Turkiye Klinikleri Journal of Biostatistics

.: ORIGINAL RESEARCH
Üç Düzeyli Hiyerarşik Modelleri Tahmin Etmek İçin Kullanılan Maksimum Olabilirlik ve Bayesçi Yöntemlerin Performansı: Karşılaştırmalı Bir Çalışma
Performances of Maximum Likelihood and Bayesian Methods Used For Estimating Three-Level Hierarchical Models: A Comparative Study
Oya KALAYCIOĞLUa
aDepartment of Biostatistics and Medical Informatics, Bolu Abant İzzet Baysal University Faculty of Medicine, Bolu, TÜRKİYE
Turkiye Klinikleri J Biostat. 2020;12(3):252-60
doi: 10.5336/biostatic.2020-78912
Article Language: EN
Full Text
ABSTRACT
 Objective: Hierarchical data with two or more levels are common in different fields of research including medical, educational, social and sports sciences. Maximum likelihood (ML) and Bayesian Markov Chain Monte Carlo (MCMC) estimation methods are widely used in regression analyses used for modelling these hierarchical data. However, the performances of these methods are not well studied for the estimation of three-level models. This paper aims at finding the optimal estimation technique under various combinations of number of clusters at second and third levels in three-level data sets. Material and Methods: A data application example is presented using a three-level dataset on football player's performance. Then, a simulation study based on the 3-level hierarchical linear model is performed for the comparison of four different maximum likelihood and Bayesian estimation approaches under various number clusters. Results: The data analysis and simulation study illustrate how strongly different estimation approaches affect the model parameter estimates, especially variance components. It is found that, if the main interest of the analysis is in the fixed part of the model, then any maximum likelihood or Bayesian method can be used, provided that the number of clusters at both levels are more than four. However, the main difference between these meth-ods occurred in estimating the random terms. Conclusion: Results of the simulation study showed that using restricted maximum likelihood method is associated with better results for both regression coefficient and variance estimates. Obtaining valid variance estimates with Bayesian MCMC estimation requires careful consideration for defining prior distributions.

Keywords: Bayesian hierarchical modelling; mixed effects model; multilevel modelling; three-level clustering
ÖZET
 Amaç: Tıp, eğitim, sosyal ve spor bilimleri dahil olmak üzere farklı araştırma alanlarında iki veya daha fazla düzeyli hiye-rarşik veriler yaygındır. Bu hiyerarşik verileri modellemek için kullanılan regresyon analizlerinde maksimum olabilirlik ve Bayesçi Markov Zinciri Monte Carlo (MZMC) tahmin yöntemleri yaygın olarak kullanılmaktadır. Ancak, bu yöntemlerin performansı üç düzeyli modellerin tahmini için yeterince çalışılmamıştır. Bu makale, üç düzeyli veri setlerinde ikinci ve üçüncü düzeylerindeki çeşitli küme sayısı kombinasyonları altında en iyi tahmin tekniğini bulmayı amaçlamaktadır. Gereç ve Yöntemler: Futbolcu performanslarına dair üç düzeyli bir veri seti kullanılarak bir veri uygulama örneği sunulmuştur. Sonrasında, dört farklı maksimum olabilirlik ve Bayesçi tahmin yaklaşımlarını farklı küme sayıları altında karşılaştırmak için, üç düzeyli hiyerarşik doğrusal modele dayanan bir simülasyon çalışması yapılmıştır. Bulgular: Veri analizi ve simülasyon çalışması, farklı tahmin yaklaşımlarının model parametre tahminlerini, özellikle varyans bileşenlerini ne kadar güçlü etkilediğini göstermiştir. Eğer analizin temel ilgi konusu modelin sabit kısmıysa, her iki düzeydeki küme sayısının dörtten fazla olması koşuluyla herhangi bir maksimum olabilirlik ve Bayesçi yöntemin kullanılabileceği sonucuna varılmıştır. Ancak, bu yöntemler arasındaki temel fark, rastgele terimleri tahmin etmede ortaya çıkmıştır. Sonuç: Simülasyon çalışmasının sonuçları, sınırlı maksimum olabilirlik yön-teminin kullanılmasının hem regresyon katsayısı hem de varyans tahminleri için daha iyi sonuçlarla ilişkili olduğunu göstermiştir. Bayesçi MZMC tahmini ile geçerli varyans tahminleri elde etmek, önsel dağılımları tanımlamak için dikkatli bir değerlendirme gerektirmektedir.

Anahtar Kelimeler: Bayesçi hiyerarşik modelleme; karma etkili modeller; çok düzeyli modelleme; üç-düzeyli kümeleme
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