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Hi Linda, I would like to fit an SEM model to a set of ordered categorical variables that have L or J shaped distribution. That is, the assumption of the latent variable be normally distributed is not valid and there are some missing (up to 2 percent) values. 1. Can the Mplus handle it properly 2. Can it accommodate formative construct. Thank you in advance haihong 


Factors with categorical factor indicators are not necessarily nonnormal. 1. Yes. 2. Yes. 

hai hong li posted on Tuesday, August 01, 2006  2:03 pm



You have written that" Factors with categorical factor indicators are not necessarily nonnormal". So if I have five point Likert scales which are extremely skewed (L or J shape), for analysis of the data, you do not make the assumption that the underlying latent variable is normally distributed. That is, you do not substitute polychoric correlations instead of the covariance to analyze the data. Is this correct? Thanks haihong 


There are 2 things here. First, the fact that your Likert scale items are skewed does not mean that the factor must be skewed. Take the example of very extreme attitude items where most people disagree. This gives skewed items, but the factor may still be normal. The observed nonnormality may simply be due to extremeness of the item wording. Second, the default assumption in Mplus is that the factor is normal. It does not have to be normal, however, if you work with mixture modeling. As for your last sentence, you may be interested in the following statement from our short courses: Note that by assuming normal factors and using probit links, ML uses the same model as WLSMV. This is because normal factors and probit links result in multivariate normal u* variables. For model estimation, WLSMV uses the limited information of first and secondorder moments, thresholds and sample correlations of the multivariate normal u* variables (tetrachoric, polychoric, and polyserial correlations), whereas ML uses full information from all moments of the data. 

Xu, Man posted on Thursday, July 12, 2012  11:23 am



Could I please follow up on this lead? If one's latent variable is not normally distributed, regardless of continuous or ordinal items. For example, most psychiatric screening instruments aren't normally distributed and binary cutoff is often applied on sum scores. It is now quite often to apply CFA for these ordinal item level data (either as continuous under RML or as ordinal under WLSMV), and use these latent factor as predictor or outcome in SEM. I was wondering what the implication is for the results based on this kind of measurement model and is there a way to deal with this? Thanks! 


Not sure what your major concern is  perhaps it is that you don't believe your latent variable is normally distributed. You seem to take a nonnormal distribution of a psychiatric screening instrument as an argument that the corresponding latent variable is not normal, but per my discussion above, I don't think that necessarily follows. For a related, recent article, see Wall, M. M., Guo, J., & Amemiya, Y. (2012). Mixture factor analysis for approximating a nonnormally distributed continuous latent factor with continuous and dichotomous observed variables. Multivariate Behavioral Research, 47:2, 276313. where a nonnormal latent variable is obtained using the Mplus mixture approach. 

Xu, Man posted on Friday, July 13, 2012  2:17 am



yes, I am worried that the factor is not normally distributed. I checked the data I have (3 instruments). One of them has factor scores normally distributed (as you say). The other two are not. I declared all items to be categorical with the default estimator. On the item level, I found the instrument has normal distribution has more items that are normally distributed. The two that have got skewed factor score distributions have mostly highly skewed items. Does it mean that the latent variable approach is not suitable for the other two intruments. Thank you for the paper. I shall read it. 


Take a look at slide 117 of out Topic 2 handout and you can see why an observed score can be nonnormal at the same time as a latent score being normal. A latent variable can be normal and still give a nonnormal estimated factor score distribution. This is because of items that don't capture the tails of the factor distribution well, for instance too easy or too hard items. I would guess that the issue you are concerned with is likely of less importance than other aspects of your modeling. 


Transforming skewed data in ESEM Hello! In my ESEM model I try to estimate relationship between two variables: explanatory variable is positevely skewed (likert scale on preference) and dependent variable is negatively skewed (consumption data). I wonder if I need to use logtransfromation to normilze the data or it is enough to use WLSMV  estimator and skip transformations? Another reason for choosing WLSMVestimator is that I have some other categorical variables in the model. Thanks, Jazgul 


I would not transform variables unless that makes the linearity specification more realistic. The MLR estimator handles nonnormality, or if you don't want to assume continuous variables, then WLSMV takes care of it. 


Thanks! If I have both continous and dichtomous variables in the model, can I specify both estimators? /Jazgul 


You cannot use more than one estimator in an analysis. Both weighted least squares and maximum likelihood can be used for a model with a combination of continuous and dichotomous variables. 


Hello Linda, I was hoping you can help me. I am new to Mplus but I have read many discussions here and also the Mplus manual. However, I am failing to find a way how I can improve my CFA model fit (incremental stats used already). I am trying to fit 2 factor CFA ( 2 latent variables). I am using factor indicators that were calculated as the means of specific items measured on a likertscale. Thus the factor indicators are not normally distributed (histogram and normality tests support my assumption). On the top of the nonnormality,my data possesses quite a large number of missing data and imputation method nor transformation did not improve the normality. I have tried to use WLSMV as well as WLS and getting errors stating that I have no categorical variables present (I can not select the factor indicators as categorical as they are not integers). When I use MLM or MLR I get an error message that I have to use listwise deletion which is impossible due to the amount of missing values. I have used FIML so far but the model fit is poor when I evaluate all fit indices other than chisquare (TLI= 0.8). Have you got any suggestions? Thank you in advance for your help. Lucie 


Try an EFA to see if the two factors you are specifying are supported by the data. You could consider going back to the original items. 


If I have data that is not normally distributed  reaction time, so likely exGaussian, would it be appropriate to use Mplus 7.2's features for skew and kurtosis to model this? Or are there properties of the distribution that limit the feasibility of applying the new features to such a distribution? Is it possible to predict the amount of skew with an exogenous variable? If so, how? Thanks, Steve 


If you have a continuous DV that is nonnormal and without floor or ceiling effects, why don't you try out the new 7.2 features. If you have an exogenous variable, the nonnormal specification is on the residual of the DV. This means that the DV can be nonnormal due to both a nonnormal exogenous variable and a nonnormal residual. 

Daniel Lee posted on Monday, June 30, 2014  9:08 pm



Hi Dr. Muthen, Thank you for responding to my questions. I appreciate you, and your team, very much! While conducting factor analysis, I had a few questions in mind: 1) Items in my latent variable are scaled differently (ordinal & dichotomous). Could I use WLSMV in this scenario? If so, is there anything else I need to be aware of in terms of conducting my analyses the right way? 2) If I elect to use WLSMV estimation, do I need to conduct a test (Tech13?) for multivariate normality? I would guess no since WLSMV uses a probit function...but I just wanted to make sure! 3) Lastly, are there any diagnostics I should be aware of when conducting a CFA using WLSMV? Thank you so much Dr. Muthen!!! Best, Dan 

Daniel Lee posted on Monday, June 30, 2014  9:11 pm



Hi Dr. Muthen, I'm so sorry, I had one more question that I forgot to include in the previous post! 4) If I conduct a WLSMV, can I still use FIML to treat missingness in data? Thank you so much, Dan 


1. You can use WLSMV or ML in this situation. 2. There is no test for multivariate normality. 3. TECH10. 4. No. FIML is fullinformation maximum likelihood. With WLSMV, missing data are handled using pairwise present. If you want FIML, use the ML estimator. 

Daniel Lee posted on Tuesday, July 01, 2014  11:17 am



Hi Dr. Muthen, Thank you for the response! Just one question for clarification purposes: With regards to my first question, if you have categorical and dischotomous manifest variables, wouldn't ML estimation generate biased standard errors? I always thought (I don't remember where I saw this) WLSMV was the way to go when indicators were categorical. Therefore, I'm curious as to why ML might be appropriate in this situation? Thank you so much! Your responses are always so helpful! Dan 


You can treat variables as categorical with both WLSMV and ML. You would put them on the CATEGORICAL list in both cases. 


I have a U shape distribution for one of my latent variable as if many observation were 0 many were 1 and less were in betewen. What regression type should i use? 


I don't know of any regression for a ushaped variable. 

Julie Kim posted on Thursday, August 13, 2015  10:51 am



Hello Linda I appreciate all your posts on the forum. I am conducting CFA and SEM, but before I even start, I know I have to do multivariate normality checking. When I saw univariate items, it is not normal and therefore, the data is not multivariate normal. From your posts so far, it seems like WLSMV or TECH13 take care of (?) these issue? Am I understanding correctly? In other words, if I use WLSMV or TECH13, I do not need to do anything about multivariate nonnormal data? (e.g., transformation?) Thank you so much 


Are you asking about categorical variables or continuous variables. 

Julie Kim posted on Thursday, August 13, 2015  4:32 pm



Linda, my data has both categorical (e.g., gender) and continuous (many likert) scale. But for SEM part, they are all continuous, I beleive. 


The scale of covariates like gender is not an issue in regression. Covariates can be binary or continuous and in both cases they are treated as continuous in regression. If you have likert items and they have floor or ceiling effects, a piling up of observations in the lowest or highest categories, you should treat them as categorical. It sounds like that is the case. If you treat them as categorical by putting them on the CATEGORICAL list and using either WLSMV or ML, the categorical methodology of probit or logit regression takes care of this. 

Julie Kim posted on Monday, August 17, 2015  7:10 pm



Linda ,I appreciate your answers very much. Excuse me for asking so basic questions. 1. It is my understanding so far as I analyize my data (that have categorical such as yes no question, continuous such as percentage, and many likert scale). From what you described, when I have univariate nonnormal in any kind of data, you can put "categorical" (because it's likely because of floor or ceiling effects) and use either WLSMV or ML to take care of multivariate normality. Is this true? Am I not understanding correctly? In other words, is it true I do not need to transform anything, if I use categorical and WLSMV or ML? 2. Does WLSMV/ML option take care of homoscedasticity as well? Thank you 


1. If you have a variable measured on a continuous scale like height or weight and the variable is not normally distributed, you can use an estimator that is robust to nonnormality like MLR. It is not necessary to transform the variable. You cannot put it on the CATEGORICAL list. If you have a binary or Likerttype variable, the numbers, 0/1, or 0/1/2/3/4 have no numeric meaning. They simply denote categories. Only this type of variable can be put on the CATEGORICAL list. When it is put on the CATEGORICAL list, categorical variable methodology is used. This methodology is developed to handle nonnormal distributions of frequencies across categories. 2. Not to my knowledge. 

Julie Kim posted on Tuesday, August 18, 2015  7:31 am



Linda, thank you so much. Please allow me to ask followup question. After more thought, I realize one of my latent has 3 indicators 1. yes no (0,1) categorical 2. % of women in chosen job 3. a composite score (likely score from 1060). In this case, I have mixture of continuous (2,3) and categorical (1)In this case, do you have any recommendation? I believe I can no longer use CATEGORICAL.. Thank you 


You put the categorical variables on the CATEGORICAL list. For the others, the default is to treat them as continuous. 


Hi, I am running CFAs for the scales I used in my study. One of the scales (continuous variable, 6items, 1 factor) is non normally distributed (skewness 3.2 and kurtosis 11.7). I have used MLR estimator and it still provides me with poor model fit (chi square/df ratio and RMSEA are high). How would you suggest to go about this issue? I was thinking of transforming the data, but not sure if that would be a solution. Thank you 


Maybe you have strong floor or ceiling effects. 


Thank you do much! I think it might be a strong ceiling effect. What approach would you suggest in order to run the CFA with the strong ceiling effect of one variable? Thank you! 


Censored using WLSMV is one possibility. But first I would do an EFA. 


Hello, Dr. Muthen, I have some questions about data¡¯s normality testing. Thank you in advance. In my study, I want to at first use the multiple group ESEM to test the measurement invariance and the structure invariance of 3 student groups, find out the correlations across groups, then use the latent profile analysis to explore the data¡¯s structure and take the results as references to the results of multiple group ESEM. And at this time, I come up with some questions. Question1: When should I test the observed variable¡¯s normality (9 observed variables)? I think I should undertake this step in EFA part of multiple group ESEM because the default estimator is ML in Mplus. Question2: How can I test the normality of the 9 observed variables? Is there any introduction about this? The only thing I can find is the normality testing when dealing with mixture model in CFA. Any response will be really appreciated! Wen Congcong 


Here is my program. Is it correct? TITLE: Testing nonnormality; DATA: FILE="C:/Users/dell/Documents/data.csv" LISTWISE=ON; VARIABLE: NAMES ARE cate y1y9; USEVARIABLES ARE y1y9; OUTPUT:SAMPSTAT TECH12; 


There is no need to test for normality. Just use the MLR estimator and if the SEs and chi2 are substantially different from those of ML you know that nonnormality is an issue and you should use MLR. 


Thank you very much! 


Dear Professors, I've specified an SEM with manifest as well as latent variables. As some of the indicators and manifest variables are nonnormal, I would like to calculate robust fit indices. If I understand it correctly, the MLM Estimator is based on the SatorraBentler correction, which corrects for kurtosis, but not for skewness. It that the case? Could I use a different estimator which is also (more or less) robust to skewness? Best wishes, Chris 


We recommend MLR for all kinds of nonnormality. 


Many thanks for your prompt answer. If I may ask about MLR: According to the Manual, MLR estimates are robust when used with type=complex. When I insert type=complex, however, I get the message that this command "requires a cluster variable, a stratification variable or replicate weights." Is there a way around this? 


Maybe the key is the correct interpretation of the following sentence in the Mplus User's Guide: "MLR – maximum likelihood parameter estimates with standard errors and a chisquare test statistic (when applicable) that are robust to nonnormality and nonindependence of observations when used with TYPE=COMPLEX." Does this sentence mean "MLR estimates and the chisquare test statistic are are ONLY robust to nonnormality IF used in combination with type=complex" or "MLR estimates and the chisquare test statistic are always robust to nonnormality, AND ALSO robust to NONINDEPENDENCE of observations if used with type=complex" (or something else)? 


Answer to your 3:06 post: MLR does not need Type=Complex or Cluster=. See the Estimator table in the User's Guide (look up index entry Estimator). 


MLR is robust to nonnormality. I would recommend reading web note 2 http://statmodel.com/examples/webnote.shtml#web2 and the references that are there. Here is another good reading https://www.stat.berkeley.edu/~census/mlesan.pdf 


Just one more line: if your variables are nonnormal but the model is correct, the ML estimator gives correct (consistent) point estimates (simply because the sample mean and sample variance estimates are consistent under nonnormality). The only problems that nonnormality causes is that the standard errors and chisquare testing are incorrect with ML, and this is where the HuberWhite(1980) sandwich standard errors come in (this is MLR in Mplus) and fix that problem. The HuberWhite(1980) method is now very well established everywhere and can fix other problems, see https://en.wikipedia.org/wiki/Heteroscedasticityconsistent_standard_errors or this http://www.statmodel2.com/download/webnotes/mplusnote72.pdf 


Thanks a lot for your efforts. 


Hi, When assessing the distribution of nonnormal data, I obtained the following AIC and BIC values (using a four time point growth curve model). Zero Inflated Poisson: AIC BIC 9247.5369294.937 Zero Inflated Negative Binomial: AIC BIC 9244.6439313.112 As you can see, the AIC and BIC values are lower in different models. What should I assume to be the best fitting distribution? Thanks! Hillary 


This is a case where statistics doesn't provide clear guidance (although the BIC advantage of the ZIP model is bigger than the AIC advantage of the ZINB model). 


Hello Dr. Muthen, Thank you for your response! Ok, so despite the lack of clarity, I should use the ZIP model? Hillary 


I would. But it is up to you. To make choice, you can also plot the fit of the model to the data as we show in our Short Course and also in our RMA book. 


Great, thank you for these resources. Hillary 

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