Constrained extrapolation of multistate transitions

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:

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.

Thank you!

We hope to see you all in Bilbao one day!