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Bonnie posted on Thursday, February 17, 2005  2:41 pm



I am running some SEM models with two mediators, then how can I decide whether there is mediation effect or not. When there is only one mediator, I can compare the indirect effect to direct effect to determine the mediation effect or test the product of coefficient of path b and a. Then if there are two mediators, can i also judge it by calculating the ratio of the total indirect effect to direct effect, like (b*c+d*e)/a? Thanks a lot! Bonnie 

bmuthen posted on Thursday, February 17, 2005  4:54 pm



Please search for David Mackinnon on Mplus Discussion where he discusses mediation and gives references to papers. And do a literature search on MacKinnon. 


I am running a multiple mediator model involving 1 IV, 4 Mediators, and 1 DV. Initially, these models were run using the Preacher & Hayes SAS Macro for Multiple Mediators; however, 2 Meds are dichotomous and 1 is ordinal  so I am rerunning in Mplus to reassess effects. A couple questions: 1) I initially ran the model in Mplus without indicating categorical data to serve as a crosscheck for output b/w the SAS Macro and MPlus Indirect Effects. Although results are similar, I see some differences between standard errors and resulting 95% CI. Any thoughts as to why this might be the case? I have specified to covary mediator error terms. The SAS Macro provides both normal theory tests and bootstrap tests, but the se estimates are slightly different from each (I'm using the bcbootstrap option). 2) I see from a search of the discussion postings that the logic/approach for dichotomous predictors using Model Indirect is still appropriate  any issues with ordinal data (a 5level ordinal response that is not normally distributed)? Thank you in advance. Christian 


An additional followup based on this path model (with no latent variables): when I indicate the appropriate mediators are categorical, I get a warning indicating: THE LATENT VARIABLE COVARIANCE MATRIX (PSI) IS NOT POSITIVE DEFINITE.... The problem variable is one of the dichotomous mediators. 1) Given I'm running path analysis (rather than an LV model)  should I just interpret this warning to indicate the resulting covariance matrix is not positive definite and the problem is arising from manifest variable relations? 2) Tech4 does not reveal any correlations >= to 1, and I don't see any negative residual variances in the Model Results (for my DV or the 1 continuous Mediator). Is there some additional check that can be requested to try to identify the model problem? 


1. I would guess different estimators. 2. No. Categorical data methodology is not affected by floor and ceiling effects. Please send your output and license number to support@statmodel.com. Be sure you have asked for STANDARDIZED in the OUTPUT command. 


Data is sent (in reference to my second post). In reference to my first post, might the discrepancy also be the result of the bootstrapping procedure (and resulting resampling that occurs) if the point estimates match but the standard error estimates differ somewhat? In reference to my second post, I discovered that the warning disappears if I fix the correlations of mediator error terms to zero (i.e., don't model them with a WITH statement). Preacher & Hayes indicate that these should be estimated if the product of the coefficient method is used (as fixing at "0" can affect the validity of inferences), but indicate this is not a requirement of the bootstrapping method (since their confidence intervals are not dependent on the correlation of mediator terms). However, results of my Model Indirect Statement differ significantly across these two approaches. Which is most appropriate in this instance? Christian 


Regarding your first post, SAS may be using ML and you may be using MLR, which would give different SEs. I don't know why the bootstrap procedure of SAS would give different results; as you say when the point estimates are the same the difference lies in the bootstrap procedure. An ordinal mediator is still ok for indirect product effects since this considers its underlying latent response variable. Regarding your second post, the model with correlated mediator residuals should be used a priori because it is likely that you have correlated, leftout predictors of the mediators. Otherwise you will get a seriously misspecified (and probably misfitting) model. You have a high correlation between 2 of your mediator residuals: YCHPROV2 WITH YCHWITH3 0.775 and this correlation being much higher than the others may cause the nonposdef res corr matrix. That may not be a serious problem given that your sample size is rather low  a larger sample might give posdef. But you may ask why this 0.775 res corr occurs  that's a lot of common correlated leftout causes for these 2 mediators. 


I've tried a few different approaches to address the 'not positive definite' warning, including: (a) dropping one or the other of the offending binary mediators, (b) creating an ordinal variable that combines the binary mediators in a logical progression, (c) switching the 3rd ordinal mediator for a conceptually similar variable that is continuous, and (d) removing the correlated mediator residuals from the model. The later is the only solution that removes this warning (but the results are different  with the correlated residual model behaving more like the results I get when I treat all the variables as continuous (i.e., don't specify that 2 mediators are binary). Is there some other diagnostic option or modeling solution that I should examine? If not, is there any guidance/literature that can indicate when such a warning is "serious" and when it may indicate a problem that is "explainable" (e.g., may be the result of sample size issues/concerns)? Relatedly  is there any simulation research that examines the performance of categorical mediators and correlated error  just wondering to what extent this issue is the result of issues related to the type of data? Thank you, Christian 


It sounded to me like two of the mediators were similar and therefore might share leftout covariates to a large degree  leftout covariate influence ending up in the residual. So one solution is to include more relevant covariates for those mediators to reduce the residual (and hopefully the residual correlation). Another is to remove one of the two highly correlated mediators. 


I wonder what approach Mplus uses to compute the standard errors of indirect effects. 1. Is it a Sobel approach or some other formula? 2. How does the computation work for multivariate designs with various mediators? I would be glad for any information and/or references to the literature. Thanks in advance! Best regards, Oliver Arránz Becker 


1. We use the Delta method. In most cases it coincides with the Sobel approach. 2. We use the Delta method. Try Googling Delta method standard errors. I don't know a reference offhand. 


Dear Linda and Bengt, I think it is so wonderful that Mplus offers an option for testing mediation effects involving more than one mediator using the Delta Method. I thought the article published by Taylor, MacKinnon, & Tein (2008) in Organizational Research Method might be a good reference. I tested indirect effects using VIA commend ( dy1 VIA ln7 mal;) and found that 3 mediation pathways were significant out of 8 possible mediation pathways. May I go by the p level .05 OR should I worry about using some kind of adjustment for a possibility of inflated Type I error due to conducting multiple significance tests? Thank you! Jungmeen 


Whenever you do more than one test, this should be taken into consideration by being conservative about your pvalues. I don't know of any rule for your situation. 

Emily Yeend posted on Tuesday, August 24, 2010  3:12 am



Hi, I'm running a mediation model containing one independent variable and one outcome variable (both continuous). The effect of the IV is suggested to be mediated by two mediators (one binary and one continuous). I have allowed for correlation between the residuals of the mediators which I find to be nonsignificant. I have a few questions. Firstly, am I right in thinking that this correlation can now be dropped from the model or should it still be retained even though nonsignificant. Secondly, I find that one of the proposed mediators does not actually carry an indirect effect, nor does the IV influence it. So my model now involves one mediator and a covariate of the outcome (the old mediator). This means I now have two exogenous variables (IV and covariate / old mediator) which Mplus will automatically allow to correlate. Does this then mean that the correlation I've specified between the residuals of the "mediators" is now actually the correlation between the mediator and the covariate? And presumably this nolonger needs to be in the model. I hope that makes sense! Many Thanks, Emily 


1. I would retain even nonsignificant correlations. 2. This correlation does not need to be in the model. 

Linda posted on Tuesday, September 21, 2010  1:16 pm



I am building a path model with three mediators using an intervention data. I have x, m1, m2, m3, and y. m1 on x; m2 on x; m3 on x; m3 on m1 m2; y on m3 x; Does Mplus provide % variance explained by each mediator variable on the outcome? Will this be, the residual variance, rsquare? 


An Rsquare is provided for each dependent variable using its independent variables. For example, m3 has three predictors, x, m1, and m2. 


Dear Dr. Muthen, Regarding the SE for indirect effects: In a posting it is written, that Mplus uses the delta method, not the Sobel test. Im somewhat confused about the difference. For example MacKinnon (2008: 52) gives the SE as SE² = a²s²(b) + b²s²(a) and writes, that computer programs (Mplus, Lisrel, EQS) use this formula, based on the delta method. And as I know, this is also the SE for the Sobel test. Or does Mplus use the formula SE² = a²s²(b) + b²s²(a) + s²(a)s²(b) best regards Christoph Weber 


The MacKinnon (2008) book describes the Sobel method and the delta method for the indirect effect a*b in Section 4.14. See especially page 92. The delta method uses formula (4.27) with an added covariance term between the a and b estimates (see second line below 4.27). For some models, such as the mediation model for continuous observed variables, the covariance term is zero so that the delta method simplifies to the formula of (4.27). I believe (4.27) is what is referred to as the Sobel method. Mplus uses the delta method in Model Indirect and also in Model Constraint. 


P.S. The zero covariance term is exactly zero when the ML estimator is used, but only approximately zero when MLF or MLR are used. 


Dear Dr. Muthen! What is the best way to test mediation models with SEM? I am testing a 2wave latent difference score mediation model (2 mediators). The problem is (just for t1, but similar for the full model): First, I run a full mediation model. The paths from the independent latent F1 to the two latent mediators (M1 and M2) are significant and also the paths from the mediators to the latent dependent var F2 are significant. There are also sign. indirect effects.Then, I use the chi²difference test to compare the full mediation model with a "partial mediation" model (+direct paths from F1 on F2). The results favour the full mediation model. > this approach is similar to the ones outlined by Holmbeck (1997) or Cole and Maxwell (2003). But in the partial mediation model, there are no signficant effects (F1>F2, M1>F2, M2>F2), although the standardized effects are quite "high" (about 0,2). I get big SE due to collinearity of the between F1, M1 and M2. Following, the indirect effects are also not significant. Would I follow other recommondations like the ones outlined in Iacobucci et al (2007) and test direct and indirect paths simultanously in the first step, I would conclude that there is no mediation. I would appreciate any help! Christoph Weber 


This question should be posted on a general discussion forum like SEMNET. 


I'm running multiple mediation models and asking for the model indirects. In the output I'm not seeing the Rsquared that will tell me how much each mediated effect contributes to the model. Is there code I need to add in to get this? Thanks! 


The Rsquare cannot be divided up because the components are related. 


Thank you. Just to clarify, is there a way to see which mediator contributes the most to the most to the model? 


No. 


Dear Dr Muthen, I am trying to test a model that consists of 3 IVs, 3 MVs, 1 DV and gender as moderator. I ran the Mplus program, which I am very new to its environment, as follow: F1 ON F2 F3 F4; F2 ON F5 F6 F7; F3 ON F7; F4 ON F5 F6; MODEL INDIRECT: F1 IND F2 F5 F6 F7; F1 IND F3 F7; F1 IND F4 F5; and I got this warning *** WARNING in MODEL INDIRECT command There is an indirect effect involving a path between the following variables, but no indirect or direct path exists in the model. Indirect effect: F5 IND F6 *** WARNING in MODEL INDIRECT command There is an indirect effect involving a path between the following variables, but no indirect or direct path exists in the model. Indirect effect: F6 IND F7 2 WARNING FOUND IN THE INPUT INSTRUCTIONS Are the commend and procedures right? and is normal to get such warning? What is the best way to utilize Mplus for mediational testing? Your assistance is highly appreciated. Best regards Ibrahim 


F1 IND F2 F5 F6 F7; implies your model command has F7 ON F6; F6 ON F5; It does not. 


Really it implied that. Thank you so much for your prompt reply, Dr. Linda. One more small question, Is there any need for bootstrapping? 


You can use the BOOTSTRAP option. 


Thank you once again. Dr. Linda, Is this input correct to test the hypothesized model? F1 ON F2 F3 F4; F2 ON F5 F6 F7; F3 ON F7; F4 ON F5 F6; MODEL INDIRECT: F1 IND F2 F5; F1 IND F2 F6; F1 IND F2 F7; F1 IND F3 F7; F1 IND F4 F5; Best Wishes 


Try it out and see. 

Helen Coo posted on Monday, January 23, 2017  8:23 am



Hello: We are trying to do a path analysis in which the outcome and all 6 mediators are binary variables. This will be a secondary analysis of an existing dataset. There is quite a lot of missing data, so we’re using ML as the estimator. The statistical consultant told me that I should allow the mediator residuals to covary. When I tried to do so using the WITH statement (e.g. m1 WITH m2; and does this set the mediators themselves to covary, rather than the residuals?), I got the following error message: *** ERROR in MODEL command Covariances for categorical, censored, count or nominal variables with other observed variables are not allowed. For covariances between categorical variables, specify PARAMETERIZATION=RESCOV in the ANALYSIS command. Problem with the statement: m1 WITH m2. When I then specified PARAMETERIZATION=RESCOV in the Analysis command, I got this message: *** ERROR in MODEL command Categorical variables are not allowed on the lefthand side of an ON statement for PARAMETERIZATION=RESCOV. Problem with: xxx. I am at a loss as to how to allow the mediator residuals to covary when using binary variables and ML. Help! And thanks… 


There are several issues here  and all are explained in our new book: http://www.statmodel.com/Mplus_Book.shtml First, there are special issues for defining indirect and direct effects when you have binary mediators. Second, you have to decide whether you think the binary mediator itself is the substantively best motivated version or if it is an underlying continuous latent response variable. Third, with multiple mediators that are binary, the question is if mediatorspecific effects can be evaluated. Fourth, ML is difficult computationally when you want the residuals among binary DVs to correlate. To simplify matters I would use either WLSMV or Bayes estimation. 


Dear Dres Muthen, My question concerns correlations between mediators. My model looks like this: y on M1 M2 x1 x2; M1 on x1 x2; M2 on x1 x2; Is the default setting that residual variance between the mediators (M1 and M2) are allowed or is this residual variance set to 0 by default? 


Looking at the UG page 518 it looks like they are set to 0 as the default because they are not the ultimate DVs  but you can easily check to see what you get. And you can easily add a WITH statement if it isn't there already. 


Thank you very much for the prompt response! If I may ask about the background for this default setting: Often mediators are likely to be not only exclusively correlated via the predictors, but also because of unknown third variables. In this sense, I see this case analogously to correlations between ultimate DVs. May I ask whether there is a reason to treat mediators differently from ultimate DVs in this respect? Many thanks again for your help. 


It can be hard to foresee if the model is identified with correlated residuals when the variables involved predict other variables, so to not default to an unidentified model these correlations are set to zero to be on the safe side  and the output shows that this is done. 

Tor Neilands posted on Tuesday, September 11, 2018  2:33 pm



Dear Bengt, Does (or will) Mplus support causal direct, indirect, and total effects for models containing 2 or more mediating variables via MODEL INDIRECT? I have several colleagues here who are interested in fitting such models and obtaining the causal effects. Thanks and best wishes, Tor Neilands 


No, then you would have to use Model Constraint as in the article which uses Mplus: Nguyen, T.Q., WebbVargas, Y., Koning, I.K. & Stuart, E.A. (2016). Causal mediation analysis with a binary outcome and multiple continuous or ordinal mediators: Simulations and application to an alcohol intervention. Structural Equation Modeling: A Multidisciplinary Journal, 23:3, 368383 It may be something we want to add later. 


Interpreting results: How does one interpret these two? Effects from PSST1 to MWBT1 Total 1.681 0.123 13.696 0.000 Total indirect 0.163 0.046 3.575 0.000 Specific indirect MWBT1 ERQT1 PSST1 0.163 0.046 3.575 0.000 Direct MWBT1 PSST1 1.518 0.122 12.457 0.000 AND Effects from FAMT1 to MWBT1 Total 0.156 0.080 1.948 0.051 Total indirect 0.033 0.020 1.672 0.095 Specific indirect MWBT1 ERQT1 FAMT1 0.033 0.020 1.672 0.095 Direct MWBT1 FAMT1 0.123 0.075 1.627 0.104 The model is justidentified. 


We need to see the full output  send to Support along with your license number. Indirect effects with multiple mediators is discussed in our Short Course Topic 1 video and handout on our website. 

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