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

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 Modelling (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.