Assessing an Alternative for 'Negative Variance Components': A Gentle Introduction to Bayesian Covariance Structure Modeling for Negative Associations Among Patients With Personalized Treatments

摘要

The multilevel model (MLM) is the popular approach to describe dependences of hierarchically clustered observations. A main feature is the capability to estimate (cluster-specific) random effect parameters, while their distribution describes the variation across clusters. However, the MLM can only model positive associations among clustered observations, and it is not suitable for small sample sizes. The limitation of the MLM becomes apparent when estimation methods produce negative estimates for random effect variances, which can be seen as an indication that observations are negatively correlated. A gentle introduction to Bayesian covariance structure modeling (BCSM) is given, which makes it possible to model also negatively correlated observations. The BCSM does not model dependences through random (cluster-specific) effects, but through a covariance matrix. We show that this makes the BCSM particularly useful for small data samples. We draw specific attention to detect effects of a personalized intervention. The effect of a personalized treatment can differ across individuals, and this can lead to negative associations among measurements of individuals who are treated by the same therapist. It is shown that the BCSM enables the modeling of negative associations among clustered measurements and aids in the interpretation of negative clustering effects. Through a simulation study and by analysis of a real data example, we discuss the suitability of the BCSM for small data sets and for exploring effects of individualized treatments, specifically when (standard) MLM software produces negative or zero variance estimates. Translational Abstract Clustered data are so common in many fields, that it is well-known that multilevel models are helpful in the analysis of clustered data (e.g., students within classroom, employees within companies, and observations in individuals). In psychotherapy research, a multilevel analysis closely aligns with the clinical practice and arises naturally, when studying questions pertaining to individual change in a repeated measures design. As individuals differ in how they react to (similar) events, modeling the heterogeneity of their responses provides a key avenue for modeling individual effects. However, for personalized interventions, increased heterogeneity can be expected among individuals over time: Therapists will succeed better in personalization (the intervention) for some than for others, and individuals will respond differently to a personalized treatment. When the effectiveness of the personalized treatment given by the therapist differs across individuals, associations among measurements across individuals can even be negative. This apparent contradiction requires a different multilevel modeling parameterization to be able to model negative associations among clustered measurements, because the multilevel model will always describe positive associations. We present Bayesian covariance structure modeling (BCSM) for assessment of negative associations between measurements, a novel modeling technique that can deal with the issues of individual differences. BCSM provides a way to explore individual differences in treatment effects next to identifying a main treatment effect. With BCSM individual treatment effects can be identified, and it can be explored who will or will not benefit from the treatment.