4
$\begingroup$

I have made three generalized linear models, one with zero-inflated Poisson, the second with negative binomial, and the third with binomial conditional distributions. I am now trying to interpret the results, and have tried to back transform the estimates by taking the exponent of the estimates for the models with Poisson and neg. binomial, and know that I should use inverse logit function on the binomial but that is still on the to-do list.

Is it correct to use the natural exponent? And is it possible to get negative values? All my transformed estimates are positive so far, but how can I tell if a predictor has a negative impact if the sign is always positive?

Improve this question
$\endgroup$
4
  • 3
    $\begingroup$ The models can output a prediction on the scale of the link function. In the case of the Neg Binom and Poisson GLM, the log function is used to relate the mean to your linear predictors. So yes use the exponent to transform the fitted values. The exponent maps from the reals to the positive reals, so it isn't clear how to get negative predictions after transforming. The effects of your predictors in these cases are negative or positive on the log scale, and hence they lead to a multiplicative change in the count outcome. $\endgroup$
    – user1848065
    Commented May 10, 2023 at 12:28
  • $\begingroup$ Thank you. I still don't understand how to interpret this, though. If the log effect is negative before taking the exponent, does that indicate a negative effect on the response variable? Saying anything about the size of the effect is difficult with log coefficients, but I can't say anything about the direction of the effect after they are transformed. I'm confused $\endgroup$
    – kirchoffs
    Commented May 11, 2023 at 6:53
  • 3
    $\begingroup$ The exp of your linear predictors will yield a prediction about your conditional means of your response variable. It is better if you don't conceptualize it as a negative effect because the negative effects are on the linear predictor scale, rather than the response scale. On the response scale everything is positive. The direction of the effects are not difficult with the coefficients. If $\beta$ is your coefficient, a one unit increase will changes the expected value of the response by a factor of $e^\beta$. With log links the effects of the covariates are multiplicative. $\endgroup$
    – user1848065
    Commented May 11, 2023 at 13:55
  • $\begingroup$ Thanks again! So If the exponentiated beta = 1.5, it is a 50% positive difference in the response, but If it is 0.5, it is a 50% negative difference? $\endgroup$
    – kirchoffs
    Commented May 11, 2023 at 19:11

1 Answer 1

Reset to default
7
$\begingroup$

Assuming your Poisson and negative binomial models used the (natural) log as the link function, yes, you would transform those coefficients by exponentiating them. It is certainly possible to have a 'negative' relationship between $X$ and $Y$ such that increasing values of $X$ are associated with decreasing values of $Y$. On the scale of the linear predictor, that will show up as a negative coefficient. Note that on the scale of the linear predictor, everything is additive—that is, every time you go up $1$ unit on $X$, you go up $\hat{\beta}_X$ units on (the log of the mean of) $Y$. When you transform those coefficients, you are no longer on the scale of the linear predictor, and things are no longer additive—now, they are multiplicative in nature. Every time you go up $1$ unit on $X$, the former mean is multiplied by $\exp(\hat{\beta}_X)$ to get (the new mean of) $Y$. Moreover, when you exponentiate a negative number, you get a value between $0$ and $1$. To overemphasize this, you get a value less than one. So every time you multiply a number by such a coefficient, the product is smaller than the former value. Thus, you still have a 'negative' relationship between $X$ and $Y$.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.