Kajian Model Poisson untuk Pendugaan Area Kecil dengan Galat Pengukuran Pada Peubah Penyerta

Abstract

Small area estimation (SAE) is an important tool in producing better estimates through “borrowing strength” on the accompanying variables with limited sample data information. One of the challenges of the application of SAE model is handling measurement error on the accompanying variables. In the Fay-Herriot model as one of the SAE models, accompanying variables are considered free from measurement error, but in practical applications, this rarely happens. Measurement error if ignored can lead to biased and inaccurate estimates. Therefore, this study develops a small area estimation model that considers measurement error on accompanying variables and is applied to count data with overdispersion problems. The hierarchical Bayes approach in handling measurement error is applied to estimate model parameters. Simulation studies based on data distribution, sample size and sample data sources are investigated to evaluate the model in dealing with these problems. The developed methodology is then applied to the estimation of the under-five mortality rate at the regency/city level in Java. The first study develops an SAE model that considers measurement error on accompanying variables. Simulation studies are applied by evaluating the effect of the number of areas, sample size and variety of measurement errors on the accuracy of model parameter estimates. The results showed that larger sample sizes significantly reduced relative bias, especially under conditions of large measurement error variance. The findings in this study confirmed that the model consistently produced more stable and accurate estimates, especially under conditions of larger sample sizes and small measurement error variance. The second study developed and evaluated three small area estimation models for Poisson distributed data. The three models, namely SAE PMM, SAE NBMM and SAE ZINBMM were applied to estimate the under-five mortality rate (U5MR) in Java Island using the hierarchical Bayes approach. The results of the U5MR data exploration showed that there were excess zeros and overdispersion, so the SAE PMM model was less able to handle these problems. This can be seen from the highly fluctuating estimated values and residual analysis. The SAE NBMM model with dispersion parameters expands the flexibility of the model in reducing instability in the estimated results. Model evaluation showed that SAE NBMM produced lower RMSE and RSE compared to SAE PMM. The SAE ZINBMM model, which combines the ability of Negative Binomial to handle overdispersion and zero-inflated components to overcome excess zeros, is the best model among the other models. The ability of the zero-inflated component to estimate the additional probability that an area has a zero value that cannot be explained by the Negative Binomial distribution makes the U5MR estimation results more consistent across all regencies/cities in Java. Furthermore, the development of the aforementioned model is based on the two previous studies. Models with Poisson and Negative Binomial distributions in small area estimation are developed by applying a hierarchical Bayes approach to overcome the problems of overdispersion and measurement errors in the accompanying variables. Investigation of the effect of data characteristics on estimation accuracy is designed by applying different sample sizes and data sources, especially for variables containing measurement errors and variables of interest. Model performance is also evaluated when the model ignores and considers measurement errors. The results show that models that consider measurement errors with large samples, SAE ME PMM and SAE ME NBMM perform better than models that ignore measurement errors. Residual analysis shows that the confidence intervals for models that ignore measurement errors tend to be wider, indicating a higher level of uncertainty in parameter estimation. The coverage rate (CR) value in the SAE ME PMM and SAE ME NBMM models is over-coverage, especially in the Negative Binomial model. This can occur because the model captures data diversity. Of all the proposed models, the SAE ME NBMM model is the best model with lower MSE and ARB. The results of the ANOVA test show that population distribution, measurement error and sample size have significant differences in performance. The application of the developed model shows that the model on a large sample, SAE ME NBMM2 has more stable and accurate U5MR estimation results compared to other models with smaller RMSE and RSE values. Maternal education, health facilities and health workers show a significant negative relationship indicating that improving the quality of accessibility of health facilities can reduce Under-five mortality rates. Conversely, poverty and slums have a positive relationship that reflects socio-economic challenges in certain regencies/cities. In conclusion, handling measurement errors and overdispersion through a hierarchical Bayes approach and combining data sources from various surveys are important findings in the small area estimation model. This approach can improve the accuracy of the estimator, making it a very relevant tool for policy makers and researchers who need quality statistical data for small areas.