Circumplex Models With Ordinal Data

Abstract

Circumplex models arrange personality or mood variables around a circle such that the relative locations of these variables reflect their correlations. Although circumplex models were originally developed for continuous variables, their applications often involve ordinal variables. To accommodate ordinal variables, we propose estimating Browne’s circumplex correlation structure with polychoric correlations instead of Pearson correlations. We consider ordinary least squares estimation due to its computational robustness with polychoric correlation matrices. We illustrate the newly developed method with an empirical study that involves 18 binary variables and 12,108 participants. We also conduct a simulation study to explore its statistical property and compare it with maximum likelihood estimation with polychoric correlations, ordinary least squares estimation with Pearson correlations, and maximum likelihood estimation with Pearson correlations. The new method provides essentially unbiased point estimates and more satisfactory confidence intervals than other methods.

Keywords:

• Circumplex models
• ordinal data
• polychoric correlations

Notes

1 Adding a weight matrix to the center of Equation (7)(7) $$f_{\text{OLS}}=(\mathit{r}-\mathit{\rho }(\mathit{\gamma }))\prime (\mathit{r}-\mathit{\rho }(\mathit{\gamma })).$$(7) changes it to a weighted least squares discrepancy function. Although weighted least squares estimation is theoretically superior to OLS estimation, several researchers (Forero et al., 2009; Yang-Wallentin et al., 2010) reported that OLS estimation performs more satisfactorily in practice for factor analysis with ordinal variables.

2 The development is available as a user-friendly R package “CircumO.”

3 We also constructed confidence intervals. We include the confidence intervals in the online support file.

4 The ${\chi }^2$ statistic with PC_OLS-B is 1058.26. These test statistics with PC_OLS-A and PC_OLS-B lead to essentially the same RMSEA.

5 We describe the corresponding parameter values in an online support file.

6 The convergence rates of PC_OLS-B are not included because they are identical to those of PC_OLS-A.

7 We present more details (nonconvergences and Heywood cases) in the online support file.

8 Because the point estimates of PC_OLS-A and PC_OLS-B are identical, we only present those of PC_OLS-A.

9 The unsatisfactory empirical coverage rates of PS_OLS and PS_ML are mainly due to low empirical coverage rates for the communality indices. The unsatisfactory empirical coverage rates of PC_ML are due to low empirical coverage rates for all parameters. We report the empirical coverage rates of all parameters in an online support file.

10 We could understand this phenomenon by decomposing deviations of sample estimates from population values into two parts: random sampling error and “model error” introduced by nonsymmetric distributions. Random sampling error decreases as the sample size increases but such “model error” stays the same at different sample sizes. At small samples, random sampling error could dominate such “model error.” Consequently, the statistics of “T4” and “T5” are close to their reference distributions. At larger samples, such “model error” will eventually dominate random sampling error and the test statistics will be larger than ${\chi }^2$ distributions.

11 We include the figures comparing the CDFs of the five test statistics against the reference distribution in online support file.

12 Note that these blocks consist of blocks of symmetric matrices because ${\mathit{P}}_x$ is symmetric. The second-order partial derivatives $(\frac{{\partial }^2\mathit{\rho }(\mathit{\gamma })}{\partial {\mathit{\gamma }}_i\partial {\mathit{\gamma }}_j})$ in Equation (21)(21) $${\mathbb{A}}_{ij}={\left(\Delta \prime \Delta \right)}_{ij}+\left(\frac{{\partial }^2\mathit{\rho }(\mathit{\gamma })}{\partial {\mathit{\gamma }}_i\partial {\mathit{\gamma }}_j}\right)\prime \left(\mathit{r}-\mathit{\rho }(\mathit{\gamma })\right).$$(21) are nonduplicated elements of $\left(\frac{{\partial }^2{\mathit{P}}_x}{\partial {\mathit{\gamma }}_i\partial {\mathit{\gamma }}_j}\right).$