Interpretation of products of paths (and Est.) when multiple link functions are combined in a single model

Interfaces brms
2

My understanding of OLS style regressions is that in this case the x -> y estimates should capture change in y as a function of change of 1 in x, whereas the x -> m estimates should capture change in the log-odds of m = 1 as a function of change of 1 in x . Is this still true in the brms output, i.e. do different rows in the Estimate column have different concrete interpretations?

That’s correct. The paths: x -> y and m -> y are interpreted like linear regressions, and the path x -> m is interpreted like a logistic regression (i.e., the estimates are logits and can be exponentiated to give odds ratios).

As a corollary, given the model specified above, what is the correct way to concretely interpret products of paths given the estimates may be measured on different scales? For example, to test whether the indirect effect ( x -> m -> y , or ab in mediation terminology) is negative I would use the method, hypothesis(fit, 'y_m*m_x< 0') . Is this still correct and how would I interpret the resulting CI in terms of unit change in y as a function of change in x ?

Assessing indirect effects with a categorical mediator is a tricky subject, because the outcome of the first path (x -> m) is a probability, whereas the predictor of the second path (m -> y) is the category itself. This makes it harder to assess mediation, because the mediator is assessed on a different scale in each path. There are two more complex methods for handling this: using a latent underlying variable for the binary mediator as both predictor and outcome, or using causally-defined effects (more info in this paper).

However, a simpler approximation is to standardise both paths before computing the indirect effect.

The x -> m pathway is standardised by:

b_{xm} * SD_x / SD_m

Where SD_x is the observed standard deviation of the predictor and SD_m is the standard deviation of the binary outcome when analysed with a logit link, given by:

SD_m = \sqrt{b_{xm}^2 + SD_x^2 + \pi^2/3}

The m -> y pathway is standardised similarly:

b_{my} * SD_m / SD_y

But in this case SD_m is the observed standard deviation of the binary mediator.

Once you standardise both paths, you can then compute the indirect effect as usual