Model Hazard Additif Untuk Data Kejadian Berulang
Triastuti Wuryandari, Dr. Gunardi, M.Si
2023 | Disertasi | S3 Matematika
Model regresi untuk data survival antara lain adalah model hazard multiplikatif yang sering disebut model Cox dan model hazard additif. Model Cox mensyaratkan proportional hazard. Jika proportional hazard sulit dipenuhi, maka salah satu alternatifnya digunakan model hazard additif. Ada 2 jenis model hazard additif yaitu model hazard additif Aalen dan model hazard additif Lin Ying. Pada model Aalen, koefisien regresinya tergantung waktu sedangkan pada model Lin Ying, koefisien regresinya konstan. Pada data survival, seringkali dijumpai kejadian lebih dari sekali dalam setiap individu yang disebut kejadian berulang (recurrent event). Data kejadian berulang juga bisa dimodelkan dengan model hazard multiplikatif maupun model hazard additif. Penelitian ini membahas model hazard additif Lin Ying untuk data kejadian berulang. Parameter yang diestimasi pada model adalah fungsi hazard baseline kumulatif dan koefisien regresi. Metode estimasi parameter yang digunakan adalah metode partial likelihood dengan pendekatan counting process. Estimasi koefisien regresi pada model Lin Ying diselesaikan dengan cara menentukan persamaan skor yang meniru persamaan skor pada model Cox, sehingga estimasinya dapat langsung dicari karena diperoleh model yang closed form. Studi kasus dalam penelitian ini diambil dari penelitian Lind, dkk tentang pengaruh jenis kelamin, pemberian supplemen dan tingkat pendidikan ibu terhadap kekambuhan sakit diare pada bayi. Data penelitian diambil pada bulan Juli 1997 hingga Mei 1999 di Purworejo Jawa Tengah. Berdasarkan pengolahan data, diperoleh hasil bahwa jenis kelamin dan tingkat pendidikan ibu berpengaruh terhadap kekambuhan penyakit infeksi pada bayi, sehingga berpengaruh terhadap tumbuh kembang bayi.
Regression models for survival data include the multiplicative hazard model which are often called the Cox model and the additive hazard model. The Cox model requires a proportional hazard. If the proportional hazard is fulfill, then an alternative to the Additive Hazard Model. The additive hazard model measures difference risk to the effect of covariate in absolute term while multiplicative hazard measures it excess risk in relative term. There are 2 types of additive hazard models, namely the Aalen additive hazard model and the Lin and Ying additive hazard model. In the Aalen model, the regression coefficient is time dependent while in the Lin and Ying model, the regression coefficient is constant. In the survival data, there are often more than one event in each individual. Events that occur more than once for an individual are called recurrent events. The recurrent event data can be modelled by multiplicative hazard and additive hazard model. This research discusses Lin and Ying additive hazard model for recurrent event data. The parameter estimation use partial likelihood with counting process approach. The estimation of the regression coefficient in the Lin and Ying model is solved by determining the score equation that mimics the score equation in the Cox model, so that the estimate can be searched immediately because obtained a closed form model. The estimator in the additive hazard model for recurrence data have properties are asymptotically normal and consistent. The case study in this research was taken from research by Lind, et al on the influence of the sex, supplements and edclass on the baby's growth and development. Data was taken from July 1997 to May 1999 in Purworejo, Central Java. Based on data processing, the results obtained show that gender and edclass influence the recurrence of infectious diseases in babies, thus affecting the baby's growth and development.
Kata Kunci : survival, additive hazard models, recurrent event, partial likelihood, score equation, counting process.