Negative Binomial Regression Equation

Question

I know that for a univariate linear regression the predictions are generated like: $$\hat{y} = \beta_0 + \beta_1 x $$

And for a univariate logistic regression the predictions are generated like:

$$\hat{y} = \dfrac{1}{1+e^{-(\beta_0 + \beta_1 x)}}$$

So my question is, what is the equation for a univariate negative binomial regression, specifically one fitted by R's nb.glm function? My guess is that it's something like:

$$\hat{y} = \binom{k+\theta-1}{k} \bigg(\frac{1}{1+e^{-(\beta_0 + \beta_1 x)}}\bigg)^k\bigg(1-\frac{1}{1+e^{-(\beta_0 + \beta_1 x)}}\bigg)^\theta$$

But then I don't know how to make sense of the parameters $k$ and $\theta$

I feel I've reached the limits of my google-fu, so any help would be much appreciated.

Answer 1

You're specifying the conditional distribution as negative binomial, but the exact functional form depends on the link function, it's not inherent in the fact that it's negative binomial.

This is distinct from logistic regression where the logistic is the link function being used, the conditional distribution is binomial and other links may be used with a binomial (i.e. logistic regression is really a binomial GLM with a logit link).

The default link for the function you call - the one that's used if you don't specify a link, and a common choice - is the log link (as can be seen from the help on the function).