Using robust variance estimation in mixed models: a review

Abstract

Presence of Clusters/ sub-groups within datasets is a common phenomenon in statistical data analysis. Examples include repeated measures data, longitudinal data, hierarchical data, and etc. The shared feature in such datasets is that observations within a group are related / similar to each other. Data of this kind is termed as correlated data or non-independent data. When analyzing such data, the methods of analysis should CS01 – CS13 120 | P a g e not rely on the assumption of independence which is a dominant assumption in statistics. Robust Variance Estimation which is often nicknamed as Sandwich Variance Estimation (SVE) is a method of variance estimation initially proposed by Peter J Huber in 1967 to correct the estimation of standard errors of miss-specified models, i.e. in models that are being fitted incorrectly. These miss-specifications/errors may be due to various reasons such as incorrect distributional assumptions, assuming linear relationships for non-linear data, assuming independency for correlated data and etc. This methods gained more popularity with its derivation in linear regression by H. White in 1980 where he demonstrate its usage for independent, heteroscedastic errors in linear regression models where the miss-specifications was not due to independence but due to errors being heteroscedastic. In contrast, with correlated data modeling, SVE requires to cater for heteroscedatsic, non-independent data. Hence, SVE is being a method for adjusting the standard errors of model parameters; it had been extensively used in correlated data analysis for obtaining standard errors that are adjusted to the correlation of the data where the adjustment made by SVE doesn’t rely on the model being fitted to the data. In olden days, when statistical models were not developed for correlated data, models assuming independence were fitted for non-independent data and the model standard errors were adjusted by using SVE. The literature had emphasized that SVE has provided improved inferential results in correlated data analysis in the absence of statistical models for correlated by improving the functionality of independency assumed models fitted for correlated data. In addition to the classical SVE, various adjustments for the classical SVE had been developed for various data scenarios such as small sample data, data with auto-correlation and etc. Lately, specialized statistical models were developed for correlated data such as Mixed Models and Generalized Linear Mixed Models (GLMMs). Since these models are defined for correlated data, the model parameter estimates and standard error estimates are resultant to the correlation exist in the data. Therefore, the necessity of SVE in such models was at argument by authors in the literature. Mixed models are defined in such a way that clusters/groups that impose correlation to the data is being introduced to the model as random effects that follow a particular statistical distribution (Gaussian, Gamma, t-distribution) where the linear predictor of mixed models consists of a component that represent the grouping/clustering in the data. More over the literature consists of few authors that had demonstrated probable miss-specifications of Mixed Models despite they are defined for correlated data. These miss-specifications are mainly due to the disparity between the correlation structure of the data and the way the random effects are defined in Mixed Models. Upon the identification of miss-specifications of such hybrid models, adoption of SVE in GLMMs becomes remedial since SVE is meant for improving miss-specified models. Though SVE was initially proposed for correcting the standard errors of maximum likelihood estimates, it can be used for parameter estimation methods which obtain parameter estimates by equating the estimation function to zero which doesn’t necessarily be a derivative of a log-likelihood. Thus, SVE are feasible with Mixed Models hich mostly accommodate pseudo likelihood methods in parameter estimation.