Constrained extrapolation of multistate transitions
Rustam Tursun-zade, Tim Riffe
Motivation:
Surveys used for calculation of multistate transition probabilities or HLE often have truncated age ranges.
SHARE covers ages 50 to 110, but EU-SILC only has data for ages 17 to 85.
Compositional nature of transition probabilities makes it difficult to extrapolate them with naive methods due to their constrained structure.
Empirical data example:
(a) Empirical SHARE
(b) Empirical EU-SILC
Figure 1: Empirical transitions, SHARE and EU-SILC
Standard approaches:
Imach software (Maximum Likelihood Computer Program using Interpolation of Markov Chains) - great tool, but notoriously difficult to use.
Multistate transition probabilities approach - fast and efficient method, easily modified for changes in workflow making it a great tool for EDA, does not require external software, but may be less accurate.
Methods:
We fit the multistate model (multinomial logistic) to the SHARE data to calculate the parameters.
We compare the obtained results with those from the Imach, to check for systematic errors. To do so we plot the models vs. the empirical (interpolated for Imach) transition probabilities.
We truncate age range (by age 80) and extrapolate transitions up to the age 110.
A thoughtful fact:
Transformed transition probabilities from multinomial logistic turn out linear. This transformation is largely identical to the compositional ALR \(alr(x)_i = ln\frac{x_i}{x_D}\)
This transforms compositional simplex \(alr:S^D\) to \(R^{D-1}\) and allows for a neat reparametrization of the model from multinomial to compositional linear regression.
The results then can be sent back by inverse ALR.
ALR transformation:
ALR transformed transitions from multistate model with SHARE data
Imach and multistate model:
Comparison of Imach and Multistate model fit
Multistate vs ALR:
Extrapolation of truncated (age 80) data from model. Traditional approach vs. ALR
Discussion:
Modal age at death is often higher than 85 and not all people at high ages are unhealthy.
We can derive more information from the survey using the multistate health expectancies rather than prevalence.
Eurostat closes the HLE with QALY indicator obtained with from EU-SILC that follows a Sullivan approach with constant prevalence for closing of a lifetable.
There are better ways of doing so, for example the UN method that needs more age range.
Discussion pt. 2:
Imagine we only have a figure of transition probabilities for a subset of ages generated by Imach or by multistate model.
We can parse it and extrapolate to generate the full age range with the ALR transformation.
We can disregard the estimation of parameters in this case.
Example:
Crimmins, E. M., Hayward, M. D., Hagedorn, A., Saito, Y., & Brouard, N. (2009). Change in disability-free life expectancy for Americans 70 years old and older. Demography, 46, 627-646.
Conclusion:
ALR transformation is a neat re-parametrization of multinomial model.
Our multinational model currently fits worse compared to Imach, but can be used to extrapolate the constrained data, especially for the initial research purposes, like covariate selection and EDA.
The further improvement of the model could be made, for example the incorporation of missing data handling.
