Contributions to Estimation of Polychoric Correlations

ABSTRACT

This research concerns the estimation of polychoric correlations in the context of fitting structural equation models to observed ordinal variables by multistage estimation. The first main contribution of this research is to propose and evaluate a Monte Carlo estimator for the asymptotic covariance matrix (ACM) of the polychoric correlation estimates. In multistage estimation, the ACM plays a prominent role, as overall test statistics, derived fit indices, and parameter standard errors all depend on this quantity. The ACM, however, must itself be estimated. Established approaches to estimating the ACM use a sample-based version, which can yield poor estimates with small samples. A simulation study demonstrates that the proposed Monte Carlo estimator can be more efficient than its sample-based counterpart. This leads to better calibration for established test statistics, in particular with small samples. The second main contribution of this research is a further exploration of the consequences of violating the normality assumption for the underlying response variables. We show the consequences depend on the type of nonnormality, and the number and location of thresholds. The simulation study also demonstrates that overall test statistics have little power to detect the studied forms of nonnormality, regardless of the ACM estimator.

Keywords:

• Categorical data
• structural equation modeling
• Monte Carlo methods
• polychoric correlation

Article Information

Conflict of Interest Disclosures: The author signed a form for disclosure of potential conflicts of interest. The author did not report any financial or other conflicts of interest in relation to the work described.

Ethical Principles: The author affirms having followed professional ethical guidelines in preparing this work. These guidelines include obtaining informed consent from human participants, maintaining ethical treatment and respect for the rights of human or animal participants, and ensuring the privacy of participants and their data, such as ensuring that individual participants cannot be identified in reported results or from publicly available original or archival data.

Funding: This work was not supported by a grant.

Role of the Funders/Sponsors: None of the funders or sponsors of this research had any role in the design and conduct of the study; collection, management, analysis, and interpretation of data; preparation, review, or approval of the manuscript; or decision to submit the manuscript for publication.

Acknowledgments: The author would like to thank the anonymous reviewers and Associate Editor for their comments on prior versions of this manuscript. The ideas and opinions expressed herein are those of the author alone, and endorsement by the author's institution is not intended and should not be inferred.

Notes

1 The parametric bootstrap differs from the so-called “naive” bootstrap (Bollen & Stine, 1992), where the observed data are resampled with replacement.

2 Recently, Jin and Yang-Wallentin (2017) studied estimation of the polychorics under this model when h_ν( · ) is specified as a t or skew-normal density.

3 We do not consider the asymptotic distribution of the thresholds in this research. Since we assume that no constraints are placed on the thresholds, the asymptotic covariances among the thresholds, and between the correlations and thresholds, do not impact the test statistics studied here.

4 The popular “WLSMV” estimation option in Mplus actually corresponds to DWLS estimation and the T₂ statistic, using the Muthén (1984) sample-based ACM estimate.

5 The values for λ and ψ were found using a nonlinear minimizer, after specifying all other values.

6 The values for ω and φ were found using a nonlinear minimizer, after specifying all other values.

7 The R code used to compute is available upon request from the author.