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3-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:

\[\lambda_{k, s}(t) = \text{exp}(\boldsymbol{\beta}_{s}'\mathbf{X}_k) \cdot H_{k, s}(t)\]

In the above forumlation, \(\boldsymbol{\beta}\) contains coefficients of transition type \(s\) to be estimated, and \(\mathbf{X}\) contains the covariates of individual \(k\). Furthermore, we assume the Markovaian property where \(H_{k, s}(t) = 1\).

Three Models

Static Model

\[ \text{ln}\lambda_{k, s}(t) = \beta_s + \gamma^{\text{age}}_s x_k(t) + \gamma^{\text{female}}_s F_k \]

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

\[ \text{ln}\lambda_{k, s}(t) = \beta_s + \gamma^{\text{age}}_s x_k(t) + \gamma^{\text{female}}_s F_k + \gamma^{\text{time}}_s t \]

where \(t\) is the time trend. Note that \(t=1\) corresponds to the year 1998-1999.

Frailty Model

\[ \text{ln}\lambda_{k, s}(t) = \beta_s + \gamma^{\text{age}}_s x_k(t) + \gamma^{\text{female}}_s F_k + \gamma^{\text{time}}_s t + \alpha_s \psi(t) \]

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

Parameters

The module uses cox hazard model parameters estimated from external research studies. The parameters of the two following studies are included in the package:

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

  • China Chinese Longitudinal Healthy Longevity Survey (parameter name: china_CLHLS)