Self-Normalized, Score-Based Tests of Parameter Heterogeneity in Mixed Models

Score-based tests have been used to study parameter heterogeneity across many types of statistical models. This chapter describes a new self-normalization approach for score-based tests of mixed models, which addresses situations where there is dependence between scores. This differs from the traditional score-based tests, which require independence of scores. We first review traditional score-based tests and then propose a new, self-normalized statistic that is related to the previous work by Shao and Zhang (J Am Stat Assoc 105(491):1228–1240, 2010) and Zhang et al. (Electron J Stat 5:1765–1796, 2011). We then provide simulation studies that demonstrate how traditional score-based tests can fail when scores are dependent and that also demonstrate the good performance of the self-normalized tests. Next, we illustrate how the statistics can be used with real data. Finally, we discuss the potential broad application of self-normalized, score-based tests in mixed models and other models with dependent observations.