Abstract. Maximum likelihood estimates in GLMM are often difficult to be obtained since the calculation involves high dimensional integrals. It is not easy to find analytical solutions for the integral so that the approximation approach is needed. In this paper, we discuss several approximation methods to solve high dimension integrals including the Laplace, Penalized Quasi likelihood (PQL) and Adaptive Gaussian Quadrature (AGQ) approximations. The performance of these methods was evaluated through simulation studies. The 'true' parameter in the simulation was set to be similar with parameter estimates obtained by analyzing a real data, particularly salamander data (McCullagh & Nelder, 1989). The simulation results showed that the Laplace approximation produced better estimates when compared to PQL and AGQ approximations in terms of their relative biases and mean square errors.

The standard small area estimator, the empirical best linear unbiased predictor (EBLUP), estimates small area parameters by way of linear mixed models. The EBLUP assumes normal and independent random small area effects as well as normal and independent random sampling errors. Under these assumptions, the variable of interest also follows a normal distribution. In practice, however, the above assumptions are often violated. The variable of interest is often non-normal and highly skewed, and the small areas are frequently spatially dependent. In this paper, we propose the spatial empirical Bayes predictor (SEBP) of the small area mean of a positively skewed variable of interest in the presence of spatial dependence among the random small area effects. We assume that the variable of interest follows a normal distribution after a log transformation and that its log transform is linked to some auxiliary variables by a nested error regression model. The SEBP is derived under the log-transformed nested error regression model. By way of simulation, we show that compared to its alterna-

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