## 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] 1.556431
> exp( qnorm(c(0.25, 0.75)) * sigma ) - 1
[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{s*^{2}/2} and variance *(exp(s*^{2})-1) exp(s^{2}).
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 *s*^{2}/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.