how can I obtain specifities on the two levels (between and within) when doing a multilevel cfa with ordinal observed variables? Do I have to change the parameterization or is it just a question of freeing parameters? If I've to change the parameterization, is there a short explanation of the differences between theta and delta parameterization?
MODEL RESULTS section: WITHIN - estimates for BY and WITH statements, S.E., Std StdYX - variances for latent factors BETWEEN - estimates for BY and WITH statements, S.E., Std StdYX - variances for latent factors - thresholds - residual variances (all fixed to 0)
R-SQUARE: - all within R² - all between R² (all fixed to 1)
RESIDUAL OUTPUT section: - univariate distr. fit - bivariate distr. fit
Then the technical outputs follow. That's all.
bmuthen posted on Monday, December 19, 2005 - 11:24 am
Ah yes, you are doing 2-level factor analysis of categorical items, which means that the WLSMV estimator is not available and you are using ML estimation which uses a logistic link (as opposed to the probit link of the WLSMV estimator). Note that with ML and logistic link, the Theta/Delta parameterization distinction is not made. In your ML run, the within-level residual variances are those of the logistic density, namely pi-square divided by 3 (see text books on logistic regression), where pi = 3.14. Because this is the value they always have, we don't print them, but this is the value used in the within-level R-square computations. R-square = 1 on between because the between-level residual variances are zero by default.
But if the residual variance is (pi^2)/3 for all items, the R² should not differ ...
So does it mean, the TOTAL variance of an item equals (pi^2)/3 and the explained variance is R²*(pi^2)/3?
As in a statement above, I'm interested in the computation of communalities. Via (explained by factor)/(total variance) I'd like to show the contribution of a within factor compared to a between factor.
Hope it's getting clearer now ;-)
bmuthen posted on Monday, December 19, 2005 - 11:47 am
R-square varies across items as a function of the loadings varying across the items. Compare with regular regression - the slopes influence the R-square.
Well, for the computation of ICCs I already used pi²/3 as total item variance, since it's the only way to obtain ICCs with ordinal variables. But there I thought it was the TOTAL variance. So when building say a factor model, the unexplained variance (which I would call residual variance then) should decrease - or is the total variance increasing by lambda*(factor variance)?
I think, I got it. To get the total variance of the variable, I add up the factor contributions plus pi²/3, so the residual is always the same, but it's proportion in the total item variance differs and will increase with larger contributions of factors.
bmuthen posted on Monday, December 19, 2005 - 6:39 pm