Message/Author 


I am attempting to test a moderated mediation model where X>M>Y (all observed variables) is moderated by w, a binary variable (gender). I know how to examine this with a multiple group approach but am attempting the approach described in Preacher, Rucker & Hayes (2007) in Figure 2 model 5 where W influences paths a and b of the mediation chain. With some help from Preacher & Hayes, I have modeled this as pasted below. However, all of the estimates SEs, ps, CIs are exactly the same for all of the paths with the exception of X>M and M>Y. This seems very odd to me. Can you point me to what I might be doing wrong? Thank you. ANALYSIS: BOOTSTRAP = 1000; MODEL: M WITH MW; M ON X (a1) W XW (a3); Y ON M (b1) X W XW MW (b2); MODEL CONSTRAINT: NEW(eff1 eff2); eff1=(a1+a3*1)*(b1+b2*1); eff2=(a1+a3*2)*(b1+b2*2); OUTPUT: CINTERVAL(bcbootstrap); 


Can you send your input, data, output, and license number to support@statmodel.com. 


I should have seen this earlier. The label b2 after x, w, ex, and mw holds all of those regression coefficients equal. It should be stated as: Y ON M (b1) X W XW MW (b2); 


Thank you. The results make much more sense now. Now it is only the M on W and M on MW results that are identical. Does it make sense that this would be the case? Might this be because the MW crossproduct involves a binary variable (coded 1 & 2)? 


I think you mean M on w and Xw. It is the same issue. The label needs to be on the next line. M ON X (a1) W XW (a3); 


Ah, yes. All looks well now. Thank you again for the help. 


Drs. Muthen, I am attempting to fit a Moderated Mediation model with simple (single level) survey data. Specifically, I wish to examine the moderating influence of a dichotomous variable (Mod) effect coded (1, +1) on the 'A', 'B', and 'Cprime' paths as seen in Edwards and Lambert (2007; Model H). I realize that the multigroup function of Mplus may be used in this specific instance(e.g. a categorical moderator), however I would like to retain the more traditional 'cross product' approach for two reasons: 1) To replicate OLS estimates for all paths, 2) I am often interested in similar models with continuous moderators where multigroup analysis is not appropriate. IV and Mod were centered prior to importing into Mplus, and product terms were created using a DEFINE statement. I use the following code: MODEL: Med on Mod IV IVxMod; ! 'A' Path Y on Mod IV IVxMod Med ModxMed; ! 'B' and 'Cprime' Paths MODEL INDIRECT: Y IND Med IV; The resulting model has df = 1 and a significant ChiSq. Examination of RESIDUAL matrix suggests a notable covariance between Med and ModxMed. Adding a covariance parameter results in a notpositive firstorder derivative matrix. I have run into this problem before in the past, and I have a hunch that it stems from the fact that Med is an endog variable. Any advice? Thanks 


If you obtain standard errors, the message most likely comes from the fact that the mean and the variance of your binary variable are not orthogonal. You can ignore the message if this is the case. 

Hong Deng posted on Thursday, May 10, 2012  8:10 pm



Dear Drs. Muthen, I’m trying to test a moderated mediation model with nested data. Data for all my variables were collected from individual and the level of interest is individual too. Basically it is a 111 mediation model (xmy)with a level 1 moderator (w). It should be an easy one if my data isn’t nested in several clusters. Is it possible to test such a model in Mplus? How should I write my Syntax. Thanks very much. 


Moderation can be handled by a multiple group analysis if the moderator is categorical or by creating an interaction term between the moderator and the variable being moderated. You can use TYPE=COMPLEX to take the nonindependence of observations into account. Modify Example 3.11 and add CLUSTER to the VARIABLE command and TYPE=COMPLEX to the ANALYSIS command. 


I am attempting to conduct a moderated mediation analysis where all variables are latent variables formed with continuous indicators. In my model, the independent variable (X) functions as a moderator of the b1 path (the path from the mediating variable to outcome). This model is illustrated in Preacher, Rucker & Hayes (2007)  Figure 2, Model 1. I get an error message that says: MODEL INDIRECT is not available for TYPE=RANDOM. 1. Is it possible to do moderated mediation with a latent interaction variable? 2. Any guidance as to how to generate syntax for this model would be greatly appreciated. Best, Elizabeth 


This type of model is discussed in Section 3 and Section 5 of Muthén, B. (2011). Applications of causally defined direct and indirect effects in mediation analysis using SEM in Mplus. which is on our web site under Papers, Mediational Modeling. As stated in Section 3, Model Indirect cannot be used for this type of model. Section 5 shows how to define the direct and indirect effects instead using Model Constraint. 


Dear Drs. Muthen, I am currently running a moderated mediation model (X > A > Y with the pathway from A to Y being moderated by gender) using the following syntax: Y on X; A on X(Med1); Y on A(Med2); Y on Gender A*Gender; MODEL CONSTRAINT: NEW(Indir1); Indir1=Med1*Med2; This model revealed significant moderated mediation; however, correlations suggested that moderation could also be expected on the pathway between X and A. When I checked for moderation of this pathway it also revealed significant moderated mediation. I've been told that presenting a model with both moderated pathways is inappropriate and that I need to check to see which model is best; however I am unsure of how to do this. Do you have any advice or syntax for such a problem? Many thanks in advance, Paula 


I would suggest a 2group analysis with gender as the grouping variable. You can then easily test if the two paths are the same or different across gender. I can't see why a model with both paths differing across gender would be inappropriate. 

Luisa Rossi posted on Thursday, October 25, 2012  4:32 am



Dear Drs Muthen, I have been running some mediation analyses and found that the x>y relationship was mediated by m. I would now like to see if the a and b paths of the indirect models are different depending on whether respondents score high or low on a number of personality measures. So far, I have used multiple group comparisons with difftest to assess this. I am now thinking there may be a better way to do it but I am not sure. Can you help? Thanks, Luisa 


Why are you dissatisfied with your approach? Is it because your moderator is continuous and you are categorizing it? See Example 3.18 in the Version 7 Mplus User's Guide on the website for another approach to moderated mediation. 

Luisa Rossi posted on Thursday, October 25, 2012  10:08 am



Hi Linda, Thanks for the tip! I will look at the example. I am unsure whether multiple group comparison is the most effective way to look at moderated mediation or whether reviewers may criticize it. Thanks again! 


There are two ways to look at moderation. One is multiple group and the other to create an interaction variable. 

Luisa Rossi posted on Tuesday, October 30, 2012  7:03 am



Hi Linda, I work with survey data so I had to impute my dataset to account of missing data. I noticed that difftest cannot be used with imputed data so I am thinking about using the interaction variable approach. The example you recommended above seems to be based on path analysis (?) as it is able to use the define option after the usevariable one. I am working with latent variables so I first need to define them  however, mplus is not happy for me to use 'define' within the 'model' option. What can I do? 


With latent variables, you should use the XWITH option for interactions. DEFINE is for observed variables. You don't include it in the MODEL command but above or below it. 

Luisa Rossi posted on Wednesday, October 31, 2012  5:15 am



Thanks Linda! AS I am interested in the potential moderating role of w on path a and b, I adapted the model 5 example in Preacher, Rucker, and Hayes (2007) but I used the XWITH command to obtain the two interaction terms mw  m XWITH w; xw  x XWITH w; and included these interaction terms in the model as they suggest: MODEL: y on m (b1) x w mw (b2) xw; m on x (a1) w xw (a3); I find that there is no evidence for a moderating role of either mw or xw (Ps>.05). However, when I had run the multiple group comparisons using w as the grouping variable I had found that it moderated path a (not b)... I am slightly confused on why this might be... can you help? are the 2 approaches not 100% comparable? 


These two approaches should yield identical results. Please send the two outputs and your license number to support@statmodel.com. 

C. Lechner posted on Monday, January 14, 2013  7:58 am



Dear Drs. Muthén, I am testing a moderated mediation model where the moderator is a latent variable; i.e., there is a latent interaction using the XWITH command involved. Because this requires TYPE=RANDOM, the STANDARDIZED output is not available in these analyses. However, I would like to report Rsquare values for my outcomes. > Is there any way to obtain Rsquare values for these analyses? Many thanks in advance! Clemens 


See the FAQ Latent Variable Interactions on the website. 

C. Lechner posted on Thursday, January 17, 2013  7:38 am



Thanks, Linda. I have a followup question: Using the formulas provided in the FAQ, I programmed myself an Excel spreadsheet that calculates Rsquare, change in Rsquare, and standardized path coefficients. It reproduces the numbers from your example on p. 6 in the FAQ sheet perfectly well. However, I wonder how one would generalize the equations from the FAQ sheet to include covariates. In my model (otherwise identical to Fig.2 on p.7), all three latent variables are regressed on a set of covariates. > How do I get the total variances of the latent variables in this case? My approach was to sum all the product terms of the squared regression weights and the variance of the respective predictor, plus the residual variance of the latent variable (which I get in the output). E.g., for a latent variable eta2, regressed on eta1 and covariates x1 and x2, where b1 to b3 denote regression coefficients and zeta2 the residual variance of eta2, the total variance would be obtained by computing: Var(eta2) = b1^2 * Var(x1) + b2^2 * Var(x2)+ b3^2 * Var(eta1) + Var(zeta2). However, this seems to systematically underestimate rsquare. > Have I overlooked anything? > Is there any way to directly get the total variance of a latent variable that is regressed on a set of covariates in the Mplus output? 


You need the variances of the two factors in the interaction. I think you are using the residual variances. If the variances are not available in TECH4, you will need to compute them. 

C. Lechner posted on Thursday, January 17, 2013  9:42 am



TECH4 is unavailable for TYPE = RANDOM. I tried to compute the variances of all latent variables in the way described above. Referring to the notation in Figure 2 in the FAQ sheet, extended by two covariates x1 and x2 for each latent variable, I compute Var(eta1) = b11^2 * Var(x1) + b12^2 * Var(x2)+ ß^2 * Var(eta2) + Var(zeta1) Var(eta2) = b21^2 * Var(x1) + b22^2 * Var(x2)+ Var(zeta2) Var(eta3) = b31^2 * Var(x1) + b32^2 * Var(x2)+ ß1^2 * Var(eta1) + ß2^2 * Var(eta2) + 2*ß1*ß2*Cov(eta1,eta2)+ß3^2*Var(eta1*eta2) + Var(zeta3) where eta1 is regressed on eta2 (coefficient ß), eta3 is regressed on eta1 (coefficient ß1), eta2 (coefficient ß2), and their interaction (coefficient ß3), all three latent are regressed on the covariates x1 and x2 (coefficients bij), zeta_i denote residual variances for the latent variables; and where Var(eta1*eta2) = Var(eta1)*Var(eta2)+[Cov(eta1,eta2)]^2 and Cov(eta1,eta2) = ß*Var(eta2). > Is this correct? There must be something missing. Rsquares are substantially smaller than the ones Mplus computes for a model without interaction. 


In your eta1 equation you have to express eta2 in terms of the x's so that eta1 is written as a function of these x's. You also have to take into account that the x's are correlated and add a term like for the et1 equation: 2*Cov(x1, x2)*b11*b12. 

C. Lechner posted on Friday, January 18, 2013  1:04 am



Thanks you, Bengt – I think the covariance term is what I had overlooked. I'll add it and see whether the numbers add up to something that makes sense then. As this involves a lot of manual computation when more covariates are involved, I wonder whether there is any easy workaround? E.g., could one simply estimate the factor variances in a measurementpart only model (without the covariates and structural paths) and use those as input for the calculations of standardized parameters and rsquare in the final model? I'm afraid that would bias the variance estimates, wouldn't it? 


It might be too approximate. 

ywang posted on Friday, March 29, 2013  12:49 pm



Dear Drs. Muthen: I have a question about how to decide the signficance of the moderated medaition. M is a mediator and W is a moderator (gender, 0 and 1). For the following input: ANALYSIS: BOOTSTRAP = 1000; MODEL: M ON X (a1) W XW (a2); Y ON M (b1) X W XW MW (b2); MODEL CONSTRAINT: NEW(eff1 eff2); eff1=a1*b1; eff2=(a1+a2)*(b1+b2); Should we test the indirect effect of a2*b1 and a1*b2 in order to conclude the signficant moderated mediation? Thank you very much in advance! 


Seems like you have set it up right. One thing you want to test is the difference between eff1 and eff2. To test significance of the moderation in the mediation, don't you want to test that a2*(b1+b2) is significant? Both the difference and this last term can be given NEW names so you get ztests. 


Hello! I wanted to ask a question about a post from the first author on this post regarding a model he created from your 2007 Conditional Indirect Effects Paper, working from "Model 5". In line with the model, I am confused about the role of eff1 and eff2. Having performed previous mediations with the eff1 = a*b, this makes sense to me based on the M ON (X & XW) and Y on (M & MW) paths. I am confused about the eff2 role of multiplying each by 2  what would this be representing? Thank you! Leslie MODEL CONSTRAINT: NEW(eff1 eff2); eff1=(a1+a3*1)*(b1+b2*1); eff2=(a1+a3*2)*(b1+b2*2); OUTPUT: CINTERVAL(bcbootstrap); 


One and two standard deviations above the mean of zero for the moderator. 


Hello, I am attempting to complete a moderated mediation in which my predictor and mediator variables are latent and I have a categorical and latent outcome. The format follows Preacher’s Model 5 in which the moderator impacts both paths a and b. The moderator variable has 4 categories (Black, White, Hispanic, Other). Would it be best to utilize the multiple group function, comparing all 4 racial groups together? Though, I am unsure of how to complete group comparisons from there. Or would it be better to run models with several dummy codes representing the moderator? 


I would suggest doing multiplegroup analysis, letting the a and b paths vary. 


I found most other researchers would calculate the critical ratios of differences(CRD) by dividing the difference between two estimates by an estimate of the standard error of the difference (Arbuckle, 2003). A CRD greater than 1.96 indicates that there was a significant difference between the two parameter estimates at p < 0.05.However, they usually do this in the Amos software. However, in the Mplus result part, I cannot find the standard error of the difference. I am wondering how can you do this through the results provided by Mplus? 


My question is, I want to do multigroup analysis to identify whether the path coefficients differ significantly between east and west. We compared the first model, which allows the structural paths to vary across cultures, with the second model, which constrains the structural paths across cultures to be equal to examine the cultural differences. All the other paths (i.e., factor loadings, error variances and structure covariances) were constrained to be equal. However, I found the factor loading are still different in the Mplus result part. Are there other things that need to be constrained equally? By the way, do you have any sample code for me to do such a multiple group comparison of the mediation model? Thanks. 


You can use MODEL TEST to do the CRD test. See the user's guide. See the Topic 1 course handout on the website under multiple group analysis to see the inputs for testing for measurement invarinace. 

JOEL WONG posted on Tuesday, October 08, 2013  5:28 pm



I am attempting to test a first and second stage moderated mediation model. I have one predictor, one outcome, one mediator, and one moderator (W). W is hypothesized to moderate the relationship between the predictor and the mediator and the relationship between the mediator and the outcome. This model is described as model 58 in Hayes' PROCESS manual  see http://mres.gmu.edu/pmwiki/uploads/Main/process.pdf Does anyone know the Mplus syntax for this model? Preacher et al. (2007) provides the Mplus syntax for several moderated mediation models, but it doesn't include this model. Thank you. 


UG ex 3.18 describes the case of a moderator Z that moderates the influence of the predictor X on M and X on Y. That involves creating X*Z in Define. A plot of the effects and their confidence bands are obtained by LOOP. So that's the first part of your question. The second part is the moderation of the M>Y relationship. This calls for creating M*Z and regressing Y on it. LOOP could be used here as well. 


Hello, Thank you for your previous suggestion in running my moderated mediation with 2 latent predictors, continuous mediator, and binary outcome. I have 4 groups, and have tested the model using the indirect command and bootstrapping (with BC confidence intervals). To compare the effect size of the indirect effects across groups. Would it be acceptable to examine whether the confidence intervals between groups overlaps as a means of testing for significant differences? Or is there a better way to test whether the effect sizes are significantly different (e.g. running a difftest where I constrain paths a and b and compare them to a nonconstrained model)? Also, when I run the mediation for the whole sample, the mediation is significant. However, when I run the mediation using multigroup, the indirect effect is no longer significant in any of my four groups. Would it be safe to say that this could be due to sample size issues (as, I have already established measurement invariance in my predictors) 


You should use DIFFTEST. Yes, lower power could be the reason for this. 

Laura Baams posted on Tuesday, October 29, 2013  9:02 am



Hi, I am running a bootstrap multigroup mediation model with observed variables. Two predictors (x1, x2), two mediators (m1, m2) and one outcome (y). There are 5 groups for the multigroup part. I have compared model fit of a model in which I constrain all paths to be equal across groups, and a model in which they are not equal, and variations of this. The model with the best fit, is the one where groups 1 and 2 are constrained to be equal, and group 3 and 4 are constrained to be equal. I need to report the standardized estimates, but these are not equal for groups 1 and 2, or 3 and 4, while the unstandardized estimates are. From other posts I understand that Mplus does not constrain standardized estimates, does this mean I cannot report standardized estimates in this case? Is there a way I can still obtain standardized estimates (that are equal across groups 1 and 2; and 3 and 4)? Thanks so much! 


The standardized coefficients are standardized using different standard deviations for each group. This is why the coefficients are different. It is not because they are not constrained to be equal in the analysis. 


Dear Drs Muthén, I am running a moderated mediation model based on Preacher, Rucker & Hayes (2007) model 3: the path from mediator m to y is moderated by w. The output shows that all paths are significant at p =.000. However, the fit indexes are as follows: AIC 195.379 BIC 221.996 SampleSize Adjusted BIC 171.948 ChiSquare Test of Model Fit Value 12.001 Degrees of Freedom 2 PValue 0.0025 RMSEA Estimate 0.358 90 Percent C.I. 0.182 0.565 Probability RMSEA <= .05 0.004 CFI 0.958 TLI 0.852 ChiSquare Test of Model Fit for the Baseline Model Value 243.003 Degrees of Freedom 7 PValue 0.0000 SRMR 0.311 Can I consider that my model is significant, relying on the significance of the paths? Thank you for your input, Patricia 


Model fit is not assessed using the significance of the model parameters. The fit statistics indicate that leaving out the two paths represented by the two degrees of freedom create a lack of fit. 

Hannah Lee posted on Sunday, March 30, 2014  2:17 pm



Hello, I am trying to run a moderated mediation model (Similar to Model 2 as Hayes and Preacher 2007 shows). The IV, Moderater are latent constructs and I am unable to use the MODEL INDIRECT with the latent interactions. So I referred to Muthen (2011) sections 3 and 5 using BOOTSTRAP=1000; and OUTPUT: cinterval(bootstrap); But I keep getting the error message that bootstrap cannot be used with TYPE=RANDOM. How can I get the indirect/total effects? ==================================== ANALYSIS: TYPE = RANDOM; bootstrap = 1000; MODEL: x1 BY CPI1CPI3; x2 BY CPC1CPC3; w BY CCC1CCC3; m BY NPA1NPA5; d BY Perf1 Perf6; x1xw  x1 XWITH w; x2xw  x2 XWITH w; d ON m (b1) x1 x2 w x1xw x2xw c1 c2; m ON x1 (a1) x2 (a2) w x1xw (a3) x2xw (a4) c1 c2; MODEL CONSTRAINT: new (ind wmodval); wmodval = 1; ind=(a1+a3*wmodval)*b1; ind=(a2+a4*wmodval)*b1; OUTPUT: SAMPSTAT STAND; cinterval(bootstrap); *** ERROR in ANALYSIS command BOOTSTRAP is not allowed with TYPE=RANDOM. 


Drop the bootstrap request. If your sample size is not small it is unlikely that bootstrap SEs would be very different from regular ML SEs. 

Hannah Lee posted on Sunday, March 30, 2014  7:07 pm



Thank you, Dr. Muthen. Would N=201 sample size be sufficient? Some of the tvalues for the indirect effects come out to be borderline significant. I was wondering if this may be improved if I were able to bootstrap. 


I would not worry unless the sample size is less than 100. I would be conservative as far as significance goes given that you are not doing a single test but several tests. 

CW posted on Tuesday, April 22, 2014  10:53 am



I have used the following syntax to test a moderated mediation model: y ON m (b); m ON x (gamma1) z xz (gamma2); However, I am now realizing that this did not include a test of y on x, which seems like the typical c path in a mediation model. Can you explain to me why did is not present in this syntax, and is it necessary? Thank you! 


Whether you include a direct effect in your model is up to you. Your research hypothesis should determine this. 

Alvin posted on Thursday, May 01, 2014  10:31 pm



Hi there, I estimated a moderated mediation model (similar to that of Hayes and Preacher). All looks good except I did not get fit statistics in my output, even though I did it without bootstrap. Can you advise? many thanks Here's the relevant bit of the syntax: Analysis: Type = random; ALGORITHM=INTEGRATION; Model: pte by conflict family witness access child; adapt by safety bonds justice roles exist; stress by basic health dispa safe; ptexad  pte xwith adapt; whodas12 on pte (b1) adapt stress ptexad; stress on pte (a1) adapt ptexad (a3); Model constraint: new (ind wmodval); wmodval = 1; ind=(a1+a3*wmodval)*b1; Information Criteria Akaike (AIC) 9305.490 Bayesian (BIC) 9484.270 SampleSize Adjusted BIC 9319.462 (n* = (n + 2) / 24) 


Chisquare and related fit statistics are not available when means, variances, and covariances ore not sufficient statistics for model estimation. This is the case for numerical integration. 

Alvin posted on Friday, May 02, 2014  7:48 pm



Thanks very much Dr Muthen for the quick reply. I could not do it without numerical integration however. Is there a way to get fit statistics without using numerical integration. If not, how do I know if my model fits? In terms of interpreting the outcome, is it correct to say that the putative moderator did not moderate the mediating relationship between the IVs and DV, as the interaction product (ptexad) was not significant? 


When there are no absolute fit statistics, nested models can be tested using 2 times the loglikelihood difference which is distributed as chisquare. Nonnested models with the same set of dependent variables can be compared using BIC. If the interaction is not significant, there is no moderation. 

Alvin posted on Wednesday, May 07, 2014  1:27 am



Thanks. Another question is the output does not give standardized coefficients as this was not allowed under Type=random analysis .. Is there a way to obtain standardized estimates as I'm interested in the effect sizes 


Because the variance of y varies as a function of x with TYPE=RANDOM, we do not give standardized estimates. 

Alvin posted on Wednesday, May 07, 2014  5:44 pm



Thanks Dr Muthen  it probably doesn't make sense then to standardize the estimates of the effect size, does it? Also, I notice boostrap can not be used to construct CIs in this case with a randominterceptrandomslope model. Why is that? 

Alvin posted on Wednesday, May 07, 2014  6:06 pm



A further question Dr Muthen is, I notice in model 5 by Preacher et al, the value of w at which the conditional indirect effect of x is estimated is set to 1. Is this an accepted practice? The authors refer to the JohnsonNeyman (Jn) technique where a region of significance or a range of values of w is used in lieu of using arbitrary conditional values of w. Is this method used in analysis of moderated mediation models? 


The variance of y varies as a function of x for any random effect. In this case, standardization is not done. If you add ALGORITHM=INTEGRATION; to the ANALYSIS command, I think you can use BOOTSTRAP. 1 is probably one standard deviation below the mean. This is a common choice. See Example 3.18 where you can get a plot that shows a region of significance. 

Alvin posted on Thursday, May 15, 2014  11:41 pm



Thanks Dr Muthen. I wanted to look at the significance level of a particular set of indirect paths (beta1 x beta2) in the mediation part of the moderated mediation model, as this path (beta2) dropped out (not significant) after building the moderators into the model. Of interest is whether this could be due to power issues (as I have a small sample size). In randomeffect analysis however Mplus does not allow modelindirect command. Is there an alternative way to request estimates of indirect/direct effects in this case? 


See Slides 8085 of the Topic 7 course handout on the website. 


I have an environmental observed variable of maltreatment, one observed impulsivity mediator, and a latent antisocial variable as my outcome. I'm interested in using genotype as a moderator. The problem is my moderator (genotype) is categorical and my maltreatment variable is categorical (no maltreatment, emotional abuse, physical abuse). I understand that the common practice in SEM is to use multiple group comparisons. This is my problem and confusion. Recent recommendations by GxE experts call for all covariates to be included as interaction terms in your model. For example, I will have race and age as covariates in my SEM (full partial). However, it's now recommended that you should also add "race x environment" and "age x environment" as covariate terms in addition to having them be "main effect" terms. The Age x environment term is easy to add as a covariate because age is continuous. Race is not so easy, because it's categorical. Thus I'm not sure how to "control" for these other moderation effects that include categorical variables. Do I just do a massive multiple group comparison? I've considered dummy coding the categorical moderator variables, so that I can create interaction variables and add everything in one SEM rather than do many group comparisons. That way I can appropriately add covariates where I want (full partial). Is that bad practice? Any advice? 


Seems like (1) dummy coding categorical variables and creating interactions is a relatively simple way to go. (2) Multiplegroup modeling is also fine, but you would either have to combine that with interaction terms or do groups as maltreatment x genotype (so 6 groups). Approach (1) allows only different regression effects while approach (2) also allows different (residual) variances for DVs. You may want to ask this type of general modeling question on SEMNET. 

Alvin posted on Monday, July 28, 2014  4:54 pm



Dear Dr Muthen, following a mediated moderation model, I ended up with unstandardized estimates. I was wondering since standardization isn't available in randomeffect models, are there alternative ways for standardizing the estimated coefficients manually? I've previously obtained unstandardized estimates from a similar model but without interaction terms, and they look similar to the one with interaction effects. Does it make sense to obtain standardized effects in this case? this is a fair question that I anticipate reviewers may ask. Many thanks 


What are your Model command statements that give rise to a random effect model? 

Alvin posted on Monday, July 28, 2014  5:34 pm



Here's the relevant bit of the syntax. The model was estimated using algorithmintegration. Many thanks Model: pte by conflict family witness access child; adapt by safety bonds justice roles exist; stress by basic health dispa safe; ptexad  pte xwith adapt; stxad  stress xwith adapt; whodas on pte (b1) adapt stress (c1) ptexad stxad (b2); stress on pte (a1) adapt ptexad (a3) stxad with stress; whodas on dsmany; Model constraint: !plot(indirect); !loop(mod,1,2,0.1); new (ind wmodval); wmodval = 1; ind=(a1+a3*wmodval)*(b1+b2*wmodval); new(ac); ac = a1*c1; 


Hi! I have problems when computing moderated mediation with 2 moderators (no problems if I run the analysis separately for each moderator): MAXIMUM LOGLIKELIHOOD VALUE FOR THE UNRESTRICTED (H1) MODEL IS 10455.691 THE ESTIMATED COVARIANCE MATRIX IS NOT POSITIVE DEFINITE AS IT SHOULD BE. COMPUTATION COULD NOT BE COMPLETED. PROBLEM INVOLVING VARIABLE VBFS_1. THE CORRELATION BETWEEN VBFS_1 AND ZETA_2 IS 1.000 THE PROBLEM MAY BE RESOLVED BY SETTING ALGORITHM=EM AND MCONVERGENCE TO A LARGE VALUE. Syntax for the analysis with 2 moderators. analysis: type=random; model: wo_1 by … ; gro_1 by … ; vbf_1 by … ; int1  plum_1 XWITH vbf_1; plum_1 on la_1 ze_1 wo_1 (a1) gro_1 (a2) vbf_1 (a3); ze_2 on plum_1 (b1) la_1 ze_1 int1; int2  plda_1 XWITH vbf_1; plda_1 on la_1 ze_1 wo_1 (a4) gro_1 (a5) vbf_1 (a6); leng_2 on plda_1 (b2) la_1 ze_1 int2; MODEL CONSTRAINT: NEW (ind1 ind2 ind3 ind4 ind5 ind6); ind1 = a1*b1; ind2 = a2*b1; ind3 = a3*b1; ind4 = a4*b2; ind5 = a5*b2; ind6 = a6*b2; Can you help me? Thanks a lot. 


You should ask yourself why THE CORRELATION BETWEEN VBFS_1 AND ZETA_2 IS 1.000 Two variables that correlate one cannot both be used in an analysis. They are not statistically distinguishable. 


Alvin  you are using XWITH. For standardization with XWITH, see our FAQ: Latent variable interactions. 

Alvin posted on Tuesday, July 29, 2014  6:55 pm



Many thanks Dr Muthen. Can I just clarify, following the equations set out in your FAQ, in order to calculate the standardized coefficient of say beta1(direct effect)  beta1 by square root of V (n1) where V is the variance of latent factor 1 and n1 is ? 

Pieter Baay posted on Wednesday, July 30, 2014  12:10 am



Dear Linda, I'm performing two moderated mediation analyses (one with observed categorical moderator, one with latent continuous mod) and I wonder which moderator value(s) to specify in the MODEL CONSTRAINT. For the categorical moderator, I wrote: X BY x1x4; M BY M1M5; Y BY Y1Y8; XM  X XWITH MOD; Y ON M (a) X; M ON X (gamma1) mod XM (gamma2); MODEL CONSTRAINT: new(pinde pindeN test); pinde = a*(gamma1+gamma2*1); pindeN = a*(gamma1+gamma2*0); test = pinde  pindeN; Is this correct? And what would this look like for a continuous, latent moderator? In previous posts (e.g. by Hannah Lee and Alvin), a moderator value of 1 was specified. Could you please explain what the 1 represents (especially given their construction of latent variables)? Thanks so much! 

annared posted on Wednesday, July 30, 2014  5:18 am



Dear Linda, Thanks for your quick answer. If I use wo_1 instead of vbf_1 to build the interactions (int1 and int2) I get the same error message (i.e. that the correlation between wo_1 and ze_2 is 1.0). However, in both cases the correlation is not 1.0 when I run the analysis without int2. So I guess I have something wrong with my syntax? Thank you! 


Alvin  are you referring to the bottom of page 7? Two lines from the bottom has Sqrt{V(eta_1)}, where eta_1 refers to Figure 2. 


Pieter  that input looks correct. For a latent moderator you have to use the estimated factor variance and look at two values such as 1 SD below vs 1 SD above the factor mean of zero. 


Annared  send your two outputs to support. 

Zaijia posted on Friday, August 01, 2014  8:17 pm



Should the result of indirect effects from Mplus be the same as PROCESS? 


PROCESS(ORS) specified the hardware you the analysis uses and has nothing to do with estimated effects, only how long it will take to compute them. 

Zaijia posted on Saturday, August 02, 2014  7:31 pm



Hi, Dr. Muthen, thank you for your response. But I didn't mean the processor. I mean the Macro developed by Hayes for mediation analysis. Should the results from that macro be the same as Mplus? 


I am not familiar with Hayes' macro. If M and Y are continuous and you use ML I would assume you get the same results. 

Zaijia posted on Sunday, August 03, 2014  1:08 pm



Yes, I found that when Y is dichotomous the results are different. Why does that happen? Another question is how to add interaction term into Mplus?For example, I have 2 dichotomous IV, one mediator, one DV and I want to test the moderated mediation effect, what will the syntax be like? 


Question 1: Take a look at the paper on our website: Muthén, B. & Asparouhov T. (2014). Causal effects in mediation modeling: An introduction with applications to latent variables. Forthcoming in Structural Equation Modeling. Question 2: See the second bullet of our Mediation page http://www.statmodel.com/Mediation.shtml 

Zaijia posted on Sunday, August 03, 2014  2:13 pm



That's really helpful!Thank you so much! 

Zaijia posted on Sunday, August 03, 2014  2:19 pm



What does this following syntax in your mediation page mean? MODEL INDIRECT: y MOD m z (2 2 0.1) xz x; 


See the Version 7.2 Mplus Language Addendum on the website with the user's guide. 

Pieter Baay posted on Tuesday, August 05, 2014  12:14 am



Thank you for the suggestion to use + 1SD and 1SD above the factor mean of zero, based on the factor variance. I've got two followup questions: 1) I think I should use the factor variance of the latent interaction effect rather than the factor variance of the moderator. Is this correct and if so, how can I obtain its value? 2) Is the mean of factors (such as XM) automatically set to zero or should I add a statement to do so? Or, alternatively, should I use the mean /+ 1 SD as moderator values? Thanks again for your help! 


1) No. And, this value is not a parameter, so not available. 2) The mean of a latent variable interaction is not a parameter and should not be referred to. For moderator values, see the Latent Variable Interaction FAQ. 

Alvin posted on Wednesday, August 06, 2014  10:41 pm



Many thanks Dr Muthen. Can I ask  my model is a probit model (with a categorical outcome variable, the others are continous) would it be possible to specific a logit model to derive ORs? Thanks v much for your help! 


You can obtain a logistic model by using maximum likelihood estimation. WLSMV gives a probit model. 


I am working on a multilevel mediated moderation model. Technically that should be the same as a moderated mediation (like Model 2 in Preacher, Rucker & Hayes, 2007). The indpendent (Intr) and dependent variable (Irri) in the model have been measured every day over five days (Level 1). Both the moderator (Age) and the mediator (WM) are traits (Level 2). I know that a crosslevel interaction usually is modelled by regressing the moderator on the random slope. But I need to treat the crosslevelinteraction as a variable. Can I use the DEFINEcommand for a crosslevelinteraction, and what should I do with the random slope? Or maybe there is another solution? USEVARIABLES ARE Intr Age WM Irri iact; MISSING = All(99); CLUSTER IS id; BETWEEN ARE Age WM; DEFINE: iact = Intr *Age; ANALYSIS: TYPE IS TWOLEVEL RANDOM; MODEL: %WITHIN% s  Irri ON Intr; %BETWEEN% Intr Age WM Irri; Irri ON WM(b1) Intr Age iact; WM ON Intr(a1) Age iact(a3); MODEL CONSTRAINT: NEW (ind wmodval); wmodval=10; ind=(a1+a3*wmodval)*b1; OUTPUT: TECH1 TECH8 CINTERVAL; Your help is appreciated! 


If you have %within% s  y on x; %between% y s on w; s@0; Fixing the residual variance s@0 is the same as working with an observed variable xw = x*w. You see this is you write out the formulas (or look at slide 45 of our Topic 7 handout). In practice you have to fix at a small value such as 0.01. 


Thanks for your quick answer. I checked the handout, and the examples in the users guide, but still I am puzzled how to model the indirect effect between a crosslevelinteraction (IV = Level 1, Mod = Level 2, DV = Level 1), and a Level 2 Mediator? What I want to test is a mediated moderation. 

Shiny posted on Saturday, August 16, 2014  2:39 pm



I am testing a moderated Mediation model, with categorical DV. Moderator=mod. Mediators=m. I want to see how mod interacts with the IV (x) and influences the path a (similar to Preacher et al 2007 model 2) The input in mplus is as follow: USEVARIABLES ARE x1 x2 x3 mod m1 m2 m3 y1 y2 y3 y4 y5; CATEGORICAL =y4 y5; ANALYSIS: TYPE = RANDOM; MODEL: xw BY x1 x2 x3;!IV mw BY m1 m2 m3;! Mediator y1w BY y1 y2;!DV1 y4 ON y3 (E); DV3 on DV2 y5 ON y1w(D); y3 ON mw (C); y1w ON mw(B); mw ON xw (A); Here starts with the moderator: y3 y1w ON mod xwmod;! Regress dv1 2 on interaction and moderator mw ON mod;! Regress mediator on moderator xwmod  xw XWITH mod; mw ON xwmod (A2);! Regress mediator on interaction MODEL CONSTRAINT: NEW(AB BD ABD wmodval AC CE ACE); AB = (A+A2*wmodval)*B; BD = B*D; ABD = (A+A2*wmodval)*B*D; wmodval =1; AC = (A+A2*wmodval)*C; CE = C*E; ACE = (A+A2*wmodval)*C*E; OUTPUT: TECH1 CINTERVAL; This Syntax is based on Preachers script. Is it correct? do i need to Regress the distal DVs (y4 y5)on moderator and interaction? Since bootstrap is not possible with Random, can I rely on the CI without Boot?! Thx! 


This looks correct. You may also want to use LOOP plotting (see our Mediation topic in the left column of the home page). I would regress y4 and y5 on moderator and interaction. Bootstrap is available with Random in Version 7.2. 

Shiny posted on Monday, August 18, 2014  3:30 pm



Thank you Dr. Muthen. I looked at the Output. before moderator was added, all Regression and indirect effects were significant. Adding interaction and moderator, neither interaction nor moderator and indirect effect significant. Does it mean there is no moderated Mediation effect? The Mediation model: Xw (latent variables) mw( latent mediator)  y1w (latent)  y5 (categorical) Xw mw y3  y4 (binary) Interaction is xw * mod, named as xwmod Y4 ON Y3: estimate =0.594 p= 0.054 Y5 ON Y1w: 0.316 p=0.046 Y3 ON Mod(moderator):0.075 p= 0.310 y3 on Mw(mediator):3.260 p=0.000 y3 on Xwmod(interaction):0.027 p=0.503 Y1w ON mw (mediator) : 2.891 p=0.000 y1w on Xwmod(interaction) 0.022 p= 0.417 y1w on Mod(moderator): 0.105 p= 0.026 mw ON xw: 0.181 p= 0.004 mw on Xwmod: 0.006 p= 0.783 mw on mod:0.037 p= 0.110 New/Additional Parameters estimate p AB 0.506 0.032 BD 0.913 0.072 ABD 0.160 0.133 WMODVAL 1.000 1.000 AC 0.571 0.047 CE 1.937 0.088 ACE 0.339 0.167 Because I have categorical DV, by Default estimator is MLR, but bootstrap does not Support MLR. When I used WLSMV, then I cannot use Random. What can I do to get bootstrap function? Thx! 


It's a tricky situation. You have y4 binary and y5 categorical, which means that you need to carefully consider how indirect and direct effects are computed; see Muthén, B. & Asparouhov T. (2014). Causal effects in mediation modeling: An introduction with applications to latent variables. Forthcoming in Structural Equation Modeling. Your model is more complex because of the multiple indirect effects. You may get away with considering the latent response variables y4* and y5* as the relevant DVs which is ok with MLR. I would not worry about bootstrapping unless your sample size is low. If your mod variable can be categorized you can use multiplegroup modeling instead of XWITH and then do this using Bayes where no bootstrapping is ever needed. 

Shiny posted on Monday, August 25, 2014  2:08 pm



In your paper (page 8) it said: a*b effect is only valid for the underlying continuous latent response variable y*, not for the observed categorical outcome y itself. Does it mean I can not use MODEL CONSTRAINT and specify NEW (ab) and ab=axb, if I use original categorical y? Must binary y be converted into y*? 2. If so, how shall i modify in d original syntax? Shall I write something like: CATEGORICAL =y4 y5; DEFINE y4 = y4* y5=y5*??? Then change the original path y4 (binary) on y3 into y4* on y3? 3. Can I use ANYLSIS: ESTIMATOR IS WLSMV AND BOOTSTRAP instead of bayes in multiple groups? 4. I did a multiple Group Analysis to test if the mediation effect differs across two Groups(Group H, Group L). Input: MODEL CONSTRAINT: NEW(AHBHDH ALBLDL) AHBHDH=AH*BH*DH;! Indirect effect in group H ALBLDL = AL*BL*DL; !indirect effect in group L DIFF = AHBHDH  ALBLDL; Output (Bootstrap 5000) AHBHDH 0.531 0.425 0.376 0.166 0.030 0.006 0.042 ALBLDL 0.320 0.241 0.199 0.047 0.032 0.054 0.093 DIFF 11.874 6.308 4.536 0.884 0.311 0.520 1.160 I can see in group H, there is an indirect effect at 95% CI, yet this indirect effect does not exist in group L. However the difference test includes 0. does the indirect effect differs across groups? 


2. You can still use a*b when the continuous latent response variable is considered the distal outcome. 3. Yes. 4. No. Think of it this way. You have 2 estimates and you have zero that you can line up as: 0.166 0.047 0 The value 0.166 is significantly smaller than zero but not significantly smaller than 0.047 because 0.047 is closer to 0.166 than zero is. 

Shiny posted on Monday, August 25, 2014  3:03 pm



Thank you very much for your prompt and helpful answer, Dr. Muthen. I Need a bit more Explanation of the 1st Point from your side. To simplify, let me just use a simple Mediation model(xmy). If I use a categorical y, what can I do in mplus to test Mediation? Use IND function but not the a*b? If I want to use a*b, according to your answer, it shall be y*? Would you be so Kind to give me a clue how to create y* in mplus? Thank you indeed! 

Shiny posted on Wednesday, August 27, 2014  12:37 pm



Dear Dr. Muthen, could you pls tell me how to code y* (continuous latent Response variable) in mplus? I have an observable binary y (not latent). Shall i simply replace y with y*? I have not found related script. Thank you for your help! 


You don't have to create y*. If you use WLSMV then a*b refers to the effect on y*. So all you have to do is use WLSMV. 

Shiny posted on Wednesday, August 27, 2014  3:18 pm



That is great to know, since I did use WLSMV already. Is the coefficient a probit Regression weight or a normal Regression weight (if this is a y*)? In multiple Group Analysis, can I do the same wald test or chi difference test, working with constrained or free Parameters of y*, like treating it as a continous variable, once WLSMV is specified? Thank you very much for your helpful comment! 


Q1. a*b is a probit regression coefficient when you consider y to be the DV and it is a linear regression coefficient when you consider y* to be the DV; so the same coefficient plays two roles. Q2. Yes 

Shiny posted on Wednesday, August 27, 2014  4:43 pm



Thank you for your answers, I really appreciate them! 

Winnie Yang posted on Tuesday, November 11, 2014  4:59 am



Dear Drs Muthen, I am running a mediated moderation model. HP(X)>EP(M)>JS(Y). age as the moderator (W), HP, EP and JS are all latent variables, age is collected by asking participants to fill in their age ,so I believe that would be an observed variable.I have 4 questions below: 1. When creating a interaction variable, do I use XWITH command? I know that XWITH is suitable for latent variables only, while HP is a latent variable, Age is not. would XWITH still be right? 2. Assuming that I should use XWITH, I have the following syntax, do they look correct? ANALYSIS: estimator = ml; type = random; ALGORITHM=INTEGRATION; MODEL: HP BY HP101HP106; EP BY EP101 EP102 EP103 EP104R EP105 EP106R; JS BY JS101JS103; HCPxAGE  HCP XWITH AGE; JS ON EXP(B1) HCP AGE HCPxAGE; EXP ON HCP(A1) AGE HCPxAGE (A3); MODEL CONSTRAINT: new(IND W); W=1; IND=(A1+A3*W)*B1; 3. In terms of interpretation, if IND shows to be significant (p <.05), does that suggest then that W is a significant moderator? Or I would need to look at other areas for determination? 4. My model is actually more complicated than this. As a whole, I have 4 outcome variables and 2 independent variables and 2 mediators for each IV. Can I run it altogether then?Thanks! 


1. XWITH can be used for 2 latents or 1 latent and 1 observed. 2. Looks ok, but don't say AGE HCPxAGE (A3); because you can have only one parameter labeled per line. 3. Look at our website under Mediation (left margin of the home page)and you find the document Loop plot for ex 3.18 Make sure you understand this before going on with your analyses. If needed, read the Preacher, Rucker Hayes article we refer to in the UG. 4. Only after you got the simpler models under your belt. 


Hello, I am running a sem model and want to test both moderation and mediation effects as follow m = x w xw; y = x m w xw mw; Y: continuous; M: dichotomous; X and W both categorical (4 modalities each); Using multi group I have an estimate (of X and M) for each category of W, although it s not clear to me how to proceed to test XW and MW (i guest i should compare estimates of X (and estimates of M) among the W'subgroups)? Could you please give me some insights? Thank you 


With W as the grouping variable you want to test that the slopes of Y on M and M on X are the same across groups; that's testing the moderation. Make sure that the Y intercept is different across groups and that the M intercept is different across groups. Note also that a dichotomous M calls for special considerations. See the paper on our website Muthén, B. & Asparouhov T. (2014). Causal effects in mediation modeling: An introduction with applications to latent variables. Structural Equation Modeling: A Multidisciplinary Journal, 22(1), 1223. DOI:10.1080/10705511.2014.935843 You may also/instead consider WLSMV estimation where M* instead of M is the relevant mediator (see my 2011 paper referred to in the paper above to explain this). 


thank you very much. Yes I want to test moderation effect. But I had a mistake in my previous post. Y is actually categorical (5 categories). As suggested I used the WLSMV estimator. To test Y intercept I defined a restricted model in which the paths are constrained to be equal across the groups for X, M and W. M on x w; Y on m (1) x (2) w (3); ! As y is categorical I fixed each threshold as follow [Y$1@2 Y$2@3 Y$3@4 Y$4@5]; 1 I am not sure if the syntax is right? 2 should I only fix the Y intercept? Or both? Then I used DIFFTEST to compare the unconstrained model and the restricted one. Thank you 


Sorry to post it twice it seems like it didnt work the first time. thank you very much. Yes I want to test moderation effect. But I had a mistake in my previous post. Y is actually categorical (5 categories). As suggested I used the WLSMV estimator. To test Y intercept I defined a restricted model in which the paths are constrained to be equal across the groups for X, M and W. M on x w; Y on m (1) x (2) w (3); ! As y is categorical I fixed each threshold as follow [Y$1@2 Y$2@3 Y$3@4 Y$4@5]; 1 I am not sure if the syntax is right? 2 should I only fix the Y intercept? Or both? Then I used DIFFTEST to compare the unconstrained model and the restricted one. Thank you 


I don't understand your setup. We talked about a multigroup analysis, grouping on W. But your input doesn't show that, nor any interactions with W. And I don't know why you fix the Y thresholds. I think the thresholds change across groups as the default so no need to mention them. 


thank you as i defined w as a grouping variable, i remove the interaction XW and MW in the equations. Obviously it was not the good thing to do. I am not sure how to proceed to test "Y intercept and M intercept" across groups as suggested. could you please give me some more details thanks again 


hi, here is my syntax : usevariables are y x m w; grouping is w (0=never 1=one 2=two 3=always) ; categorical are m y; Analysis: type=mgroup; difftest=deriv.dat; !from the unconstrained model ; estimator=WLSMV; model: M on x w; Y on m (1) x (2) w (3); how could i improve it to thank you 


When you have Grouping = w... you should not include w in the Model statements. See Topic 1 of our course on our website to get the basics. Ignore the intercepts for now. 

Riza Casidy posted on Wednesday, March 04, 2015  7:10 am



Hi I am attempting to use the multiple group analysis in Mplus to test for moderated mediation with a categorical moderator, I would like to test whether an indirect effect differs between two subsamples (group a & group b). The IDV, DV, and MV are all latent, continuous variables For each group, I requested the indirect effect by using the model constraint command. GROUPING IS UNI (0=group a 1=group b); MODEL: PS by Q6_1 Q6_2 Q6_3; SAT by Q8_1 Q8_2 Q8_3; AO by Q18_1 Q18_2 Q18_3; SAT ON PS; PS ON AO; MODEL DEAKIN: PS by Q6_1 Q6_2 Q6_3; SAT by Q8_1 Q8_2 Q8_3; AO by Q18_1 Q18_2 Q18_3; SAT ON PS (p1a); PS ON AO (p2a); MODEL CONSTRAINT: NEW (ind_1); Ind_1 = p1a*p2a; MODEL UNIMELB: PS by Q6_1 Q6_2 Q6_3; SAT by Q8_1 Q8_2 Q8_3; AO by Q18_1 Q18_2 Q18_3; SAT ON PS (p1b); PS ON AO (p2b); MODEL CONSTRAINT: NEW (ind_2); Ind_2 = p1b*p2b; MODEL CONSTRAINT: NEW (diff); diff = ind_1ind_2; Is my method above correct? Did I miss anything? Do I have to constrain the latent variable as well? How do I do that? I would like to be sure. Thank you. 


You also need to make sure you have a sufficient degree of measurement invariance across groups for your latent variable indicators. 


Hi there, I'm trying to run the following model: A > B > C Where A>B and B>C are both moderated by D and A = Categorical (3 different conditions) B = Continuous (mediator) C= Continuous (outcome) D = Continuous (moderator) Could you advise on what the input should be? Thank you! 


Create products of variables (interactions) using DEFINE. Assuming 3 categories of A and therefore 2 dummy variables A1, A2: A1D = A1*D; A2D = A2*D; BD = B*D; and add B ON AD1, AD2 in the B regression and add C ON BD in the C regression. 

Jordan posted on Tuesday, April 21, 2015  8:28 pm



Hello, I am attempting to run a moderated mediation model using SEM. I have been using the XWITH command to create the interaction term, and asking for indirect effect estimates using the MODEL CONSTRAINTS, as below. However, I do not know what to input for the question marks. I would like to see the indirect effect conditional upon the moderator at 1SD, mean, and +1SD; but since my moderator is latent, I'm not sure what to input. I tried coding like this: [moderator] (zmean); moderator (zvar); ...and then simply inputting "zmean+sqrt(zvar)", for example to get +1SD. ...but the model wouldn't estimate the mean...so this doesn't appear to work. Can you tell me what I should input in place of the question marks below if my moderator is latent and I"m using the XWITH command? MODEL CONSTRAINT: new (ind1ind3); ind1=(a1+a3*(?)) * b1; ind2=(a1+a3*(?)) * b1; ind3=(a1+a3*(?)) * b1; 


Unless you have multiple groups or time points, latent variables have mean zero. The SD is the square root of the modelestimated latent variable variance. Taken together, that gives you the ? 

Jordan posted on Wednesday, April 22, 2015  4:33 pm



Got it. Makes sense. Thanks for the help! 


Hi I am testing a ML mediated moderation, first link moderated by a group (aggregated) var. I might be mislabeling the coefficients. Assuming a1 is the relationship IV(x) to med(m); a2 mod(w) to med; a3 is the interaction on med; b1 is the my effect; cdash1 is the direct effect xy; I run the model below but model constraint results suggest this is not right. Also, is it ok to use the coefficients obtained in the between part of the model? I assumed so given that an L2 moderator can only influence the between level variance of an L1 variable... Usevariables = x m w y; Cluster = team ; between = w ; within = ; Analysis: Type = Twolevel random; Estimator = ML; MODEL: %Within% Sm ON x; y ON m! SHOULD THIS BE (b1) instead? %Between% S ON w (a3);! CORRECT? CTF ON w (a2); [S] (a1);! CORRECT? [y] (b0); y ON m (b1) ; y ON x (cdash1); S WITH x; MODEL CONSTRAINT: NEW(LOW_w HIGH_w CIND_LO CIND_HI TOT_LO TOT_HI dif1 dif2); LOW_W = 2.77; HIGH_W = 3.74; ! cond ind CIND_LO = a1*b1 + a3*b1*LOW_W; CIND_HI = a1*b1 + a3*b1*HIGH_W; dif1 = CIND_HI  CIND_LO; ! cond tot TOT_LO = CIND_LO + cdash1; TOT_HI = CIND_HI + cdash1; dif2 = TOT_HI  TOT_LO; 


SORRY, in the post above, by CTF is meant 'm', so %Between% ... CTF ON w (a2); should read %Between% ... m ON w (a2); 


One question is at which level you want the mediation  what does your research question say? If level2 is the focus, why not say %Within% y on m x; m on x; %Between% y on m (b1); m on x (a1) w(a2); inter  x XWITH w; m on inter (a3); etc. 


Thank you for your very swift reply, x is a level 1 var, w is level 2 var, so I was trying to examine the crosslevel interaction, but in essence I would like to understand the underlying logic to model both between and within variance when testing for cross level interactions so i can best decide when one or the other is most suited * In any case, I have tried to test the proposed syntax, but obtained the following error; *** ERROR in MODEL command Interactions declared on the between level require numerical integration. Add ALGORITHM=INTEGRATION to the ANALYSIS command. I have added this ALGORITHM=INTEGRATION, and obtained then the following error *** ERROR in MODEL command The XWITH option is not available for observed variable interactions. Use the DEFINE command to create an interaction variable. Problem with: INTER  X XWITH W  what would you suggest? * finally, in the syntax proposed, should the second line in the between read 'w on m (a2)'? Otherwise I am missing something. Many thanks, 


w (a2) is a continuation of the m on line above  look at the the semi colon. For your question, send output to support. See also slides 8085 of our Topic 7 short course handout. 


apologies, had missed that absent ; in the previous line.. I have already gone through the slides and associated recorded video, as well as chapter 9 in the manual but am afraid i was still unable to figure this out  I have sent the material and license. Thank you very much, claudia 


Dr. Muthen, I would like to test moderated mediation in which sex moderates the B path (asdom3>fsdom3) in the mediation: group>asdom3>fsdom3. I also want the model to estimate the mediation hifreq2>asdom3>fsdom3. hifreq2, sex and group are binary. Indicators of latent variables are continuous. Is my desired model possible? If so, is adding MODEL CONSTRAINT commands for indirect effects to the existing model below (which only includes the interaction) the recommended syntax? Thank you. CLASSES = c(2); KNOWNCLASS IS c (group=0 group=1); Model: %overall% asdom3 by as23_r as24_r as25_r as26_r as27_r as28_r; fsdom3 by fs23_r fs24_r fs25_r fs26_r fs27_r fs28_r; as23_r as24_r as25_r PWITH fs23_r fs24_r fs25_r; as26_r as27_r as28_r PWITH fs26_r fs27_r fs28_r; int  sex XWITH asdom3; fsdom3 ON hifreq2 sex asdom3 int; asdom3 ON hifreq2 sex; sex with hifreq2; [asdom3@0]; [fsdom3@0]; %c#2% fsdom3 ON hifreq2 sex asdom3 int; asdom3 ON hifreq2 sex; sex with hifreq2; [asdom3]; [fsdom3]; [as24_r fs27_r]; as23_r as24_r as25_r PWITH fs23_r fs24_r fs25_r; as26_r as27_r as28_r PWITH fs26_r fs27_r fs28_r; Analysis: Type=Mixture Random; Algorithm=integration; 


I think it could be more easily done by creating a 4class Knownclass c variable using the combination of group and sex. Then you have full control over how the slopes and intercepts should vary across those 4 categories. 


I am unsure how the indirect effect of group or sex should be tested in this recommended approach because once considered as a KNOWNCLASS variable, group and sex will not appear in the %overall% model syntax which seems necessary for MODEL CONSTRAINT to request the indirect effects. I am a beginner and do really appreciate the extra clarification. Thank you again. 


If you are a beginner, perhaps it is easier to not get into mixture Knownclass at all. Use XWITH to get "int" as you have done for the m>y moderation by sex, use sex as a binary x variable and use "group" as a binary x variable. Then you get your moderated indirect effect as usual: a*b+g*a*z, where a and b are the usual ones, g is the slope of fsdom3 regressed on int and z is the sex moderator (I hope I got that right; this is essentially Model 3 of Preacher et al, 2007). 


Thank you. Is this still viable if I am also interested in testing the sex X asdom3 interaction across groups (3 way interaction)? I don't know why my model below receives the new error "One or more variables have a variance greater than the maximum allowed of 1000000." Model: asdom3 by as23_r as24_r as25_r as26_r as27_r as28_r; fsdom3 by fs23_r fs24_r fs25_r fs26_r fs27_r fs28_r; as23_r as24_r as25_r PWITH fs23_r fs24_r fs25_r; as26_r as27_r as28_r PWITH fs26_r fs27_r fs28_r; sXas  sex XWITH asdom3; sXasXg  group XWITH sXas; fsdom3 ON hifreq2(Beta4); fsdom3 ON sex (Beta2); fsdom3 ON group (Beta5); fsdom3 ON asdom3(Beta1); fsdom3 ON sXas (beta3); fsdom3 ON sXasXg(Beta6); asdom3 ON hifreq2(gamma2); asdom3 ON sex(gamma1); asdom3 ON group(gamma3); sex with hifreq2 group; hifreq2 with group; !Model constraint: ! NEW (INDhif INDg); ! INDhif=(gamma2*Beta1)+(gamma2*Beta3)...; ! INDg=.... 


Please send output with this question to support. 


Hello, I am doing a moderated mediation (code below) with categorical moderator (ethnicity) split into 5 dummy variables. CISRHigh is a binary mediator, anyserv is a binary outcome. WLMSV estimation is used. yw1 to yw5 (mediator*ethnic group) created under the DEFINE command. Can you tell me how to test the difference between indirect effects using the model constraint command? Thanks, Dharmi Analysis: Type = RANDOM; Model: CISRhigh on positive; CISRhigh on lacksupp; CISRhigh on White; CISRhigh on Irish; CISRhigh on BlackCaribbean; CISRhigh on Bangladeshi; CISRhigh on Indian; xw1  lacksupp XWITH White; xw2  lacksupp XWITH Irish; xw3  lacksupp XWITH BlackCaribbean; xw4  lacksupp XWITH Bangladeshi; xw5  lacksupp XWITH Indian; xw6  positive XWITH White; xw7  positive XWITH Irish; xw8  positive XWITH BlackCaribbean; xw9  positive XWITH Bangladeshi; xw10  positive XWITH Indian; CISRhigh on xw1; CISRhigh on xw2; CISRhigh on xw3; CISRhigh on xw4; CISRhigh on xw5; CISRhigh on xw6; CISRhigh on xw7; CISRhigh on xw8; CISRhigh on xw9; CISRhigh on xw10; anyserv on CISRhigh White Irish BlackCaribbean Bangladeshi Indian; anyserv on yw1; anyserv on yw2; anyserv on yw3; anyserv on yw4; anyserv on yw5; 


You label the parameters in the MODEL command, forexample: Model: CISRhigh on positive; CISRhigh on lacksupp; CISRhigh on White (p1); CISRhigh on Irish (p2); CISRhigh on BlackCaribbean; CISRhigh on Bangladeshi; CISRhigh on Indian; and create a difference parameter in MODEL constraint: MODEL CONSTRAINT: NEW (diff); diff = p1  p2; 


In your case, you would need to label the parameters involved in the indirect effects and create the two indirect effects in MODEL CONSTRAINT along with the difference parameter. 


Thanks Linda  I have done that now. However, I have just realised that I am using MLR estimation because WLSMV is not available when using XWITH for latent variable interactions. Does this mean that I need to change my model setup because my mediator is binary? I have read the forum regarding binary mediator being treated differently when used as independent/ dependent variable when using MLR. Thanks, Dharmi 


The combination of XWITH and a binary mediator is tricky:  XWITH is only available for ML  ML with a binary mediator needs to use the new counterfactuallydefined direct and indirect effects  the counterfactuallydefined effect option is not available with XWITH (although it is available for latent variables) I'm not sure what to suggest until we've generalized this. 


Dear Dr. Muthen, I want to investigate if a mediation is moderated by a dichotomous variable (coded 0 and 1). I opted for the multiple comparisons option and both the a and bpath are significant for each group, with a moderation on the bpath. As I am interested in indirect effects as well, I ran model 5 from Hayes (2007, Figure 2) with the syntax below. My results indicate no apath moderation and a bpath moderation which is in line with the multiple group comparison. However, both a1 and b1 are not significant anymore. This seems odd to me, as both analyses should yield in the same findings? Many thanks for your help! Model: M ON X (a1); W XW (a3); Y ON M (b1) X W XW MW(b2); MODEL CONSTRAINT: NEW(eff1 eff2 modval1 modval2); modval1 = 0.00; eff1=(a1+a3*modval1)*(b1+b2*modval1); modval2 = 1.00; eff2=(a1+a3*modval2)*(b1+b2*modval2); output: stand; cinterval (bcbootstrap); 


Hello, Thanks for the information. Do you think it is worth doing a multigroup mediation analysis instead? I haven't done multigroup analysis before, so I am unsure as to whether it would allow a CFA + mediation analysis (binary mediator) with WLSMV estimation. Thanks, Dharmi 


I ran into the same problem as posted above, in such that my mediated moderation using the syntax by Preacher and Hayes (binary moderator) seemingly yields different results than a multigroup approach. Could you point me in a direction where to search for the problem? I add the syntax underneath. Thank you and kind regards, Joyce I S  mECBIis1@0 mECBIis2@1 mECBIis3@2; I WITH S; S ON Con; MECBIIS1 ON I; MECBIIS3@0; Im Sm  M_NEGC1@0 M_NEGC2@1 M_NEGC3@2; Im WITH Sm; Sm ON Con (a1); S ON Sm (b1); M with Sm; ConxM with Sm; MxSm  Sm XWITH M; S ON MxSm (b2); MODEL CONSTRAINT: new (ind wmodval); wmodval = 1; ind=a1*(b1+b2*wmodval); 


Lisa, I don't know what you mean by "multiple comparisons". I thought you meant multiple group, but it doesn't seem like it. Regarding a1, b1 significance, note that these are main effects and behave differently when you have the interaction. 


Dharmi, WLSMV can do this, but the mediator will be the continuous latent response variable behind the binary observed variable. Not sure how you think multiplegroup replaces XWITH. 


Joyce, Send the 2 outputs that you compare to support along with license number. 


Thanks Bengt. At the moment, I am using XWITH for an interaction between latent variable and ethnic group for a moderated mediation. By using multigroup, I could get rid of this interaction in the moderated mediation and estimate a multigroup model instead. Dharmi 


It was not clear that one variable was categorical in the interaction. You can approach this as a multiple group analysis using ethnicity as the grouping variable. 


In the new MODEL INDIRECT language for moderated mediation you provide an example (3.18) for moderation of the X > M path: MODEL: y ON m xz z; m ON z xz x; If the M > Y path is moderated, would this be the correct model: MODEL: y ON m mz z; m ON x; or would it be: MODEL: y ON m mz z x; m ON x; Thanks! 


In the new, October 2015, Version 7 User's Guide which is on our website, ex 3.18 has for your first model the more common approach y ON m x z; m ON x z xz; and if you want to moderate the M>Y path I would say: y ON m mz z x; m ON x; (in the last line you may want to add z in case it also influences m). 


Many thanks! A follow up question: Release 7.4 handles ordinal mediators using the IND command but the moderated mediation approach in my previous question seems to not accept an ordinal M. Is there a workaround for moderated mediation with a categorical/ordinal mediator? 


Yes, the counterfactual MOD is not yet available for ordinal mediator. You would have to use WLSMV or Bayes viewing the mediator as a continuous latent response variable, not using MOD, but using Model Constraint in line with UG ex 3.18. 


Someone asked me for help recently with an example where they have a pure moderator of the relation between a predictor and a mediator. Let Y be the outcome, X be the predictor, M be the mediator, and Z the moderator. In my mind, if Z is a pure moderator of the relation between X and M, then the specification should be: Y on X M; M on X XZ; However, in all the examples they found, like 3.18 in Mplus, as well as in Hayes (2015), Preacher et al. (2007), and Bengt's example above, the syntax is Y on X M Z; M on X Z XZ; It seems like in this case, Z is not purely a moderator but also has direct effects on M and Y, although in some articles it is illustrated as simply being a pure moderator, so maybe I'm missing something. Of course I don't want to steer this person wrong and I'm not really familiar with this literature, but am I correct to say that in the first example, Z is a pure moderator (between X and M) whereas in the second example it is a moderator with additional direct effects on M and Y? 


I don't think there are any firm rules here. The statement M on X Z XZ; uses the idea that if the interaction XZ is needed then the main effects X and Z should also be included. Perhaps that is often needed in practice in the sense that the effect of Z is significant. Including Z in the statement Y on X M Z; is perhaps less convincing, but again Z may often be significant. 

Sam R posted on Tuesday, December 01, 2015  8:37 am



Hi there, I'm trying to test moderated mediation using the approach described in Preacher, Rucker & Hayes (2007) in Figure 2 model 2 so that I can test moderation in path A. My model is a manifest and the moderator is a binary 0/1 variable. I'm a graduate student and still learning so bear with me, but I've done some searching on the forum before drafting syntax. My input is as follows: usevariables = W_AG X_SCI M_S1 Y_Rat; missing = all(999); Grouping = W_AG (0=Low 1=High); analysis: type = general; model: Y_Rat on M_S1(b1_Low); Y_Rat on X_SCI; M_S1 on X_SCI (b2_Low); model constraint: New (ind_Low); ind_Low = b1_Low*b2_Low; model: Y_Rat on M_S1(b1_High); Y_Rat on X_SCI; M_S1 on X_SCI (b2_High); model constraint: New (ind_High); ind_High = b1_High*b2_High; model test: ind_Low = ind_High; Output: sampstat stdyx; This input produces an error in model constraint command (label is ambiguous) for the label B1_Low I know this is likely an issue with where I've placed the parameter label but any input would be greatly appreciated  Thank you for all of our work updating with these posts! 


I think the issue may be that you say Model: twice, whereas you should say Model low: Model high: or follow UG ex 5.15. Also, with a binary moderator you can do this more easily as a singlegroup analysis with a dummy covariate and using Define to create a product. 

Chan Chen posted on Tuesday, January 19, 2016  10:31 pm



Hi, here is my moderated mediation model categorical dependent variable: USEVARIABLES = x y m w mw; CATEGORICAL = y; DEFINE: xw = x*w; Analysis: estimator = WLSMV; BOOTSTRAP = 10000; MODEL: Y on M (b1); Y on X (cdash); M on X(a1); M on W (a2); M on XW (a3); MODEL CONSTRAINT: IND_LOWw = a1*b1 + a3*b1*LOW_W; IND_MEDw = a1*b1 + a3*b1*MED_W; IND_HIw = a1*b1 + a3*b1*HIGH_W; TOT_LOWw = IND_LOWw + cdash; TOT_MEDw = IND_MEDw + cdash; TOT_HIw = IND_HIw + cdash; PLOT(LOMOD MEDMOD HIMOD); LOOP (XVAL, 0,8,0.1); LODMOD = IND_LOWw*XVAL; MEDMOD = IND_MEDw*XVAL; HIMOD = IND_HIw*XVAL; PLOT: TYPE = PLOT2; OUTPUT: Cinterval (bcbootstrap); But the output showed an error as "Variances for categorical outcomes can only be specified using PARAMETERIZATION=THETA with estimators WLS, WLSM, or WLSMV.” As you could see in the model that I have identified the estimator as WLSMV; why could the model not coverage? Thank you! 


Please send the output and your license number to support@statmodel.com. 


Note also that these effects are not the optimal ones for a binary Y. See instead the paper on our website: Muthén, B. & Asparouhov, T. (2015). Causal effects in mediation modeling: An introduction with applications to latent variables. Structural Equation Modeling: A Multidisciplinary Journal, 22(1), 1223. DOI:10.1080/10705511.2014.935843 

Chan Chen posted on Friday, January 22, 2016  7:26 am



thank you so much! is it possible to model a moderated mediation model with categorical dependent variable. If so, is there any topic note or paper I could reference? Thank you! 


Moderated mediation is described in that paper. 


To clarify your earlier response (Bengt O. Muthen posted on Friday, November 13, 2015  10:22 am): When you say use WLSMV or Bayes viewing the mediator as a continuous latent response variable, you mean not specifying the binary M as a categorical variable, correct? Somehow, when M is included in the list of categorical variables using WLSMV (using TYPE=COMPLEX for clustered data) the mod med LOOP PLOT makes sense, but when treated as a continuous variable the resulting plot looks very strange. 

chan chen posted on Friday, February 05, 2016  8:41 am



Thanks Shiko BenMenahem, I only specified Y as categorical variable, others are all continuous, and then I used WLSMV as the estimator since Y is categorical. My mediator is a continuous variable with 7 anchor point. 


Thanks, but the question was addressed to Dr. Muthén. 


You should specify M as categorical. 


Dear Dr. Muthen, I wanted to investigate a moderated mediation by model 14 of hayes. The total index of moderated mediation and the interaction(int1) appear to be nonsignificant. However, when I take a closer look to the conditional indirect effect, there are some significant specific conditional indirect effects. More specifically, the mediation effect is significant for a certain moderator value, but not for the other. Hence, I am wondering if I could still argue that some effects do present moderated mediation and discuss them? Thanks in advance, Emma 


You can easily do that model in Mplus and get a moderation plot with confidence intervals that show you for which moderator values you have a significant effect. 

Margarita posted on Thursday, February 25, 2016  6:36 am



Dear Prof. Muthén, I am new to moderated mediation, and although I managed to run the analysis in Mplus I am not quite sure if I am interpreting the results correctly. Are you aware of any references about how to interpret the Mplus output? Thank you 


The Mplus output for moderated mediation gives the usual results that are discussed in articles such as by Preacher and in books such as by Hayes. See also our Mediation page: http://www.statmodel.com/Mediation.shtml FYI  the Mplus team is wrapping up a book on mediation so later this year you'll have more guidance. If you have a specific question about the output, please ask. 

Margarita posted on Thursday, February 25, 2016  3:22 pm



Prof. Muthén thank you for your reply. I look forward to the book. I have a question, if you have the time. My understanding is that when the interaction terms are not statistically significant then we can conclude that there is not a moderation. Is that correct? I get nonsignificant interactions but some of the conditional indirect effects are significant. Is such a discrepancy expected? Thank you for your time 

Margarita posted on Thursday, February 25, 2016  3:38 pm



Apologies for reposting, I thought it would save you some time. Following my previous question, to be more precise, the bc confidence intervals showed that for some moderator values there is a significant effect. My question is, even though the interaction term was nonsignificant can I still conclude that for some moderator values there is actually an effect? Thank you again 


Yes, I think you can still conclude that for some moderator values there is an effect based on the CIs. The interaction term is often not significant due to low power unless the sample size is very big, so I think it can be dangerous to throw it out. 

Margarita posted on Saturday, February 27, 2016  10:45 am



That's great, thank you. May I ask one last question: If the model fit is poor, in my case x2 (9) = 2133.29, p < .001 CFI = .00 TLI = 1.73 RMSEA = .81 (.78.84), p <.001 would you suggest still reporting the statistically significant effects? I really appreciate your help. 


If you have moderation involving the mediator the usual tests of fit do not apply. If you are not in that moderation case, the bad fit precludes interpreting the results. 


Dear Prof. Muthen, maybe you can help me further with a question regarding model 14, as well? My data indicates a nonsignificant effect b2 between the moderating variable V and the outcome variable Y. Indirect effects and interaction effects are all significant. Is that mentioned nonsignificant effect a problem for my model? Thanks for your help, best regards from Germany. Michael 


Not a problem. The main effect of V on Y cannot be interpreted separately from the effect on Y by the M*V interaction. 


Hello I have some general questions regarding the indirect effects you obtain with the model indirect command: 1) To obtain reliable information on the significance of the indirect effects, do I have to specify a bootstrap analysis (under the analysis command) or can I interpret the pvalues directly from the output without performing bootstrap analysis? 2) If bootstrapping is preferred and I use the MLR estimator in my analyses, should I rerun my model with ML so that I can examine the indirect effects under bootstrapping? 3) what exactly is the significance test for indirect effect based on? All the best Johan 


You should use bootstrapping, for instance Bootstrap = 1000; and in the Output command you should say Cinterval(bootstrap); and then judge significance of the indirect effect by whether or not the bootstrap confidence interval covers zero or not. Often, the symmetric confidence intervals from MLR are similar to the bootstrap intervals but it is better to go by the bootstrap intervals. 


Thanks for your quick reply! Johan 


Hello, I am trying to test for differences in indirect effects following the procedures described by Hayes in his 2015 MRB paper. The moderator is a categorical variable with only two levels. One of the covariates only varies within one of the two levels that makes up the moderator variable, but it is critical to control for that covariate in the group in which it does vary. How can you set up the model to avoid errors related to no variance but still get a statistical comparison of indirect effects across levels of the moderator. (specifically, if I include the main effect term AND the interaction term in the model, there is zero variance, but including only one or the other isn't a problem, but obviously not completely controlling for that covariate). Thank you! 


Hello (reposting), I am trying to test for differences in indirect effects following the procedures described by Hayes in his 2015 MRB paper. The moderator is a categorical variable with only two levels. One of the covariates only varies within one of the two levels that makes up the moderator variable, but it is critical to control for that covariate in the group in which it does vary. How can you set up the model to avoid errors related to no variance but still get a statistical comparison of indirect effects across levels of the moderator. (specifically, if I include the main effect term AND the interaction term in the model, there is zero variance, but including only one or the other isn't a problem, but obviously not completely controlling for that covariate). Thank you! 


I think this is a good question for SEMNET. 

Eric Deemer posted on Thursday, May 12, 2016  12:07 pm



Hello, I'm also trying to determine which values of a latent moderator I should use to test the significance of an indirect effect. Dr. Muthen, in an April 2015 post you said, "Unless you have multiple groups or time points, latent variables have mean zero. The SD is the square root of the modelestimated latent variable variance. Taken together, that gives you the ?" Does this mean I could, say, do a 1factor CFA of the latent moderator, estimate the residual variance, and subtract the residual variance from 1 to get the explained variance? Then take the square root of the latter to get the SD? Thank you. Eric 


No. Your 1factor CFA gives an estimated factor variance. Take the sqrt of that to get the SD. 


Great, thanks so much. Eric 


Dear Dr. Muthen I am testing a longitudinal panelmodel (multigroup). First I arrived a parsimonious unigroup model and than tested whether paths were similar between subgroups (males/females). All testing of constraining went okay. However, if I constrain an indirect effect to be equal for subgroups, the results (xsquare and df)are similar to the model in which the indirect effect is freely estimated. Does Mplus allow for testing of constraints for indirect effects? I hope you have the time to answer this Gabriël 


How do you constrain the indirect effect to be equal across groups? 


Dear Dr. Muthen In my multigroup analyses, I first pasted the script of the unigroupmodel (containing constraints). Second i specified the grouping variable and than the models for both males and females. Than i started comparing paths between the male and female models by placing constraints. I gave the indirect effect in the model for males the same value (01) as in the model for females. model indirect: C_INT IND B_PV A_GNC (01); C_EXT IND B_PV A_GNC; C_INT IND B_GNC A_PV; C_EXT IND B_GNC A_PV; C_GNC IND B_PV A_INT; C_GNC IND B_PV A_EXT; C_PV IND B_GNC A_INT; C_PV IND B_GNC A_EXT; 


In your statements model indirect: C_INT IND B_PV A_GNC (01); the label (01) that you give is ignored. You can't test that way. You can use Model Constraint to express the indirect effect in each group and express the difference in indirect effects  that then gets a SE and thereby a ztest of the difference. 

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