Conditional Normality - Categorical V... PreviousNext
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 Chris M Manuel posted on Friday, October 11, 2013 - 2:58 pm
Based on my understanding of the Technical Appendices and both 1984 and 1997 Muthen papers, conditional normality is a requirement in order to calculate the conditional mean and variance of the latent variable (y*) when we have non normal covariates (x).

My question: What happens when we don't have (x) variables in our model? For example, what happens when we conduct a EFA or a CFA, in which we only have the observed variables y? Are the observed variables (y) treated as the covariates (x)? Or does the WLSMV approach correct for not having these (x) covariates?
 Bengt O. Muthen posted on Friday, October 11, 2013 - 5:46 pm
When you don't have covariates, the normality assumption is for the y* variables. This is Case A in Muthen 1984.
 Chris M Manuel posted on Tuesday, November 19, 2013 - 9:34 pm
Thank you for your response. I have some follow up questions:

1. In a no covariate setting, how are the unconditional probit probabilities calculated? (equations 52 and 53 in the tech appendix) Do we just integrate over the range of the threshold to infinity?

2. Why do we need to calculate the bivariate probability on page 10 (tech appendix)? For a categorical variable with 4 outcomes, would there be 5 probability expressions?

3. What is the unconditional expectation of y*? Is it just the delta*pi_0? (equations 50) What does pi_0 represent (defined as the premultiplied vector on page 15)

4. How does standardization (in terms of parameters, page 19) work in terms of EFA/CFA?
 Bengt O. Muthen posted on Friday, November 22, 2013 - 12:41 am
1. Uncond'd probabilities just use the normal distribution function. So, yes, (52) and (53).

2. WLSMV works with bivariate information in order to get estimated latent correlations. With a single categorical variable with 4 categories there would be 4 probabilities.

3. Yes delta*pi_0. With x's, pi_0 is the intercept in the probit regression of y on x, but without x's pi_0=0 since we already have threshold parameters. You may get more out of reading Web Note 4:

http://www.statmodel.com/download/webnotes/CatMGLong.pdf

4. EFA doesn't need standardization because a correlation matrix is analyzed and the factors have variances 1. CFA follows pages 15-16.
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