I used some categorical variables as predictors to a negative binomial model. The dependent variable is numerical. I used glm.nb in R and the results show relative coefficients of one category respective to another category.

Then I tried using lm.beta to standardize the coefficients, but still the results are relative. How can I interpret a positive versus negative coefficient here?


I'm going to answer this using a Poisson model, which is precisely a negative binomial model without overdispersion, because the math will be simpler. The poisson model predicts the probability of observing $y_i$ to be a particular non-negative discrete number $$P(y_i|X) = \dfrac{\exp(-\lambda_i)\lambda_i ^{y_i}}{y_i!}$$

The conditional mean of this distribution $\lambda_i$. $$E[y_i|x_i] = \lambda_i = \exp(x_i\beta)$$ $$\log \lambda_i = x_i\beta$$ The conditional variance of the poisson model is also $\lambda_i$, but the variance of the negative binomial model is $\lambda_i + \alpha \lambda_i$. This is the only practical difference between the two models for the purposes of this answer.

This is effectively a log-linear model. So the marginal effect of $x$ on $\lambda$ can be shown as

$$\dfrac{\partial E[y|x]}{\partial x} = \dfrac{\partial\lambda_i}{\partial x} = \exp(\beta)$$

So if you have a negative $\beta$ for a dummy variable $x$, you can say that "on average, $x$ lowers the expected value of $\log(y)$ by $\beta$*100 percent."

  • 4
    $\begingroup$ You can also get a multiplicative effect by exponentiating the coefficients. For example if $\beta_D=.43$, then folks with $D=1$ are expected to have $\exp \{.43\} \approx 1.54$ times higher outcome (holding everything else constant). This is the incidence rate ratio interpretation. $\endgroup$ – Dimitriy V. Masterov Jan 24 '13 at 20:47

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