Skip to content

5-State Model

General Principle

The Cox Proportional Hazards Model is used to estimate the hazard rate or transition intensity for different transitions. The general form of the transition intensities for an individual \(k\) and transition type \(s\) is:

\[\lambda_{k, s}(t)=\exp \left\{\beta_{s}+\gamma_{s}^{\prime} w_{k}(t)+\alpha_{s} \psi(t)\right\}\]

In the above forumlation, \(\beta_{s}\) is the baseline log-intensity for transition type \(s\), \(w_{k}(t)\) contains the age and gender of the individuals, \(\gamma_{s}^{\prime}\) measure the sensitivity to age and gender, \(\psi(t)\) is a latent process that captures the uncertainty, and \(\alpha_{s}\) measures the sensitivity to the latent process.

Three Models

Static Model

\[\ln \left\{\lambda_{k, s}(t)\right\}=\beta_{s}+\gamma_{s}^{\text{age}} x_{k}(t)+\gamma_{s}^{\text {female}} F_{k},\]

where \(x_k(t)\) represents the age for the \(k\)th individual at time t and \(F_k = 1\) if the \(k\)th individual is female. The coefficients are parameters to be estimated.

Trend Model

\[\ln \left\{\lambda_{k, s}(t)\right\}=\beta_{s}+\gamma_{s}^{\text{age}} x_{k}(t)+\gamma_{s}^{\text {female}} F_{k}+\phi_{s} i,\]

where \(i\) is the index number of the interview indicating the time trend. Note that \(i=1\) corresponds to the interview year 1998, and the interview is taken every two years. Therefore, i=(interview year - 1998)/2 + 1.

Frailty Model

\[\ln \left\{\lambda_{k, s}(t)\right\}=\beta_{s}+\gamma_{s}^{\text{age}} x_{k}(t)+\gamma_{s}^{\text {female}} F_{k}+\phi_{s} i+\alpha_{s} \psi_{i},\]

where \(\psi_{i}\) captures the stochastic latent factor. In our model, we use a random walk: \(\psi_{i} = \psi_{i-1} + \epsilon_{i}\) with \(\epsilon_{i} \sim \text{N}(0, 1)\).

Parameters

The module uses cox hazard model parameters estimated from external research studies. The paper that this module refers to is available at: https://www.cepar.edu.au/publications/working-papers/multi-state-model-functional-disability-and-health-status-presence-systematic-trend-and-uncertainty

The estimated parameters of the static, trend, and frailty models from the above study with the US HRS data are embedded in the module:

  • US Health and Retirement Study (parameter name: US_HRS_5)