On the Estimation of Controllability Metrics in Ordinal Vector Autoregressive Models

The quantification of symptom importance in psychological disorders is a central problem in clinicalscience. Many diverse methods have been proposed to tackle this problem, and one such technique isthrough the application of control theory to psychological time series. In this approach, the evolution ofmultiple time-varying indices (such as symptoms and behaviors) across time is treated as a dynamicalsystem and symptom importance is quantified through the controllability gramian of the underlyingsystem. Existing work, however, assumes that individual indices are measured on continuous scales.This is in contrast to the inherent ordinal nature of most psychological measures. In this regard, it islargely unknown how effectively we can recover the true underlying gramian when ordinal measures areof interest, especially in small sample regimes. In this work, we study the relevance and recoverability of the controllability gramian obtained byestimating (ordinal) vector autoregressive models in an SEM framework. This is done in 3 parts. Inthe first part, we demonstrate that controllability metrics obtained from the gramian provide distinctinformation vis-a-vis traditional symptom metrics obtained from graph theory. Secondly, we studied ifordinal modeling techniques using polychoric correlations improve upon a naive treatment of ordinaldata as continuous variables when estimating the controllability gramian. Lastly, we examined how wellwe can recover the difference in gramian values in pre-post intervention designs. Results indicate that,unless the length of the observed time-series is unreasonably large, the use of polychoric correlationsprovided little benefit relative to the naive approach of assuming continuous responses. Furthermore, wefound that accurate estimation of differences in the gramian in a pre-post setting can be challenging, withthe results varying depending on the metric used.