On Enhancing Plausibility of the Missing at Random Assumption in Incomplete Data Analyses via Evaluation of Response-Auxiliary Variable Correlations

Abstract

A procedure for evaluating candidate auxiliary variable correlations with response variables in incomplete data sets is outlined. The method provides point and interval estimates of the outcome-residual correlations with potentially useful auxiliaries, and of the bivariate correlations of outcome(s) with the latter variables. Auxiliary variables found in this way can enhance considerably the plausibility of the popular missing at random (MAR) assumption if included in ensuing maximum likelihood analyses, or can alternatively be incorporated in imputation models for subsequent multiple imputation analyses. The approach can be particularly helpful in empirical settings where violations of the MAR assumption are suspected, as is the case in many longitudinal studies, and is illustrated with data from cognitive aging research.

Keywords:

• auxiliary variable
• confidence interval
• incomplete data set
• longitudinal study
• missing at random
• missing data

ACKNOWLEDGMENTS

We are grateful to C. K. Enders, R. J. A. Little, G. A. Marcoulides, and K.-H. Yuan for valuable discussions on missing data analysis. Thanks are also due to P. B. Baltes, F. Dittman-Kohli, and R. Kliegl for permission to use for illustration purposes data from their project “Aging and Fluid Intelligence.”

Notes

¹ The threshold of .40 should not be considered a rule of thumb, but only a rough current guide about desirable possibly minimal strength of (bivariate) correlation between response and a potentially effective AV (cf. Enders, 2010).

² The explanatory variables x₁, … , x_p are assumed in the rest, for simplicity, to be measured without error, but the following method is straightforwardly extended to the case of fallible independent measures by (a) introducing for appropriate subsets of them latent variables indicated by the measures in those subsets, and (b) regressing the response(s) on these latent variables (e.g., Bollen, 1989).

³ The two procedures in this article for evaluation of semipartial and bivariate correlations of candidate AVs with response variables(s) are best used in tandem with the group difference examination approach in Raykov and Marcoulides (2014) to identify AVs that might well enhance substantially the MAR plausibility and likely statistical power. These are variables that (a) show pronounced mean and possibly variance differences across the groups with present and with missing data on an outcome(s) of concern, as identified by the method in Raykov and Marcoulides (2014); and also (b) correlate markedly with the response, conditionally or unconditionally, on the explanatory variables under consideration, as found with the procedures of this article. In this sense, the latter procedures and that in Raykov and Marcoulides (2014) could be considered complementary to each other, rather than competitors or rivals, in the search for effective AVs for inclusion in following ML or MI analyses of an incomplete data set.