourth assumptions have been used elsewhere when 2n covers fewer than 5 years, and occasionally for longer interval.

In this method, the first and second assumptions are used to estimate the death rates of active and disabled status. The estimation is based on minimizing the errors of using the modeled death rates to describe the death rates of the active and disabled combined population. After estimating death rates of active or disabled population, however, transition rates cannot yet be estimated without knowing how the number of active or disabled population changes with age. Using the third and fourth assumptions to describe such changes, transition rates are estimated. Assuming the rates of death and transition remain constant for the lifetime of a synthetic cohort, the first problem is solved and the ALE is calculated to describe the current risks of mortality and disability. After estimating the transition rates, which describe the age-specific disability risks, this method provides a summary measure of disability risk in the way similar to the use of life expectancy to summarize the risk of death.

For demonstrating the method, examples are provided using the US data. In these examples, the age-status-specific death rates, the age-specific rates of net transition from active to disabled status, the ALE and DLE based on current risks of mortality and disability, and the summary measure of disability risk are estimated. These estimates, at national level, have not been provided previously by either the prevalence method or the multistate model.

Section snippets

The mortality model

Let the death rate and the disabled proportion of population aged x to x+n years, obtained from vital statistics and census data, be m(x) and d(x), respectively. For this age group, let the death rates of active and disabled population be ma(x) and md(x), respectively, and be described by the following model:ma(x)=exp[c1+c0(x−s)],md(x)=exp[c2+c0(x−s)].In , , coefficient c0 makes death rate increase with age exponentially from a starting age s, according to the Gompertz assumption. That the

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This research was funded by the Morrison Institute for Population and Resource Studies at Stanford University, and by the NIA-funded P30 Center on the Economics and Demography of Aging at the University of California at Berkeley. I thank particularly Marcus Feldman, Shripad Tuljapurkar and Ronald Lee for their generous support. I also thank Tom Burch, Jean-Marie Robine, Robert Schoen, and Zheng Wu for their relevant comments. I am solely responsible for any error.

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