Multiple imputation of missing covariate values in multilevel models with random slopes: a cautionary note

Abstract

Multiple imputation (MI) has become one of the main procedures used to treat missing data, but the guidelines from the methodological literature are not easily transferred to multilevel research. For models including random slopes, proper MI can be difficult, especially when the covariate values are partially missing. In the present article, we discuss applications of MI in multilevel random-coefficient models, theoretical challenges posed by slope variation, and the current limitations of standard MI software. Our findings from three simulation studies suggest that (a) MI is able to recover most parameters, but is currently not well suited to capture slope variation entirely when covariate values are missing; (b) MI offers reasonable estimates for most parameters, even in smaller samples or when its assumptions are not met; and (c) listwise deletion can be an alternative worth considering when preserving the slope variance is particularly important.

Biases in multilevel analyses caused by cluster-specific fixed-effects imputation

“…The dummy imputation could be appropriate when the clusters and ICC are large and when the focus is on the regression coefficients only. The first paper to also consider random slopes was published by Grund et al (2016). The authors evaluated the performance of two multilevel imputation strategies and listwise deletion under various settings.…”

“…To summarize, while all these articles cover imputation strategies for hierarchical data, they are subject to three important limitations: They only consider random intercept models (Reiter et al, 2006;Andridge, 2011;van Buuren, 2011;Drechsler, 2015;Enders et al, 2016;Zhou et al, 2016;Taljaard et al, 2008;Lüdtke et al, 2017), they only rely on simulation studies to evaluate the impact of different imputation approaches (Reiter et al, 2006;van Buuren, 2011;Enders et al, 2016;Zhou et al, 2016;Taljaard et al, 2008), or they do not evaluate the cluster-specific fixed-effects imputation approach as an alternative to the multilevel imputation model (Grund et al, 2016). Our contribution to the literature is that we analytically generalize the findings regarding the cluster-specific fixed-effects imputation compared to the multilevel imputation model by considering a setting with (arbitrarily many) cluster-specific variable dummies.…”

“…V obs is the subset of V containing those observations for which the outcome Y is observed. Note that we assume that all explanatory variables are fully observed or that missing values in these variables have been imputed in previous steps, as missing values in an explanatory variable can cause biases in some parameter estimates (Grund et al, 2016). Lastly,σ ε 2 andδ are the ordinary least squares estimates for σ 2 ε and δ.…”

“…To our knowledge, the impact on random effects if fixedeffects models with cluster-specific slopes are used for imputation has not yet been studied analytically, despite the demand for such research (Drechsler, 2015;Lüdtke et al, 2017;Grund et al, 2016). Our paper closes this research gap by comparing cluster-specific fixed-effects imputation and multilevel imputation and generalizing the evaluations to all types of random coefficient models.…”

“…In educational research, it is often the random effects themselves (or derivatives, such as the intra-class correlation) that are of particular interest when measuring the school effect (Lenkeit, 2012;Nye et al, 2004;McCaffrey et al, 2004b). The impacts on random effects were addressed in later papers but the authors either only focused on random intercept models (Drechsler, 2015;Lüdtke et al, 2017;Zhou et al, 2016), or the evaluations were limited to running simulation studies without analytical derivations to identify which factors influence the bias observed in the simulation studies (van Buuren, 2011;Enders et al, 2016;Grund et al, 2016). In random intercept models, it is assumed that within a cluster, the average intercept deviates from the global intercept by a cluster-specific random value.…”