Germán Rodríguez
Survival Analysis Princeton University

## Log-Normal Frailty

An alternative choice of frailty distribution is the log-normal. In Stata you may fit this model via the Poisson trick because `xtreg` lets you select gamma or log-normal random effects, whereas `streg` implements gamma and inverse Gaussian frailty. In R `coxph()` implements gamma frailty by adding `frailty()` to the model formula, whereas `coxme()` assumes log-normal random effects.

One advantage of log-normal frailty is that we can view the log of frailty as `σ z where z ~ N(0,1) is a standard normal random variable. This means that we can interpret the parameter σ, the standard deviation of shared frailty, as a regression coefficient for a normally-distributed random variable representing unobserved family characteristics, just like the β's are coefficients for observed characteristics.

For the Guatemalan data we have been discussing, a piecewise exponential model (which regretably does not converge in R), produces an estimate of σ of 0.442. Exponentiating this we obtain exp(0.442) = 1.556. This means that a one standard deviation difference in unobserved family characterisitcs is associated with 55.6% higher risk at any age. We can also look at the interquartile difference in risks, using the quartiles of the normal distribution:

```. mata sigma = 0.4423953

. mata exp(sigma)
1.556430876

. mata exp( invnormal( (0.25, 0.75) ) * sigma ) :- 1
1              2
+-------------------------------+
1 |  -.2579889141    .3476887597  |
+-------------------------------+
```
```> sigma <- 0.4423953

> exp(sigma)
 1.556431

> exp( qnorm(c(0.25, 0.75)) * sigma ) - 1
 -0.2579889  0.3476888
```

So we see that families in the lower quartile have 26% lower risk, and families in the upper quartile have 35% higher risk, than families at the median. These results are similar to those obtained under gamma frailty (29% lower and 36% higher risk).

Another important advantage of log-normal frailty is that it extends easily to more than two levels and to more general random-intercept and random-slope models, as we will see in the multilevel course.

A disadvantage, however, is that the unconditional survival distribution does not have a closed form, unlike the case of gamma frailty. To do calculations similar to those of the previous sections we would have to use Gaussian quadrature to integrate out the random effect.

A technical note: when log-frailty is N(0,s2) frailty itself has mean exp{s2/2} and variance (exp(s2)-1) exp(s2). The fact that the mean is not one affects only the constant. If the model does not have a constant, the baseline hazard is shifted by s2/2 = 0.0979 in the log-scale. Once you take this into account the baseline hazards for the piecewise exponential models using gamma and log-normal frailty are extremely similar. The variance of frailty works out to be 0.2629, which is similar to the value of 0.2142 in the previous model.