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A method for calculating the implied no-recovery three-state transition matrix using observable population mortality incidence and disability prevalence rates among the elderly

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Abstract

The most accurate estimation of transition probabilities for a multi-state model of health status requires longitudinal data. However, for many countries such data are usually not available. Instead, population level mortality incidence and disability prevalence rates are often all that can be accessed. In this paper, for a three-state no-recovery model (with states healthy, disabled, dead), using simple mathematical derivations, we propose a framework to estimate the age- and gender-specific boundaries within which each of the transition probabilities should fall. We then provide two methods for estimating unique transition probabilities—a least squares procedure and a method based on the ‘extra mortality’ factor proposed by Rickayzen and Walsh (Br Actuarial J 8(2):341–393, 2002, https://doi.org/10.1017/s1357321700003755). We also show the acceptable range for the ‘extra mortality’ factor given the mortality and disability data. Furthermore, we provide a critique of the method proposed by Van der Gaag et al. (Demogr Res 32:75, 2015), as their estimates can fall outside the acceptable boundaries. Finally, we estimate life and health expectancies, as well as premium rates for a life care annuity and a disability annuity using our derived transition probabilities.

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Notes

  1. An extension to the more general three-state, with-recovery model is provided in “Appendix C” of this paper.

  2. Appendix C provides the model formulation with the no-recovery assumption relaxed.

  3. In the Appendix A we have provided data sources for \( q_{x} \) and \( \pi_{x} \) for different world regions.

  4. Ecological inference refers to making inferences about the numbers in each cell of a cross-table given the marginals of the cross-table.

  5. We note that the functional form of the extra mortality formulae is not very flexible. Moreover, later literature has used the same functional form proposed by Rickayzen and Walsh (2002), as seen in (Leung 2004; Rickayzen 2007; Hariyanto et al. 2014a, b) while only adjusting the value of M. Here, we propose a way to generate the value of M that is consistent with the observed population mortality incidence and disability prevalence data under the condition that a researcher/practitioner intends to employ the method originally proposed by Rickayzen and Walsh (2002).

  6. For a given sub-population, the value of M is not age-dependent in the studies mentioned above, i.e., the studies employ a single M for all ages. Here, we propose a method of estimating the value of M that can be applied to all ages for a given sub-population.

  7. Please refer to “Appendix B” for the results for Australian males.

  8. Assuming no one lives past age 100, the results are truncated at age 100. In Table 5, we have excluded the health expectancy for persons who already have a disability as their health expectancy will be 0 years under a three-state model without recovery.

  9. A comprehensive guide to computing actuarial values of health state-dependent annuity is given by Pitacco (2014).

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Appendices

Appendix A: Data sources for \( \varvec{q}_{\varvec{x}} \) and \( \varvec{\pi}_{\varvec{x}} \)

Mortality data \( \left( {q_{x} } \right) \)

Compared to disability prevalence data, mortality data is more readily available. The World Health Organisation (WHO) maintains a data repository containing the demography and mortality data for most countries in the world. Similar information is also available through the Department of Economic and Social Affairs of the United Nations (DESA—UN). There are many instances where only abridged life tables (at age groups of five or ten) are available. The interested reader may refer to Shryock et al. (1973) and Kostaki and Pnausis (2001) for summary of the interpolation techniques used in expanding the abridged life tables.

Prevalence data \( \left( {\pi_{x} } \right) \)

The availability of disability prevalence data varies considerably around the world. The statistics concerning disability can be obtained through numerous databases maintained by many different international and national organisations. The joint publication by WHO and the World Bank, World report on disability provides valuable insights to many disability-related issues as well as the country-specific disability data sources. The United Nations Statistics Divisions (UNSD) maintain a data repository (DISTAT) with the disability prevalence rates around the world. One must take note that the disability prevalence rates are not directly comparable across countries because of the differences in the concepts and methods used to measure disability. Most countries adopt the disability definition specified by the Washington Group on Disability Statistics in their census. Surveys that are targeted to measure disability, on the other hand, tend to record disability status in granular details by adopting the classification standards of limitation on activities of daily living (Noelker and Browdie 2013).

The following provides a (non-exhaustive) list of the sources of disability data:

Region: Asia–pacific

The United Nations Economic and Social Commission for Asia and the Pacific (UNESCAP) works on improving the disability measurement and statistics in the Asia–pacific region. Their published work, Disability at a Glance 2015, offers country-specific disability information and data sources in addition to its emphasis on the employment of persons with disabilities. For example, the Australian disability prevalence data is provided by the Australian Bureau of Statistics (ABS)—Survey of Disability, Ageing and Cares (SDAC) and the Indian disability prevalence data is provided by the Office of the Registrar General and Census Commissioner, India—Census 2011.

Region: Europe

The disability prevalence data for the European Union is available through Eurostat, the statistical office of the European Union. Eurostat maintains a database with disability prevalence data gathered from various sources including the EU Labour Force Survey (LFS), EU Statistics on Income and Living Conditions (SILC) and European Health Interview Survey (EHIS).

Region: North America

The disability statistics of the United States are relatively abundant. The interest reader can refer to Centers for Disease Control and Prevention for a list of surveys and sources that collect data of disability status. The disability prevalence data for Canada is available through the Canadian Survey on Disability.

Region: Latin America and Carribbean

Social Panaroma of Latin America, a publication by the Economic Commission for Latin America and the Caribbean (ECLAC), studies the social gaps in the region of Latin America and the Carribbean. The report also provides high level summaries of country-specific disability prevalence rates and the sources (see Fig. 23 in the referred publication).

Region: Africa

There are limited studies of disability prevalence on a national scale in African countries in general. The disability prevalence rates of South Africa are available through the report by Statistics South Africa. The disability prevalence rates of Kenya are available through the Kenya National Bureau of Statistics, and also Global Disability Rights Now, an organisation devoted to safeguard the civil rights of people with disability. The interest reader can refer to the WHO and UNSD for disability prevalence rates for other countries.

Appendix B: Results for Australian males

Figure 7 provides the boundaries for Australian Males. A comparison with Fig. 3 shows that the boundaries for Australian males are slightly wider than those obtained for Australian females. This results in higher RMSE for Australian males using the two proposed point estimate methods (Table 3).

Fig. 7
figure 7

The boundaries and point estimates of transition probabilities versus the (true) transition probabilities—Australian males

Appendix C: Three-state with-recovery model

One of the prominent advantages of the multi-state model is its ability to simulate transitions between different health states. While assuming no recovery from a disabled state in a three-state model simplifies the process of estimating transition probabilities from the population mortality incidence rates and disability prevalence rates, it limits the ability of the model to allow for recovery transition from a disabled state to a healthy state. The suitability of such assumption remains controversial.

In this Appendix, we extend the model discussed in “Overview of the model and estimation of the transition boundaries” section to estimate the boundaries for all of the six transition probabilities \( \left( {p_{x}^{aa} ,p_{x}^{ai} ,p_{x}^{ad} , p_{x}^{ia} ,p_{x}^{ii} , p_{x}^{id} } \right) \) when the no-recovery assumption is relaxed. The notable changes apply to Eqs. 4 and 10. Specifically, the updated equations are as follows:

Equation (4):

$$ \begin{array}{*{20}l} {p_{x}^{aa} + p_{x}^{ai} + p_{x}^{ad} = 1} \hfill \\ {p_{x}^{ia} + p_{x}^{ii} + p_{x}^{id} = 1} \hfill \\ {p_{x}^{dd} = 1} \hfill \\ \end{array} $$
(19)

Equation (10) can be rewritten as the following:

$$ p_{x}^{ia} = \frac{{\left( {1 - \pi_{x + 1} } \right) \times \left( {1 - q_{x} } \right)}}{{\pi_{x} }} - \left( {\frac{{1 - \pi_{x} }}{{\pi_{x} }}} \right)p_{x}^{aa} $$
(20)

Under the assumption of no-recovery, we have \( p_{x}^{ia} = 0 \) and the value of \( p_{x}^{aa} \) can be calculated algebraically by solving Eq. (20). Under the with-recovery model, we have \( p_{x}^{ia} \ge 0 \) and the value of \( p_{x}^{aa} \) can no longer be determined algebraically.

To determine the boundaries of \( p_{x}^{ia} \) and \( p_{x}^{aa} \), we first employ an iterative procedure to find the maximum boundary of \( p_{x}^{ia} \) (note that the minimum boundary of \( p_{x}^{ia} \) is 0). For an arbitrary value of \( p_{x}^{ia} \), we can calculate the corresponding \( p_{x}^{aa} \) using Eq. (20). This then provides us enough inputs to estimate the boundaries for the remaining four transition probabilities. The procedure is identical to the method described in “Overview of the model and estimation of the transition boundaries” section except that one of the stickiness criteria (\( p_{x}^{ii} > p_{x}^{ia} \)) is no longer trivial (see “Overview of the model and estimation of the transition boundaries” section condition d). For an inadequately high value of \( p_{x}^{ia} \), one or more criteria in “Overview of the model and estimation of the transition boundaries” section (conditions a, b, c and d) cannot be met simultaneously. As the transition probabilities must satisfy these conditions, we can narrow down the range of \( p_{x}^{ia} \) iteratively and obtain the maximum value (the maximum boundary) of \( p_{x}^{ia} \) such that the all criteria are met simultaneously.

Given the boundaries of \( p_{x}^{ia} \), the boundaries of \( p_{x}^{aa} \) can be determined using Eq. (20)—note that \( p_{x}^{aa} \) is maximised when \( p_{x}^{ia} \) is minimised and vice versa. Taking the two pairs of the boundary values of \( p_{x}^{aa} \) and \( p_{x}^{ia} \), we obtain two sets of boundaries for the remaining four transition probabilities. The union of these sets is the final boundaries of the six transition probabilities in a three-state model with recovery.

Note, that given the interlinked relationship of transition probabilities discussed in “Overview of the model and estimation of the transition boundaries” section, once one of the four \( \left( {p_{x}^{ad} , p_{x}^{ai} ,p_{x}^{ii} , p_{x}^{id} } \right) \) is known in a three-state no-recovery model, the remaining three can be easily evaluated knowing any one of the four probabilities. In order to estimate this, the common assumption employed in the literature is various forms of the extra mortality experienced in the disabled state, which is equivalent to assuming either the values of \( p_{x}^{ad} \) and \( p_{x}^{id} \) or some explicit relationship between them (see Eq. 14). However, in a three-state with-recovery model, there are three equations, and just one such assumption is insufficient. In order to solve the system of equations, a further assumption needs to be made on the relationship between the unknowns in either of Eq. 13 or Eq. 20. Hence, in the discussion of generating the point estimates from the estimated boundaries, we only discuss the least squares method, which does not rely on any additional explicit assumption to be made about extra mortality or recovery rates.

Least squares estimation

To obtain point estimates of the transition probabilities, we can employ a least squares procedure such that all six of the estimated transition probabilities are as close to the middle of the range as possible by minimising the total squared distance from the mid-point of the range for all ages.

In a three-state with-recovery model, it is interesting to note that once two of the six transition probabilities are known the remaining can be calculated algebraically (note, the chosen two must not be a pair from the same equation for example \( p_{x}^{aa} \) and \( p_{x}^{ia} \) from Eq. 20, they have to come from two separate equations). The algorithm to arrive at the transition probabilities via the least-squares method is as follows:

Step 1:

For an age\( x \), start with an arbitrary pair of transition probabilities, say\( p_{estimate}^{ia} \)and\( p_{estimate}^{id} \). Using\( p_{estimate}^{ia} \) and \( p_{estimate}^{id} \)and Eqs. (13), (14), (14) and (20) calculate\( p_{estimate}^{aa} ,\;p_{estimate}^{ai} ,\;p_{estimate}^{ad} \)and\( p_{estimate}^{ii} \).

Step 2:

Given\( p_{max}^{gh} \)and\( p_{min}^{gh} \)find the mid-point,\( p_{mid - point}^{gh} \)for all six transition probabilities.

Step 3:

Calculate the distance from the mid-point for each of the six transition probabilities, for example,\( p_{dis}^{id} = p_{estimate}^{id} - p_{mid - point}^{id} \). Take the sum of squares of the six distances.

Step 4:

Repeat the above by changing the starting value of the pair of\( p_{estimate}^{ia} \)and\( p_{estimate}^{id} \)., until the total squared distance is minimised for age\( x \).

Step 5:

Repeat Steps 14 for all ages.

Least squares and recovery assumption

Depending on the population mortality incidence rates and disability prevalence rates, the estimated upper bound of the recovery transition rate (\( p_{x}^{ia} \)) could be a value that is too high (e.g. a recovery rate of 50%). In this situation, the method proposed in “Least squares estimations” section tends to overestimate \( p_{x}^{ia} \). If, however, one has strong prior information about recovery rates, this information could be incorporated in the procedure in “Least squares estimations” section by setting \( p_{estimate}^{ia} \) to be a predetermined value or a predetermined trend. To illustrate the potential improvement of embedding additional recovery assumptions, we compare the RMSEs (Table 7), the estimated values of life expectancy (Table 8) and health expectancy (Table 9) using the least squares methods with age-independent recovery rate of 5%.

Table 7 RSME of the least square estimation methods for a three-state model with recovery
Table 8 The point estimates (and percentage difference from actual results) of truncated life expectancy for Australian Females at ages 65, 75 and 85
Table 9 The point estimates (and difference from actual results) of health expectancy for Australian Females at ages 65, 75 and 85

The true recovery rates for Australian females decrease as age increases, from 8% to 0.15%, with an average of 2.55% over age 65 and 100. Although a constant recovery rate of 5% does not reflect the true recovery, it is a significant improvement over the least squares method without any additional assumption, with a lower RMSE and, on average, lower estimation errors on both life and health expectancy. In practice, it is known that recovery rates tend to decrease with age. Hence, a simple formula using a decreasing linear function for \( p_{x}^{ia} \), for example,

$$p_{x}^{ia} = 7.5\% - \frac{{\left( {x - 65} \right)}}{{\left( {100 - 65} \right)}} \times 5\%$$

may produce better estimates of the transition probabilities. An obvious extension of this work could involve using a generalised function for recovery rates.

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Lim, W., Khemka, G., Pitt, D. et al. A method for calculating the implied no-recovery three-state transition matrix using observable population mortality incidence and disability prevalence rates among the elderly. J Pop Research 36, 245–282 (2019). https://doi.org/10.1007/s12546-019-09226-9

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