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 Xuan Huang posted on Saturday, May 12, 2007 - 8:20 pm
Dear Professors:
I am using Mplus for my dissertation projects. Now I need to do EFAs and CFAs. The variables are ordinal variables. These variables have non-normality problem. Some of them have floor and ceiling effects problems. I am wondering whether I can still treat them as continuous variables and use MLR to handle with the non-normality. Or I have to treat them as categorical variables. Could you introduce some references?
Also my professor suggests that I may treat the variables as censored and use MLR. What do you think abut the suggestion?
Thanks a lot.
May
 Linda K. Muthen posted on Sunday, May 13, 2007 - 9:21 am
You don't say how many ordered categories you have in your variables. But if it is 9 or less and you have floor and ceiling effects, I would treat the variables as categorical. Censoring is usually applied to continuous variables with a censoring at one or both ends of the continuum. See the following papers and references therein:

Muthén, B. & Kaplan D. (1985). A comparison of some methodologies for the factor analysis of non-normal Likert variables. British Journal of Mathematical and Statistical Psychology, 38, 171-189.

Muthén, B. & Kaplan D. (1992). A comparison of some methodologies for the factor analysis of non-normal Likert variables: A note on the size of the model. British Journal of Mathematical and Statistical Psychology, 45, 19-30.
 Xuan Huang posted on Monday, May 14, 2007 - 5:05 pm
Dear Professors:
Thank you so much for your prompt help. Could you help me with another question? I did a one –factor CFA analysis with eight ordinal variables. My variables have 7 ordered categories and these variables have strong ceiling effects. According to you recommendation, I treat the variables as categorical and use WLSMV estimator. My results look like this:
÷2(14) =93.036, CFI=.962, TLI=.986, RMSEA=.143, WRMR=.823.
I could not figure out why the RMSEA is so poor while CFI, TLI, and WRMR are good. Based on the results above, can I make a conclusion that the CFA model is acceptable? Where can I find reference on criterion of good model fit for categorical variables?Thank you very much.
 Linda K. Muthen posted on Tuesday, May 15, 2007 - 8:20 am
With categorical outcomes, I would look at CFI, TLI, and chi-square. I don't see your chi-square p-value. There is a dissertation on our website by Yu that looks at fit statistics for categorical outcomes.
 Alexandre Morin posted on Tuesday, February 23, 2010 - 8:21 am
Hi,
I am conducting MG invariance testing of an ESEM model with the MLR estimator (but similar results arise with CFA models) and the CFI increases (gets better) when I add the constraints of equal loadings. i.e. CFI increases subtantially while moving from configural invariance (.972) to loadings invariance (.985).
Aren't CFI supposed to be monotonic ? The other fit indices "behave" correctly altough those controling for parcimony also rise slightly, which is alright.

I am sure there are no errors in the inputs.
This appears to be due to the uses of the MLR estimator. Under ML, the CFIs are fine : .985 (config) versus .983 (loadings).

I did previously encounter similar issues using AMOS bootstrap chi square (Bollen-Stine).
 Tihomir Asparouhov posted on Tuesday, February 23, 2010 - 9:39 am
CFI is not monotonic. It has a penalty for the number of parameters in the model. If adding constraints does not affect the chi2 value then CFI will increase.
 Alexandre Morin posted on Tuesday, February 23, 2010 - 10:02 am
Hi Tihomir,
The chi square does change:

(1) configural invariance: chi square = 250.855 (df = 120, scaling = 0.861, logl = 25253.469, free parm = 258, logl scaling factor = 1.666) CFI = .972
(2) Loadings invariance:
chi square = 264.866 (df = 192, scaling = 1.124; logl = -25294.403, free parm = 186, logl scaling factor = 1.706), and CFI = .985
(3) scaled LRT test = 52.390 (df = 72, ns)
However, this only occurs with MLR.

With ML, the CFI and chi squares follow each other. In other applications, we only encountered this pattern with robust estimators.
(1) configural: chi2 = 215.902 (df = 120), cfi = .985
(2) loadings inv: chi2 = 297.770 (df = 192), CFI = .983
 Tihomir Asparouhov posted on Tuesday, February 23, 2010 - 10:48 am
If the chi2 change (14) is smaller than the DF change (72) the CFI will be bigger for the more restricted model. Robust MLR chi2 are generally smaller then ML chi2 so this is why you are seeing this more often for robust estimators but ML is not immune to this as well.

Apart from that you have the scaling of 0.861 (which is off course the cause of the difference between ML and MLR) and possibly the key you are looking for. I do not know how to interpret scaling smaller than 1. In simulated data the scaling is usually bigger then 1 and the more non-normal the data is the bigger the scaling. So I suspect that scaling smaller then 1 may be related to weakly identified model and may be we have to take that into account somehow, but I can not provide you with any specific information or references on this. Seems like a hard topic. You might want to try a simulation study and see what you get.
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