SAS/STAT^® software can perform a factor analysis on binary and ordinal data. To fit a common factor model, there are two approaches (both known as Latent Trait models):

• The first approach is to create a matrix of tetrachoric correlations (for binary variables) or polychoric correlations (for ordinal variables). Use the OUTPLC= option in PROC CORR (or use the POLYCHOR macro documented in SAS Note 25010, "Create a polychoric correlation or distance matrix"). You can then do an exploratory (p-values are not valid) analysis by using PROC CALIS or PROC FACTOR.

• The second approach for binary variables, in the context of item response theory, is the Rasch model. See the Rasch model section of SAS Note 30333, "FASTats: Frequently Asked-For Statistics".

If you want a principal component analysis of binary data, then use PROC CORRESP or PROC PRINCOMP. Whether you use CORRESP or PRINCOMP depends on whether you are interested in Euclidean or in chi-squared distance. You can also use PROC PRINQUAL, but if your data is all binary, then PRINQUAL gives the same results as PRINCOMP.

References

• Andrich, D. 1988. Rasch Models for Measurement. Sage University Paper on Quantitative Applications in the Social Sciences, 07-068. Beverly Hills: Sage Publications.
• Bartholomew, D. 1987. Latent Variable Models and Factor Analysis. London: Charles Griffin & Company Limited.
• van Rijckevorsal, J. L. A., and J. de Leeuw, eds. 1988. Component and Correspondence Analysis. Chichester, UK: John Wiley & Sons.

See also KB0056893, "Can PROC CALIS analyze categorical data?"