In statistics, polychoric correlation is a technique for estimating the correlation between two theorised normally distributed continuous latent variables, from two observed ordinal variables. Tetrachoric correlation is a special case of the polychoric correlation applicable when both observed variables are dichotomous. These names derive from the polychoric and tetrachoric series which are used for estimation of these correlations. These series were mathematical expansions once but not anymore.
Applications and examples
This technique is frequently applied when analysing items on self-report instruments such as personality tests and surveys that often use rating scales with a small number of response options (e.g., strongly disagree to strongly agree). The smaller the number of response categories, the more a correlation between latent continuous variables will tend to be attenuated. Lee, Poon & Bentler (1995) have recommended a two-step approach to factor analysis for assessing the factor structure of tests involving ordinally measured items. This aims to reduce the effect of statistical artifacts, such as the number of response scales or skewness of variables leading to items grouping together in factors.
- Mplus by Muthen and Muthen 
- polycor package in R by John Fox 
- psych package in R by William Revelle 
- POLYCORR program
- PROC CORR in SAS (with POLYCHORIC or OUTPLC= options) 
- An extensive list of software for computing the polychoric correlation, by John Uebersax 
- package polychoric in Stata by Stas Kolenikov 
- "Base SAS(R) 9.3 Procedures Guide: Statistical Procedures, Second Edition". support.sas.com. Retrieved 2018-01-10.
- Lee, S.-Y., Poon, W. Y., & Bentler, P. M. (1995). "A two-stage estimation of structural equation models with continuous and polytomous variables". British Journal of Mathematical and Statistical Psychology, 48, 339–358.
- Bonett, D. G., & Price R. M. (2005). "Inferential Methods for the Tetrachoric Correlation Coefficient". Journal of Educational and Behavioral Statistics, 30, 213.
- Drasgow, F. (1986). Polychoric and polyserial correlations. In S. Kotz and N.L. Johnson (Eds), Encyclopedia of Statistical Sciences, Vol. 7. New York, NY: John Wiley, pp. 68-74.