Kajian Evaluasi Dampak Overdispersi pada Respon Nol Berlebih dan Keragaman Tak Teramati dalam Analisis Data Cacahan

Abstract

Model regresi paling sederhana yang biasa digunakan dalam memodelkan data cacahan adalah regresi Poisson. Namun, regresi Poisson memiliki asumsi equidispersi yang harus dipenuhi yaitu rataan dan ragam peubah respon bernilai sama besar. Sementara itu, kondisi yang umum ditemukan pada data cacahan adalah peubah respon yang memiliki ragam jauh lebih besar dibandingkan rataannya. Kondisi ini menyebabkan overdispersi pada regresi Poisson karena tidak terpenuhinya asumsi equidispersi. Overdispersi dapat terjadi karena beberapa hal, diantaranya karena respon nol berlebih dan adanya keragaman tak teramati. Penelitian ini bertujuan mengevaluasi dampak overdispersi karena respon nol berlebih dan keragaman tak teramati pada analisis regresi Poisson. Penelitian dilakukan melalui kajian simulasi dan studi kasus. Data simulasi dibangkitkan dari 27 kondisi dengan kombinasi parameter ukuran contoh, proporsi nilai nol, dan asal sebaran dari peubah respon. Sementara itu, kematian balita akibat Pneumonia di Pulau Jawa digunakan sebagai studi kasus pada penelitian ini. Evaluasi dilakukan dengan membandingkan kemampuan prediksi dan pengujian hipotesis model regresi Poisson dengan delapan model regresi alternatif lainnya. Kemampuan prediksi model dibandingkan menggunakan nilai RMSE dan bias relatif penduga koefisien regresi sedangkan kemampuan pengujian hipotesis model dibandingkan dengan nilai ϕ dan penduga galat baku koefisien regresi. Hasil evaluasi berdasarkan kajian simulasi menunjukkan bahwa dampak overdispersi pada analisis regresi Poisson terlihat pada kemampuan pengujian hipotesis model terutama pada pendugaan galat baku model yang menjadi underestimate. Kesalahan pendugaan galat baku pada pemodelan regresi Poisson berpengaruh pada hasil pengujian hipotesis yang menjadi cenderung lebih mudah menolak hipotesis nol atau cenderung lebih mudah menyimpulkan bahwa suatu peubah berpengaruh signifikan pada peubah respon. Sementara itu, dampak kondisi overdispersi tidak terlalu terlihat pada sisi kemampuan prediksi model regresi Poisson, terlihat dari nilai RMSE dan bias relatif penduganya yang bernilai kecil. Selanjutnya, menggunakan perbandingan nilai AIC, terpilih tiga model terbaik yaitu model ZINB, ZINBMM, dan ZIGPMM. Ketiga model tersebut kemudian diterapkan pada studi kasus kematian balita akibat Pneumonia di Pulau Jawa tahun 2021. Berdasarkan ketiga model regresi terbaik, pengujian hipotesis menunjukkan bahwa persentase balita bergizi kurang terbukti memberikan pengaruh yang signifikan terhadap kasus kematian balita akibat Pneumonia di Pulau Jawa.

The simplest regression model commonly used in modeling count data is Poisson regression. However, Poisson regression has an equidispersion assumption that needs to be fulfilled, which is the mean and variance of the response variable must be the same value. Meanwhile, in count data, it is common to find that the response variables have larger variations than their mean. This condition causes overdispersion in Poisson regression because it violates the equidispersion assumption. Overdispersion can occur because of excess zero response and unobserved variability. This research aims to evaluate the effect of overdispersion due to excess zero response and unobserved variability. The research was conducted through simulation and case studies. The simulation study used generated data from 27 conditions with a combination of three parameters which are sample size, proportion of zero values, and the distribution of the response variable. Meanwhile, the under-five children deaths due to Pneumonia on Java island were used as a study case. The evaluation was done by comparing the predictive and hypothesis-testing ability of the model. The predictive ability was compared using the RMSE value and relative bias of the regression coefficient estimator while the hypothesis-testing ability was compared with the ϕ value and standard error estimator of the regression coefficient. Based on the simulation study, the evaluation result shows that the effect of overdispersion in count data analysis can be seen in the hypothesis testing model ability, especially in estimating the standard error of the regression coefficient which became underestimated. The underestimated standard error of the regression coefficient will affect the results of the hypothesis testing model which will make the test more easily to reject the null hypothesis or more easily to conclude that a variable has a significant effect on the response variable. Meanwhile, the effect of the overdispersion condition in the Poisson regression model is hardly seen in terms of the predictive model ability, as can be seen from the RMSE value and the relative bias of the regression coefficient estimator is relatively small compared to other models. Furthermore, using a comparison of AIC values, the best three regression models were selected, which are ZINB, ZINBMM, and ZIGPMM. Later, the best three regression model was used to analyze the under-five children deaths due to Pneumonia on Java Island in 2021st. Based on the best three regression model, the hypothesis test shows that the percentage of under-five children with lack of nutrition significantly affects the number of under-five children deaths due to Pneumonia on Java Island.