Multistate Actuarial Models of Functional Disability

Abstract

Long-term-care (LTC) costs are expected to significantly increase over the coming decades as the Baby Boom generation nears retirement. Recent policy discussions in the United State have focused on expanding the private LTC insurance market so as to alleviate some of the pressure on public programs. An important and fundamental input to the pricing of LTC insurance products is a set of age- and sex-specific functional status transition rates that can flexibly take into account alternative benefit trigger specifications. We apply generalized linear models to evaluate disability transitions for individuals in old age based on a large sample of U.S. elderly. We estimate a multistate model for LTC insurance applications and find significant differences in disability rate patterns and levels between our set of estimates and those separately estimated using an earlier approach developed by the Society of Actuaries. Our results suggest that the elderly face a 10% chance of becoming LTC disabled only at ages past 90, rather than in their 80s. Furthermore, age patterns of recovery are found to differ significantly between the sexes. We also show that these estimates of transition probability are sensitive to the definition of “LTC disability,” which has implications for the design of benefit triggers for private and public LTC insurance programs.

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Notes

The Task Force was set up in 1991 at the request of the National Association of Insurance Commissioners to address the valuation needs for LTC insurance.

The NLTCS is a longitudinal survey of a random sample of Medicare enrollees and aims to study changes in the health and functional status of older Americans (aged 65+). The survey began in 1982, with follow-ups in 1984, 1989, 1994, 1999, and 2004. Society of Actuaries (1995) utilized data from the first two waves only, and Robinson (1996) utilized data from the first three waves.

The “transition intensity approach” refers to the method of deriving the differential equations for calculating transition probabilities from transition intensities (see Haberman and Pitacco 1999).

Note that the period of 90 days is not a requirement for a 90-day elimination period. Instead, it means that the policyholder has to obtain certification from a licensed health care practitioner that he or she is expected to continue to meet the ADL trigger for the next 90 days.

Befor the 1998 wave, the survey question on ADLs was inconsistent across waves in terms of specific question wording, number of activities assessed, and response collection (see Fong et al. 2014). Data were obtained from the RAND HRS file version L.

As in Renshaw and Haberman (1995), we consider only linear predictors for graduation purposes here. See Forfar et al. (1988) for a discussion of nonlinear predictors.

This is akin to the uniform distribution of death assumption conventionally adopted in life table computations, and is consistent with prior applied studies (e.g., Reuser et al. 2009) that have used HRS data to analyze ADL disability transitions.

Note k + 1 is the total number of unknown parameters.

See the Appendix for mathematical formulas of the residual deviance statistic.

In the likelihood-ratio test, the test statistic is asymptotically distributed as a chi-square with degrees of freedom equal to the difference in number of parameters between the two nested models.

The numbers of exposure years in the 50–54 and 95–100 age groups are relatively smaller than those for other age groups, so care is taken in interpreting any results pertaining to these ages.

For a given age x, the crude (or raw) transition intensity is the raw transition count divided by the number of central exposure years. Since the waiting time to transition follows an exponential distribution, the crude transition intensity is the maximum likelihood estimator of the true transition intensity.

We compared the AICC and BIC criteria values of four GLMs (the Poisson GLM with log-link, the binomial GLM with complementary log-log link, binomial GLM with logit-link, and binomial GLM with probit-link) for each of the four transition intensity of interest and find no substantive differences (results not reported in detail here).

Overdispersion is given by φ > 1, implying that the conditional variance of the response variable increases more rapidly than its mean. Much less common, although also possible for count data, is the case of underdispersion. See Appendix for a mathematical representation of the dispersion parameter.

Total variance of the responses is thus given by the sum of these two components. The covariance matrix is employed to compute the variance of the fitted transition rates using the delta method. Simulations are used to generate the unconditional distribution of the responses.

Most of these earlier studies employed a definition of disability, which is not specific to LTC ADL triggers. For example, some studies measure only lower-body disability while others define disability as having just one or more ADLs.

For example, also using recent waves of HRS data, Fong et al. (2014) report that the prevalence rate of 2+ or more ADL disabilities is 30% for females and 22% for males in the 85–89 age group.

The SSA cohort tables 1900– 1940 are sourced from Bell and Miller (2005). For males, the weights applied based on sample composition are 0.3% (1900 table), 6% (1910), 19% (1920), 32% (1930), and 43% (1940). For females, the weights are 1%, 8%, 20%, 28%, and 42% respectively.

The solid and dashed gray lines reflect mortality rates for the nondisabled and the disabled (2 or more ADL disabilities) respectively. To convert the eight-state transition probabilities in Chandler (2007) to three-state probabilities, we use long-run stationary weights.

Unlike in Robinson (1996) where data observation periods are of different lengths, our data are split into uniform two-year periods. Consequently, the maximum likelihood estimates derived from aggregating the data by observation period is identical to those derived using six separate observation periods.

This is done to adhere as closely as possible to the seven initial health states given in Robinson (1996). Including the “death” state; there are five states in our replica of the Robinson model. To enable comparability with the GLM estimates, we then apply long-run weights to convert the Robinson five-state transition probabilities into three-state transition probabilities. In a separate analysis, we directly derive the Robinson three-state transition probabilities instead based on a three-state setup. We find no significant differences between the two sets of Robinson transition probabilities and thus present only results from the former.