Correlation structure in hierarchical linear modelling: An illustration with the therapeutic alliance

Abstract

Previous studies have found an association between therapeutic alliance and treatment outcome, but only recently have researchers begun to analyse time-lagged relationships between session-to-session measures of alliance and outcomes with hierarchical linear modelling (HLM). HLM assumes simple correlation structures between any two measurements from the same client. In this paper, we suggest that this assumption might be problematic. Session-to-session measurements of outcomes (Outcome Questionnaire-10.2) and alliance (Working Alliance Inventory) in a sample (N = 63) were used to perform HLM analyses to test time-lagged (lag +1) relations between outcomes and alliance in both directions. A first set of analyses replicated the models consistently used in the literature, whereas a second set of models considered a correlation structure as a function of time. A correlation independent of time distance resulted in a bidirectional influence between alliance and outcomes (the model commonly used in the literature), but when considering a correlation structure as a function of time, only the outcomes were predictive of alliance. Considering a more complex correlation structure as a function of time seems to be an important analytical strategy for addressing the issue of variability in within-client measurements over time. This study highlights how the misspecification of a statistical model, namely, not considering a time-dependent correlation structure of the response variable, may lead to misleading findings in HLM studies. This is particularly relevant in process–outcome research, such as studies analysing the impact of therapeutic alliance on clinical outcomes.