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I am somewhat familiar with various ways of testing mediation for factors in different types of regression analysis. (I'm using R and currently working with a multilevel binary logistic regression.) But now I have a situation in which I'd like to test whether one interaction between factors mediates another, and I'm not sure how this could be done properly.

To give a simplified example of what I am interested in doing:

I have a multilevel binary model using student characteristics to predict pass/fail for students who are in a control versus experimental group.

Let's say that the "intervention" appears to affect women more strongly than men, because the interaction gender*intervention is significant. But then adding in a number of co-variates (and their interactions with the intervention), results in a decrease in the magnitude of the coefficient and the significance of the fenale*intervention interaction, suggesting that once we control for these co-variates and their interactions with the intervention, differences between how the intervention "affects" men and women are no longer significantly different.

I would like to be able to say something about mediation, and I understand how to test the individual factors for mediation of the gender*intervention interaction, but what if there is another interaction, such as (hours spent on childcare)*intervention, which I think may mediate the gender*intervention interaction? Is there a way to test whether the first interaction mediates the second one?

EDIT:

As requested, here is a simple example equation which I think explains what I want to do. For the purposes of simplicity, I am specifying this as a simple binary logistic regression model instead of a multilevel binary logistic regression model.

Let's suppose there are three IV being used as factors: intervention = whether student was in control or experimental group gender GPA

And the DV is whether the student passed or failed.

And let's suppose we consider the following models (I'm just listing the factors here, without coefficients or error terms, for ease of readability):

M1: OUTCOME = INTERVENTION + GENDER + INTERVENTION*GENDER

M2: OUTCOME = INTERVENTION + GENDER + GPA + INTERVENTION*GENDER + INTERVENTION*GPA

Suppose in M1 that INTERVENTION*GENDER was significant, so that women benefited significantly more than men from the intervention.

Then supposed that INTERVENTION*GENDER was not significant (and the female*experimental coefficient had a smaller magnitude) in M2, and we suspect that this is because INTERVENTION*GPA mediates INTERVENTION*GENDER.

What I would like to know is if there is a way to test whether or not INTERVENTION*GPA mediates INTERVENTION*GENDER for these two models....

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    $\begingroup$ In general, there is nothing stopping you from specifying 3-way or 4-way or N-way interactions, except a general desire for model parsimony. I, for one, would have a better time understanding what is going on with your question if you could give an equation representing your model, and a clearer exposition of the hypotheses that you want to test. Another thing: having lots of interaction terms can often lead to multicollinearity, which can make main effects on subsets difficult to interpret, require joint tests, and other exotic approaches to teasing out effects. $\endgroup$ – generic_user Jul 9 '14 at 15:38
  • $\begingroup$ Thanks, @ACD, for your response. I have edited my post to provide more explicit models, and I hopes that helps to clarify my question? $\endgroup$ – cww Jul 9 '14 at 15:56
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I see a few potential questions here:

  1. Does the intervention have a significant effect?
  2. Does the intervention have a differential effect by gender?
  3. Does the intervention have a differential effect by GPA?
  4. Does the intervention have a differential effect by gender, controlling for the effect of GPA, and does the effect of GPA in mediating the intervention vary by gender?

Seems like a case where you want a 3-way interaction: $$ pr(pass) = \Lambda\left[\alpha + \beta_1 I + \beta_2 Ge + \beta_3GPA + \beta4(I\times Ge)+ \beta5(I\times GPA)+ \beta5(Ge \times GPA) + \beta6(I\times Ge \times GPA)+ \epsilon \right] $$ where $\Lambda$ is the logit link.

If indeed the interaction is mediated by GPA, then $\beta_6$ will be significant.

Interpreting such a model is of course more complicated that interpreting a model with fewer terms. I'd probably rely a lot on predict(model,type='link',se.fit=TRUE, newdata=newdata), and then transforming by the logit to get effects for subsets, and confidence intervals.

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