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I have a dataset with a large number of Yes/No responses. Can I use principal components (PCA) or any other data reduction analyses (such as factor analysis) for this type of data? Please advise how I go about doing this using SPSS.

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What made you consider PCA specifically as oppose to discriminant analysis? – Chris Simokat Oct 2 '11 at 19:07
See also this question – ttnphns Jul 3 '15 at 7:01
See also: – ttnphns Dec 10 '15 at 8:37

The question of dichotomous or binary variables in PCA or Factor analysis is eternal. There are polar opinions from "it is illegal" to "it is alright", through something like "you may do it but you'll get too many factors". My own current opinion is as follows. First, I deem that binary observed variable is descrete and that it is improper to treat it in any way as continuous. Can this discrete variable give rise to factor or principal component?

  • Factor analysis (FA). Factor by definition is a continuous latent that load observable variables. Consequently, the latter cannot be but continuous (or interval, more practically speaking) when enough loaded by factor. Also, FA, due to its linear regressional nature, assumes that the rest - not loaded - part, called uniqness, is continuous either, and so it comes that observable variables should be continuous even when loaded slightly. Thus, binary variables cannot legislate themselves in FA. However, there are at least two ways round: (A) Assume the dichotomies as roughened continues underlying variables and do FA with tetrachoric - rather than Pearson - correlations; (B) Assume that factor loads a dichotomous variable not linealrly but logistically and do Latent Trait Analysis (aka Item Response Theory) instead of linear FA. Read more.

  • Principal Component Analysis (PCA). While having much in common with FA, PCA is not a modeling but only a summarizing method. Components do not load variables in the same conceptual sense as factors load variables. In PCA, components load variables and variables load components. This symmetry is because PCA per se is merely a rotation of variables-axes in space. Binary variables won't provide continuity for a component by their own selves, but the continuity can be provided by angle of rotation which can appear any. Thus in PCA, and in contrast with FA, you can get quite continuous latents (rotated axes) with purely binary variables (unrotated axes) - angle is the cause of continuity. It is debatable whether it is legal to compute mean for binary variables (if you take them as truly categorical features). If to do PCA on those, it makes sense not to center the variables prior it.

    A related question about PCA.

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About the connection between FA on binary items and IRT models (1- and 2-PL), here are two articles that might be interesting: Takane & de Leeuw, On the relationship between item response theory and factor analysis of discretized variables, Psychometrika (1987) 52(3):393; and a more recent one, Kamata & Bauer, A Note on the Relation Between Factor Analytic and Item Response Theory Models, SEM (2008) 15:136. – chl Dec 9 '11 at 13:18
@chl Useful links. Many thanks. – ttnphns Dec 9 '11 at 13:24

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