statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, polychoric correlation{{Cite web, url=https://support.sas.com/documentation/cdl/en/procstat/65543/HTML/default/viewer.htm#procstat_corr_details14.htm, title=Base SAS(R) 9.3 Procedures Guide: Statistical Procedures, Second Edition, website=support.sas.com, language=en, access-date=2018-01-10 is a technique for estimating the

correlation In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics ...

between two hypothesised

normally distributed In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...

continuous

latent variable In statistics, latent variables (from Latin: present participle of ) are variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or measured. Such '' latent va ...

s, from two observed ordinal variables. Tetrachoric correlation is a special case of the polychoric correlation applicable when both observed variables are

dichotomous A dichotomy () is a partition of a set, partition of a whole (or a set) into two parts (subsets). In other words, this couple of parts must be * jointly exhaustive: everything must belong to one part or the other, and * mutually exclusive: nothi ...

. These names derive from the polychoric and tetrachoric series which are used for estimation of these correlations.

Applications and examples

This technique is frequently applied when analysing items on self-report instruments such as

personality test A personality test is a method of assessing human personality construct (psychology), constructs. Most personality assessment instruments (despite being loosely referred to as "personality tests") are in fact introspective (i.e., subjective) self ...

s and surveys that often use

rating scale A rating scale is a set of categories designed to obtain information about a quantitative property, quantitative or a Qualitative data, qualitative attribute. In the social sciences, particularly psychology, common examples are the Likert scale, L ...

s 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 Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observe ...

for assessing the factor structure of tests involving ordinally measured items. Kiwanuka and colleagues (2022) have also illustrated the application of polychoric correlations and polychoric confirmatory factor analysis in nursing science. 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. In some disciplines, the statistical technique is rarely applied however, some scholar

have demonstrated how it can be used as an alternative to the Pearson correlation.

Software

* Mplus by Muthen and Muthe

* polycor package in R by John Fo

* psych package in R by William Revell

* lavaan package in R by Yves Rossee

* semopy package in Python by Georgy Meshcheryako

* PRELIS
POLYCORR program
* PROC CORR in SAS (with POLYCHORIC or OUTPLC= options

* An extensive list of software for computing the polychoric correlation, by John Uebersa

* package polychoric in Stata by Stas Koleniko

References

* 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 Kotz, Samuel, Narayanaswamy Balakrishnan, Campbell B. Read, Brani Vidakovic & Norman L. Johnson (Eds), ''Encyclopedia of Statistical Sciences'', Vol. 7. New York, NY: John Wiley, pp. 68–74. *Kiwanuka, F., Kopra, J., Sak-Dankosky, N., Nanyonga, R. C., & Kvist, T. (2022). "Polychoric Correlation with Ordinal Data in Nursing Research". ''Nursing research, 10.1097/NNR.0000000000000614''. Advance online publication. https://doi.org/10.1097/NNR.0000000000000614.

External links

The Tetrachoric and Polychoric Correlation Coefficients
Summary statistics Covariance and correlation Summary statistics for contingency tables Latent variable models

Applications and examples

Software

See also

References

External links