Message/Author 

Andy Cohen posted on Thursday, July 12, 2007  8:40 am



I am conducting a 2 level analysis in which I would like to include interactions between two main effect variables in the within portion of my analysis. I am defining the variables using the DEFINE command (e.g. IntA_B = A * B). The underlying variables for the interaction terms need to be group mean centered. I have already specified group mean centering for these variables in the VARIABLE command (as that is necessary for the use of the TWOLEVEL option in the ANALYSIS command, but am wondering if the DEFINE command will use the original or centered form of the variables. Thanks. 


The transformations in the DEFINE command are done before the centering. 


Dear Dr. Muthen, I wonder, whether  as a consequence of the above  the newly defined interaction terms should be listed under the CENTERING command? Does the order of operations (first computing the interaction terms, then centering) affect the results? 


Yes, the order of operations matters. Any transformations using DEFINE should be done first and the data saved. The centering should be done on the saved data. 

C. Lechner posted on Friday, February 17, 2012  6:38 am



Ok, thank you very much for your answer! May I ask two additional questions: #1: I suppose the same would apply to interactions between a latent variable and a manifest variable computed using the XWITH command? I would first compute the interaction, save it, and then center it along with the other variables in the model? #2: Assume I have a multilevel model with two predictors and an interaction between the two on level 1. One of the two predictors that interact have a random effect, the other is treated as a fixed effect. The interaction thus has to be treated as a random effect as well. However, do BOTH predictors that are part of the interaction have to be treated as random, or will it suffice to treat one as random and the second one as fixed (as I would assume)? Technically, both works fine, because in a regression or path model, Mplus will treat these interactions as any other variables. But is it correct? 


1. Using XWITH, you would center the manifest variable in Define say, but do that in the same onestep analysis. The latent variable has mean zero in most models and need not be altered. 2. I don't see that the choice of fixed or random for each of the variables has any implication for their interaction. 


Just so that I'm clear, it seems like there is no way use the CENTER command to group mean center a set of variables and then use them in a DEFINE statement in the same procedure. For example: variable: names = AgencyID Gender Age T employ enroll engage housegb incany totsup infsup formsup anysup anyinf anyform; cluster = AgencyID; missing are all .; usevar = Gender Age T incany formsup Intx; categorical are incany; within = T formsup Intx; between = Age Gender; center = grand mean (Age) group mean (T formsup); Define: Intx = T*formsup; This would compute the Intx variable before group mean centering T and formsup and this isn't what I want. Is there any way around this? The manual states that the CLUSTER_MEAN option also cannot be used with subsequent DEFINE statements. So I guess that leaves me with using the SAVEDATA command to save the group means, then running another procedure using those saved variables to compute the group mean centered values. Save that data for a final time, and run a third procedure calculating the interaction term with the saved, group mean centered values. Is that correct? Or did I add in an extra step somewhere. Thanks for all your help! 


It sounds like you are correct. 


Hi, I am running a twolevel model to test group differences before and after an intervention. I'm entering my own time variable to represent the number of days since baseline (see sample script below). I noticed that Mplus is automatically centering my time variable. Is there a way to not center it? I would like the baseline (T1) to = 0, as this is more meaningful. Thanks! DATA WIDETOLONG: WIDE = DV_T1 DV_T2 DV_T3  T1 T2 T3 ; LONG = DV  timeB ; IDVARIABLE = person ; REPETITION = time ; Variable: Names are ID group DV_T1 DV_T2 DV_T3 T1 T2 T3 ; Usevariables are group DV timeB person ; Cluster = person ; Within time timeB; Between = group ; Missing are .; ANALYSIS: TYPE = TWOLEVEL RANDOM; MODEL: %WITHIN% s  DV on timeB ; %BETWEEN% DV s on group ; DV with s ; 


The REPETITION option assigns to the variable time consecutive numbers starting with zero. 

Melvin C Y posted on Friday, January 25, 2013  4:44 am



In my two level sem model, I specified the following (there''re more variables but I''ll keep it simple)  Within is x1 x2; !very low ICC. % within% WL1 by x1 x2 x3; Dep on WL1; % between% BL2 by x3; X3@0; Dep on BL2;  Fit statistics are fine except for L2 srmr (above1.0). When I add groupmean centering to x1 and x2 in a subsequent run, the L2 srmr improved substantially. What could be the reason for this? Should I center? I placed x1 and x2 at within as their ICCs were very low. Thanks.. 


I would feel more comfortable with other fit indices for twolevel modeling. Stay with chisquare, RMSEA, and CFI. 

Kirill Fayn posted on Thursday, May 30, 2013  12:45 am



Hello, i am trying to run my first MLM on mplus and am having difficulty centring my level one variables. The model and the error is below: USEVARIABLES ARE Interest Cope1 Nov1 ZOpen ZInt; WITHIN = Cope1 Nov1; BETWEEN = ZOpen ZInt; MISSING ARE all (9999); CLUSTER = subject; DEFINE: CENTER Cope1 Nov1(GROUPMEAN); ANALYSIS: TYPE = TWOLEVEL RANDOM; MODEL: %WITHIN% IntCop  Interest ON Cope1; !need to make these factors IntNov  Interest ON Nov1; %BETWEEN% Interest IntCop IntNov ON ZOpen ZInt OUTPUT: TECH8 SAMPSTAT; *** ERROR in DEFINE command Error in assignment statement for CENTER Could you please help. The syntax seems to be right so I am guessing I can't centre these variables for some reason. Thanks in advance for your time. Best regards, Kirill 


What version of Mplus are you using? If it is earlier than Version 7, the CENTERING option was in the VARIABLE command. If it is Version 7 or later, please send the output and your license number to support@statmodel.com. 

Katerina Gk posted on Wednesday, October 09, 2013  4:42 am



Hi, I have 5factor model(job sat.) and selfeff.( 3factor model).I want to aggregate by school the observed variables of the job sat. and selfeffi. in the between level. If I use CENTERING = GRANDMEAN (x) is enough to understand that I need to aggregate at between level?the observed variable are the same in two levels.... Missing are all (999); CLUSTER IS sxoleio; DEFINE: CENTER = e1..a1..(GRANDMEAN) ANALYSIS: TYPE IS TWOLEVEL ; ESTIMATOR = WLSMV; MODEL: %within% er1_w by e1@1... ; er2_w by e7@1... ; a1_w by a3@1 ...; a2_w by a1@1 ...; er1_w ON a1_w; .... %between% er1_b by e1@1...; er2_b by e7@1...; a1_b by a3@1 ... ; a2_b by a1@1 ...; er1_b ON a1_b; .... OUTPUT: standardized; Thank you very much Katerina 


If you want an aggregated variable on the between level, use the CLUSTER_MEAN option of the DEFINE command to create it. See Example 9.1 where using this variable versus a latent variable decomposition of the individuallevel variable is discussed. 

Katerina Gk posted on Wednesday, October 09, 2013  11:55 am



Thank you for your help!! 


I am running the following random intercept multilevel model DEFINE: CENTER IS x (GRANDMEAN); ANALYSIS: TYPE IS TWOLEVEL; MODEL: %WITHIN% y ON x; %BETWEEN% y ON x; Is it correct that in such a situation, where x is also used at the between level, x is basically groupmean centered at the within level although I specified GRANDMEAN in the Define section? Is there a way to overrule this procedure and use grandmean centering at Level 1? Many thanks in advance for your time. Ute 


You can put x on the WITHIN list and then create a clusterlevel variable for x on between using the CLUSTER_MEAN option of the DEFINE command. See Example 9.1. 


Hi, In relation to the issues of centering in multilevel models raised above, I have two questions: 1. If any transformations (e.g., interaction between observed variables) are made before centering, it means that the interaction term does not use standardized scores of the products, which violates a basic requirement for the computation of any interaction term. How can I bypass this problem? Or is it not a problem? 2. When using grandmean centering (or no centering) for my withinlevel predictors, the fit of the model is considerably higher than when using groupmean centering. What could be the reason? Is there a preferable centering method for withinlevel predictors? Thank you very much in advance! Best, Paris 


1. It is standard to center, not standardize, variables before creating an interaction between them. 2. See the Raudenbush and Bryk book. This is a complex topic. 

Zen Goh posted on Friday, August 01, 2014  7:16 am



I'm running a 111 modmed model, and have a generic question about centering. (1) Why do we only center the X but not the M(mediator), as we do using HLM software? In HLM software, X and M are specified as groupcentered predictors, so that only the withinlevel relationships are apparent. In Mplus, I'm only allowed to center X but not M. (see code and error message below) (2) how does the lack of centering the mediator in MPlus affects the results and interpretation? Thanks! 

Zen Goh posted on Friday, August 01, 2014  7:18 am



NAMES = clust Gender Age Child WLoad WFCts WFCemo LSat MgSup; USEVARIABLES ARE WLoad WFCts LSat Gender Age Child MgSup; MISSING ARE ALL (1); WITHIN = WLoad; BETWEEN = Gender Age Child MgSup; CLUSTER= clust; Define: CENTER MgSup (GRANDMEAN); CENTER WLoad (GROUPMEAN); ANALYSIS: TYPE = TWOLEVEL RANDOM; MODEL: *** WARNING in MODEL command Variable on the lefthand side of an ON statement in a  statement is a WITHIN variable. The intercept for this variable is not random. Variable: WFCTS *** ERROR in MODEL command Withinlevel variables cannot be used on the between level. Withinlevel variable used: WFCTS *** ERROR in MODEL command Withinlevel variables cannot be used on the between level. Withinlevel variable used: WFCTS *** ERROR in MODEL command Withinlevel variables cannot be used on the between level. Withinlevel variable used: WFCTS *** ERROR The following MODEL statements are ignored: * Statements in the BETWEEN level: WFCTS ON MGSUP WFCTS ON GENDER WFCTS ON AGE WFCTS WITH S1 LSAT WITH WFCTS 


Variables on the BETWEEN list cannot be used in the within part of the model. They are measured on the cluster level. This is not related to centering. Read Example 9.1. It goes over all of the multilevel options. Example 9.2 shows a random slope model with a crosslevel interaction. 

Zen Goh posted on Friday, August 01, 2014  12:53 pm



I think I might not have been clear in my question. My mediator (not moderator) variable is also a within, level1 variable  why do we not center this as well? Would this not affect the interpretation of the results? 


You can center mediators as well. The error message refers to something else. 


Hi, I am running a 111 path model (using manifest variables) with nonindependence in my DV (high ICC1; performance rated by common supervisor). In reading about centering (e.g., Enders & Tofighi, 2007), it is clear I need to groupmean center my predictors to prevent betweengroup variance biasing my results. However, I wonder why we don't groupmean center the DV to "purge" out betweengroup variance in the DV and only include withingroup variance? In books and in papers I generally find that the DV is not centered, but in the case of nonindependence, doesn't the DV then include a lot of "noise" and how exactly does Mplus deal with the DV then? E.g., is defining the DV at the WITHIN level an option? Are there other options to take care of this? Thank you very much for your advice, Tina 


You model betweengroup variance in multilevel modeling. There is no need for anything else. 


Dear Muthén, I appreciate the new define functions in version 7.2, where order of commands has significance. In a multilevel regression model where y(within) is regressed on x1(groupmeancentered (GMC)) and x2(GMC), should the interaction term for x1 and x2 (x1*x2) be computed before or after the GMC? That is, to first calculate x1*x2, and then GMC x1, x2, and x1*x2, OR calculate x1(GMC) and x2(GMC), and then calculate x1*x2 as x1(GMC)*x2(GMC). 1) y(within) on x1(GMC) x2(GMC) x1*x2(GMC) OR 2) y(within) on x1(GMC) x2(GMC) x1(GMC)*x2(GMC) (say that, for example, y is lung cancer, x1 is smoking, and x2 is working with asbestos). Hope you can help me with this conundrum, Eivind Ystrom 


You can get wider input on general centering matters from Multilevelnet. 

Angela posted on Friday, October 09, 2015  1:55 pm



Hello, I am running a model using type=complex and I would like to look at some interaction effects. I read on the discussion board that when running a twolevel model, centering should occur after creating the interaction variables in the define statement. However, I am not looking at effects across levels, so I was not sure if this applied to my model. At what point should I center my variables when looking at interaction effects in a multilevel model? Thanks! 


Type=Complex is not a 2level model but a singlelevel model. With singlelevel models it is not required to center variables, but if it is done it should be before creating the interaction. 

JLuk posted on Sunday, December 13, 2015  7:29 pm



CrossLevel Interaction in Multilevel ZIP Model 1. I am running a twolevel zip model, with drinking being the outcome. I'm interested in testing a crosslevel interaction. Can this be done in Mplus (both for the zeroinflation and count part)? 2. I adapted syntax from example 9.2, with the following key changes:  VARIABLE: a zip model is specified by "count is drink(i)"  MODEL: %Within% s  drink on x; si  drink#1 on x; %Between% drink s on w xm; drink with s; drink#1 si on w xm; drink#1 with si; Does this look right? 3. Is the plotting function for crosslevel interaction (second part of example 9.2) robust while using ZIP? Thanks a lot for your help! 


1. Yes, I believe so. 2. Looks right. 3. Yes, use the style of http://www.statmodel.com/download/Crosslevel%20effect%20plotted%20by%20LOOP.pdf 

JLuk posted on Tuesday, December 15, 2015  6:40 pm



Great! Thank you so much, Dr. Muthen! 

JLuk posted on Monday, December 21, 2015  12:41 pm



CrossLevel Interaction in Multilevel ZeroInflated Model 1. In running a crosslevel interaction, I compared using ZIP vs. ZINB models. It appears that the multilevel ZIP model ran, but the ZINB model did not. It gave the following error message: *** FATAL ERROR Internal Error Code: GH1006. An internal error has occurred. Please contact us about the error, providing both the input and data files if possible. I think using ZIP model would be justifiable given similar BIC. However, I'm just curious why the model did not run with ZINB. 2. In the syntax in the previous post:  MODEL: %Within% s  drink on x; si  drink#1 on x; %Between% drink s on w xm; drink with s; drink#1 si on w xm; drink#1 with si; If I would like to include other covariates in the withinlevel, do I simply specify: %Within% s  drink on x covar1 covar2; si  drink#1 on x covar1 covar2; 3. I'm using Mplus on a Mac and am realizing that the plot function may not work on Mac. Is that still the case with the latest version? If so, any alternative resources that you'd recommend to probe the interaction? Thank you! 


Please send files to Support along with your license number. The Mac plots are listed on our website. 


I have a question related to your Dec 9, 2011 2:22 post. I am running a multilevel regression model and getting different ICCs depending on which predictor variables are in my model (e.g., models with and without a moderator variable). In your previous post, you explained: "Usually the ICC changes with the covariates due to this misspecification where a covariate is on the within list but actually it is not a within level variable because it hasn't been centered. You can use this command to fix this misspecifcations: centering=GROUPMEAN(x); for all x variables that are on the within= list. The only x variables I have not been centering are a binary treatment variable (treatment = 1 vs control = 0) and my interaction term because both have a meaningful zero. If I center these variables, I believe I will no longer be able to easily interpret the intercept as the score for control participants with average (due to centering) scores on all other predictors. What is your recommendation in this situation? Which ICCs should I report and why? 


We have a FAQ explaining why ICCs change across models: "Icc changes from one model to another" One approach is to to use the ICCs from a Type = Twolevel Basic analysis. You may also want to ask this general analysis question on Multilevelnet. 


Hello, I recall reading that it has become possible to order define commands so that centering occurs before computation of interactions. However, I cannot seem to find this information in the user's guide or on the discussion board. Could you please speak to how I can do this, if possible? Thanks! 


See the DEFINE command in the current user's guide on the website. Page 578. 


I am conducting a multigroup analysis and wanted to center the age variable on the mean age for each group for more meaningful interpretation of the intercepts. However, my data is also complex survey data so I am using TYPE=Complex, but I know the default for CENTER (GROUPMEAN) is to use the cluster mean. Because I wanted to instead center age on the mean for the groups of the multigroup analysis (not the cluster mean), I used the following command DEFINE: CENTER age (GROUPMEAN urban) where urban is the grouping variable that denotes urban/rural groups. However, I got the following warnings: *** WARNING in DEFINE command Specification for the CENTER function with GROUPMEAN includes extra information that will be ignored. Extra information given below: GROUPMEAN URBAN *** ERROR Categorical variable DEP contains less than 2 categories in Group 0. Is it possible to specify a different groupmean variable than a clustering variable for complex data (i.e. in the case of a grouping variable for multigroup analysis)? It's not essential that I center age, but I just thought I would try it. 


Just say Define: age = age  x; where x is a value that you want to use to center age. 


Thanks that is straightforward. Can I do something like Can I add "if" statements to the command to condition it on something else? For instance Define: age= agex if <grouping>=0 age= agey if <grouping>=1 where grouping represents the variable that designates the different groups 


Yes. See the DEFINE command in the user's guide for the correct specification. 

Frank Egloff posted on Thursday, December 14, 2017  10:42 am



Dear Drs. Muthen, I have question regards what Mplus is doing when running this multilevel model: %within% iw sw  tv12Slow@0 tv22Slow@1 tv32Slow@2 tv42Slow@3 tv52Slow@4 tv62Slow@5 tv72Slow@6 tv82Slow@7; tv12Slow  tv82Slow (1); b  sw on iw; %between% ib sb qb tv12Slow@0 tv22Slow@1 tv32Slow@2 tv42Slow@3 tv52Slow@4 tv62Slow@5 tv72Slow@6 tv82Slow@7; tv12Slow  tv82Slow@0; ibxcos  ib xwith cos; sb on cos IB ibxcos; b on cos; Are “iw” and “sw” automatically centered at group mean? Thank you for your answer! Frank 


The means of iw and sw are zero. See for instance the output shown on on website ofr UG ex 9.12. 


Dear Dr. Muthen, thank you for your answer! I found in ex 9.12 that the mean of x (which is the level 1 predictor) is 0.021 (which is almost zero) which is the grand mean (not the group mean, which might possibly still differ). I actually got a request from a reviewer who stated (drawing on Enders & Tofighi, 2007) that level 1 predictors (as "iw" (latent variable) in my case) need to be centered at group mean to allow for an unbiased crosslevelinteraction parameter. I agree. Thus, in my case I am particularly interested if the group means of “iw” are zero. When I look at the factor scores of my model and calculate the “iw”group means by hand, they are almost zero (but not exactly zero: e.g. 0.023, 0.053, 0.074 – which are close to zero values considering an “iw”standard deviation of 4.87). 1. What is Mplus doing (is group mean centering applied to lower level latent predictor variables by default)? 2. Why are there these small deviations from zero in group means (assumed group mean centering was applied)? 3. Do you know a source to cite which states what Mplus is doing? Thank you so much again! 


1. E(iw)=E(sw)=0. This is how you specified the model. I would not characterize this as centering. It is simply how you specified the model. There is no hidden procedure here. 2. Factor scores are based on the data in each row. I wouldn't really expect the average to be zero. If you have a population with mean zero and you draw a sample from that population the sample mean is not zero. In some very simple models this can indeed happen (models with explicit solution for the factor scores). This is not a simple model so I would not expect this to happen. Another argument why you should not expect the average factor score to be exactly zero is this. If the factor scores are supposed to average exactly to zero, then they should do so also in each cluster  but they don't  the model parameter estimates are based on the entire population and would not be able to do that for very cluster. Among other reasons I would not expect to see zero average is missing data and unbalanced design. 3. Muthén, B. & Asparouhov, T. (2009). Growth mixture modeling: Analysis with nonGaussian random effects. In Fitzmaurice, G., Davidian, M., Verbeke, G. & Molenberghs, G. (eds.), Longitudinal Data Analysis, pp. 143165. Boca Raton: Chapman & Hall/CRC Press. http://statmodel.com/download/ChapmanHall06V24.pdf 

Frank Egloff posted on Thursday, December 21, 2017  5:54 am



Dear Dr. Asparouhov, thank you very much for your answers. I have a request concerning your answer to question 1. Actually, my problem is that from the Mplus syntax (that I specified) I still have difficulties to conclude exactly what it does. Since you explained that E(iw)=E(sw)=0, I can conclude that these two variables have a mean of zero in some way  but I do not know if iw and sw add to zero within of their groups. I think it would help me al lot if you could tell me if the following equations (focus on level 2) represent the model I specified. If yes, is “Muthén, B. & Asparouhov, T. (2009). Growth mixture modeling: Analysis with nonGaussian random effects” the appropriate source to cite? If no, could you please write down the level 2 equations that correspond to my model? Level 1: Ytij = PI0ij + PI1ij * (timetij) + PI2ij * (timetij)squared + Etij Level 2: PI0ij = ß00j + R0ij PI1ij = ß10j + R1ij R1ij = ß20j + ß11j * R0ij + R2ij Level 3: ß00j = GAMMA000 + U00j ß10j = GAMMA100 + GAMMA101 * (COSj) + GAMMA102 * ß00j + GAMMA103 * (COSj * ß00j) + U10j ß11j = GAMMA200 + GAMMA201 * (COSj) + U20j PI2ij = GAMMA300 Your help is very much appreciated! Merry Christmas! 


That reference is the correct reference. The equations are correct with the exception of one thing: you have to delete ß20j. According to the model E(iw)=E(sw)=0 not just for the entire population but also within each cluster. 


Dear Dr. Asparouhov, thank you very much for your kind answer! I understood that when specifying… %within% iw sw  tv12Slow@0 tv22Slow@1 tv32Slow@2 tv42Slow@3 tv52Slow@4 tv62Slow@5 tv72Slow@6 tv82Slow@7; tv12Slow  tv82Slow (1); b  sw on iw; (see full input above) …Mplus is doing automatically… Level 2: PI0ij = ß00j + R0ij PI1ij = ß10j + R1ij R1ij = ß11j * R0ij + R2ij …which expresses that… …R1ij is predicted by R0ij. My second question is for a source that confirms that Mplus is doing this when we use this specification. So I looked into your recommendation (Muthén, B. & Asparouhov, T. (2009). Growth mixture modeling: Analysis with nonGaussian random effects). ...unfortunately I could not find any equations or textual descriptions that would confirm that Mplus is doing “R1ij = ß11j * R0ij + R2ij” when specifying “b  sw on iw;” (not even when looking into papers cited in this work). Could you help me to find such a statement? Your help would be again, very much appreciated! With kind regards Frank 


I would say that you should refer to page 755 from Mplus user's guide. That page explains the Mplus language regarding s  Y on X i.e. it gives the language specification for how random slopes are specified in Mplus. 

Frank posted on Friday, January 12, 2018  3:01 am



Dear Dr. Asparouhov, again, thank you very much for your kind answer! I looked into the Users’ Guide (p. 755) and found that there is a relatively general description of what Mplus does when we specify: s  y ON x; “s is a random slope in the regression of y on x where y is a continuous dependent variable and x is an independent variable.” So actually from this I cannot be sure to infer that it is actually the residuals of x and y that are regressed on each other. Nevertheless, I looked into example 9.2. (p.277) and found a promising additional statement (sentence two): "The random slope s is defined by the linear regression of the dependent variable y on the observed individuallevel covariate x. The withinlevel residual variance in the regression of y on x is estimated as the default." Does this actually say that it is the residuals of y and are regressed on the residuals of x? Your help would be again, very much appreciated! With kind regards Frank 


Your understanding is correct. We do not use that language however. Instead we use this. Every variable is decomposed as a within and between components Y=YW+YB You are calling YW the residual. In the Mplus framework we call this the within component of the variable Y, but it is the same things. In your case IW and SW are the withing level components of the random intercept and slope. If you are still unsure about what Mplus does I would recommend that you conduct a little montecarlo study. Generate data according to your model (and your understanding of what the model is), you can of course generate such data in Mplus, but probably you want to generate it somewhere else where you program the generation yourself. Generate large sample then run the model through Mplus and verify that all parameters are recovered by the Mplus estimation. 


The connection between the Mplus notation and that of multilevel growth modeling is laid out in detail in our short course Topic 8 video and handout on our website, starting with slide 58. 

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