Polychoric correlation

# Polychoric correlation

Discussion

Encyclopedia
In statistics
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, polychoric correlation is a technique for estimating the correlation
Correlation
In statistics, dependence refers to any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence....

between two theorised normally distributed continuous latent variable
Latent variable
In statistics, latent variables , are variables that are not directly observed but are rather inferred from other variables that are observed . Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models...

s, from two observed ordinal variable
Level of measurement
The "levels of measurement", or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled "On the theory of scales of measurement"...

s. 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, mathematical expansions once, but no longer, 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
Personality test
-Overview:There are many different types of personality tests. The most common type, the self-report inventory, involves the administration of many questions, or "items", to test-takers who respond by rating the degree to which each item reflects their behavior...

s and surveys
Statistical survey
Survey methodology is the field that studies surveys, that is, the sample of individuals from a population with a view towards making statistical inferences about the population using the sample. Polls about public opinion, such as political beliefs, are reported in the news media in democracies....

that often use rating scale
Rating scale
A rating scale is a set of categories designed to elicit information about a quantitative or a qualitative attribute. In the social sciences, common examples are the Likert scale and 1-10 rating scales in which a person selects the number which is considered to reflect the perceived quality of a...

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
Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved, uncorrelated variables called factors. In other words, it is possible, for example, that variations in three or four observed variables...

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.