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

Abstract

Partial association refers to the relationship between variables $Y_1,Y_2,\dots ,Y_K$ while adjusting for a set of covariates $\mathit{X}=\{X_1,\dots ,X_p\}$. 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 $\mathit{X}$, 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 $\varphi$ to quantify association strength. As opposed to polychoric correlation, $\varphi$ 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 $\varphi$ 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.

KEYWORDS:

• Covariate adjustment
• Multivariate analysis
• Partial regression plot
• Polychoric correlation
• Rating data
• Surrogate residual

Acknowledgments

The authors thank Editor Hongyu Zhao, an anonymous associate editor, and two referees for constructive comments.