Dial M̂ for Monotonic: A Kernel-Based Bayesian Approach to State-Trace Analysis

Abstract

A monotonic state-trace implies that a single latent factor is sufficient to explain the joint variation between two outcome variables across a set of conditions. There are, however, few methods available for assessing how much evidence a sample of data provides about whether the variables are truly monotonically related or not. We present a model that allows researchers to estimate the statistic M̂ which reflects the amount of evidence a dataset provides about whether two outcome variables are jointly monotonically related. This model is based on modeling the covariation between outcome measures in terms of a kernel function, which allows for computation of the latent derivatives of each outcome variable with respect to the other. M̂ is the posterior odds that these derivatives are all of the same sign and are thus monotonic. Simulations show that M̂ discriminates between monotonic and non-monotonic state traces and an example illustrates how the model can be applied to both continuous and binomial data from between-subjects, within-subjects, or mixed designs.