How should coefficients (intercept, categorical variable, continuous variable) in a negative binomial regression model be interpreted? What is the base formula behind the regression (such as for Poisson regression, it is $\ln(\mu)=\beta_0+\beta_1 x_1 + \dots$)?
Below I have an example output from my specific model that I want to interpret, where seizure.rate is a count variable and treatment categorical (placebo vs. non-placebo).
Call:
glm.nb(formula = seizure.rate2 ~ treatment2, data = epilepsy2,
init.theta = 1.499060952, link = log)
Deviance Residuals:
Min 1Q Median 3Q Max
-2.3504 -0.8814 -0.4627 0.4279 1.8897
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.0750 0.1683 12.332 <2e-16 ***
treatment2Progabide -0.4994 0.2397 -2.084 0.0372 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for Negative Binomial(1.4991) family taken to be 1)
Null deviance: 71.220 on 57 degrees of freedom
Residual deviance: 66.879 on 56 degrees of freedom
AIC: 339.12
Number of Fisher Scoring iterations: 1
Theta: 1.499
Std. Err.: 0.362
2 x log-likelihood: -333.120
It's the exponential of the sum of the coefficients: seizure.rate2= exp(2.0750-0.4994*treatment2Proabide) =exp(2.075)*exp(-0.4994*treatment2Proabide)
or you can use the code names(YourModelname) This code will give you output of the names and you can look at fitted.values to give you the predicted values. I occasionally do this as a double check to see if I wrote out my formula correctly.