Can linear and logistic regression coefficients be combined using an inverse variance weighted average? Suppose I have coefficients from several studies, some are linear regressions and some are logistic regressions. They measure the same construct, but because of the nature of the experiments (i.e., using continuous vs binary responses), they were analyzed with linear vs logistic regressions.
Is it okay to combine the coefficients from these studies using an inverse variance weighted average?
(If it matters, the regression models were all mixed effects models using Bayesian parameter estimation.)
Edit: I should have mentioned that all predictors and outcome variables were scaled in the manner recommended by Gelman et al. (2008). This meant that all linear DVs were scaled to have a mean of 0 and a standard deviation of 0.50.
 A: Inverse-variance weighting makes sense for combining parameter estimates that are in the same units. The problem is that linear and logistic regression coefficients are in different types of units. Per unit change in the same predictor, a linear regression coefficient represents the associated change in a numeric outcome, while a logistic regression coefficient is the change in log-odds of a binary outcome. So any average of a logistic regression coefficient and a linear regression coefficient wouldn't make sense.
There might be some methods from meta-analysis that somehow combine information from such fundamentally different types of studies to provide some overall measure of "importance," but I'm not familiar with them and the combination wouldn't be an average coefficient in any event. Perhaps you could do a crude combination by converting linear-regression results to predictions for an equivalent logistic regression based on a cutoff in the linear regression response variable. But offhand there would seem to be multiple problems with that approach.
It looks like this difficulty in combining different studies on the same phenomenon is another argument against dichotomizing continuous variables.
