I've got a question regarding the intersection of mediation and instrumental variables. I want to test if a randomized treatment variable (T) has a direct effect on outcome (Y), as well as an indirect influence on the outcome via a mediator (M). However, the mediator M was not randomized, so I cannot test a causal, only correlational relationship between M and Y. In other words, I cannot be sure whether M -> Y or Y ->M.
However, I have an instrument that is correlated with mediator M, but not with outcome Y. As far as I know, this would allow me to test for a causal impact (given additional assumptions) of M on Y, via 2SLS, as far as I know.
Doeser anyone know if and if yes how the mediation model, and the instrumental variable approach can be combined? How can I test whether T affects Y via M, provided that I have a valid instrument for M.
Literature suggestions would be highly appreciated.
Hi Hendrik,
I see two possibilities: a) Use 2SLS with the first stage regressing M on X and the instrument (W) and the second stage regression Y on X and M. The essential restriction of a conditional independence of Y and W given X and M.
b) Specify a SEM with X having effects of M and Y; W having effects on M. Then test (either by the chi-square test of this restricted model or by the chisq difference test of this model and a model in which the error covariance cov(e.M, e.Y) correlates if endogeneits is present. If yes, keep the error covariance estimated. If not re-fix it.
Best,
Holger
Hi Holger,
thanks for the reply. I will try that out.
Do you maybe have a thought on this procedure: 2SLS basically takes the predicted values of the variable that needs to be instrumented from a regression on the instrumental variable (first step) and uses these as the predictor for the outcome variable in the second step. What if I used these predicted values instead of the observed values in the normal mediation analysis (e.g. SEM, bootstrapped mediation, or Baron and Kenny approach, etc.)?
Dear Hendrik,
if you use the predicted values for M (from the regression of the first stage. In the first stage you should regress M on the instrument and also on X) , you get consistent estimates, while the estimate are biased if you use the observed values of M (in case that M suffers from endogeneity).
The idea is that you isolate the variation in M that cannot be explained by X and the instrument. This variation is the one that is correlated with your error term. The remaining should not be correlated with the error term as X and Z are by asummption independent of the error term.
I hope this helps
Best regards,
Florian
Hey guys, as far as I remember, doing 2sls manually, by using the predicted values of M in a separate model leads to consistent estimate but the standard errors are wrong.
Best,
Holger
Hi Hendrik,
thank you for posting this question. I am also still not entirely clear how 2SLS is combined with mediation analyses. But the approach that you suggests (using the predicted values of M) seems valid. I had a look into the following article:
Shaver, J. M. (2005). Testing for mediating variables in management research: Concerns, implications, and alternative strategies. Journal of management, 31(3), 330-353.
The author writes:
"To summarize, in the 2SLS procedure, (a) OLS is used to estimate Equation 2, (b) predicted values of M from Equation 2 replace M in Equation 3, (c) least squares are used to estimate coefficients of Equation 3, and (d) the standard errors of these estimates are calculated."
This approach is what you suggests, right?
The one thing that still confuses me is what instrumental variable(s) to use in predicting M? Since your x (= treatment) is randomly assigned, it is exogenous and also relates to M. So I wonder if you could just use X (your treatment variable) as instruments to estimate M. Then, you use the predicted values of M in your final OLS regression.
This is at least what seems to be suggested here:
Antonakis, J., Bendahan, S., Jacquart, P., & Lalive, R. (2010). On making causal claims: A review and recommendations. The Leadership Quarterly, 21(6), 1086-1120.
"4.2.1.3. Finding instruments. Finally, one of the biggest challenges that researchers face when attempting to estimate instrumental
variable models has to do with where to find instruments. In the case of an experiment, where the modeler wishes to establish
mediation, the modeler will have the perfect instrument/s: the manipulated variables. As long as the model is estimated correctly
(with the cross-equation disturbances of the endogenous variables correlated), then the causal mediation influence can be
correctly identified."
I am not sure why Antonakis also adds "with the cross-equation disturbances of the endogenous variables correlated" because 2SLS or correlating the disturbances/error terms across equestions (SEM) seem to be two alternative approaches for dealing with endogeneity. Either doing 2SLS or SEM (with correlated error terms) alone should be sufficient, I guess.
Hi Folks,
1) Yes, in an experiment, the treatment can act as an instrument, if
a) the treatment is only transmitted via M (and has no direct effect). This is the usual exclusion criterion).
b) The effect is strong. That's the usual relevance criterion.
2) If the mediator shall be tested for endogeneity (which should be the case), the disturbances have to be correlated - either in a TSLS regression or a SEM (via correlating the error terms/disturbances).
That is, if the mediator is endogeneous, including an instrument will still imply a biased indirect effect of the treatment because the effect of the mediator is biasd, if the disturbances are not allowed to correlate.
And yes, SEM or TSLS lead to the same results.
Best,
Holger
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