Hierarchical Likelihood Methods for Small Area Estimation with Beta-Binomial Response and Measurement Error in the Auxiliary Variable

Abstract

This research is motivated by developing a model on small area estimation. Small Area Estimation (SAE) is a statistical method to estimate parameters in a subpopulation with a small number of samples or none. The SAE model can be applied to data with continuous and discrete interest variables. Research on binary data is prone to Overdispersion. According to Wagner et al., Overdispersion can be overcome using a mixed distribution of Beta-Binomial. Rao and Molina developed a Beta-Binomial model based on SAE by estimating the proposed parameters using the Bayesian approach. According to Lee and Nelder, estimating the parameters of the mixed model with the Beta-Binomial hierarchy can be done through Hierarchical Likelihood (HL) and Adjusted Profile Hierarchical Likelihood (APHL). This method is claimed to be better than the Bayes approach analytically. In addition, the properties of small area estimators, such as bias and MSE, are conditionally derived from the additional information assuming that the auxiliary variables are measured without error. However, when the auxiliary variables used in the model are measured by error, the small area estimator that ignores the error may be worse than the direct estimator. In the regression model, measurement error in the auxiliary variables is known to bias the estimation of model parameters and lose the power to detect interesting relationships between variables. In a small area model, using survey data for the auxiliary variables means that the data can be measured with error. The researchers are interested in developing a Beta-Binomial model for a small area estimation without and with measurement errors on the auxiliary variables. Parameter estimation be done through HL and APHL. The proposed models are called SAE-BB-HL, SAE-BB-APHL, and SAE-ME-BB-HL. Furthermore, the proposed models are evaluated and investigated through a simulation study. The empirical study used illiteracy rate data from Bengkulu Province and East Java Province in 2021. The simulation results of fixed effect, dispersion, and overdispersion estimators of SAE-BB-HL are indicated to be biased. However, in the majority, the MSE value of the HL estimator was lower than that of the direct estimator of MSE. From this value and the bias value of the HL estimator, which is smaller than the direct estimator, it can be said that the SAE-BB-HL model can improve the precision of proportion estimation. This study also carried out the proportion estimation in a small area. The estimator of the proportion (p ̂_i^HL) from several areas has the same tendency. In other words, the SAE-BB-HL model has good flexibility. The difference between the SAE-BB-APHL and SAE-BB-HL models lies in the estimation of the fixed effect parameter β_z in the proposed model is done through the maximization of the APHL function. MSEP values were analyzed using analysis of variance (ANOVA) based on a complete factorial design with five factors. The results showed that all the main factors had a significant effect on the MSEP score. Moreover, the highest interaction was statistically significant. Tukey's HSD analysis was then performed to analyze the different influences of each significant interaction factor. There were 48 treatments tested. The SAE-BB-APHL model with area size m = 15, the variance of random effect σ_v^2=2, sample 2% in dataset 1 has the smallest MSEP value. The simulation results show that the variance of estimates of the parameter of SAE-BB-APHL model is smaller than the SAE-BB-HL model. The variance of dispersion and overdispersion of SAE-BB-APHL is smaller than SAE-BB-HL. However, these estimators are indicated to be biased. In real study, Bengkulu City has the average of sub-districts with lower illiteracy prediction values compared to other Regencies. In contrast, Kepahiang District has the highest illiteracy rate in Bengkulu Province. SAE-BB-APHL model is also applied to illiteracy rate data in East Java Province. In the SAE-ME-BB-HL model, the variance of the fixed effect estimates of SAE-ME-BB-HL are smaller than SAE-BB-HL. These results imply that the SAE-ME-BB-HL model tends to increase the precision and accuracy of proportion estimation. In conclusion, SAE-ME-BB-HL is better than SAE-BB-HL. The empirical data, reveals that the predictive value of the illiteracy rate at the sub-district level in Bengkulu Province using the SAE-ME-BB-HL model is greater than the estimate of the SAE-BB-HL model. The highest average illiteracy rate is in Kepahiang district. While the lowest average illiteracy rate is the Bengkulu City. Meanwhile, the result of the prediction of illiteracy rate in East Java shows that the districts located in Madura Island and several districts in the eastern part of East Java have a relatively high prediction of illiteracy rate. In this study, only one covariate with measurement error was used. In future research, it is necessary to develop the SAE-ME-BB-HL model by using more than one covariate with measurement error. In addition, future research will be more challenging if the MSE estimation of the proportion is done analytically. In empirical studies, the prediction of proportions using the proposed models tends to overestimate. Although in the simulation study, this tendency was not detected. In future research, it is necessary to select better auxiliary variables and random factors.