Polychoric correlation measures the correlation between two unobserved, continuous variables that have a bivariate normal distribution. Information about each unobserved variable is obtained through an observed ordinal variable that is derived from the unobserved variable by classifying its values into a finite set of discrete, ordered values (Olsson, 1979; Drasgow, 1986). Polychoric correlation between two observed binary variables is also known as tetrachoric correlation.
The polychoric correlation coefficient is the maximum likelihood estimate of the product-moment correlation between the underlying normal variables. The range of the polychoric correlation is from –1 to 1. Olsson (1979) gives the likelihood equations and the asymptotic standard errors for estimating the polychoric correlation. The underlying continuous variables relate to the observed ordinal variables through thresholds, which define a range of numeric values that correspond to each categorical level. PROC CORR uses Olsson’s maximum likelihood method for simultaneous estimation of the polychoric correlation and the thresholds.
PROC CORR iteratively solves the likelihood equations by using a Newton-Raphson algorithm. The initial estimates of the thresholds are computed from the inverse of the normal distribution function at the cumulative marginal proportions of the table. Iterative computation of the polychoric correlation stops when the convergence measure falls below the convergence criterion or when the maximum number of iterations is reached, whichever occurs first.
The CORR procedure computes two types of testing for the zero polychoric correlation: the Wald test and the likelihood ratio (LR) test.
Given the maximum likelihood estimate of the polychoric correlation and its asymptotic standard error , the Wald chi-square test statistic is computed as
The Wald statistic has an asymptotic chi-square distribution with one degree of freedom.
For the LR test, the maximum likelihood function assuming zero polychoric correlation is also needed. The LR test statistic is computed as
where is the likelihood function with the maximum likelihood estimates for all parameters, and is the likelihood function with the maximum likelihood estimates for all parameters except the polychoric correlation, which is set to 0. The LR statistic also has an asymptotic chi-square distribution with one degree of freedom.