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

Area of Science:

• Statistics
• Statistical Modeling
• Correlation Analysis

Background:

• The ratio between point-polyserial and polyserial correlations is constant for bivariate normal distributions.
• This constancy is assumed in many statistical models for mixed-type data.

Purpose of the Study:

• To assess the departure from correlation constancy when moving away from bivariate normal distributions.
• To investigate how varying marginal distributions and copulas affect this constancy.

Main Methods:

• Examined combinations of marginal distributions (normal, uniform, exponential, Weibull) and copulas (Gauss, Frank, Gumbel, Clayton).
• Varied the distribution of the discretized variable to evaluate the impact on correlation ratios.
• Quantified the magnitude of deviation from the constancy condition.

Main Results:

• The constancy condition is frequently lost with non-normal margins and dependence structures.
• Highly asymmetrical marginals combined with tail-dependent copulas significantly disrupt the correlation ratio constancy.
• In some cases, linear correlation unexpectedly increased, contrary to typical assumptions.

Conclusions:

• Existing simulation techniques and statistical models for mixed-type data, assuming a linear relationship between point-polyserial and polyserial correlations, require cautious application.
• The findings suggest a need to reappraise current methodologies due to the potential for significant deviations from expected correlation behavior.