Assessing Partial Association Between Ordinal Variables: Quantification, Visualization, and Hypothesis Testing

Abstract

Partial association refers to the relationship between variables Y 1 , Y 2 , … , Y K while adjusting for a set of covariates X = { X 1 , … , X p } . To assess such an association when Y k ’s are recorded on ordinal scales, a classical approach is to use partial correlation between the latent continuous variables. This so-called polychoric correlation is inadequate, as it requires multivariate normality and it only reflects a linear association. We propose a new framework for studying ordinal-ordinal partial association by using Liu-Zhang’s surrogate residuals. We justify that conditional on X , Y k , and Y l 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 nonlinear or even nonmonotonic 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 partial regression plots and three-dimensional 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. Supplementary materials for this article are available online.