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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.

Dave
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  • The coefficients are in different scales. Linear regression coefficients are the change in numeric outcome per unit change in a predictor. Logistic regression coefficients are the change in log-odds of binary outcome per unit change in a predictor. So in what way do you wish to combine them? – EdM Jul 22 '20 at 18:00
  • @EdM My goal is to generate a single value that combines all of the coefficients. I had thought to use this approach: https://en.wikipedia.org/wiki/Inverse-variance_weighting – Dave Jul 22 '20 at 18:05

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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.

EdM
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  • Thanks for your answer @EdM. I should have noted 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 standard deviation of 0.50. Does this change your answer at all? – Dave Jul 22 '20 at 19:38
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    @Dave no. With such scaled DVs linear regression coefficients are in units proportional to standard deviations of the DV per unit change of predictor. That would help if one study reported a distance outcome in miles while another one reported it in millimeters, but it doesn't get around the fundamental difference between coefficients representing a continuous outcome and those representing the log-odds of a binary outcome. – EdM Jul 22 '20 at 19:49
  • Gotcha! Thanks for the help! – Dave Jul 22 '20 at 21:29