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The Effect of Altering ADL Thresholds on Active Life Expectancy Estimates for Older Persons

¹ Department of Sociology and Office of Population Research, Princeton University, Princeton, New Jersey.
² Department of Sociology and Center for Demographic Studies, Duke University, Durham, North Carolina.

Address correspondence to Scott M. Lynch, Department of Sociology, Princeton University, Princeton, NJ 08544. E-mail: slynch{at}princeton.edu

	Abstract

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Objectives. Research on disability and active life expectancy (ALE) has often criticized the measurement of disability but has rarely empirically investigated the effect of changing measurement. The purpose of this study was to determine whether altering the number of activities of daily living (ADLs) required to consider an individual "disabled" affects population-based ALE estimates after considering parametric uncertainty and sampling error.

Methods. The authors develop a Bayesian approach to estimating multistate life tables for a three-dimensional state space, using data on community-dwelling older adults from the 1989 and 1994 National Long Term Care Survey analytic files. Empirical confidence intervals for ALE are compared across 6 models using successively higher ADL cutoffs for defining individuals as being disabled.

Results. After considering sampling and other errors in the estimation of transition probabilities, the authors found that altering the threshold for measuring disability has relatively little effect on ALE estimates, especially with higher ADL-level thresholds and at older ages.

Discussion. The implications of the results include that disability measurement, including altering the definition of being disabled and possibly expanding the state space of a model, may not affect population-based estimates of ALE.

Active life expectancy (ALE), or remaining years of life spent healthy or nondisabled, has been examined extensively by gerontologists, demographers, and others (e.g., Crimmins, Hayward, & Saito, 1996; Crimmins, Saito, & Ingegneri, 1989, 1997; Land, Guralnik, & Blazer, 1994). Whereas most attention has been focused on either the predictors of functional impairment and the relationship between limitations and mortality (Crimmins, Hayward, & Saito, 1996; Johnson & Wolinsky, 1993; Maddox & Clark, 1992; Maddox, Clark, & Steinhauser, 1994; Manton, Corder, & Stallard, 1997) or prevalence/incidence rates of disability (Crimmins et al., 1989, 1997; Manton, 1988; Manton, Corder, & Stallard, 1997), very little attention has been paid to the empirical assessment of disability measurement.

Disability is often defined in terms of the inability to perform certain tasks that are necessary for independent living. Operationalizing this definition is difficult, however. At the conceptual level, several researchers have criticized various aspects of the measurement of disability. Glass (1998) has noted that researchers have measured disability via different tenses. He argued that hypothetical (can respondent do x), experimental (did respondent do x), and enacted (does respondent do x) tenses of disability items should, and do, produce different results. Others have argued (and found) that individuals do not follow a simple unidirectional path from healthy to disabled to deceased (e.g., Glass, Seeman, Herzog, Kahn, & Berkman, 1995; Land, Guralnik, & Blazer, 1994), the implications of which include reconsidering the usefulness of simple cross-sectional examination of prevalence rates (see Verbrugge, 1991). Still others have argued that the process of disability is just that—a process—that makes "disability" distinct from morbidity and functional limitation (Johnson & Wolinsky, 1993; Nagi, 1976; Verbrugge & Jette, 1994) and adds another layer of difficulty in defining and measuring disability, and thus ALE. Finally, some have argued that, among older persons, disability is multifaceted and not adequately captured by standard summary scales, requiring considerable post hoc refinement of initial measures (e.g., Manton & Stallard, 1994).

Although numerous researchers have discussed the conceptualization of disability and its subsequent measurement, discussions of the shortcomings of disability measurement are not empirical tests regarding the effects of altering the measurement or operationalization of disability. Relatively few empirical examinations of the effects of disability measurement have been undertaken. Jette (1994) found that measuring limitations as "a difficulty" rather than as "a need" resulted in considerably higher rates of disability. Others have identified problems surrounding question wording of functional status (see Glass, 1998). Glass (1998) empirically examined this possibility by testing different tenses of disability questions. He found that these differing tenses in fact produce substantially different levels of disability.

We can find no studies that have examined the impact of modifying the operationalization of disability in terms of changing the threshold at which one is considered disabled. A test of such variation is important because much of the conceptual debate concerns defining disability itself and distinguishing disability from impairment and functional limitation (Verbrugge & Jette, 1994), but little attention is paid to how much limitation makes a person disabled, and most models require such a threshold to be specified in the production of estimates of ALE.

In the literature, there are two typical approaches to labeling an individual as disabled: (a) choose an arbitrary ADL threshold or (b) develop a highly sophisticated state space. The former approach appears to assume that there is either a strong theoretical rationale for choosing a particular threshold or that the choice of threshold matters little. The latter approach assumes that measuring disability is sophisticated enough that a complex state space is needed to produce an accurate picture of disability. Studies following the former approach—using activities of daily living (ADLs) as measures—with relatively simple state spaces (e.g., healthy, disabled, deceased) often define being disabled as having at least one ADL limitation (e.g., Branch et al., 1991; Land et al., 1994), reflecting the original conceptualization of dependence. Others in the same tradition have expanded the state space to include more than one level of disability in an effort to capture greater dynamics of transitions in and out of disabled states (e.g., Crimmins, Hayward, & Saito, 1996; Wolfe & Laditka, 1997). Studies following the latter approach have avoided using an arbitrary ADL cutoff by producing an even larger and more complex state space, with individuals allowed to have partial membership in multiple "pure" states (e.g., Manton & Land, 2000a, 2000b).

In this study, we examine how changing the ADL cutoff for defining individuals as disabled affects ALE after sampling and parametric uncertainty are considered—something other research has not examined—using a method that produces empirical confidence intervals for ALE. We vary the operationalization of disability by altering the number of ADL limitations required to be considered disabled and examine whether altering this definition affects estimates of ALE. The only study to date that we can find that has conducted a similar investigation constructed status-based total life expectancy estimates for each state in a complex state space to determine whether total life expectancy is state dependent (see Manton & Land, 2000b). Our approach uses population-based ALEs derived from a community sample for a relatively simple state space, an approach more common in the literature. Furthermore, whereas Manton and Land's (2000b) approach produced estimates of total life expectancy for each state, our concern is simply whether altering the requirements for being in a particular state affects estimates of ALE.

	Data and Methods

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Sample and Variables
The data used in this study are from the National Long Term Care Survey (NLTCS) analytic file. The NLTCS is a panel study, whose members are drawn from a list frame consisting of all Medicare beneficiaries, conducted in 1982, 1984, 1989, 1994, and 1999. Persons reaching age 65 are screened into the sample on the basis of positive responses to very liberal disability items: if an individual shows any sign of disability (e.g., instrumental activities of daily living [IADLs] or ADLs), she/he is included in the survey. However, a subset of persons found to be nondisabled in 1989 were followed up in 1994. The NLTCS is a multistage sample, with new sample members added at each wave to replace decedents. Exit from the study is by death or attrition only; individuals are not excluded from follow-ups by design. By 1994, the analytic file had a cumulative total of 35,848 sample members. In order to make this dataset a community survey (for subsequent comparisons to other surveys), we exclude all persons from the NLTCS who were institutionalized in either 1989 or in 1994. Thus, the sample includes only community-dwelling adults who were alive and interviewed in 1989. We exclude persons who were institutionalized between 1989 and 1994 for two reasons. First, we are unsure of their disability status in 1994. It is possible, for example, that some individuals became less disabled in terms of ADL count but simply had exhausted sources of informal care. Second, in most surveys, individuals who move to institutions between waves would most likely be lost at follow-up; thus, our sample here may be more consistent with those of other studies.

In the analytic file there were 10,318 persons eligible for interview in 1989. By 1994, 5.7% of the sample (585 persons) had attritted. Logistic regression analyses of these missing indicated that the missing were more likely to be male (odds ratio = 1.37, p <.05) and more likely to be disabled in 1989 (odds ratio = 1.10, p <.05), probably indicating inability to locate some respondents who died or became institutionalized. After deleting the missing, the final sample consisted of 9,733 persons. Given the small proportion missing, we have not made any imputation, as imputation may bias the results and gain us little additional statistical power (Allison, 2002). Additionally, we argue that the missing do not affect our results here because our comparisons are made between models within a standard final sample. The final sample was 62.1% female and 91.0% White, with only slight variation in these proportions across age groups, starting states, and models.

We use six standard ADL measures of limitation, including eating, bedding, toileting, transferring, dressing, and bathing. The NLTCS asks a very detailed array of questions to determine whether a respondent shows any limitation, including whether the respondent has a problem with an ADL task, uses special equipment to assist with an ADL task, receives help for an ADL task, or needs help with an ADL task. If the individual indicated "yes" to any of these questions, she/he was considered limited on the item. We measure age using 5 age groups: 65–69, 70–74, 75–79, 80–84, and 85+. ALEs are produced for each age group. We use sample weights in the analyses to compensate for the complex design of the NLTCS. However, results obtained from unweighted data did not differ appreciably from those we report.

Model
We use multistate life table methods to estimate ALEs, because individuals transition between disabled and healthy states, rather than proceeding sequentially in a healthy–disabled–deceased direction (Manton, 1988; Verbrugge, 1991). Existing approaches to estimating multistate life tables at best only allow for construction of approximate confidence intervals around point estimates for state expectancies. Here, we develop a Bayesian method for obtaining empirical confidence intervals, so that uncertainty can be explicitly considered in the process of evaluating whether changes in the operationalization of disability affect ALE estimates.

Assume that there are k - 1 states an individual can be in at Wave 1 of a study, with k possible states at Wave 2 (the kth state being "deceased"). Then, given a starting state, the possible transitions into the remaining states constitute a multinomial variable of dimension k - 1, with each dimension having 2 levels (experiences transition or does not). A multinomial likelihood function can be established with uncertainty about individuals' locations within cells integrated out in model estimation. Alternatively, we can assume individuals have latent propensities to make transitions out of their current state to one of the k - 1 remaining states between waves. As in a standard single dimensional probit model, we can assume these propensities are distributed as truncated N(0,1) variables (see e.g., Johnson & Albert, 1999); however, given the multidimensional nature of the problem, we can assume that individuals' vectors (Z_i) of transition propensities are distributed as truncated multivariate normal (TruncMVN) variables:

Here, Z_i is a k - 1 dimensional vector of propensities for the ith individual. The Z are distributed as truncated multivariate normal random variables, with the mean vector, µ, determined by a regression on a vector of covariates: Z_i(a) = X_i(a)'b(a), across all dimensions (a). The truncation point for the latent variables is defined at 0 in all dimensions, with the individual's propensities drawn from the cell region of the multinomial contingency table that represents the observed transition (e.g., an individual who transitions into death is truncated to the region below 0 for all other state dimensions but above 0 for the death dimension). The dimensionality of the vector of transition propensities (and hence regression equations) is k - 1, reflecting the omission of remaining in the current state as a reference category. (Each dimension of the multinomial has 2 categories, with the 0, 0 ... 0 cell representing no transition, and any cell with more than one 1 having a structural 0 count). The regression coefficients and the X are parenthetically subscripted, reflecting that the dimension of X, and their effects, may vary across equations.

The relationships between the latent propensities that are unaccounted for by the model are expressed in the covariance matrix . The off-diagonal elements are assumed to be negative and large, given that, in this data, individuals can only transition to a single state.

With a completed MVN distribution of latent propensities, an augmented likelihood function for the other parameters in the model (b, ) is multivariate normal:

where the vector, = [_i,1 ... _i,k-1] contains the residuals [_i,a = Z_i(a) - X_i(a)b(a)]. Weights are included in this portion of the model by recognizing that each individual may contribute more or less than a single observation to the likelihood function. Thus, each observation's contribution is raised to the n_i power, where n_i is the sample weight for person i. To make the analysis fully Bayesian, we adopt the reference prior for a multivariate normal likelihood function (we tried other priors as well and observed little change in the results; see Gelman, Carlin, Stern, & Rubin, 1995).

In this research, we have two starting states (healthy and disabled) and three ending states (healthy, disabled, and deceased), across five age groups (65–69, 70–74, 75–79, 80–84, and 85+). With this number of states, the dimensionality of the problem reduces to a pair of bivariate probit regressions on five dummy covariates representing age (no intercept). Age is dummied to make the model nonparametric. One model predicts transitions to disabled and deceased states from the healthy state; the other predicts transitions to healthy and deceased states from the disabled state. In each case, remaining in the starting state is omitted as the reference.

	Estimation

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In Bayesian statistics, parameters are treated as random quantities and hence can be represented via probability distributions. The goal of Bayesian analysis is to obtain information about the distributions of parameters, given the data at hand (the posterior distribution; see Box & Tiao, 1973; Gelman et al., 1995; Lee, 1989). Markov chain Monte Carlo (MCMC) methods constitute a class of approaches to generating samples from posterior distributions of parameters, from which inference can be made (see Gilks, Richardson, & Spiegelhalter, 1996).

A flexible approach to generating samples from posterior distributions for parameters is provided by the Metropolis-Hastings (MH) algorithm, which consists of the following steps:

1. 0. Establish starting values for the parameter (⁰).
   
2. Simulate a candidate parameter ^c from a proposal density ().
   
3. Evaluate the posterior density at ^c and ^t-1, where t-1 references the previous iteration, and form the ratio R = p(^c)(^t-1|^c)/p(^t-1)(^c|^t-1). Here (a|b) is the probability that a would be proposed when the current state of the chain is b, compensating for an asymmetric proposal density, if one is used. If not, R = p(^c)/p(^t-1).
   
4. Compare this result to a U U(0, 1) random number. If R > U then accept the candidate (set ^t = ^c). Otherwise, reject the candidate (set ^t = ^t-1). Return to Step 1 until an adequately large sample has been obtained.
   

In this research, we use a hybridized Gibbs/MH algorithm, specified as follows:

1. 0. Establish starting values for the parameters b and .
   
2. Simulate the latent propensities from TruncMVN distributions as discussed above. To be more specific, an individual Z vector is drawn until it satisfies the truncation requirements discussed above. (This is a Gibbs sampling step.)
   
3. Given these latent data, the augmented likelihood function is now the multivariate normal likelihood discussed above—conduct MH steps to update b and over several additional iterations.
   
4. Return to Step 1.
   

A number of early iterations of the algorithm need to be discarded (a "burn-in" period), because it takes some time for the stationary distribution of the Markov chain to converge on the posterior distribution of interest. The remaining iterates can be treated as a sample from the joint posterior distribution of the parameters. For these analyses, we let the algorithms run for 2 million iterations, saving every 200th observation and retaining the last 1,000 such observations.

Life Table Calculations
Computing multistate life tables requires transition probabilities as input. Each of the 1,000 MCMC iterates represents a single realization from the distribution for b and . A set of transition probabilities can be computed using multivariate normal integrals of the form:

This integral provides the probability of experiencing transition j, given the covariates and the regression parameters. The innermost integral refers to the transition of interest, whereas the -j refers to all other transitions, and x(.) refers to the covariates included in the equation for transition (.). Note that the integration is over a multivariate normal density with 0 mean vector, but with covariance matrix estimated via the algorithm discussed above.

With a completed set of probabilities, life table quantities can be computed as follows:

This equation states that the number of persons in state j at time t + 1 is equal to the number of individuals in state j at time t, plus the sum of the decrements from other states m (m j) into state j, less the sum of the decrements from state j to other states. The decrements to and from state j are calculated using the transition probabilities calculated above multiplied by the number of persons in the respective states at time t. From the ls, life/state expectancies can be calculated in standard fashion (see Palloni, 2000).

	Results

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The weighted number and percentage of the sample experiencing each transition at each ADL level cutoff by age group is presented in Table 1. As expected, the healthy group is substantially larger than the disabled group. Furthermore, with each increase in the ADL level required to become classified as disabled, the disabled group is reduced in size. Also, the number transitioning to death increases for the healthy-start group and decreases for the disabled-start group with increasing ADL cutoff level. This is expected because increasing the ADL level at which one is classified as disabled necessarily increases the size of the healthy starting population, and therefore, increases the number of healthy individuals transitioning to death.

View larger version (17K): [in this window] [in a new window]   Figure 1. Empirical confidence intervals for active life expectancy (ALE) by age group and activity of daily living (ADL) level used to define being disabled. Empirical probability intervals for each ADL level are indicated with vertical lines. The mean ALE for each is represented by the dot

	Discussion and Limitations

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In this research, we have examined whether varying the level at which we define individuals to be disabled affects estimates of ALE. Our results indicate that when one considers variability in estimates of ALE, little difference in ALE estimates is observed. In addition, we obtained these results using one of the largest datasets possible, with a sample size of just under 10,000 observations. In smaller datasets, therefore, it is probable that the differences would be entirely nonexistent. For example, if the dataset contained 1,000 observations rather than 10,000, the gap for persons aged 65–69 between the upper bound of the interval for the 1-ADL threshold model and the 6-ADL threshold model would be only approximately 1.5 years. Among the other age groups, the differences would become nonexistent. Furthermore, our intervals do not consider additional sources of variability in life expectancy that would increase the width of these intervals even further (see Wolfe and Laditka's, 1997, simulation study, which argues that there is a difference between estimating variability in life expectancy and sampling variability around a point estimate, on which most classical inference relies).

Our finding of little difference is important, because it indicates that although point estimates for ALE may vary if the threshold is altered, sampling and other errors may be enough to overwhelm these differences. This means that different surveys may yield different estimates for ALE that would be accounted for by sampling and other errors and not necessarily measurement differences, in terms of the stringency with which being disabled is defined.

Before discussing the implications of these findings for future research, we should pause to ask why we found what we found. We think there are both substantive and methodological explanations for the findings. Substantively, it is typically assumed, given the severity of standard ADL items, that any one ADL limitation is sufficient for operationalizing disability (e.g., Branch et al., 1991; Land et al., 1994). Terminal drop and terminal decline perspectives help frame this argument, in which both perspectives refer to the process of near-death loss of functioning. These perspectives suggest that persons near death suffer an irreversible level of disability that leads ultimately to death. Research has found some support for these hypotheses (e.g., in reference to cognitive impairment, see Aronson et al., 1991; Eagles et al., 1990; Johansson & Zarit, 1997; in terms of self-rated health, see Ferraro & Kelley-Moore, 2001; in terms of comprehensive nutritional/physical-functioning/cognitive measures, see Egbert, 1996).

Terminal drop and terminal decline differ subtly in that terminal drop implies a temporally shorter process (a drop) to death; whereas terminal decline implies a longer (though also irreversible) process to death. In this sense, terminal drop can be seen as a microlevel version of Fries's (1980) compression of morbidity hypothesis. Fries argued that as life expectancy increased, the survival curve would rectangularize as it reached the edge of the (possibly finite) human lifespan. Improvements in health and medicine would then compress morbidity into the last few years of life. Thus, persons would live longer, healthier lives until they declined rapidly to death.

Our results may be explained by the terminal decline perspective, because whereas persons experiencing terminal decline should have similar and somewhat lengthy ALEs, persons experiencing terminal drop should have ALEs that are both similar and short. Here, we found relatively large ALE estimates in all models, regardless of the threshold used to define disability. Yet, the consistency of these estimates across especially the latter three cutoffs implies that relatively few reverse transitions occur prior to death. We temper our reliance on this explanation, however, by noting that a true test of terminal decline/drop would require comparing status-based expectancies, as in Manton and Land (2000b).

Methodologically, altering the thresholds on a simple state space (e.g., healthy/disabled/deceased) is relatively similar to expanding the state space (healthy/slightly disabled/ ... /very disabled/deceased). In the former case, raising the bar for defining disability reduces the likelihood for reverse transitions and makes it more difficult for persons to transit out of the healthy state. In the latter case, whereas expanding the state space increases the propensity for transitions, it does not necessarily increase the propensity for individuals to transit out of the healthy state, unless the state space is expanded below 1 ADL limitation. What our results demonstrate, very simply, is that 1–6 ADL limitations may not require much differentiation; we may, however, need to differentiate the healthy group into mild impairment groups, but that is also an empirical question. To date, it has often simply been assumed that such differentiation is necessary (e.g., Crimmins et al., 1996).

In sum, we suggest that complex state spaces may not be necessary to produce a reasonably accurate picture of ALE. However, there are a number of questions that remain. First, we use only ADL limitations in this study. Additional tests should be conducted using IADL measures and other measures of limitation and functional status. Second, the effects of varying the difficulty level within limitation items should be examined (e.g., does measuring difficulty on any one ADL—such as transferring—as "needs assistance with transferring" vs. "has a problem with transferring" result in significantly different ALE estimates?). The ADLs in the NLTCS are measured via a comprehensive set of questions. In many studies, however, only a single question is used to assess level of functioning. Such limited questioning may yield different results than those found here. Future research should simultaneously examine the differences between altering the dimensionality of the state space and altering the thresholds for states in the state space. These two factors are related in some sense. Raising the threshold on any particular item (making it more difficult to respond affirmatively to a disability item) may be akin to altering the number of positive responses required to be considered disabled—both have the same effect of increasing or decreasing the threshold for considering and individual to be disabled. Our results suggest that complex state spaces may be unnecessary in obtaining adequate estimates of ALE, but they beg the question of whether excessive concern over item difficulty is warranted.

	Acknowledgments

 
The NLTCS is sponsored by the National Institute on Aging and conducted by the Duke University Center for Demographic Studies.

The first two authors contributed equally to this work. Scott Brown is now at the Carolina Population Center, University of North Carolina, Chapel Hill.

	Footnotes

 
Decision Editor: Charles F. Longino, Jr., PhD

Received for publication May 2, 2002. Accepted for publication September 30, 2002.

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