# Gamma distributed random effects

 Code: liver_gamma.tpl Data: liver_gamma.dat Initial values: liver_gamma.pin Expected Results: liver_gamma-expected-results.par

### Model description

It is customary to use normally distributed random effects, but in some situations other distributions than the normal are required. We shall here illustrate the use of gamma distributed random effects in ADMB-RE.

The problem with non-Gaussian random effects is that the Laplace approximation underlying ADMB-RE may be inaccurate. To avoid this we start out with N(0,1) distributed random variables (r.v.), which are transformed into gamma distributed r.v., via the inverse cumulative distribution function for the gamma distribution. The steps are:

u ~ N(0,1) is the underlying random effect.
z = F(u), is uniformly distributed, where F() is cumulative distribution function of u.
g = G_inv(z), where G_inv() is the inverse cumulative distribution function of the target gamma distribution.

As a result, g will be a variable, with a gamma distribution, that can be used in the model.

Nelson et al (2006, Sec 4.1) consider a survival model with a gamma distributed random effect. They fit the model using SAS NLMIXED, but face problems when trying to fit the model using adaptive Gaussian quadrature, which is the default in SAS NLMIXED. The box to the left gives an implementation of the model in ADMB-RE, which has no difficulties in fitting the model with adaptive Gaussian quadrature (via the command line option -gh 50). Hence, it appears that ADMB-RE is numerically more stable than SAS NLMIXED

The following table compares the results from Nelson et al (Table 2, using non-adaptive Gaussian quadrature) with ADMB-re (50 quadrature points in adaptive Gaussian quadrature). It is seen that the results are very similar.
 Nelson et al. ADMB-RE theta_1 0.2173 0.2173106 b0 -2.6154 -2.61536 b_trt -0.8005 -0.800516 b_hrt 0.6920 0.691962

### References

Nelson KP, Lipsitz, Garrett, Fitzmaurice, Ibrahim, Parzen and Strawderman (2006). Use of the Probability Integral Transformation to Fit Nonlinear Mixed-Effects Models With Nonnormal Random Effects. Journal of Computational & Graphical Statistics, Vol. 15, No. 1, pp.39-57