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Base SAS Procedures Guide: Statistical Procedures
 

The CORR Procedure

Polyserial Correlation

Polyserial correlation measures the correlation between two continuous variables with a bivariate normal distribution, where one variable is observed directly, and the other is unobserved. Information about the 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, Drasgow, and Dorans 1982).

Let X be the observed continuous variable from a normal distribution with mean mu and variance sigma squared, let Y be the unobserved continuous variable, and let rho be the Pearson correlation between X and Y. Furthermore, assume that an observed ordinal variable D is derived from Y as follows:

upper D equals StartLayout Enlarged left-brace 1st Row 1st Column d Subscript left-parenthesis 1 right-parenthesis Baseline 2nd Column normal i normal f upper Y less-than tau 1 2nd Row 1st Column d Subscript left-parenthesis k right-parenthesis Baseline 2nd Column normal i normal f tau Subscript k minus 1 Baseline less-than-or-equal-to upper Y less-than tau Subscript k Baseline comma k equals 2 comma 3 comma ellipsis comma upper K minus 1 3rd Row 1st Column d Subscript left-parenthesis upper K right-parenthesis Baseline 2nd Column normal i normal f upper Y greater-than-or-equal-to tau Subscript upper K minus 1 EndLayout

where d Subscript left-parenthesis 1 right-parenthesis Baseline less-than d Subscript left-parenthesis 2 right-parenthesis Baseline less-than ellipsis less-than d Subscript left-parenthesis upper K right-parenthesis are ordered observed values, and tau 1 less-than tau 2 less-than ellipsis less-than tau Subscript upper K minus 1 are ordered unknown threshold values.

The likelihood function for the joint distribution (X, D) from a sample of N observations left-parenthesis x Subscript j Baseline comma d Subscript j Baseline right-parenthesis is

upper L equals product Underscript j equals 1 Overscript upper N Endscripts f left-parenthesis x Subscript j Baseline comma d Subscript j Baseline right-parenthesis equals product Underscript j equals 1 Overscript upper N Endscripts f left-parenthesis x Subscript j Baseline right-parenthesis upper P left-parenthesis upper D equals d Subscript j Baseline vertical-bar x Subscript j Baseline right-parenthesis

where f left-parenthesis x Subscript j Baseline right-parenthesis is the normal density function with mean mu and standard deviation sigma (Drasgow 1986).

The conditional distribution of Y given upper X equals x Subscript j is normal with mean rho z Subscript j and variance 1 minus rho squared, where z Subscript j Baseline equals left-parenthesis x Subscript j Baseline minus mu right-parenthesis slash sigma is a standard normal variate. Without loss of generality, assume the variable Y has a standard normal distribution. Then if d Subscript j Baseline equals d Subscript left-parenthesis k right-parenthesis, the k Superscript t h ordered value in D, the resulting conditional density is

upper P left-parenthesis upper D equals d Subscript left-parenthesis k right-parenthesis Baseline vertical-bar x Subscript j Baseline right-parenthesis equals StartLayout Enlarged left-brace 1st Row 1st Column normal upper Phi left-parenthesis StartFraction tau 1 minus rho z Subscript j Baseline Over StartRoot 1 minus rho squared EndRoot EndFraction right-parenthesis 2nd Column normal i normal f k equals 1 2nd Row 1st Column normal upper Phi left-parenthesis StartFraction tau Subscript k Baseline minus rho z Subscript j Baseline Over StartRoot 1 minus rho squared EndRoot EndFraction right-parenthesis minus normal upper Phi left-parenthesis StartFraction tau Subscript k minus 1 Baseline minus rho z Subscript j Baseline Over StartRoot 1 minus rho squared EndRoot EndFraction right-parenthesis 2nd Column normal i normal f k equals 2 comma 3 comma ellipsis comma upper K minus 1 3rd Row 1st Column 1 minus normal upper Phi left-parenthesis StartFraction tau Subscript upper K minus 1 Baseline minus rho z Subscript j Baseline Over StartRoot 1 minus rho squared EndRoot EndFraction right-parenthesis 2nd Column normal i normal f k equals upper K EndLayout

where normal upper Phi is the cumulative normal distribution function.

Cox (1974) derives the maximum likelihood estimates for all parameters mu, sigma, rho  and tau 1, …, tau Subscript k minus 1. The maximum likelihood estimates for mu and sigma squared can be derived explicitly. The maximum likelihood estimate for mu is the sample mean and the maximum likelihood estimate for sigma squared is the sample variance

StartFraction sigma-summation Underscript j equals 1 Overscript upper N Endscripts left-parenthesis x Subscript j Baseline minus x overbar right-parenthesis squared Over upper N EndFraction

The maximum likelihood estimates for the remaining parameters, including the polyserial correlation rho and thresholds tau 1, …, tau Subscript k minus 1, can be computed by an iterative process, as described by Cox (1974). The asymptotic standard error of the maximum likelihood estimate of rho can also be computed after this process.

For a vector of parameters, the information matrix is the negative of the Hessian matrix (the matrix of second partial derivatives of the log likelihood function), and is used in the computation of the maximum likelihood estimates of these parameters. The CORR procedure uses the observed information matrix (the information matrix evaluated at the current parameter estimates) in the computation. After the maximum likelihood estimates are derived, the asymptotic covariance matrix for these parameter estimates is computed as the inverse of the observed information matrix (the information matrix evaluated at the maximum likelihood estimates).

Probability Values

The CORR procedure computes two types of testing for the zero polyserial correlation: the Wald test and the likelihood ratio (LR) test.

Given the maximum likelihood estimate of the polyserial correlation ModifyingAbove rho With caret and its asymptotic standard error normal upper S normal t normal d normal upper E normal r normal r left-parenthesis ModifyingAbove rho With caret right-parenthesis, the Wald chi-square test statistic is computed as

left-parenthesis StartFraction ModifyingAbove rho With caret Over normal upper S normal t normal d normal upper E normal r normal r left-parenthesis ModifyingAbove rho With caret right-parenthesis EndFraction right-parenthesis squared

The Wald statistic has an asymptotic chi-square distribution with one degree of freedom.

For the LR test, the maximum likelihood function assuming zero polyserial correlation is also needed. If rho equals 0, the likelihood function is reduced to

upper L equals product Underscript j equals 1 Overscript upper N Endscripts f left-parenthesis x Subscript j Baseline comma d Subscript j Baseline right-parenthesis equals product Underscript j equals 1 Overscript upper N Endscripts f left-parenthesis x Subscript j Baseline right-parenthesis product Underscript j equals 1 Overscript upper N Endscripts upper P left-parenthesis upper D equals d Subscript j Baseline right-parenthesis

In this case, the maximum likelihood estimates for all parameters can be derived explicitly. The maximum likelihood estimates for mu is the sample mean and the maximum likelihood estimate for sigma squared is the sample variance

StartFraction sigma-summation Underscript j equals 1 Overscript upper N Endscripts left-parenthesis x Subscript j Baseline minus x overbar right-parenthesis squared Over upper N EndFraction

In addition, the maximum likelihood estimate for the threshold tau Subscript k, k=1, …, K-1, is

normal upper Phi Superscript negative 1 Baseline left-parenthesis StartFraction sigma-summation Underscript g equals 1 Overscript k Endscripts n Subscript g Baseline Over upper N EndFraction right-parenthesis

where n Subscript g is the number of observations in the g Superscript t h ordered group of the ordinal variable D, and upper N equals sigma-summation Underscript g equals 1 Overscript upper K Endscripts n Subscript g is the total number of observations.

The LR test statistic is computed as

minus 2 log left-parenthesis StartFraction upper L 0 Over upper L 1 EndFraction right-parenthesis

where upper L 1 is the likelihood function with the maximum likelihood estimates for all parameters, and upper L 0 is the likelihood function with the maximum likelihood estimates for all parameters except the polyserial correlation, which is set to 0. The LR statistic also has an asymptotic chi-square distribution with one degree of freedom.

Last updated: April 10, 2023