Is there a principled way to estimate factor scores when you have ordinal, discrete variables.
I have $n$ ordinal, discrete, variables. If I make the assumption that underlying each response is a continuous, normally distributed variable, then I can calculate an $n\times n$ polychoric correlation matrix. I can then run a factor analysis on this matrix and get factor loadings for each variable.
How can I combine the factor loadings and the variables to estimate the factor scores. The typical ways to estimate scores would appear to require that I treat the ordinal data as interval.
I suppose I might need to dig deeper into the guts of polychoric correlation to figure out a link function.