Date:         Mon, 2 May 2005 17:33:46 +0200
Reply-To:     Stefan Pohl <stefan.pohl@WISO.UNI-KOELN.DE>
Sender:       "SAS(r) Discussion" <SAS-L@LISTSERV.UGA.EDU>
From:         Stefan Pohl <stefan.pohl@WISO.UNI-KOELN.DE>
Subject:      ML estimation for clustered data (shared frailty model)
Content-Type: text/plain; charset="iso-8859-1"

Hi all!

Sorry, I posted this message one week ago, but nobody answered to me. Today I post my message again hoping that anybody is able to give me some
good advice. I want to estimate by maximum likelihood a parametric shared gamma frailty model.

My data consist of the triple (T_ij, d_ij, Z_ij). T_ij is the event time of subject j in group i (i= 1,...,G and j=1,...,n_i).
d_ij is the censoring indicator of subject j in group i and Z_ij is a covariate of subject j in group i.

Let D_i = sum_(j=1)^(n_i) d_ij be the number of events in group i.

Then the log likelihood is given by 

llik(theta,beta) = sum_(i=1)^(G)  D_i * ln(theta) - ln[Gam(1/theta)]  + ln[Gam(1/theta + D_i)]

                        - (1/theta + D_i) * ln[1 + theta sum_(j=1)^(n_i)  H_0 (T_ij) * exp(beta Z_ij)]
                       
                        + sum_(j=1)^(n_i) d_ij * [beta Z_ij + ln(h_0 (T_ij) )]



Gam is the gamma function, theta is the variance of a gamma random variable, H_0 is a cumulative baseline hazard function (which has to be specified),
h_0 is a baseline hazard function (which has to be specified).

Which procedure in SAS should I use? The general PROC NLP? How can I take into account that my data are clustered in G groups in this procedure?
To estimate a weibull shared normal frailty model I used proc nlmixed. In proc nlmixed the data are clustered according to the subject= variable.
But the only available frailty distribution in proc nlmixed is the normal distribution.

Thank you for your help,

my best regards,

Stefan.