MODEL REGRESI GLMM-GEE-TREE UNTUK RESPON BERSEBARAN BETA DAN PENERAPANNYA DALAM ANALISIS DATA RASIO GINI DI INDONESIA

Abstract

Pemodelan data respon berupa proporsi atau rasio yang ada pada selang terbuka (0,1) dengan struktur longitudinal menghadapi kompleksitas ganda berupa heterogenitas tidak teramati antarsubjek dan korelasi galat temporal. Pendekatan konvensional seperti Generalized Linear Mixed Models (GLMM) seringkali sensitif terhadap asumsi sebaran, sementara Generalized Estimating Equations (GEE) mengabaikan prediksi spesifik pada level subjek. Metode hubungan linier parametrik global seringkali gagal menangkap pola interaksi nonlinier yang umum ditemukan dalam data empiris. Penelitian ini bertujuan mengembangkan model GLMM-GEETree untuk data respon bersebaran beta yang mengintegrasikan kekuatan prediksi GLMM, ketangguhan inferensi GEE, dan fleksibilitas algoritma pohon regresi. Metodologi penelitian ini dilaksanakan melalui empat tahapan utama. Pertama, formulasi analitik model hibrida GLMM-GEE melalui pendekatan linearisasi pseudolikelihood untuk menduga parameter efek tetap dan efek acak. Kedua, penggabungan mekanisme penduga sandwich empiris untuk mengoreksi galat baku agar kekar (robust) terhadap salah spesifikasi struktur korelasi. Ketiga, integrasi algoritma partisi rekursif (tree) yang dibangun dari galat pseudo-linear yang telah terkalibrasi untuk menangkap hubungan non-linier. Keempat, validasi kinerja model melalui simulasi Monte Carlo dan penerapannya pada data empiris rasio gini di Indonesia periode 2018-2024. Hasil kajian simulasi menunjukkan bahwa model GLMM-GEE-Tree menunjukkan kinerja yang baik. Model ini mampu mempertahankan akurasi prediksi (Root Mean Square Error) rendah setara dengan GLMM. Selain itu, model ini juga menjamin validitas inferensi (Coverage Probability mendekati level nominal 95%) setara dengan GEE, terutama pada kondisi heterogenitas dan autokorelasi tinggi sedangkan model standar lainnya mengalami penurunan performa signifikan. Algoritma pohon yang dikembangkan juga terbukti efektif dalam mengidentifikasi struktur interaksi kovariat secara tepat dan konsisten. Penerapan model pada data rasio gini provinsi di Indonesia mengungkap adanya struktur interaksi non-linier dan efek ambang batas yang signifikan antara indikator pembangunan makroekonomi terhadap ketimpangan. Model mengidentifikasi beberapa kelompok wilayah dengan pola ketimpangan yang berbeda. Dampak peubah seperti pertumbuhan ekonomi dan tingkat kemiskinan terhadap rasio gini bersifat kondisional terhadap karakteristik spesifik wilayah tersebut. Temuan ini memberikan kontribusi teoritis bagi pengembangan statistika komputasional serta kontribusi praktis dalam penyusunan kebijakan intervensi yang lebih terlokalisasi dan berbasis data (evidence-based policy)

Modeling of response data in the form of proportions or ratios at open intervals (0,1) with longitudinal structures faces double complexity in the form of unobserved heterogeneity between subjects and temporal error correlations. Conventional approaches such as Generalized Linear Mixed Models (GLMM) are often sensitive to distribution assumptions, while Generalized Estimating Equations (GEE) ignore specific predictions at the subject level. In addition, global linear-parametric relationships often fail to capture the non-linear interaction patterns commonly found in real data. This study aims to develop a GLMM-GEE-Tree model for beta dispersed response data that integrates the predictive power of GLMM, the robustness of GEE inference, and the flexibility of regression tree algorithms. The methodology of this research was carried out through four main stages. First, the analytical formulation of the GLMM-GEE hybrid model through the Pseudo-Likelihood Linearization approach to estimate the parameters of fixed effects and random effects. Second, the incorporation of the Empirical Sandwich Estimator mechanism to correct standard error to be robust against the mis-specification of the correlation structure. Third, the integration of a recursive partition algorithm (Tree) built on a pseudo-linear residue that has been calibrated to capture non-linear relationships. Fourth, validation of model performance through Monte Carlo simulations and applications to empirical data on the Gini Ratio in Indonesia for the 2018-2024 period. The results of the simulation study show that the GLMM-GEE-Tree model showed favorable performance under the evaluated scenarios. This model is able to maintain prediction accuracy (low Root Mean Square Error) equivalent to GLMM and ensure inference validity (Coverage Probability close to the nominal level of 95%) equivalent to GEE, especially in conditions of high heterogeneity and autocorrelation where other standard models experience significant performance degradation. The developed tree algorithm has also proven to be effective in accurately and consistently identifying the structure of covariate interactions. The application of the model to the Gini Ratio data of provinces in Indonesia reveals the existence of a non-linear interaction structure and a significant threshold effect between macroeconomic development indicators on inequality. This model successfully identifies several different inequalities, where the impact of variables such as economic growth and poverty rates on the Gini Ratio is conditional on the specific characteristics of the region. These findings make a theoretical contribution to the development of computational statistics as well as a practical contribution to the formulation of more localized and evidence-based intervention policies