Gamma distributed random effects |
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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 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 phace problems when trying to fit model using addaptive Gaussian quadrature, which is the default in SAS NLMIXED. The box to the left gives an implementation of the model in ADMB-RE. The ADMB-RE implementation has no difficulties in fitting the model with addaptive Gaussian quadrature (via the command line option -gh 50). Hence, it appears that ADMB-RE is numerically more stable than SAS NLMIXED ReferencesNelson 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 | |||||||||