Dial Mˆ for monotonic: A kernel-based Bayesian approach to state-trace analysis

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 estimates the statistic Mˆ which reflects the amount of evidence a dataset provides about whether two or more outcome variables are jointly monotonically related at the group level. This statistic 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. We then compare the prior and posterior probabilities that these derivatives are all of the same sign (and are thus monotonic) to obtain Mˆ. Simulations show that Mˆ 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.