Analisis Probabilistik dan Pemodelan Risiko dengan Rantai Markov, Metode MCMC, dan Regresi Logistik Ordinal pada Perkembangan Penyakit Diabetes Tipe 2

Abstract

Type 2 Diabetes Mellitus is a chronic disease that progresses gradually and often shows no symptoms in its early stages. To understand the dynamic transition of health status from normal to prediabetes and eventually to diabetes, a mathematical approach that can capture the uncertainty of the system quantitatively is required. This study develops a stochastic model using Markov Chains to represent changes in health status across three categories: normal, prediabetes, and diabetes, based on medical records of 65 patients over a five-month observation period. The transition probability matrix is used to compute the likelihood of moving between statuses, as well as to estimate the steady- state distribution and mean sojourn time. To address data limitations and complex distributions, the Metropolis-Hastings algorithm, as part of the Markov Chain Monte Carlo (MCMC) method, is applied to numerically approximate the probability distri- butions of transitions. Additionally, ordinal logistic regression is used to analyze the influence of risk factors such as age, gender, BMI, blood pressure, smoking habits, and family history on health status. The results indicate that the Markov model effec- tively maps disease progression, while the MCMC method provides flexible distri- bution estimates under complex conditions. Ordinal logistic regression reveals that age, blood pressure, and family history significantly impact the likelihood of progressing to more severe health statuses. This study demonstrates that a combined approach using stochastic modeling and risk analysis offers a comprehensive understanding of Type 2 Diabetes progression.