Recovering Latent Association from Likert-Type Responses: A Monte Carlo Comparison of Pearson, Spearman, Kendall, and Polychoric Correlations Under Discretization, Response Perturbation, and Gumbel-Copula Dependence

Likert-type responses are often analyzed as continuous scores, although their categories are ordered rather than truly metric. This study uses Monte Carlo simulation to examine how ordinal discretization, response perturbation, and non-Gaussian dependence affect four association estimators: Pearson correlation, Spearman correlation, Kendall’s τb, and latent-normal polychoric correlation. In the main latent-normal design, bivariate normal variables with known correlations were discretized into 3-, 5-, and 7-category scales using equal-probability thresholds, with n=300 and 200 replications per scenario. Responses were either left unchanged or modified by adjacent-category error and central-tendency shifts. A complementary Gumbel-copula extension used n=1000, 1500 replications, k=5,…,10 categories, and upper-tail dependence at Kendall’s τC=0.50,0.70,0.90. In the unperturbed latent-normal setting, Pearson and Spearman underestimated the latent association, especially with three categories, whereas the polychoric estimator closely recovered the latent correlation. Under response perturbation, this advantage weakened; at k=5, ρ=0.70, and 20% adjacent error, polychoric bias increased to −0.072. In the Gumbel design, the preferred estimator changed because the target changed: Kendall’s τb showed the smallest bias and mean squared error for copula-scale dependence, while polychoric correlation returned larger Pearson-type latent associations and overestimated τC. Overall, Pearson may be reasonable for many-category, symmetric observed-score questions; polychoric correlation is preferable when latent-normal assumptions are plausible; and Kendall’s τ is the natural choice for ordinal concordance or copula-scale dependence.