Review Article Volume 5 Issue 3 - February 2018 DOI: 10.19080/BBOAJ.2018.04.555662 Biostat Biometrics Open Acc J Copyright © All rights are reserved by Yahia S El-Horbaty Some Estimation Methods and Their Assessment in Multilevel Models: A Review Yahia S El-Horbaty* and Eman M Hanafy Department of Mathematics, Insurance and Applied Statistics, Helwan University, Egypt Submission: January 27, 2018; Published: February 27, 2018 *Corresponding author: Yahia S El-Horbaty, Department of Mathematics, Insurance, and Applied Statistics, Helwan University, Egypt; Email: Biostat Biometrics Open Acc J 5(3): BBOAJ.MS.ID.555662 (2018) 0069 Introduction Multilevel modeling is an approach that can be used to handle clustered or grouped data. Social sciences often involve problems that investigate the relationship between individual and society. The general concept is that individuals and the social groups are conceptualized as a hierarchical system, where the individuals and groups are defined at separate levels of this hierarchical system. There are many types of multilevel models, which differ in terms of the number of levels, type of design (random intercept, random slopes and random coefficients regression model), scale of the outcome variable (continuous, categorical), and number of outcomes (univariate, multivariate). In this article, the two-level random coefficients regression model is represented, letting the univariate outcome variable to be continuous. The parameters to be estimated in multilevel regression models are the fixed coefficients that represent the fixed part of the model. The parameters also include the variance covariance matrix of the random coefficients and the variances of the residual errors that represent the random part of the model. If the components of the random part in the model were known, the unknown fixed coefficients could be estimated using generalized least squares (GLS) estimation method [1]. Similarly, if the fixed coefficients were known, then the unknown variance components of the model could also be estimated using GLS estimation method. If both are unknown, hence we have to follow different approaches. In order to estimate the unknown parameters several procedures are discussed in Searle et al. [2]. These methods include the ANOVA method for balanced data which uses the expected mean squares approach. However, this method is difficult to deal with under unbalanced data situations. Under the latter case, Rao [3] proposed the minimum norm quadratic estimation (MINQUE) method for estimating the variance parameters that produces quadratic unbiased estimators with minimum norm (MINQUE). Maximum likelihood method (full maximum likelihood (ML) and restricted maximum likelihood (REML)), iterative generalized least squares (IGLS) and the expectation maximization (EM) estimation method represent standard methods that are used under both balanced and unbalanced data. These methods involve iterative procedures and thus the resulting estimators are not necessarily expressed in a closed form. The commonly used methods to estimate the multilevel model are ML and REML [4]. The ML estimators of the variance components do not correct for the degrees of freedom lost due to the estimation of the fixed effects. As a result, the estimates Biostatistics and Biometrics Open Access Journal ISSN: 2573-2633 Abstract Multilevel linear regression models represent a generalization of linear models in which the regression coefficients are themselves given a model whose parameters are also estimated from the data. This paper reviews multilevel random coefficients regression models with a focus on the estimation problem and its assessment. Parameter estimation for the fixed effects and the variance components are highlighted. In addition, comparisons that are made in the literature to choose among the competing methods are highlighted. This is particularly emphasized when some of the assumptions underlying the estimation methods are violated. Keywords: Multilevel models; Parametric estimation; Robust standard errors Abbreviations: GLS: Generalized Least Squares; MINQUE: Minimum Norm Quadratic Estimation; ML: Maximum Likelihood; REML: Restricted Maximum Likelihood; IGLS: Iterative Generalized Least Squares; EM: Expectation Maximization; OLS: Ordinary Least Squares; ARE: Asymptotic Relative Efficiency