How bad could it be? Worst-case bounds on bias in multistate models due to unobserved transitions

Abstract

Multistate models are often used in social research to analyze how individuals move between states. A typical application is the estimation of the lifetime spent in a certain state, like the lifetime spent in employment, or the lifetime spent in good health. Unfortunately, the estimation of such quantities is prone to several biases. In this paper, we study the bias due to the often implicitly used assumption that there are no unobserved transitions between states. This assumption does often not hold for the panel data typically used to estimate multistate models, as the states occupied by individuals are only known at specific points in time, and further transitions between panel waves are not recorded. We present partially identified estimates of the lifetime spent in a state, or worse-case bounds, which show the maximum possible level of bias due to unobserved transitions. We provide two examples studying the lifetime spent in disability (disabled life expectancy; DLE). The first example applies our methods to results on cohort trends in DLE in the U.S. taken from Crimmins et al. (2009). In the second example, we replicate findings from Mehta and Myrskylä (2017), and apply our methods to data from the U.S. Health and Retirement Study (HRS) in order to estimate the effects of health behaviors on DLE.

Keywords

identification bounds multistate model Markov model disabled life expectancy disability-free life expectancy bias worst-case bounds Health and Retirement Study

Introduction

Multistate models are often used in quantitative social research to analyze how individuals move between different states (Piccaretta and Studer 2019). Based on longitudinal data, the rates or the probabilities of transitioning between states are estimated and used to calculate quantities such as the expected lifetime spent in a specific state. Applications of multistate models include transitions between labor force states (Harris, Zhao, and Zuccheli 2021; Hayward and Lichter 1998; Lorenti et al. 2020), change of family status (Bonetti, Piccaretta, and Salford 2013; Schoen, Landale, and Daniels 2007; Studer, Struffolino, and Fasang 2018), poverty dynamics (Bernstein et al. 2018; Hale, Dudel, and Lorenti 2021), and migration (Klabunde et al. 2017; Raymer, Willekens, and Rogers 2019; Vega and Brazil 2015).

The estimation of multistate models using longitudinal data is prone to several biases. Among the sources of bias that have attracted considerable interest in parts of the literature are unobserved transitions (Hardy et al. 2005; Wolf and Gill 2009). Unobserved transitions are attributable to study design and are a property of the data used to estimate multistate models. They occur when the states that individuals occupy are recorded at fixed time intervals such as every 24 months. Because the transitions that take place during the period between observations are not recorded, some transitions might be missed.

This article describes a procedure that allows us to assess the potential bias in estimates of the lifetime spent in a state caused by unobserved transitions. Based on minimal assumptions, the method yields identification bounds, in the sense of Pearl (2015) and Manski (1993) that can be interpreted as worst-case bounds for bias. Identification bounds, or set estimates, mean that the result of the estimation procedure is not a single, specific value but an interval. Identification bounds essentially show the information that can be gleaned about the lifetime spent in a state from the data without making strong assumptions. The identification bounds we present are sharp. Tighter identification bounds can be achieved if the researcher has additional information, which can sometimes be derived from the data, or if the researcher is willing to commit to stronger assumptions.

Rather than casting the exposition in general terms, we concentrate on a particular application, life expectancy with disability (or disabled life expectancy [DLE]) and disability-free life expectancy (DFLE), as a means to develop and illustrate ideas. This will keep the discussion focused and facilitate notation. Both DLE and DFLE are important indicators of population health (Nusselder and Peeters 2006; Stiefel, Perla, and Zell 2010), especially in aging populations, among whom disability is a growing health concern. For this reason, DLE and DFLE have been extensively studied in social science and health research. Among the topics that have recently been addressed in the literature are differences in DLE by gender, ethnicity/race, and education (Crimmins and Saito 2001; Solé-Auró, Beltrán-Sánchez, and Crimmins 2015); other socioeconomic determinants (Chiu 2019; Magnusson Hanson et al. 2018; Zimmer et al. 2020); changes in DLE and DFLE as a measure of the compression of morbidity (Cai and Lubitz 2007); and the effects of health behaviors like smoking and drinking on DLE (Mehta and Myrskylä 2017). Measures similar to DLE, such as the lifetime spent with or without a certain disease or ailment, have also received considerable attention (e.g., Zimmer and Rubin 2016).

For DLE, unobserved transitions are especially problematic if they happen shortly before death (Agree and Wolf 2017; Gill et al. 2005; van den Hout and Matthews 2009; Wolf and Gill 2009; Yi, Danan, and Land 2004). Most unobserved transitions before death are from nondisabled to disabled (and then to death), as an individual’s health often deteriorates immediately prior to death, and the likelihood that the person will recover from disability shortly before death is low (Yi et al. 2004). Because a portion of the lifetime spent being disabled is therefore missed, DLE tends to be biased downward. Simulation studies have been conducted to assess the potential bias caused by unobserved transitions before death (Wolf and Gill 2009), and a correction procedure was proposed by Yi et al. (2004). The results of these efforts underline that the bias due to unobserved transitions is a cause for concern and might threaten the validity of DLE estimates. Unobserved transitions from being disabled to being nondisabled, and vice versa, that do not happen before death likely cancel each other out over a lifetime and are thus considered to be less problematic (Gill et al. 2005; Wolf and Gill 2009).

To illustrate our method, we present two examples. The first example takes estimates of DLE and DFLE published by Crimmins et al. (2009). We show that the conclusion of the original study—across cohorts, DLE did not increase but DFLE expectancy did—requires implicit assumptions. The second example is based on the U.S. HRS from 1992 to 2014, replicating and expanding on the results of Mehta and Myrskylä (2017). We calculate the DLE of groups with different health behaviors and show that the differentials in DLE found between these groups are relatively robust to bias. We also demonstrate how identification bounds can be narrowed using empirical data. For this example, the Stata code can be downloaded online from the Open Science Framework via https://osf.io/y8qz6/. In addition, we outline an additional example in the final section of the article showing how our method can be applied to topics other than DLE. This additional example uses results of a multistate model of labor market transitions taken from Lorenti et al. (2020).

Bounds on Bias Due to Unobserved Final Transitions

Notation and Setup

We consider a multistate model of disability as shown in Figure 1. Individuals can be nondisabled and disabled. They can stay in the state they occupy, or they can transition to the other state; i.e, nondisabled individuals can become disabled, and disabled individuals can recover. The model also includes the absorbing state dead. Applications of multistate models with a similar structure include transitions between the two states “in the labor force” and “out of the labor force” (e.g., Skoog and Ciecka 2010) or migration between two countries (e.g., Vega and Brazil 2015). Transitions between states can be described using transition probabilities or rates, which are estimated from longitudinal data. While not explicitly shown in Figure 1, the transition probabilities or rates can depend on age and other variables.

Figure 1.

State space and possible transitions of the multistate model of disability.

The longitudinal data used for estimating the multistate model are collected every $κ$ time units, i.e., at exact times $t = 0, κ,2 κ, . . .$ For each time t and for each individual, the state an individual is in is known but not between measurements. This means, for instance, that if an individual is nondisabled at time t and dead at time $t + κ$ , the transition between these states must have happened in the interval $(t, t + κ)$ and that the person may have transitioned from nondisabled to disabled prior to death, even though such a transition is not observed due to the design of the longitudinal study.

For each individual in the data, S_t captures the state the individual is in at the beginning of the interval $[t, t + κ)$ . Moreover, let D denote a variable that captures the interval of death. If for some individual this variable equals d, then that person died in the interval $(d, d + κ)$ ; this interval is open on the left, as we assume that individuals survive to the beginning of the interval of death, and are thus still alive at $t = d$ . S_D captures the last observed state the individual was in before dying, i.e., the state at the beginning of the interval $[d, d + κ)$ . Finally, let I be a variable capturing when transitions occurred, irrespective of the starting state and the receiving state; I can take values in the interval $(0, κ)$ . For instance, transitions are often assumed to occur mid-interval, implying $E (I) = 0.5 κ$ , where $E (\cdot)$ denotes expectation.

We assume that the multistate model and its transition probabilities are estimated correctly. That is, we assume that transition probabilities $Pr (S_{t + κ} = {s^{'}}^{'} | S_{t} = s^{'})$ , where $s^{'}$ and ${s^{'}}^{'}$ are arbitrary states, are not affected by unobserved transitions. For the description of our method, it is not important which of the several approaches to estimating transition probabilities or rates is used (e.g., Dudel 2021; Lynch and Brown 2005) or which assumptions are used to derive DLE and related quantities from the probabilities or rates such as the assumption of mid-interval transitions mentioned above or alternative assumptions.

Decomposing DLE

Let U denote a variable capturing the lifetime individuals spent in the disabled state, and let H denote a variable capturing the lifetime individuals spent in the disability-free state. For DLE, we write $E (U)$ , and for DFLE, we use $E (H)$ . Total life expectancy equals $E (U) + E (H)$ . Given estimates of transition probabilities or rates, $E (U)$ and $E (H)$ can be calculated using multistate methodology. As we outline below, this approach usually ignores transitions before death. Note that ignoring these transitions does not affect the estimate of total life expectancy; i.e., the sum $E (U) + E (H)$ is assumed to be estimated correctly regardless of whether unobserved transitions occur or not.

To show how transitions before death are usually ignored, and what degree of bias this implies, we decompose DLE in two steps. In the first step, DLE can be decomposed in the following way:

E (U) = P r (S_{D} = h) E (U | S_{D} = h) + P r (S_{D} = u) E (U | S_{D} = u),

where h stands for the nondisabled state and u represents the disabled state. That is, DLE is decomposed into the DLE of the individuals who were last observed to be nondisabled, $S_{D} = h$ , and the DLE of the individuals who were last observed to be disabled, $S_{D} = u$ . Note that here, DLE is still defined as a lifetime measure of disability and not just as DLE during the last interval of life.

DLE can be further decomposed by introducing unobserved transitions between states, after being in state S_D, and before death. For the sake of simplicity, we assume that between the last observed state S_D and death, no recovery from disability is possible. That is, there might be unobserved transitions before death from being nondisabled to being disabled but not the other way round; this assumption is not necessarily required, and we discuss how to remove it below in Removing Further Assumptions section. Let S_T capture the state a transition happens to from S_D. If for an individual $S_{D} = h$ and $S_{T} = u$ , the person was last observed to be nondisabled and then became disabled before dying. As just introduced, the combination $S_{D} = u$ and $S_{T} = h$ is ruled out. If $S_{D} = S_{T}$ , then no transition has happened. Using this notation, we can write

E (U) = P r (S_{D} = h, S_{T} = h) E (U | S_{D} = h, S_{T} = h) + P r (S_{D} = h, S_{T} = u) E (U | S_{D} = h, S_{T} = u) + P r (S_{D} = u) E (U | S_{D} = u) .

Thus, the first term on the right-hand side of equation (1) is further decomposed into the DLE of individuals who were observed to be nondisabled before death and who stayed in this state until death, $S_{D} = h$ and $S_{T} = h$ , and the DLE of individuals who were also nondisabled at the time of the last observation before death, but who became disabled before death, $S_{D} = h$ and $S_{T} = u$ .

Sharp Bounds on Bias

As S_T is not observed, the individual parts of equation (2) cannot be estimated. For this reason, it is (often implicitly) assumed in the literature that no transitions happen shortly before death, i.e., $S_{D} = S_{T}$ . Assuming that no transitions can happen between S_D and death is equivalent to assuming that $E (U | S_{D} = h, S_{T} = u) = E (U | S_{D} = h, S_{T} = h)$ . Using this assumption,

E_{N} (U) = P r (S_{D} = h, S_{T} = h) E (U | S_{D} = h, S_{T} = h) + P r (S_{D} = h, S_{T} = u) E (U | S_{D} = h, S_{T} = h) + P r (S_{D} = u) E (U | S_{D} = u),

is calculated, which differs from

E (U)

. This is implicit in time-discrete multistate models.

The bias of such models is given by the difference between (2) and (3),

\begin{array}{l} Δ_{U} = E (U) - E_{N} (U) \\ = P r (S_{D} = h, S_{T} = u) [E (U | S_{D} = h, S_{T} = u) - E (U | S_{D} = h, S_{T} = h)] . \end{array}

Writing $ρ_{U}$ instead of $P r (S_{D} = h, S_{T} = u)$ , and $δ_{U}$ instead of $E (U | S_{D} = h, S_{T} = u) - E (U | S_{D} = h, S_{T} = h)$ , equation (4) can be written as

\begin{array}{l} Δ_{U} = ρ_{U} δ_{U} . \end{array}

$ρ_{U}$ is the probability that the last observed state before death was nondisabled and that the individual became disabled before dying. $δ_{U}$ is the average time such individuals spent in the disabled state before death. As $ρ_{U} \geq 0$ and $δ_{U} > 0$ , the bias has to be nonnegative; i.e., unobserved transitions before death lead to downward bias in DLE.

$δ_{U}$ has to be in the interval $(0, κ)$ ; i.e., 0 is the lower bound of $δ_{U}$ , and $κ$ is its upper bound. This holds as individuals transitioning from h to u before death may have done so at any time from directly before death to the beginning of the interval $[d, d + κ)$ and then surviving until the end of the interval. In the former case, $δ_{U}$ approaches 0; in the latter case, $δ_{U}$ is approximately equal to $κ$ . If data on the exact time of death is available, it can be used to estimate bounds on $κ$ empirically, but such data will often not be available. For this reason, multistate models often use the mid-interval assumption or alternative assumptions. These assumptions can be used to reduce the bounds (see Tightening the Bounds Using Additional Information and Assumptions subsection).

$ρ_{U}$ has to be in the interval $[0, P r (S_{D} = h)]$ , and it is thus bounded by the probability that the last observed state before death is nondisabled. $P r (S_{D} = h)$ is an estimable quantity. Without data or additional assumptions, the bounds for $P r (S_{D} = h)$ are given by $[0, 1]$ . Note that, strictly speaking, $ρ_{U}$ is bounded by $[0, min {P r (S_{D} = h), P r (S_{T} = u | S_{D} = h), P r (S_{T} = u), P r (S_{D} = h | S_{T} = u)}]$ , as both $P r (S_{D} = h, S_{T} = u) = P r (S_{T} = u | S_{D} = h) P r (S_{D} = h)$ and $P r (S_{D} = h, S_{T} = u) = P r (S_{D} = h | S_{T} = u) P r (S_{T} = u)$ hold. However, $P r (S_{T} = u | S_{D} = h)$ , $P r (S_{D} = h | S_{T} = u)$ , and $P r (S_{T} = u)$ usually will not be known.

Worst-case bounds for the bias in DLE, $Δ_{U}$ , are thus given by $[0, κ)$ when combining the bounds for $δ_{U}$ and $ρ_{U}$ , if no additional assumptions are made. The upper bound of $Δ_{U}$ , $κ$ , is derived from the product of the upper bounds of $δ_{U}$ and $ρ_{U}$ ; the lower bound is derived in a similar way. The bounds on $Δ_{U}$ show that the maximum possible error is directly related to the granularity of measurement: i.e., if the results are measured in years, five-year intervals will lead to a potentially large bias, while monthly intervals will lead to small bias. These bounds are sharp; i.e., it is not possible to derive tighter bounds without additional information.

Bounds on Disabled and Non-DLE

Let $[Δ_{U}^{-}, Δ_{U}^{+})$ denote the bounds on the bias in DLE. Then, the bounds on DLE are simply given by $[E (U) + Δ_{U}^{-}, E (U) + Δ_{U}^{+})$ . The bias in DFLE is related to the bias in DLE, as together they equal life expectancy, $e = E (H) + E (U)$ , which is assumed to be estimated correctly. The upper bound on DLE, $E^{+} (U) = E (U) + Δ_{U}^{+}$ , implies a lower bound for healthy life expectancy, $E^{-} (H) = e - E^{+} (U) = E (H) - Δ_{U}^{+}$ . Conversely, the upper bound $E^{+} (H)$ is given by $E^{+} (H) = E (H) - Δ_{U}^{-}$ . These calculations guarantee that each additional year with disability means one less year without disability and that DLE and DFLE together yield total life expectancy. Here, the intervals can also be seen as reflecting the possible range of values for DLE and DFLE consistent with the data and without assumptions regarding unobserved transitions.

Tightening the Bounds Using Additional Information and Assumptions

The worst-case bounds presented above can be narrowed by introducing additional assumptions and by using empirical data. Whether the assumptions are appropriate will differ from case to case.

Often, the estimates of DLE and DFLE are based on the assumption that transitions occur, on average, at mid-interval or, formally, $E (I) = 0.5 κ$ . If this assumption also holds for transitions to death when the last observed state was not disabled—i.e. conditional on $S_{d} = h$ —then the additional lifetime spent in disability, $δ_{U}$ , has to be in $(0, 0.5 κ)$ . This reduces the worst-case bounds for bias in DLE, $Δ_{U}$ , to $[0, 0.5 κ)$ . Tighter bounds on $δ_{U}$ can be derived if we are willing to make assumptions about when the transition from nondisabled to disabled happened before death. Assuming that it happened, on average, at mid-interval between d and $d + 0.5 κ$ , the upper bound for $Δ_{U}$ can be reduced again, and we have $[0, 0.25 κ)$ . Note that these two scenarios are more specific than assuming $E (I) = 0.5 κ$ , in that, first, they are conditioned on $S_{d} = h$ , and, second, they apply to specific transitions.

Without making additional assumptions, the bounds for the probability that individuals who were nondisabled at the last observation and became disabled before dying, $ρ_{U}$ , are given by $[0, 1]$ . This was derived from the fact that $ρ_{U}$ is bounded from above by $P r (S_{D} = h)$ , which can in turn attain values in $[0, 1]$ . However, the upper bound of 1 seems rather unrealistic, as it assumes that shortly before death, the disabled state never is reached. But given longitudinal data, it is easily possible to calculate $P r (S_{D} = h)$ . If $P r (S_{D} = h)$ is low, the bounds on DLE will be narrow, even if $κ$ is relatively large.

A simple argument for deriving tighter bounds on $ρ_{U}$ even when $P r (S_{D} = h)$ is not known can be made as follows. In a simplified multistate model in which the transition probabilities do not depend on age and recovery from disability is not possible, $P r (S_{D} = h)$ is given by

P r (S_{D} = h) = \frac{P r (S_{t + κ} = w | S_{t} = h)}{P r (S_{t + κ} = w | S_{t} = h) + P r (S_{t + κ} = u | S_{t} = h)},

where w represents the state “dead”; if we assume that the probability of dying can be no bigger than the probability of becoming disabled, then $P r (S_{D} = h)$ is bounded from above by $0.5$ . As $ρ_{U}$ is bounded from above by $P r (S_{D} = h)$ , this would imply that $ρ_{U} \in [0, 0.5]$ . This result can be combined with the assumptions on $δ_{U}$ introduced above to arrive at $Δ_{U} \in [0, 0.125 κ)$ , i.e., using $δ_{U} \in (0, 0.25 κ)$ and $ρ_{U} \in [0, 0.5]$ . While a multistate model that does not depend on age is unrealistic, it provides a closed-form solution for $P r (S_{D} = h)$ , which can help tighten the bounds.

Applications

Trends in DLE and DFLE in the Longitudinal Studies of Aging (LSOA)

As a first example, we consider the identification bounds of the results presented by Crimmins et al. (2009). The authors studied the remaining DLE at age 70 and the DFLE at age 70 in the United States. DLE and DFLE were calculated for and compared across two cohorts using multistate life tables and data from the LSOA I and II, which were supplements to the National Health Interview Survey. Comparing the two cohorts, Crimmins et al. (2009) found that DLE remained rather stable, while DFLE increased for the younger cohort (LSOA II) by about $0.6$ years.

LSOA I was conducted in the years 1984, 1986, and 1988 and supplied data for the first cohort while LSOA II was conducted in 1994, 1996–1997, and 1999–2000 and supplied data for the second cohort. DLE and DFLE were defined as life expectancy with or without limitations in activities of daily living (ADL disability; includes difficulties in bathing, dressing, eating, getting in/out of a bed or chair, and toileting), and/or limitations in instrumental ADL (instrumental ADL disability: preparing meals, shopping for groceries, managing money, using the telephone, and doing light housework).

The point estimates of DLE and DFLE reported by Crimmins et al. (2009) are shown in the first lines of Figure 2 (DLE) and Figure 3 (DFLE). The results for LSOA I are shown in orange, and the results for LSOA II are shown in green. These point estimates are based on the assumption that there are no unobserved transitions from nondisabled to disabled.

Figure 2.

Point estimates and identification bounds of disabled life expectancy based on Crimmins et al. (2009). Point estimates are in the first line of the figure; the following lines show the identification bounds arising from different sets of assumptions, using $κ = 2$ for Longitudinal Studies of Aging (LSOA) I and $κ = 2.5$ for LSOA II. Results for LSOA I are shown in orange and for LSOA II in green. Source: Crimmins et al. (2009), own calculations.

Figure 3.

Point estimates and identification bounds of disability-free life expectancy based on Crimmins et al. (2009). Point estimates are in the first line of the figure; the following lines show the identification bounds bias from different sets of assumptions, using $κ = 2$ for Longitudinal Studies of Aging (LSOA) I and $κ = 2.5$ for LSOA II. Results for LSOA I are shown in orange and for LSOA II in green. Source: Crimmins et al. (2009), own calculations.

Applying the reasoning laid out in the previous section, lines two to five of Figure 2 and Figure 3 show the identification bounds for DLE and DFLE using different assumptions. The second line in the figures shows the bounds resulting from using $κ = 2$ for LSOA I and $κ = 2.5$ for LSOA II without additional assumptions, as described in Sharp Bounds on Bias subsection; that is, the bounds are calculated as the point estimate plus the endpoints of the interval $[0, κ)$ . The third line shows the identification bounds using the assumption that mortality occurs at mid-interval, conditional on $S_{d} = h$ , adding the endpoints of the interval $[0, 0.5 κ)$ to the point estimates. The fourth line adds the assumption of mid-interval transitions from nondisabled to disabled in the last interval before death; thus, the interval $[0, 0.25 κ)$ is added to the point estimates. The fifth line is based on additionally assuming that the probability of becoming disabled is greater than the probability of dying, as discussed in Tightening the Bounds Using Additional Information and Assumptions subsection and adding the endpoints of the interval $[0, 0.125 κ)$ to the point estimates.

The more restrictive the assumptions are, the narrower the identification regions become. For DLE, the identification regions for LSOA I and LSOA II always overlap, regardless of the assumptions made. In this context, “overlap” means that one or both of the endpoints of one identification region are between the endpoints of the other region. This generally implies that, given the assumptions used, it cannot be ruled out that the estimates have the same values. In this specific application, it means that LSOA I and LSOA II might have the same levels of DLE. If instead of DLE, the proportion of remaining life expectancy in disability is studied—i.e., DLE divided by life expectancy—similar findings emerge: For LSOA I, the bounds for the proportion of lifetime spent in disability is between $19$ percent and $21$ percent when the strongest assumptions are used (line 5 in Figure 2), while for LSOA II, the corresponding bounds are between $18$ percent and $20$ percent; thus, the bounds overlap.

The point estimates for DFLE are further apart. Nevertheless, as line 2 in Figure 3 shows, when standard assumptions are applied, the identification regions of LSOA I and II overlap. Introducing mid-interval mortality conditional on $S_{d} = h$ causes the identification bounds to become much smaller, but they still overlap. A similar pattern can be observed for introducing mid-interval transitions from nondisabled to disabled in the last interval before death although the degree of overlap is much less and would not have occurred if $κ = 2$ was used for LSOA II instead of $κ = 2.5$ . Moreover, for the DFLE estimates of LSOA I and II to be in the region of overlap, LSOA I results would have to be unaffected or almost unaffected by unobserved transitions, while the LSOA II results would have to be very strongly affected by these transitions. Still, only if it is assumed that the probability of becoming disabled is higher than the probability of dying, the bounds do not overlap anymore (fifth line of Figure 3).

The identification intervals presented in Figure 2 and Figure 3 are conditional on the sample. That is, the bounds show whether the samples are compatible with the statement that participants of LSOA I and LSOA II have the same DLE or DFLE. The uncertainty expressed in the intervals is solely due to (potential) unobserved transitions. Measuring uncertainty due to sampling variance—i.e. statistical inference—can follow in a second step. If the identification intervals overlap, and it is not possible to determine whether there are differences between the two samples, then detecting differences in the population will not be possible either. How statistical inference can be conducted for identification intervals is discussed in the next subsection.

DLE and Health Behaviors in the HRS

As a second example, we reestimate a subset of the results of Mehta and Myrskylä (2017). The Stata code is available online. Using data from the HRS, these authors studied differentials in DLE at age 50 and in DFLE at age 50 by health behaviors. Among other results, they found that individuals with advantageous health behaviors—i.e., individuals who never smoked and who were not obese—had a considerably higher DFLE than individuals with an average behavioral profile and than individuals who were engaging in adverse behaviors. While the observed differences in DLE between the advantageous behavior group and the average behavior group were small, adverse behaviors were found to be associated with increased DLE.

Our data source is the HRS. The HRS is a panel survey of U.S. residents aged 50 and older that covers a wide range of socioeconomic and health indicators. It is conducted by the Survey Research Center of the Institute for Social Research of the University of Michigan and is supported by the National Institute on Aging and the Social Security Administration. For an overview of the data, see Juster and Suzman (1995). Respondents are surveyed biannually ( $κ = 2$ ) and are followed over time. The exact date of death is also recorded and obtained from either interviews with relatives or from the National Death Index. We study individuals who were aged 50–74 in 1998 and use the corresponding follow-up data for the years 2000–2014. In total, we cover $14, 627$ individuals and $105, 786$ person-waves.

We consider health behaviors that fit what Mehta and Myrskylä (2017) called low-risk and high-risk behavioral profiles (LRB1, HRB1). The LRB1 group consists of people who had never smoked and were of normal weight or overweight at the time of the interview (body mass index, or BMI, of 20–30). The HRB1 group consists of people who had ever smoked (including current smokers) and were obese (BMI $>$ 30). As a reference, we also include results for the total population—i.e., the population average DLE—and refer to this group as average profile (AVG). Disability is defined as having a limitation in at least one ADL: walking, dressing, bathing, getting in or out of bed, and eating. For more details on the sample and the variables, see the original article.

Based on the longitudinal data for 2000 to 2014, we model transitions using discrete-time competing risk models (Allison 1982), controlling for age, gender, and health behaviors, and using sampling weights. The transition probabilities derived from these models are used in Markov chains to calculate DLE, while assuming that transitions between states occur, on average, at mid-interval. To calculate the DLE and the DFLE of the AVG, we calculate the transition probabilities at population averages of smoking and obesity.

Standard errors for the endpoints of the identification intervals can be calculated using the bootstrap resampling method, whereby for each bootstrap replication, the transition probabilities and DLE are reestimated. To construct confidence intervals, we follow Horowitz and Manski (2000). Let ${\hat{E}}^{-} (U)$ and ${\hat{E}}^{+} (U)$ denote the point estimates for the endpoints of the identification interval of DLE. B denotes the number of bootstrap replications, and ${\hat{E}}_{b}^{-} (U)$ and ${\hat{E}}_{b}^{+} (U)$ are the point estimates resulting from the bth bootstrap replication. The endpoints of the $95$ percent confidence interval are given by ${\hat{E}}^{-} (U) - γ$ and ${\hat{E}}^{+} (U) + γ$ , where the parameter $γ$ is chosen such that

min_{γ} (γ |\frac{1}{B} \sum_{b = 1}^{B} I [{\hat{E}}^{-} (U) - γ \leq {\hat{E}}_{b}^{-} (U)] I [{\hat{E}}^{+} (U) + γ \geq {\hat{E}}_{b}^{+} (U))] \geq 0.95),

where $I (\cdot)$ is the indicator function. That is, $γ$ is chosen such that the interval defined by ${\hat{E}}^{-} (U) - γ$ and ${\hat{E}}^{+} (U) + γ$ covers $95$ percent of the identification intervals from the replications. We calculate $1, 000$ bootstrap replications, resampling all individuals with the same sampling probability. Accounting for the stratified structure of the HRS in the resampling procedure leads to very similar results.

The HRS data allow us to narrow the bounds on $ρ_{U}$ and $δ_{U}$ using empirical data. First, we can estimate $P r (S_{D} = h)$ to bound $ρ_{U}$ . $P r (S_{D} = h)$ is the proportion of individuals who are observed to be in the nondisabled state before dying; we estimated it from the empirical fraction of respondents who were healthy in the survey wave before they died. We find that this proportion is roughly $55$ percent and thus rather close to the bound derived based on technical arguments in Tightening the Bounds Using Additional Information and Assumptions subsection. Second, as the date of death is available in the HRS, we can use it to bound $δ_{U}$ . Specifically, we estimate the average remaining lifetime between the last wave in which an individual was observed and death, given that the last state observed before death was nondisabled, $S_{D} = h$ . This remaining lifetime amounts to $1.23$ years, or roughly $60$ percent of the interval length, and is thus slightly longer than it is when we assume assuming mid-interval deaths. We combine the empirically derived bounds for $ρ_{U}$ and $δ_{U}$ as one of several sets of assumptions.

Results are shown in Figures 4 (DLE) and 5 (DFLE). As in the previous example, the figures show point estimates as well as several sets of identification bounds arising from different sets of assumptions. Confidence intervals are shown as dashed lines. The results in both figures are for men and women combined; results by gender are available upon request.

Figure 4.

Point estimates and identification bounds of disabled life expectancy using Health and Retirement Study data. Confidence intervals are shown as dashed lines. Point estimates in the first line of the figure; the following lines show the identification bounds bias from different sets of assumptions. Results for low-risk behavioral profile (LRB1) are in orange, results for average profile AVG are in green, and results for high-risk behavioral profile (HRB1) in purple. Source: Health and Retirement Study, own calculations.

The first line of Figure 4 shows that the point estimate of DLE of LRB1 is slightly lower than that the one of the population average ( $4.7$ years versus $5.1$ years), whereas the DLE of HRB1 is substantially higher than that of the population average ( $7.3$ years), and the confidence intervals do not overlap in line with the results presented by Mehta and Myrskylä (2017). When using the standard assumptions of multistate models, the implied identification bound of the average group overlaps with the identification bounds of the low-risk group (second line) but not with the high-risk group. However, all confidence intervals do overlap despite clearly separated point estimates. This occurs because the uncertainty expressed through the identification bounds is considerably larger than the uncertainty captured through the confidence intervals (two years vs. roughly one year).

If we are willing to accept the assumptions that transitions to death happen at observational midpoints if $S_{d} = h$ (third line of Figure 4), that individuals who became disabled shortly before death did so at the midpoint between the last observation and death (fourth line), and that the proportion of healthy individuals is bounded by $0.5$ (fifth line), we arrive at successively narrower bounds, with the last set of assumptions having nonoverlapping identification bounds and nonoverlapping confidence intervals across all groups. The bounds for the HRB1 group are always greater than the bounds for the other two groups, whereas the bounds for LRB1 and AVG do not overlap only when several assumptions are combined (fifth line), and their confidence intervals always overlap.

The identification bounds using the empirical estimates for the upper bounds of $ρ_{U}$ and $δ_{U}$ are shown in the sixth line of Figure 4. Apart from the values estimated for the bounds on $ρ_{U}$ and $δ_{U}$ , no other assumptions are used. The resulting bounds are roughly equivalent to those that are found when mid-interval transitions are assumed (fourth line of Figure 4). This means that the scenarios leading to narrower and nonoverlapping bounds for LRB1 and AVNG are not backed up by empirical data. However, these bounds cannot be ruled out, and the data are not at odds with these scenarios. Whether a researcher is willing to trust such results is a judgment call.

Figure 5 shows results for DFLE, which can be seen as the flip side of the DLE bounds. Again, we obtain results similar to those of Mehta and Myrskylä (2017). The differences between the average group and the two other groups are vast ( $- 4.2$ years to LRB1 and $5.8$ years to HRB1), and the gap in DFLE depending on whether health behaviors are advantageous (LRB1) and adverse (HRB1) amounts to $10.0$ years. With differences of such magnitude, even the identification bounds that are obtained under standard assumptions do not call into question the conclusions based on the point estimates alone. The same holds for the confidence intervals. Furthermore, if we accept all additional assumptions (fifth line), the identification bounds become a negligible factor when interpreting the results. Using empirical estimates to bound results (sixth line) leads to the same conclusions.

Figure 5.

Point estimates and identification bounds of disability-free life expectancy using Health and Retirement Study data. Confidence intervals are shown as dashed lines. Point estimates in the first line of the figure; the following lines show the identification bounds bias from different sets of assumptions. Results for low-risk behavioral profile (LRB1) are in orange, results for average profile AVG are in green, and results for high-risk behavioral profile (HRB1) are in purple. Source: Health and Retirement Study, own calculations.

Removing Further Assumptions

Unobserved Recovery

So far, we have assumed that in the last interval before death, there might be unobserved transitions from nondisabled to disabled, but not the other way round; i.e., that there is no unobserved recovery. While it has been argued in the literature that unobserved recovery is less relevant than unobserved disability (e.g., Yi et al. 2004), this assumption is not required and can easily be removed. Unobserved recovery leads to the overestimation of DLE. If both unobserved recovery and unobserved transitions to disability occur, then overestimation and underestimation will cancel each other out to some extent.

This means that the maximum possible amount of overestimation through unobserved recovery will occur if there are no unobserved transitions to disability. In this case, bias can be bounded in a manner similar to that used for underestimation, as described in Bounds on Bias Due to Unobserved Final Transitions section. It is given by

\begin{array}{l} Δ_{H} = - ρ_{H} δ_{H}, \end{array}

where $Δ_{H}$ is the bias caused by unobserved recovery; $ρ_{H}$ is the probability of being last observed in the disabled state, but then transitioning to the nondisabled state; and $δ_{H}$ is the additional time spent in the nondisabled state. Based on arguments similar to those applied in the case of underestimation, worst-case bounds for $Δ_{H}$ are given by $(- κ,0]$ , and tighter bounds can be achieved, as discussed in Tightening the Bounds Using Additional Information and Assumptions subsection.

Combining bounds on overestimation and underestimation gives the following bounds on DLE:

(E (U) + Δ_{U}^{-} + Δ_{H}^{-}, E (U) + Δ_{U}^{+} + Δ_{H}^{+}),

where $Δ_{H}^{-}$ is the lower bound of $Δ_{H}$ and $Δ_{H}^{+}$ is its upper bound. Depending on the assumptions one is willing to make, overestimation and overestimation may cancel each other out. The widest possible bounds correspond to the extreme cases in which there is either no unobserved recovery, or no unobserved transitions from the disabled to the nondisabled state, and they are given by $E (U) - κ$ and $E (U) + κ$ . Depending on the point estimate $E (U)$ and the value of $κ$ , the lower bound should be set to $max (0, E (U) - κ)$ , to ensure nonnegativity.

Intervals Other Than the Death Interval

We have focused on unobserved transitions in the last observed interval before death, given that for earlier intervals, unobserved transitions to disability and unobserved recovery likely cancel each other out (Gill et al. 2005; Wolf and Gill 2009). However, our approach could also be applied to unobserved transitions in earlier intervals, either instead of or in addition to the interval before death. The same reasoning as in Bounds on Bias Due to Unobserved Final Transitions section can be applied. Unlike in the death interval, not everyone will die in the earlier intervals, but we can allow for this by choosing $δ_{U}$ (or $δ_{H}$ ) accordingly.

Ultimately, the bounds will become too wide to be informative if too many intervals are considered at once. As an extreme case, let us assume that DLE at birth is to be estimated; i.e., DLE and DFLE sum to total life expectancy at birth. With unobserved transitions possible in all age intervals and no further assumptions applied, the bounds for DLE can range from (close to) zero to (close to) total life expectancy at birth. At the one extreme, it is possible that the individuals who are observed in the disabled state are in this state only at the time of the interview, having transitioned to disability shortly before the interview, and then recovered immediately thereafter. At the other extreme, a similar set of circumstances could apply to the healthy state. While such scenarios are rather unrealistic, they show the effect of using only a few assumptions.

Conclusions

In this article, we have presented a method that allows researchers to calculate the potential bias in multistate models due to unobserved transitions between states. Specifically, we have provided simple expressions for the bounds of the worst-case level of bias and applied them to estimates of DLE potentially biased by unobserved transitions shortly before death. Our method can easily be applied to any study of DLE and to any DLE results published in the literature, without the need to access the underlying data. This feature was illustrated by the first example we provided, which was based on results published by Crimmins et al. (2009). Our second example was based on data from the HRS. It showed how mild assumptions and empirical estimates can be combined for the calculation of identification bounds.

Our method is generally applicable to any study of the expected lifetime spent in a state of interest, and it is not limited to estimating DLE. In particular, our method can be applied to any estimate published in the literature, without the need to estimate a multistate model. For instance, Lorenti et al. (2020) provide results on the expected lifetime spent working at age 50 (working life expectancy; WLE), and they show how WLE depends on early-life socioeconomic status (SES). Their estimates are based on the biannual HRS data and a multistate model with three states: working, disabled, and not working, including retirement. One of the key findings is that men with low early-life SES have a WLE of 9.5 years. This estimate of WLE is potentially too high, as individuals who are last observed to be working and then die might transition to disability before dying. However, in this case, the bounds on bias will be low: The probability of the last observed state before death being working is relatively small because working individuals will be much younger and healthier than nonworking individuals. Even if we assume that for men with low early-life SES this probability is 25 percent and combine it with the assumption of mid-interval transitions ( $0.5 κ$ , with $κ = 2$ for the HRS), the overestimation of WLE will not be more than 0.25 years.

Our method is also compatible with different types of modeling approaches, as long as they provide estimates of the time spent in a state. This includes multistate approaches as well as descriptive analyses of real cohorts. Moreover, the narrowing of identification bounds is not restricted to the assumptions discussed in this article, and any set of assumptions that the researcher is willing to make can be used, regardless of whether they are based on substantive knowledge or empirical estimates.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Christian Dudel

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