In some applications, the response variable assumes values in the unit interval. The standard linear regression model is not appropriate for modelling this type of data because the normality assumption is not met. Alternatively, the beta regression model has been introduced to analyze such observations. A beta distribution represents a flexible density family on (0, 1) interval that covers symmetric and skewed families. In this paper, a beta Generalized linear mixed model with spatial random effect is proposed emphasizing on small values of the spatial range parameter and small sample sizes. Then some models with both fixed and varying precision parameter and different combinations of priors and sample sizes are discussed. Next, the Bayesian estimation of the model parameters is evaluated in an intensive simulation study. Selected priors improved the Bayesian estimation of the parameters, especially for small sample sizes and small values of range parameter. Finally, an application of the proposed model on data provided by Household Income and Expenditure Survey (HIES) of Tehran city is presented.

When discussing non-Gaussian spatially correlated variables, Generalized linear mixed models have enough flexibility for modeling various data types. However, the maximum likelihood methods are plagued with substantial calculations for large data sets, resulting in long waiting times for estimating the model parameters. To alleviate this drawback, composite likelihood functions obtained from the product of the likelihoods of subsets of observations are used. The current paper uses the pairwise likelihood method to study the parameter estimations of spatial Generalized linear mixed models. Then, we use the weighted pairwise and penalized likelihood functions to estimate the parameters of the mentioned models. The accuracy of estimates based on these likelihood functions is evaluated and compared with full likelihood function-based estimation using simulation studies. Based on our results, the penalized likelihood function improved parameter estimation. Prediction using penalized likelihood functions is applied. Ultimately, pairwise and penalized pairwise likelihood methods are applied to analyze count real data sets.

Introduction: Diabetes mellitus is a chronic disease, it’s prevalence is very high and is increasing recently. In this study, in addition to determining type II diabetes related factors, we compared the Generalized linear and Generalized linear mixed models.Methods: Data is related to research project to investigate the epidemiological characteristics of diabetes in adults aged 30 years and older in the province of Yazd. In this study, 2, 795 people were screened with a blood glucose test for diabetes. We for data analysis by the mixed logistic and ordinary logistic regression used the R software.Results: In this study, four variables of family history of diabetes, age, body mass index and waist circumference to hip circumference were significant in both models (p-value<.001).Job was a significant variable in the ordinary logistic regression model in level significant.1 but not significant in the mixed logistic regression model. The education, area of housing and gender not significant in neither logistic mixed model nor ordinary logistic model. According to the values of the odds ratio also, we saw quite differences between the two models. Judging from standard error of the coefficients and comparison of the their values in both models seen underestimate in ordinary logistic regression model.Conclusion: The use of Generalized linear mixed models lead to more accurate results and prevents underestimated standard error of the coefficients.

Spatial Generalized linear mixed models are used for modeling geostatistical discrete spatial responses and spatial correlation of the data is considered via latent variables. The most important interest in these models is estimation of the parameters and prediction of the latent variables. In this paper, first, a prediction method is presented. Then a Bayesian approach and MCMC algorithms are proposed. Since these models are complicated and Monte Carlo sampling is used in the Bayesian inference of these models, computation time is long. In order to resolve this problem, the Approximate Bayesian methods are considered. Finally, the proposed methods are applied to a case study on rainfall data observed in the weather stations of Semnan in 1391.

Spatial Generalized linear mixed models are usually used for modeling non-Gaussian and discrete spatial responses. In these models, spatial correlation of the data can be considered via latent variables. Estimation of the latent variables at the sampled locations, the model parameters and the prediction of the latent variables at un-sampled locations are of the most important interest in SGLMM. Often the normal assumption for latent variables is considered just for convenient in practice. Although this assumption simplifies the calculations, in practice, it is not necessarily true or possible to be tested. In this paper, a closed skew normal distribution is proposed for the spatial latent variables. This distribution includes the normal distribution and also remains closed under linear conditioning and marginalization. In these models, likelihood function cannot usually be given in a closed form and maximum likelihood estimations may be computationally prohibitive. In this paper, for maximum likelihood estimation of the model parameters and predictions of latent variables, an approximate algorithm is introduced that is faster than the former method. The performance of the proposed model and algorithm are illustrated through a simulation study.

Spatial Generalized linear mixed models are used commonly for modelling non-Gaussian discrete spatial responses. We present an algorithm for parameter estimation of the models using Laplace approximation of likelihood function. In these models, the spatial correlation structure of data is carried out by random effects or latent variables. In most spatial analysis, it is assumed that random effects have Gaussian distribution, but the assumption is questionable. This assumption is replaced in the present work, using a skew Gaussian distribution for the latent variables, which is more flexible and includes Gaussian distribution. We examine the proposed method using a real discrete data set.