We consider a model dependent approach for multi-level modelling that accounts for informative probability sampling, and compare it with the use of probability weighting as proposed by Pfeffermann et al. (1998a). The new modelling approach consists of first extracting the hierarchical model holding for the sample data as a function of the corresponding population model and the first and higher level sample selection probabilities, and then fitting the resulting sample model using Bayesian methods. An important implication of the use of this approach is that the sample selection probabilities feature in the analysis as additional outcome values that strengthen the estimators. A simulation experiment is carried out in order to study and compare the performance of the two approaches. The simulation study indicates that both approaches perform generally equally well in terms of point estimation, but the model dependent approach yields confidence (credibility) intervals with better coverage properties. A robustness simulation study is performed, which allows to assess the impact of misspecification of the models assumed for the sample selection probabilities under informative sampling schemes. SUMMARYWe consider a model dependent approach for multi-level modelling that accounts for informative probability sampling, and compare it with the use of probability weighting as proposed by Pfeffermann et al. (1998a). The new modelling approach consists of first extracting the hierarchical model holding for the sample data as a function of the corresponding population model and the first and higher level sample selection probabilities, and then fitting the resulting sample model using Bayesian methods. An important implication of the use of this approach is that the sample selection probabilities feature in the analysis as additional outcome values that strengthen the estimators. A simulation experiment is carried out in order to study and compare the performance of the two approaches. The simulation study indicates that both approaches perform generally equally well in terms of point estimation, but the model dependent approach yields confidence (credibility) intervals with better coverage properties. A robustness simulation study is performed, which allows to assess the impact of misspecification of the models assumed for the sample selection probabilities under informative sampling schemes.

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This article presents a logistic hierarchical model approach for small area prediction of proportions, taking into account both possible spatial and unstructured heterogeneity effects. The posterior distributions of the proportion predictors are obtained via Markov Chain Monte Carlo methods. This automatically takes into account the extra uncertainty associated with the hyperparameters. The procedures are applied to a real data set and comparisons are made under several settings, including a quite general logistic hierarchical model with spatial structure plus unstructured heterogeneity for small area effects. A model selection criterion based on the Expected Prediction Deviance is proposed. Its utility for selecting among competitive models in the small area prediction context is examined.

Summary The Brazilian Institute of Geography and Statistics performs an annual service survey that focuses on segments of the tertiary sector. Sample estimates for some economic activities in the north, north‐east and midwest regions of Brazil have low precision due to the sample design. Furthermore, one of the main variables of interest is considerably skewed with potential outliers. To overcome this problem, skew normal and skew t‐models are proposed to produce model‐based estimates. The small domain estimation models relate operating revenue variables to potential auxiliary variables (the number of employed people and wages) obtained from a business register. The models proposed are compared with the usual Fay–Herriot model under the assumptions of known and unknown sampling variances and its transformed log‐version under the assumption of known variances. The evaluation studies with real business survey data show that the models proposed seem to be more efficient for small area predictions under skewed data than the customarily employed Fay–Herriot model, as well as its log‐normal version.

Multivariate beta regression models for jointly modelling two or more variables whose values belong in the (0,1) interval, such as indexes, rates or proportions, are proposed for making small area predictions. The multivariate model can help the estimation process by borrowing strength between units and obtaining more precise estimates, especially for small samples. Each response variable is assumed to have a beta distribution so the models could accommodate multivariate asymmetric data. Copula functions are used to construct the joint distribution of the dependent variables; all the marginal distributions are fixed as beta. A hierarchical beta regression model is additionally proposed with correlated random effects. We present an illustration of the proposed approach by estimating two indexes of educational attainment at school level in a Brazilian state. Our predictions are compared with separate univariate beta regressions. The inference process was conducted using a full Bayesian approach.

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