## A Note on 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 characteristics 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,σ*^{2})
frailty itself has mean *exp{σ*^{2}/2} and variance
*(exp(σ*^{2})-1) exp(σ^{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 *σ*^{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.