Why does Polychoric Reduce to two Factors?

[WWGD]

Hi All,
Say we have our ordered non-continuous variables to perform Polychoric Analysis ( finding the
Polychoric correlation coefficient)
According to the theory, we will find this way, the standard Person correlation coefficient
between the underlying continuous latent variables ( Non-continuous ordered variables are
observations of these continuous latent variables; e.g., symptoms of depression, anxiety, etc.).
There is a result whereby there are just two underlying latent variables. I am just not clear
on what this means that there are just two underlying latent variables; before of, or while performing
the Polychoric analysis, we get a correlation matrix. From the correlation matrix we can do standard Factor
Analysis. But, what does it mean we just get two factors? Do we mean the correlation matrix will just have
two eigenvalues larger than 1? Or will all variables load along just two main factors?
Thanks.

 

[jim mcnamara]

I'm not understanding. I think it sounds like a binomial distribution, and I reserve the right to be wrong here. So correct me please. It seems binary to me.

If what I said is correct (from wikpedia)-

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).

https://it.unt.edu/sites/default/files/binaryfa_l_jds_sep2014.pdf

 

[WWGD]

I'm not understanding. I think it sounds like a binomial distribution, and I reserve the right to be wrong here. So correct me please. It seems binary to me.

If what I said is correct (from wikpedia)-

https://it.unt.edu/sites/default/files/binaryfa_l_jds_sep2014.pdf

My apologies Jim, I was mistaken about this, I have been told by people more knowledgeable on the topic of Polychorics. EDIT: I mean, as you said, the matrix is defined on ordinal ( basically any non-continuous variable that is not categorical) variables, it will produce a standard Pearson correlation matrix $C_{ij}$ where $c_{ij}$ is the correlation between continuous underlying variables i and j. Once we have a Pearson correlation matrix, we can do standard Factor Analysis and then we may get any number of (latent, continuous ) factors ( we will have Real eigenvalues since the correlation matrix is symmetric: $c_{ij}=c_{ji}$ ) and then we may choose on how many we select , depending on the loadings and/or whether the eigenvalues are greater than 1 ( in abs. value ). But I misunderstood that there had to be two. Thanks for the answer.