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I'm trying to make sense of the coefficients of a generalized linear model with negative binomial error structure, and a log-link function.

From Gelman and Hill, "Data Analysis Using Regression and Multilevel/Hierarchical Models", pg 111:

"The coefficients $\beta$ can be exponentiated and treated as multiplicative effects."

Such that if $\beta_1 = 0.012$, then "the expected multiplicative increase is $exp(0.012) = 1.012$, or a 1.2% positive difference ...".

My question is how do you interpret these coefficients in a model with many covariates, specifically where other coefficients have enormous standard error (i.e. non-informative/significant)?

It seems like interpreting the coefficients as multiplicative isn't as intuitive then. For example: if $\beta_1 = 10$ (and $exp(10) = 22026.47$), and the standard error is 30, then the other coefficients in the model are multiplying this huge percentage! Is it better to not interpret the coefficients as multiplicative in this context?

Any thoughts are appreciated!

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First, the coefficient is clearly non-significant and should obviously not be overinterpreted.

Second, if you have such a large coefficient then the corresponding regressor typically takes rather small values where a 1-unit change in the regressor is not realistic. Instead, imagine a 0.01-unit change might be realistic: Then the multiplicative effect of that would be $\exp(10 \cdot 0.01) \approx 1.105$, i.e., corresponding to a 10.5% increase.

It's hard to give more advice without further context.

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  • $\begingroup$ I guess my question is more specifically aimed at interpreting those coefficients that are significant, when you have others in the model that are not significant. $\endgroup$ – smccain Nov 24 '16 at 20:44
  • $\begingroup$ Also more specific to my context: I want to report the coefficients and standard errors of the significant explanatory variables, but is it useful to report them in the multiplicative-sense when other variables in the model are definitely not significant (ie. huge standard error). $\endgroup$ – smccain Nov 24 '16 at 20:48
  • $\begingroup$ I don't think this is specific to multiplicative models but is the same for all ceteris paribus interpretations of coefficients. In some communities it is common to keep non-significant effects in the model - so that one can be certain to have adjusted for these effects. In other communities it is more common to do some kind of model selection prior to interpretation of effects. $\endgroup$ – Achim Zeileis Nov 24 '16 at 21:08

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