Polychoric correlation

Polychoric Correlation is a statistical method used to estimate the correlation between two latent variables that are both assumed to be ordinal and derived from underlying continuous distributions. This technique is particularly useful in the fields of psychometrics and sociometrics, where many variables of interest are ordinal but presumed to stem from an underlying continuous scale. The polychoric correlation extends the concept of the Pearson correlation coefficient, which measures the linear relationship between two continuous variables, to the analysis of ordinal data.

Overview

The polychoric correlation assumes that the ordinal variables being analyzed arise from discretization of underlying continuous variables. These continuous variables are assumed to follow a bivariate normal distribution. The goal of the polychoric correlation is to estimate the Pearson correlation coefficient of these underlying continuous variables. This estimation is typically achieved through a maximum likelihood estimation (MLE) method, although other estimation techniques exist.

Calculation

The calculation of the polychoric correlation involves several steps. First, a contingency table is created from the ordinal data, showing the frequency of each combination of categories. Next, thresholds that define the categories for the underlying continuous variables are estimated. Finally, the correlation of the underlying continuous variables is estimated, assuming that they follow a bivariate normal distribution.

Applications

Polychoric correlation is widely used in the analysis of survey data, psychological testing, and any other area where ordinal data are common. It is particularly valuable when the researcher believes that the ordinal data represent underlying continuous factors. For example, in psychometrics, test items may be scored on an ordinal scale (e.g., strongly agree to strongly disagree), but the constructs being measured (e.g., attitudes, personality traits) are assumed to be continuous.

Advantages and Limitations

One of the main advantages of the polychoric correlation is its ability to provide a more accurate measure of the relationship between variables by taking into account the ordinal nature of the data and the assumption of underlying continuous distributions. However, the method also has limitations. The estimation of the polychoric correlation can be sensitive to the assumptions about the distribution of the underlying continuous variables. Moreover, the calculation is more complex and computationally intensive than simpler measures of association such as the Spearman rank correlation coefficient.

Software Implementation

Several statistical software packages offer routines for calculating polychoric correlations, including R (using the 'polycor' package), SAS, and Stata. These implementations typically provide options for different estimation methods and for handling various complexities of the data.

See Also

• Pearson correlation coefficient
• Spearman rank correlation coefficient
• Latent variable
• Psychometrics
• Sociometrics