Robust Mixed Model Analysis

The following sections are included:

• Dedication
• Preface
• Contents

Mixed effects models, or mixed models, have had a wide-ranging impact in modern applied statistics. See, for example, Jiang (2007), McCulloch et al. (2008), Demidenko (2013). These models are characterized by random effects that are involved in a certain way. Depending on how the random effects are involved, a mixed model may be classified as a linear mixed model (LMM), generalized linear mixed model (GLMM), non-linear mixed model (NLMM), semi-parametric mixed model (SPMM), or non-parametric mixed model (NAMMA). Due to their broad applications, there has been a growing interest in learning about these models. On the other hand, some of these models, such as LMM and GLMM, are highly parametric, involving distributional assumptions that may not be satisfied in real-life problems. It is important to make sure that a procedure of statistical analysis is robust against violation of the assumptions, or some “bad cases”. Here the word robustness is formally introduced, for the first time, and we need to make clear what it means. Generally speaking, it means the ability of a procedure to survive some unexpected situations. Life is full of surprises. The unexpected situation could be a violation of an assumption, an outlying observation, or something else. Luckily for the practitioners, there is a rich collection of methods that are currently available and robust, in certain ways. As an introductory example, consider the following.

The following sections are included:

• Method of moments
• Generalized estimating equations (GEE)
• Iterative estimating equations
• Robust estimation in GLMM
• Robust GEE
• Real-data examples
  • Hip replacement data revisited
  • Salamander-mating data
  • Epileptic seizure data
• Exercises

The opening illustrative example of Section 1.1 is a special case of inference about a non-Gaussian linear mixed model. Under such a model, the random effects and errors are assumed to be independent, or simply uncorrelated, but their distributions are not assumed to be normal. As a result, the (joint) distribution of the data may not be fully specified (up to a number of parameters). Linear mixed models without the normality assumption is practical because, in practice, the normality assumption rarely holds exactly, or even approximately. It would be useful to develop methods that are robust to violation of such an assumption. Thus, throughout this section, normality is not assumed, unless specially noted.

Part of the previous chapter (see Subsection 3.3.3) has led to the consideration of robust tests in LMM. In this chapter, we continue the coverage on this topic in much greater generality.

So far, we have been considering the types of inference such as estimation and testing. There is another type of statistical inference that has not been touched: Prediction. As noted in Jiang (2007, sec. 2.3), there are two kinds of prediction problems associated with the mixed effects models. The first kind, which is encountered more often in practice, is prediction of mixed effects; the second kind is prediction of future observation. Here in this chapter, we shall focus on prediction of the mixed effects. In this regard, the best known method is best linear unbiased prediction, or BLUP. So, before we present a new method, let us first review basic elements of BLUP.

Statistical models are key to potential gains that Statistics could make over a “simple-minded” approach. For example, a variance component model allows one to obtain a more efficient estimator of the fixed effects (regression coefficients) than the LS estimator; a SAE model makes it possible for one to “borrowing strengths”, and thus do better than direct estimators such as the (area-level) sample mean. On the other hand, as is always said, “there is no free lunch”, in the sense that such gains via statistical modeling is not risk-free for model failure, and there will be a consequence in the latter case. Therefore, the importance of careful model selection, in particular for mixed effects models, cannot be overstated…

There are other types, or aspects, of robust mixed model analysis that, so far, have not been systematically covered. However, these topics are relatively unrelated, or have some unique features of their own. Also, the literature covering each one of these topics is not as extensive as the ones covered in the previous chapters. Due to such considerations, we have put together this final chapter, with a collection of several topics. These include mixed model diagnostics, nonparametric and semiparametric methods, Bayesian analysis, outliers, benchmarking, and more about prediction. It should be noted that some of these topics may be associated, in a certain way, with topics covered in the previous chapters; yet, there are obvious reasons that justify their inclusion in the current chapter. As we have been doing all along, the focus is on robustness, or implication to robustness.

The following sections are included:

• Bibliography
• Index