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Consider a Gaussian random variable log(U), with mean mu and variance sigma^2. How can the parameter estimates of the corresponding log-normal frailty model (i.e the frailty random variable is U which follows a log-normal distribution) be obtained? The hazard distribution can assumed to be Weibull.

The parfm package (in R) assumes log(U) has a mean of zero.

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$\mu$ is fixed (to zero) to ensure identifiability of the parameters in the frailty model, similar to the zero-mean constraint for the random effects in the linear mixed model.

In other terms, $\mu$ plays the same role as the scale parameter of the Weibull baseline hazard.

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  • $\begingroup$ I suppose the identifiability condition requires E(U) = 1 which can only be satisfied if log(U)~N( -log(1+s)/2, log(1+s) ). Here, s denotes the variance of U. $\endgroup$ – johnmajimboni May 1 '19 at 6:35
  • $\begingroup$ E(U) = 1 is not mandatory. What is important is that it is fixed (rather than a parameter). In general, it is the more convenient for the sake of interpretability, though. When U follows a log-normal distribution, the typical choice is to take E(log(U)) = 0 by similarity with the linear model. $\endgroup$ – ocram May 1 '19 at 17:26

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