Specifities
Message/Author
 Florian Fiedler posted on Wednesday, December 07, 2005 - 7:48 am
Dear MPlus-team,

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?

Best regards, Florian Fiedler.
 Linda K. Muthen posted on Wednesday, December 07, 2005 - 8:24 am
By specifities do you mean residual variances?
 Florian Fiedler posted on Thursday, December 08, 2005 - 3:18 am
Yes. In my model I only obtain residual variances of the items on within level which can be interpreted as total specifities but I can't seperate them into two levels.

Or would you suggest just to compare models to argue in favour or against a certain between level factor structure?
 bmuthen posted on Thursday, December 08, 2005 - 6:35 am
You can get a between-level residual variance by mentioning this parameter on between, say

%between%
yb;

Often, however, these are close to zero and can sometimes be hard to estimate.
 Florian Fiedler posted on Monday, December 19, 2005 - 6:21 am
Actually I don't obtain any residual variances but I can compute the total specifities via 1-R² for each item ...

I'm not sure if I've understand it right.

When mentioning a variable (factor indicator) seperately on the between-level I'll obtain a R²-value as well for the between-level? So in any case I'll obtain "standardized" residual variances (1-R²)?

I tried to compute the "estimated" residual variances via:

Total item variance: Vt=(L11²*F11+L12²F12)/R²

Residual item variance: Vr=L11²*F11/Vt

Is this the right way to go?
 bmuthen posted on Monday, December 19, 2005 - 9:28 am
You should get residual variances in the output if you ask for standardized in the Output command. If not, please send your output to support@statmodel.com
 Florian Fiedler posted on Monday, December 19, 2005 - 10:07 am
Where in the output should they be found (section)?

I specified the model in the following way.

MODEL:
%WITHIN%

SK_w BY S40@1 S41*1;
SK2_w BY S42@1 S43*1 S44*.9 S45*.9;
SK_w WITH SK2_w*.4;

%BETWEEN%

SK BY S40@1 S41*1;
SK2 BY S42@1 S43*1 S44*.9 S45*.9;
SK WITH SK2*.4;
S40@0; S41@0; S42@0; S43@0; S44@0; S45@0;

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.
 Florian Fiedler posted on Monday, December 19, 2005 - 11:36 am
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.
 Florian Fiedler posted on Monday, December 19, 2005 - 12:07 pm
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)?
 Florian Fiedler posted on Monday, December 19, 2005 - 1:11 pm
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
That's right.