On the Ratio Between Point-Polyserial and Polyserial Correlations for Non-Normal Bivariate Distributions.

Abstract

It is a well-known fact that for the bivariate normal distribution the ratio between the point-polyserial correlation (the linear correlation after one of the two variables is discretized into k categories with probabilities $p_i\text{,}$ $i=1,\dots ,k$) and the polyserial correlation $\rho$ (the linear correlation between the two normal components) remains constant with $\rho \text{,}$ keeping the $p_i$'s fixed. If we move away from the bivariate normal distribution, by considering non-normal margins and/or non-normal dependence structures, then the constancy of this ratio may get lost. In this work, the magnitude of the departure from the constancy condition is assessed for several combinations of margins (normal, uniform, exponential, Weibull) and copulas (Gauss, Frank, Gumbel, Clayton), also varying the distribution of the discretized variable. The results indicate that for many settings we are far from the condition of constancy, especially when highly asymmetrical marginal distributions are combined with copulas that allow for tail-dependence. In such cases, the linear correlation may even increase instead of decreasing, contrary to the usual expectation. This implies that most existing simulation techniques or statistical models for mixed-type data, which assume a linear relationship between point-polyserial and polyserial correlations, should be used very prudently and possibly reappraised.