Assessing partial association between ordinal variables: quantification, visualization, and hypothesis testing

Partial association refers to the relationship between variables Y1,Y2,...,YK while adjusting for a set of covariates X = {X1, . . . , Xp}. To assess such an association when Yk’s are recorded on ordinal scales, a classical approach is to use partial corre- lation between the latent continuous variables. This so-called polychoric correlation is inadequate, as it requires multivariate normality and it only reflects a linear associa- tion. We propose a new framework for studying ordinal-ordinal partial association by using surrogate residuals (Liu and Zhang, JASA, 2018). We justify that conditional on X, Yk and Yl are independent if and only if their corresponding surrogate residual variables are independent. Based on this result, we develop a general measure φ to quantify association strength. As opposed to polychoric correlation, φ does not rely on normality or models with the probit link, but instead it broadly applies to models with any link functions. It can capture a non-linear or even non-monotonic association. Moreover, the measure φ gives rise to a general procedure for testing the hypothesis of partial independence. Our framework also permits visualization tools, such as par- tial regression plots and 3-D P-P plots, to examine the association structure, which is otherwise unfeasible for ordinal data. We stress that the whole set of tools (measures, p-values, and graphics) is developed within a single unified framework, which allows a coherent inference. The analyses of the National Election Study (K = 5) and Big Five Personality Traits (K = 50) demonstrate that our framework leads to a much fuller assessment of partial association and yields deeper insights for domain researchers.