Thresholds, DELTA, THETA
Thresholds, Path Analysis with Probit Regression7-03-14  4:56 pmBengt O. Muthen6
Message/Author
 Daniel Bontempo posted on Tuesday, September 27, 2005 - 5:50 pm
I am running a multi-group model with thresholds constrained and loadings free across groups, scale/residual free in non-reference group, except for one item fixed @1, finally factor var @1 in all groups and mean @0 in ref group.

There is something about differences in the standardized values when DELTA vs THETA is uses that I don't understand.

I can see why loading estimates and corresponding STD values are the same when identification fixes the factor mean to 1. I also understand why these estimates (and STD values) differ with DELTA and THETA parameterization - because the y* variances differ. Under DELTA, the y* are 1 so the STD and the STDXY both match the estimate. Also, the STDXY values are the same under DELTA and THETA because Y* differences are taken out when the standardization includes the y* as well as the latent factor variance.

However, thresholds have the opposite pattern. The estimate and STD value are the same under both parameterizations (with factor var @1 and mean@0) BUT the STDXY is the same under THETA - not DELTA, and the STDXY threshold values are not the same under DELTA and THETA. There are also small differences in the factor means, which do have the same value for estimate STD and STDXY. Unlike the loading estimates, the STDXY do not match across DELTA and THETA param.

Why does the pattern of STD and STDXY matching reverse from DELTA to THETA?

Why is there no apparent correspondance of the means and standardized thresholds under DELTA and THETA? Is it because the means and thresholds are identified/scaled under DELTA by the one constrained scale but by the one constrained residual under THETA?

DELTA (Group 2)

E BY
EPQA03_1 0.712 0.032 22.198 0.712 0.712
EPQA07_1 0.597 0.070 8.585 0.597 0.658
EPQA12_1 0.493 0.063 7.833 0.493 0.521

Means
E -0.037 0.088 -0.415 -0.037 -0.037

Thresholds
EPQA03_1\$1 -0.195 0.046 -4.260 -0.195 -0.195
EPQA07_1\$1 -0.727 0.050 -14.562 -0.727 -0.727
EPQA12_1\$1 -1.150 0.058 -19.824 -1.150 -1.150

THETA (Group 2)

E BY
EPQA03_1 1.016 0.093 10.927 1.016 0.713
EPQA07_1 0.854 0.109 7.827 0.854 0.658
EPQA12_1 0.644 0.088 7.301 0.644 0.521

Means
E -0.021 0.090 -0.234 -0.021 -0.021

Thresholds
EPQA03_1\$1 -0.267 0.067 -3.976 -0.267 -0.267
EPQA07_1\$1 -1.027 0.085 -12.029 -1.027 -1.027
EPQA12_1\$1 -1.492 0.102 -14.569 -1.492 -1.492
 Daniel Bontempo posted on Tuesday, September 27, 2005 - 6:01 pm
Correction: I said:

> Why does the pattern of STD and STDXY matching reverse from DELTA to THETA?

but it does not reverse so much as all 3 match in both DELTA and THETA.

Why are the standardized values for thresholds the same as the unstandardized?, and Why do DELTA and THETA get different values given my identification.

I still think it is the single fixed scale vs the single fixed residual, but I can't do the math.
 bmuthen posted on Wednesday, September 28, 2005 - 10:28 am
The thresholds values are the same in the raw, STD, and STDYX columns because they are not standardized. This is in line with the Version 2 Tech Appendix 3 on the Mplus web site. We haven't found a real use for standardization here, but it would be perfectly ok to standardize the thresholds. This would then be done in line with the intercept standardization shown in equation (81), that is using the SD of y*. Note that for the intercept/threshold the SD of the factor is not involved - just like in regular regression. If you do this standardization, which you can do by hand (and we will add in V4), you should find that the standardized thresholds are the same for Delta and Theta parameterizations.

The factor means in the raw column are already in standardized form since the factor variance is 1. When loadings are held invariant across groups I would expect these standardized factor means to come out the same with Delta and Theta parameterization. But with your model allowing for loading differences across groups this parameterization invariance does not follow.
 Jiebing Wang posted on Wednesday, January 13, 2016 - 11:57 am
Hello Dr. Muthen,

I did a CFA with a 22 item scale (binary items) using WLSMV estimation, and compared the one-factor model with the four-factor model. Why the unstandardized and standardized thresholds of the items are the same in one-factor model and four-factor model? Thank you!

Jiebing
 Bengt O. Muthen posted on Wednesday, January 13, 2016 - 12:41 pm
That's a feature of WLSMV having a diagonal weight matrix - the technical aspects are given in my 1978 Psychometrika article.
 Jiebing Wang posted on Wednesday, January 13, 2016 - 1:46 pm
Many thanks!