New paper on mediation analysis PreviousNext
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 Bengt O. Muthen posted on Tuesday, November 01, 2011 - 11:35 am
The following message was posted yesterday on SEMNET and is of interest to Mplus users.

SEMNET readers may be interested in a new paper:

Muthen, B. (2011). Applications of Causally Defined Direct and Indirect
Effects in Mediation Analysis using SEM in Mplus. Submitted for publication.

which is available at
together with Mplus scripts.

The paper discusses causally-defined effects for mediation analysis in an
SEM context for continuous, binary, ordinal, nominal, and count variables.
With non-continuous variables, causally-defined effects have not been
available in SEM software until now. The paper discusses maximum-likelihood and Bayesian analysis for non-normal effects, latent response variable mediators, Monte Carlo simulations for planning new studies, sensitivity analysis when violating sequential ignorability, and generalizations to continuous and categorical latent variables. Furthermore, the paper is the first to present a method for mediation with a nominal mediator. One of the illustrations is an analysis of Judea Pearl's hypothetical example with binary mediator and binary outcome that was debated on SEMNET in September.
 Bengt O. Muthen posted on Tuesday, November 01, 2011 - 11:49 am
Mplus users who do mediation analysis are urged to study the paper in the previous posting. The research in this paper elaborates and extends our previous recommendations of how to report indirect and direct effects.

Some of the analyses performed in the new paper require Mplus Version 6.12 which is scheduled to be released next week.
 Tom Hildebrandt posted on Friday, January 06, 2012 - 9:40 pm
I have just read the new paper on Mediation and had a couple of basic questions.

1) How are the DE, TIE, PIE parameters interpreted for a count or negative binomial outcome? It looks as though parameters are exponentiated before being entered into these formulas from the example code in the table 51-2.

2) I am also new to the Causal Effect modeling approach, so I will go back and read some more of the introductory papers, but I wondered what the basic change to the formulas would be with say 3 or 4 treatments as opposed to two conditions.
 Bengt O. Muthen posted on Saturday, January 07, 2012 - 6:42 am
1) Count and negbin modeling focus on the log mean (often called the log rate), whereas the casual effect formulas consider the mean (that is, the rate). Hence the exponentiation.

2) You would do the analysis jointly for all treatments and control groups, so several dummy covariates. And then use the formulas for one treatment at a time.
 Dustin Pardini posted on Tuesday, August 27, 2013 - 9:54 am
When examining mediation using parallel process growth models, several studies have looked at the relationship between X (e.g.-tx condition) and subsequent M & Y parallel process models. However, I have not been able to locate any examples for which X & M are parallel process models and Y is a future outcome.

Do you see any inherent methodological problems for a model in which the relationship between both initial status and growth of X and its relationship with the future outcome Y is thought to be mediated by the slope of M?

If such a model is plausible, does the model presented below appear to represent an adequate means for examining such a relationship?

!Growth Model

Xint Xslp | X1@0 X2@0.5 X3@1 X4@1.5 X5@2 X6@2.5;
Mint Mslp | M1@0 M2@0.5 M3@1 M4@1.5 M5@2 M6@2.5;

Xint with Mint;
Mslp with Mint;
Xint with Xslp;
Xslp on Mint;

Mslp on Xint (a1);
Mslp on Xslp (a2);
Y on Mslp (b);
Y on Xint Xslp Mint;

Model Constraint:
new (med1 med2 );
 Bengt O. Muthen posted on Tuesday, August 27, 2013 - 2:43 pm
Such modeling seems reasonable, generally speaking.

The time ordering needs to be clear - it seems odd to have Xslp on Mint if M mediates X.
 Jessica Latack posted on Monday, October 07, 2013 - 5:57 am
Hi there,

I'm brand new to MPlus so I apologize if this is extremely basic.I am attempting to decompose several significant interactions. I have one categorical predictor (sexual victimization history) and have found that my continuous moderators (life stress and social support) significantly moderate the relationship between my predictor and outcome variables (of which I have three--all are categorical). Can I do this in Mplus?
 Linda K. Muthen posted on Monday, October 07, 2013 - 11:15 am
Yes, you can do this. See Example 3.18. It is more complicated than your model but you should be able to use that as a starting point.
 Boliang Guo posted on Wednesday, March 19, 2014 - 2:43 am
2011 paper is really great!! especially for model with nominal mediator, thank. Here is also two more points in my mind for your kind attention, based on equation (16), could I say the PIE is mathematically equivalent to the traditional mediation estimate, e.g a*b? could the definition be generalized to case when x is continuous variable? i.e. X is not the treatment status but the dose of drug, so the effect of X is change in y due to unit change in x?thanks.
 Bengt O. Muthen posted on Wednesday, March 19, 2014 - 3:44 pm
I will send you a new paper that is forthcoming in the SEM journal where I lay out the counterfactual formulas for the continuous X case.
 Stig Hebbelstrup Rye Rasmussen posted on Friday, May 16, 2014 - 4:29 am
If I wanted to use the model constraint command to estimate create a sensitivity plot of a treatment outcome error correlation rather than a treatment mediator correlation is it correctly understood that I can use the estimate of beta1 in "run 23" in your 2011 paper on causal mediation analysis?

I.e. an example with a treatment an outcome and two independent variables:

outcome with treatment (cov) ;
outcome (sig) ;
treatment (sig2) ;
outcome on x1 x2;
treatment on x1 x2 ;
MODEL CONSTRAINT: plot(beta1) ;
loop(mod, -1,1,0.1) ;
NEW(rhocurl rho) ;
rhocurl=cov/(sqrt(sig)*sqrt(sig2)) ;
rho=mod ;
plot: type=plot2 ;
 Bengt O. Muthen posted on Saturday, May 17, 2014 - 11:20 am
I haven't thought about that. It is a different setting. You would have to go through the steps of Section 10 to see if that is right.
 Stig Hebbelstrup Rye Rasmussen posted on Saturday, May 17, 2014 - 12:42 pm
Hi Bengt thank you for your response. I have essentially just reproduced the script from "run24" where I use "beta1" which is the effect of the mediator on the outcome variable but do not multiply it with the effect of the treatment on the mediator to obtain the indirect effect but simply keep the "direct" effect. Would this not simply correspond to the "total" effect if it is not theorized to be a mediator but simply the total effect of an independent variable after the other independent variables are held constant, and thus obtain the sensitivity of the treatment-outcome relationship of an error correlation between the two e.g. an unobserved confounder?
 Jenna E Finch posted on Wednesday, March 23, 2016 - 8:28 am
We originally fit an SEM model with one predictor, 2 mediators at time point 1, 2 mediators at time point 2 (that are the same, therefore there is a path from time 1 to time 2) and 3 outcome variables. Reviewers recommended that we explore causal mediation, and more specifically Imai et al.’s sensitivity analyses. I found the paper (Applications of causally defined direct and indirect effects in mediation analysis using SEM in Mplus) and code for these analyses but I am unsure how to extend them to a model with multiple mediators and outcomes.

I am able to include the other pathways when I code a single pathway for the sensitivity analyses, but it won’t let me specify the indirect effects. Is there a way to code this, and particularly for calculating indirect effects through two mediators (predictor – mediator1 – mediator 2- outcome) similar to using the VIA command? Can these analyses be done with multiple mediators?

I have also noticed that when I specify a single pathway (predictor- mediator-outcome) the “INDIRECT” coefficient under New/Additional parameters is much larger than the coefficient using a simple mediation model when the rho is set to 0. Is there a reason for this?

Finally, is there code for the graphs which show how the effect is altered over a range of rho values? I saw that the LOOP command could be used, but I did not have any success in coding this.

Thank you so much for your help.
 Bengt O. Muthen posted on Wednesday, March 23, 2016 - 6:52 pm
There is an interesting paper in a 2015 issue of Amer J of Epi:

Mediation Analysis With Intermediate Confounding: Structural Equation Modeling
Viewed Through the Causal Inference Lens
Bianca L. De Stavola et al

Bianca's L and M can be seen as sequential mediators and she presents Mplus scripts.

I am not aware of sensitivity analyses with multiple mediators. I would think Imai's approach can be generalized bu I haven't tried that yet.

You can do VIA when you have multiple mediators.

For a zero rho value I would expect the same indirect effect as obtained using Model Indirect.

Mplus 7.4 automatically does a sensitivity plot when there is one mediator. In the PLOT command just say

 shaun goh posted on Friday, July 29, 2016 - 2:48 am
Dear Dr Muthens,

I have questions about the treatment-mediator as moderator model, specifically Figure 1 of 'Causal effects in Mediation modelling : An introduction with applications to latent variables' (2015) as follows

-To model Figure 1, I would need to do declare X, M(mediator), and XM as predictors, and also use 'MODEL INDIRECT : y MOD M XM X'

-I'm struggling to understand the nuances of the model. Are there texts/papers you would recommend that have used this model or give insight into the interpretation of the treatment-mediator as moderation effect?

-I believe my mediator interacts with my treatment, and am afraid it will violate the assumptions of a simple mediation model. Is this a good motivation to use this model?

Thank you for your time,
 Bengt O. Muthen posted on Friday, July 29, 2016 - 3:09 pm
This is thoroughly described in our new book, chapter 2, section 6:

The book is on back-order right now but we expect to have it available again next week.
 S.Arunachalam posted on Monday, September 19, 2016 - 5:53 am
Hello Prof. Muthen.
I am trying to get a sensitivity plot and following your suggestion dated "March 23, 2016 - 6:52 pm" above, along with your paper in the website.

I start with a simple mediation setup with (giving code excerpts below, version 7.4)
M on x1 x2;
Y on M;
Y IND x1 x2;

plot: type = sensitivity plot2;

I don't get any output of sensitivity plot. IS keyword 'MOD' necessary for automated output of sensitivity plot? In that case then I would need a moderated-mediation setup right? But I thought 'rho could tested for range of values from negative, zero, & positive in simple mediation setup as well?

The reason I ask is that, this would then force me to fit a moderated-mediation rather than a simple mediation (, which is my conceptual model) to get to this sensitivity plot. IS that a right approach?
 Bengt O. Muthen posted on Monday, September 19, 2016 - 11:47 am
MOD is not necessary but you should say

y IND m x1; ! or x2
 S.Arunachalam posted on Monday, September 19, 2016 - 11:52 pm
Thank you.

I also noticed that all 'X' variables in the regression on 'M' has to be included on the regression on 'Y' for Mplus to show me the sensitivity plots. Any specific reason for this?

i.e. for the model I noted above, where X1 and X2 are not regressed on Y, I am still not getting the sensitivity plots. But when I include them, then Mplus shows the plots. Kindly advice why is the case?

This would mean that that I must always test for partial mediation and cannot assume complete mediation?
 Bengt O. Muthen posted on Tuesday, September 20, 2016 - 10:19 am
Try fixing the slopes in the regression of Y on X1 and X2 to zero.

By the way, I think you want to say "Y regressed on X1, X2" rather than the other way around.
 S.Arunachalam posted on Tuesday, September 20, 2016 - 9:49 pm
Dear Prof. Muthen.

I did try this. Unfortunately I don't get sensitivity plots.

Given that I have several control variables in the regression of M; i.e. M on x1 x2....x5 and that I MUST add all of these in the regression of Y to get sensitivity plots, makes the model less parsimonious, too clumsy, consumes degrees of freedom and makes model-fit indices worse. Further most of these control variables are insignificant with their beta estimates close to zero.

Any workaround on this please? Or there is a theory from Imai, Pearl or others that I can cite to defend that ALL variables in the regression of M, must be there in Y? Many thanks in advance.

p.s.: Yes, I meant Y on x1 x2 . My apologies for the confusion please.
 Bengt O. Muthen posted on Wednesday, September 21, 2016 - 10:29 am
You are right that our current implementation of the sensitivity plot for a case like

y = b1*m + b2*x + b3*c + e1
m = gamma1*x + gamma2*c + e2

requires free b2 and b3. I don't think this restriction is necessary and we might change that in the future. But in fact you can do your own sensitivity plot by repeated runs of the type shown in the book for Table 3.26. Here you don't have to bring in b2 or b2 and still get a sensitivity plot for the indirect effect.

Having said that, I think that in mediation analysis one wants to explore both direct and indirect effects. It is probably rare to have a strong hypothesis about zero b2 and b3 coefficients. Instead, you want to report which are significant and which are not. But if you have that strong hypothesis there is another way to explore the concern about mediator-outcome confounding than sensitivity plots - you can use Instrumental Variable estimation. This is described in Section 3.4 in our book.
 Bengt O. Muthen posted on Wednesday, September 21, 2016 - 1:27 pm
Looking at the Imai and Muthen (2011) formulas more closely, I don't think it is correct to modify the approach in Table 3.26. I vote for bringing in all covariate effects - or use IV estimation (which is not problem-free).
 S.Arunachalam posted on Thursday, September 22, 2016 - 9:57 am
Dear Prof. Muthen. Thanks a lot for your detailed directions and advice. Truly appreciate it.

As for IV estimation, can I not directly specify the correlation between residuals of M and Y, if I have at least one X variable that doesn't have a direct path to Y? I am referring to this method as noted here

Or should I follow the book steps (page 155-156) of taking ratio of the two slopes?

So for IV estimation is sensitivity analysis not necessary? I am confused now comparing IV vs. sensitivity analysis, if I Can directly specify the residual correlation of M & Y. When should I go for sensitivity analysis and when should I go for IV estimation? Kindly advice please.
 Bengt O. Muthen posted on Thursday, September 22, 2016 - 3:06 pm
You can try both approaches you mention: IV and the freeing of the residual covariance. I don't know how they compare. In both cases, the drawback is that you have to rely on a direct effect being zero.

With IV estimation the sensitivity analysis is not necessary.
 S.Arunachalam posted on Friday, September 23, 2016 - 4:59 am
Thank you very much Prof. Muthen.
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