Gamma distributed random effects 


Model descriptionIt 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 ADMBRE.The problem with nonGaussian random effects is that the Laplace approximation underlying ADMBRE 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 ADMBRE, which has no difficulties in fitting the model with adaptive Gaussian quadrature (via the command line option gh 50). Hence, it appears that ADMBRE is numerically more stable than SAS NLMIXED The following table compares the results from Nelson et al (Table 2, using nonadaptive Gaussian quadrature) with ADMBre (50 quadrature points in adaptive Gaussian quadrature). It is seen that the results are very similar.
ReferencesNelson KP, Lipsitz, Garrett, Fitzmaurice, Ibrahim, Parzen and Strawderman (2006). Use of the Probability Integral Transformation to Fit Nonlinear MixedEffects Models With Nonnormal Random Effects. Journal of Computational & Graphical Statistics, Vol. 15, No. 1, pp.3957 