R: Estimating Three Conditional Cumulative Incidence Functions...

conditional_cif_b {semicmprskcoxmsm}

R Documentation

Estimating Three Conditional Cumulative Incidence Functions Using the General Markov Model Conditional on Random Effect

Description

conditional_cif_b estimates the cumulative incidence function based on the MSM illness-death general Markov model conditional on the fixed random effect b.

Usage

conditional_cif_b(res1,
                  t1_star,
                  b)

Arguments

`res1`	The output from `em_illness_death_phmm_weight`, the general Markov model result.
`t1_star`	Fixed non-terminal event time for estimating CIF function for terminal event following the non-terminal event.
`b`	Fixed random effect value.

Details

Similar as cif_est_usual, after estimating the parameters in the illness-death model \lambda_{j}^a using IPW, we could estimate the corresponding conditional CIF under fixed b:

\hat{P}(T_1^a<t,\delta_1^a=1 \mid b) = \int_{0}^{t} \hat{S}^a(u \mid b) d\hat{\Lambda}_{1}^a(u \mid b ),

\hat{P}(T_2^a<t,\delta_1^a=0,\delta_2^a=1 \mid b) = \int_{0}^{t} \hat{S}^a(u \mid b) d\hat{\Lambda}_{2}^a(u \mid b),

and

\hat{P}(T_2^a<t_2 \mid T_1^a<t_1, T_2^a>t_1 \mid b) = 1- e^{- \int_{t_1}^{t_2} d \hat{\Lambda}_{12}^a(u \mid b) },

where \hat{S}^a is the estimated overall survial function for joint T_1^a, T_2^a, \hat{S}^a(u) = e^{-\hat{\Lambda}_{1}^a(u)} - \hat{\Lambda}_{2}^a(u) . We obtain three hazards by fitting the MSM illness-death model \hat\Lambda_{j}^a(u) = \hat\Lambda_{0j}(u)e^{\hat\beta_j*a} , \hat\Lambda_{12}^a(u) = \hat\Lambda_{03}(u)e^{\hat\beta_3*a} , and \hat\Lambda_{0j}(u) is a Breslow-type estimator of the baseline cumulative hazard.

where S(t \mid b;a) = \exp[- \int_0^{t} \{ \lambda_{01} (u)e^{\beta_1a + b} + \lambda_{02} (u )e^{\beta_2a + b} \} d u ] = \exp \{- e^{\beta_1a + b} \Lambda_{01}(t) - e^{\beta_2a + b} \Lambda_{02} (t ) \}

Value