When time since entry makes the time scale, and covariates may be time-varying, then the hazard for the under Cox model is ithsubject whereis the covariate vector at time subject and, withbeing the failure or tfor i, censoring time for subject i. For recurrentdata, let be the number of the event occurring, and be the time for the recurrence, and denote the termination time of subject kth k=1, …,, i.Andersen andGill (1982)extendedthe Cox model by assuming that recurrent events occur randomly and the numbers of events occurring in non-overlappingintervals are independent. Thehazard for subject the recurrence under Andersen iexperiencing kth and Gill’s(AG)model is whereis the covariate vector at the recurrence for subject at time and, kth i t, i.e. remains to be 1 as long as the last event occurs at or after time , the risk set t.At contains those not experiencing their last event or censoring prior to . Since covariates may be changed over time, their hazards are evaluated accordingly. Hence, the partial likelihood under the AG model is Notice that the above function doesn’t depend on the baseline hazard function, which becomes . Clearly, the distinction between the Cox and the AG model is that once experiencing the first event or censoring, the subject is excluded from all subsequent risk sets under the Cox model, while the subject is contained in all risk sets up to the one corresponding to the last occurrence under the AG model. Therefore, it is not appropriate to analyzerecurrent survival data via fitting the classic Cox model. The log partial likelihood function of the AG model is . The maximum likelihood estimates () are obtained by simultaneously solvingthe following partial derivative equations: q =0, form=1,…,q. The observed information matrix is the negative of the matrix of the second derivative of the log partial likelihood, denoted as , with the (m,r)th element being Just like Cox’s model, estimates of are found numerically by or some other numerical method. For large samples, approximates to a normal distribution with mean vector , and variance matrix. q-variate
When time since entry makes the time scale, and covariates may be time-varying, then the hazard for the ithsubject under Cox model is whereis the covariate vector at time tfor subject i, and, withbeing the failure or censoring time for subject i. For recurrentdata, let be the number of the event occurring, and be the time for the kth recurrence, k=1, …,, and denote the termination time of subject i.Andersen andGill (1982)extendedthe Cox model by assuming that recurrent events occur randomly and the numbers of events occurring in non-overlappingintervals are independent. Thehazard for subject iexperiencing the kth recurrence under Andersen and Gill’s(AG)model is whereis the covariate vector at the kth recurrence for subject i at time t, and, i.e. remains to be 1 as long as the last event occurs at or after t.At time , the risk set contains those not experiencing their last event or censoring prior to . Since covariates may be changed over time, their hazards are evaluated accordingly. Hence, the partial likelihood under the AG model is Notice that the above function doesn’t depend on the baseline hazard function, which becomes . Clearly, the distinction between the Cox and the AG model is that once experiencing the first event or censoring, the subject is excluded from all subsequent risk sets under the Cox model, while the subject is contained in all risk sets up to the one corresponding to the last occurrence under the AG model. Therefore, it is not appropriate to analyzerecurrent survival data via fitting the classic Cox model. The log partial likelihood function of the AG model is . The maximum likelihood estimates () are obtained by simultaneously solvingthe following q partial derivative equations: =0, form=1,…,q. The observed information matrix is the negative of the matrix of the second derivative of the log partial likelihood, denoted as , with the (m,r)th element being Just like Cox’s model, estimates of are found numerically by or some other numerical method. For large samples, approximates to a q-variate normal distribution with mean vector , and variance matrix.