[R] Joint modelling of survival data

[ryusuke]

Hi Dr Terry,

Thank you for your reply.

Step(1) ----- Lets assume Generalized Poisson model (GLM) as basic model
where constant hazards ratio as time goes by. Below are two correlated GLM.
   X_ij = Poisson( lambda_1 = \gamma * \alpha_i * \delta_j )
   Y_ij = Poisson( lambda_2 =                   \alpha_j * \delta_i )
   X_ij { 0, 1, 2 } and Y_ij { 0, 1, 2 }
   Where i is not equal to j , \alpha and \delta are unknown parameters.
   mean of  production between \alpha with \delta constraint to 1 (will be
random effects in below survival model)
   \gamma is a constant parameter (might be an intercepts in GLM)

   Therefore we need to make data size from n to be 2*n to get the
coefficient value of \alpha and \delta , as well as \gamma.


Step(2) ----- A Cox proportional hazards model.
   \lambda(t) = \lambda_0(t) * exp( X * \beta )
   \lambda_0(t) is baseline hazards function , X is covariate , \beta is
coefficient value.

   If I would extend static hazards ratio from Step(1)  :-
   \lambda_1k(t)  = exp( \gamma * \alpha_ik * \delta_jk )
   \lambda_2k(t)  = exp(                * \alpha_jk * \delta_ik )
   Where k is a group, and between groups are all independence. k = 1,2,3...
n (n is data size in Step1)

   Below are fixed effects, X_ij { 0, 1, 2 } and Y_ij { 0, 1, 2 } will below
9 parameters. Then 9 coefficient values for \lambda_1(t) and also 9 for
\lambda_2(t) where X_00 for both \lambda_1(t) and \lambda_2(t) as 1:-
     01) hazards ratio during X_ij = 0 & Y_ij = 0    (terms as \lambda_00
with factor( X_00 ))
     02) hazards ratio during X_ij = 0 & Y_ij = 1    (terms as \lambda_01
with factor( X_01 ))
     03) hazards ratio during X_ij = 0 & Y_ij = 2    (terms as \lambda_02
with factor( X_02 ))
     04) hazards ratio during X_ij = 1 & Y_ij = 0    (terms as \lambda_10
with factor( X_10 ))
     05) hazards ratio during X_ij = 1 & Y_ij = 1    (terms as \lambda_11
with factor( X_11 ))
     06) hazards ratio during X_ij = 1 & Y_ij = 2    (terms as \lambda_12
with factor( X_12 ))
     07) hazards ratio during X_ij = 2 & Y_ij = 0    (terms as \lambda_20
with factor( X_20 ))
     08) hazards ratio during X_ij = 2 & Y_ij = 1    (terms as \lambda_21
with factor( X_21 ))
     09) hazards ratio during X_ij = 2 & Y_ij = 2    (terms as \lambda_22
with factor( X_22 ))


Step(3) ----- Fit correlated random effects into proportional hazards model.
Due to 
   \lambda_1(t) = \lamda_0(t) * exp( X * \beta ) exp( Z * b )
   \lambda_2(t) = \lamda_0(t) * exp( X * \beta ) exp( Z * b )
   \lambda_0(t) is baseline hazards function , X are covariates include {
X_00, X_01... X_22 }.
    b ~ N(0, A)
    \beta = fixed effects coef X = covariate matrix for fixed effects
    b= random effects coefs, Z= covariate matrix for random effects 

   *** Question : (1) How do I fit above \gamma , \alpha and \delta as
random effects?
                               (2) Under this situation, personally believe
that \gamma should be intercept where require a parametric survreg() model
but not coxme(). However I am not sure am I right? Since \lambda_1(t) and
\lambda_2(t) are sharing same \alpha and \delta coefficient values but only
\lambda_1(t) has extra \gamma value...

Thank you.


Best,
Ryusuke


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