# Mediation model: is it wrong?

From time to time it crosses my mind that the mediation model so widely used in psychology area of research is not a good model, because it 'works' when two variables are correlated.

During my mathematics studies I was taught that multicollinearity causes instability of $\breve{\beta}$ estimators, and that's the source of my confusion. If coefficients are unstable, so the whole concept is dubious. At least for me.

I even wrote to a psychologist who broadly employs and teaches this model about the collinearity and instability asking if wouldn't it be good to check VIF or something like that. He replied briefly:

One of the paradoxes of mediation analysis is that the more strongly correlated X and M, the less powerful the test on the paths from M to Y and from X to Y.

But I remained unconvinced... I know that the bootstrap is used, but I'm not sure if it addresses that (instability) issue. I'm wondering if, among you all, there are people who share the same doubts? Or is there a solution that I don't know about?

## 1 Answer

Instability in this context means that it is easier for a single (group of) observation(s) to be influential. So after you have checked for such influential cases (e.g. looking at the dfbetas), instability is no longer an issue, because you then know whether or not such influential cases exists and you have made a decision on how to deal with that. If you see in a textbook "multicollinearity causes instability" you should read that as "if you have multicollinearity check extra carefully for influential observations".

So, to answer your question: mediation analysis is OK if you are somewhat careful when checking whether your model is appropriate.

In practice, the correlations in the mediation models I am familiar with (I am from sociology) are so low, that multicollinearity would not be a big problem at all.

• but what about that instability causes that XtX matrix is not invertible? This argument was exactly written in my textbook. – Lil'Lobster Dec 19 '14 at 17:32
• If that matrix is not invertible, i.e. you have perferct coliniearity, than no software will give you estimates. So you know when that is the case, and that is thus not a problem. In the empirical applications of these models I know of, the degree of multicoliniearity is very far away from perfect, so that is another reason to not worry about it. In short: empirically it does not happen, and if you have bad luck enough that it happens to you, you know it and won't accidentily publish wrong results. – Maarten Buis Dec 19 '14 at 20:21