Heterogeneous Correlation Matrix

is a correlation matrix between a number of variables of different data type.

	W Binary	X Categorical	Y Ordinal	Z Numerical
W	tetrachoric	polychoric	polychoric
X	polychoric	polychoric	polychoric
Y	polychoric	polychoric	polychoric	polyserial
Z			polyserial	Pearson

The ‘hetcor’ function looks at each pair of variables in a data frame and computes the appropriate

heterogeneous correlation for each pair based on the type of variables which make up each pair.

binary variables <-> tetrachoric correlation is a special case of the polychoric correlation
categorical variables<->polychoric correlation
Ordinal variables<-> polychoric correlations
Numerical variables <->Pearson correlations
numeric and ordinal data<->polyserial correlations

The polychoric correlation coefficient, ρ, is used for correlation when the data consist of observations from two ordinal variables, each having an arbitrary number of response categories.

The polychoric correlation coefficient estimates the correlation between two unobserved bivariate normal variables assumed to underlie the observed ordinal variables.

The polychoric correlation coefficient is a generalization of the tetrachoric correlation coefficient, a statistic used to estimate correlation based on two dichotomous variables.

Under assumptions, the polychoric correlation provides a correlation estimate that is entirely free of the attenuation caused when two normally distributed variables are “crudely categorized”-that is, when they are reduced to sets of ordinal categories.

http://methods.sagepub.com/reference/encyc-of-research-design/n317.xml

https://cran.r-project.org/web/packages/polycor/polycor.pdf