Polychoric correlations

Some researchers suggest using an alternative to the Pearson correlation when correlating two ordinal categorical variables. Polychoric (or tetrachoric in the case of two binary variables) correlations assume that the continuous measure underlying the categorical variables is normally distributed.

If this is the case then it has been shown ( for example in Homer, P and O’Brien, RM (1988)) that polychorics more accurately estimate the correlation between pairs of categorical variables. These correlations are not estimated in SPSS but may be estimated using [:FAQ/spsspoly:this syntax] from R and incorporated into SAS using this [:FAQ/saspol:SAS macro] which can be incorporated into SPSS versions 16 and above or using a PC programme freely downloadable from JS Uebersax’s website [http://ourworld.compuserve.com/homepages/jsuebersax/xpc.htm) here.] A help guide is included showing an example of its use. Tetrachoric correlations may be computed using a [:FAQ/tetra: spreadsheet.]

The polychoric correlations can then be typed into a correlation matrix which is entered using syntax into SPSS by inputting the correlation matrix directly rather than the raw data. An example correlation input file for four variables, V1, V2, V3 and V4, is given below. Column one identifies the rows as either containing sample sizes or correlations, column two contains the variable names, the remaining columns give the sample sizes and correlations for the four variables.

N       20.0000000      20.0000000      20.0000000      20.0000000

CORR V1  1.0000000        .3360000        .1617943       -.1944917

CORR V2   .3360000       1.0000000       -.0763708        .1959513

CORR V3   .1617943       -.0763708       1.0000000       -.0093707

CORR V4  -.1944917        .1959513       -.0093707       1.0000000

This data file can then be directly inputted into a factor analysis by running the following syntax into a syntax window.

FACTOR MATRIX IN(COR=*) /MISSING=LISTWISE
/ANALYSIS=V1 V2 V3 V4
/PRINT=CORRELATION 
/PLOT=EIGEN 
/EXTRACTION=ML
/ROTATION=OBLIMIN .

References

Olsson, U. (1979) Maximum likelihood estimation of the polychoric correlation coefficient. Psychometrika, 44(4), 443-460.

Homer, P and O’Brien, RM (1988) Using LISREL models with crude rank category measures. Quality and Quantity, 22, 191-201