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We are testing two CFA models with continuous latent factors that are represented by 11 binary indicators. The first model tested a one factor model for the 11 observed variables, the second model tested 2 factors for the 11 observed variables. We obtained the following results: 1 factor chisquare= 28.3 , df=19, p=.0787. 2 factor chisquare= 28.8 , df=19, p=.0686. (n=195). We are wondering why the degrees of freedom did not change for the two factor model since new parameters were being estimated. In other CFA's using the same items but with a different sample, we saw a decrease of 1 df when the two factor model. 


You are using the WLSMV estimator. The degrees of freedom are not calculated in the regular way. The formula can be found on page 358 of the Mplus User's Guide. If you use WLS or WLSM, you will see the degrees of freedom that you expect. 

twan posted on Monday, September 17, 2001  8:27 am



How do you compute the degrees of freedom for the twolevel CFA model with unbalanced cases? 

bmuthen posted on Monday, September 17, 2001  9:25 am



You subtract the number of parameters in your model (H0) from those of the unrestricted model (H1), where H1 considers a covariance matrix on both the between and within level, and a mean structure when this is involved. Same for balanced and unbalanced data. 

Anonymous posted on Friday, March 19, 2004  12:09 pm



I conducted CFAs using all categorical (dichotomous and ordered categorical) data using the default estimator (which I believe is robust weighted least squares). This worked well but I see that my df are not what I expected and that the chisq cannot be easily used in a deltachisq test. One colleague (and your above comment) suggest that using WLS as estimator might help. Is WLS appropriate for my data and, if so, why is robust weighted least squares the default (i.e., does it have an advantage)? If I use WLS, can the df and the chisq I get be used in a deltachisq test (i.e., will they be the df and chisq that is "expected" or normally distributed) that does not require Appendix 5 of the manual? 

bmuthen posted on Friday, March 19, 2004  2:41 pm



Experience has shown that WLS does not work well unless the model has few variables (say less than 10) and you have a very large sample size (say several thousand). WLSMV works much better. WLS can be used for chisquare difference testing in the usual way. Version 3 of Mplus offers a way to do chisquare difference testing using WLSMV. 

Anonymous posted on Friday, March 19, 2004  4:59 pm



This is a followup to the above question and to L.Muthen's response above. I see that df for CFA are not calculated in the "regular way" when one uses WLSMV. I know that the formula for this is in the manual. Can you help me understand, in words, what is going on with the df in CFA with WLSMV? I am submitting a paper to a journal and there is some question about why my df are not what the Reviewer expected. I am looking for a way to eplain this in a footnote and in my R&R letter to the Editor. 


The chisquare is adjusted so that the pvalue for the chisquare distribution is correct. The degrees of freedom follow from that. With WLSMV chisquare what is of interest in the pvalue. 


I've just started using Mplus. I did a CFA (4 factors) on a set of test scores. The test scores were scored dichotomously and also polytomously (0/1/2). When model fit was examined the Chi Square df for the baseline model was 101 for the dichotomously scored data and 96 for the 0/1/2 data. Why is the df different even though the model specification is the same for the two scoring methods? 

Thuy Nguyen posted on Thursday, April 07, 2005  12:40 pm



For such a model, the WLSMV estimator is the default. The degrees of freedom for WLSMV are not computed in the usual way. You should see a note next to the degrees of freedom in the output that states ** The degrees of freedom for MLMV and WLSMV are estimated according to a formula given in the Mplus Technical Appendices at www.statmodel.com. See degrees of freedom in the index. This applies to both the regular model and the baseline model. 

Anonymous posted on Wednesday, April 20, 2005  1:49 pm



I am using a WLSMV estimator to test a simple one factor model with 8 indicators on a very small sample (85 obs), but am not getting a chisquare in my output. I am only getting a Weighted Root Mean Square Residual for a test statistic. Is there a problem with my sample size that is causing this? How can I get the other fit statistics? 


You should send the full output and your license number to support@statmodel.com. I do not know why you are not getting a chisquare from the information you are providing. 

Anonymous posted on Thursday, September 22, 2005  9:53 am



How do I get the delta parameter value from the EFA? Thanks in advance. 


If by the delta parameter you mean the scale factor that is a parameter in weighted least squares estimation for categorical outcomes, it is not a free parameter in single group analysis. It is fixed to one. It is only a free parameter in multiple group analysis or growth modeling. 

HWard posted on Monday, December 19, 2005  9:48 am



I am currently running a justidentified model of DIET measured by continuous 'fruit', 'veg', and 'grains', and would like to gain some degrees of freedom in order to obtain fit statistics. 'Fruit' has been estimated freely. At the moment, the chisquare, RMSEA, and SMR statistics are all 0.0000. In the code below, i have fixed the variance of the latent variable to 1 VARIABLE: NAMES ARE VEG FRUIT GRAINS SEXNUM; USEVARIABLES ARE VEG FRUIT GRAINS; MISSING ARE ALL (999); MODEL: DIET by fruit* veg GRAINS; DIET@1.0; But I would also like to fix the residual path coeffiecients for the error associated with each observed variable to 1  can anyone suggest to do this? Thank you! 


fruit@1; for example fixed the residual variance of fruit to one. 

HWard posted on Monday, December 19, 2005  1:56 pm



that's what I had thought  however, in doing so, it appears that the parameter estimates (not the resiudal variances) have been set to one. MODEL RESULTS Estimates S.E. Est./S.E. DIET BY FRUIT 1.000 0.000 0.000 VEG 1.000 0.000 0.000 GRAINS 1.000 0.000 0.000 Variances DIET 1.000 0.000 0.000 Residual Variances VEG 0.437 0.032 13.552 FRUIT 0.868 0.048 18.099 GRAINS 0.663 0.040 16.560 


Then you must have fixed the values in the BY statement and not fixed the residual variances, for example, MODEL: diet BY fruit* veg grains; diet@1; fruit@1 veg@1 grains@1; 


I am conducting a multigroup CFA on the CESD Scale using LISREL. To test the invariance of the factor structure across 4 groups, I am using the chisquare difference test. What I have noticed (this is expected) is that, whereas the critical value remains constant, the value of this test statistic increases with increasing sample size (across groups). At the same time I have also noticed that on testing, for instance, the equality of factor loadings across groups, the LISREL program sometimes yields contradictory output: The factor loadings are indicated to be equal across groups but the chisquare difference test is significant. My question is: How do I proceed when this happens? Should I rely on the chisquare difference test or on the other output indicating that the factor loading are equal? 


There is a description of testing measurement invariance in Chapter 13 of the Mplus User's Guide that is on the website. It comes after the section on multiple group modeling. You may find this helpful. If there is something you don't understand about your LISREL output, you should send the question to LISREL technical support. 


Prof Muthen, A reviewer on my manuscript is requesting information on how the df is computed in the WLSMV estimator for Mplus. In your previous reply to a posting from 2001, you stated that this formula can be found on p 358 of the user's guide. I am using version 4 and the accompanying user's guide has information about logistic regression parameterization on p 358, not on degrees of freedom estimation for WLSMV. Also, as instructed in the output, I looked under degrees of freedom in the index of the user's guide, but degrees of freedom is not listed as a topic in the index. Could you please advise as to where specifically this formula can be found? 


See the Technical appendices posted on our web site. The topic is covered in the tech apps up until V2: Appendix 4, page 20, eqns 109, 110. 

Alex posted on Thursday, August 30, 2007  9:36 am



Greetings, Following up on the last question (a year ago), we a doing tests of invariance on a CFA model with categorical indicators (WLSMW) using the meanstructure option. We test the invariance of the model first according to gender (2 groups) and then (in a separate analysis) according to language (2 groups). In both analyses we use the same steps/constraints and the same model (always in two groups) but we do not obtain the same DFs at corresponding steps. The reviewer does not like it. I understand that the DF are not calculated in the usual way in WLSMW and looked at the appropriate section (4) of the technical appendix (with highly technical formulas). Would you care to suggest a simple (not technical) explanation to adress this criticism ? Thank you very much 


The chisquare and degrees of freedom are adjusted to obtain a correct pvalue with WLSMV. Only the pvalue should be interpreted. I would not report the degrees of freedom or chisquare from WLSMV. It is just confusing. If you want to report traditional degrees of freedom, run the model using WLS or WLSM. 

Emma posted on Wednesday, May 11, 2011  10:47 pm



I have a similar question to the one above, in that I have run two separate sets of tests of invariance using WLSMV. In one set, the df values increase, as I expected, as more parameters are held invariant. However, in the other set, the the df values decrease as more parameters are held invariant. I realise that df is not calculated in a straightforward way in WLSMV, but does this pattern of decreasing dfs indicate that I've made an error somewhere? Additionally, in the set of analyses in which the dfs decrease, (a) the chisquare values decrease and the fit indices improve as more parameters are held invariant, while (b) the DIFFTEST chisquare values for successive pairs of models are positive and significant (which I'm told reflects a loss of fit as constraints are added). While I appreciate that the calculation of chisquare values is complicated, and that in version 5 they can't be interpreted directly, I was again wondering if this points to an error in my input files. Thanks. 


Prior to Version 6, the degrees of freedom and chisquare test statistic were adjusted to obtain a correct pvalue. Only the pvalue should be interpreted. For chisquare difference testing, the DIFFTEST option should be used. 


Hi, Within my simulation study, I would like to calculate bias and the relative chisquare (chisquare/df) within a measurement invariance framework using WLSMV. My first question is whether it is appropriate to compute and interpret these statistics. If yes, what is the easiest way to calculate bias for the overall chisquare and the chisquare difference test (DIFFTEST) within Mplus? My concern is that you say only the pvalue should be interpreted. Thanks, Tom 


With the current Mplus version, the WLSMV's chisquare is related to the usual degrees of freedom so that not only the pvalue is interpretable. But I would say that in a simulation study you want to focus on the reject proportion: With a correct model you want to get the 5% Type I error, and with a misspecified model you want to get high rejection rate. By the way, chisquare/df is closely related to RMSEA. 


Hi, I am trying to run a CFA and find 4 latent variables from 4 indicators. I constrained the factor loadings of 2 latent factors: F1 by x1 (a); F1 by x2 (b); F2 by y1 (a); F2 by y2 (b); F3 by x1 y1; F4 by x2 y2; I also constrained the correlations between the top two factors and the bottom two factors to be zero. I was noted that "THE DEGREES OF FREEDOM FOR THIS MODEL ARE NEGATIVE" and I assume this is due to the fact that i'm trying to find 4 latant factors from 4 indicators. Is there anything I can do to fix the problem? Thank you very much Dana 


With only 4 indicators you can only identify 4*3/2=6 parameters among the loadings, the factor variances, and the factor covariances. It is hard to get 4 factors out of this. You would have to impose more restrictions on your 4factor model, or reduce the number of factors. 

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