Applying Conditional Distributions to Individuals: Using Latent Variable Models

This study proposes a new method to interpret individual results of psychological test batteries. The Mahalanobis distance is a commonly-used measure of how unusual an individual’s profile of scores is compared to a population of score profiles. In models in which there is a set of predictors and a set of dependent variables (e.g., cognitive abilities predicting academic abilities), it is useful to distinguish between a profile of dependent scores that is unusual because its profile of predictor scores is unusual and a profile of dependent scores that is unusual even after controlling for the predictors. The conditional Mahalanobis distance measures the unusualness of a profile shape after controlling for a set of predictors. In psychological assessments, one only has access to observed scores, but the goal is to understand a person’s profile of latent construct scores. Factor score estimates can be calculated, but with measurement error. Using simulations studies, I investigate the accuracy of the conditional Mahalanobis distance when it is used with estimated factor scores. The conditional Mahalanobis distance used with factor scores is more accurate when the factor scores are accurately measured (i.e., the factor loadings are high or the number of indicators increases) and less accurate when the constructs in the latent structure model are highly correlated. I created an R package to assist researchers and practitioners who wish to use the conditional Mahalanobis distance. I illustrate its use with several case studies