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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