# [R] Theta from negative binomial regression and power_NegativeBinomiial from PASSED

Ivan Krylov kry|ov@r00t @end|ng |rom gm@||@com
Fri Sep 15 08:27:36 CEST 2023

```On Fri, 15 Sep 2023 01:51:27 +0000
"Sorkin, John" <jsorkin using som.umaryland.edu> wrote:

> What is theta, and how does it relate to the parameters of the
> negative binomial distribution?

Plugging the p (the success probability) and the r (the number of
successes until the experiment is stopped) from the Wikipedia article
(where they are defined in terms of mean mu and variance sigma^2)
together with the variance from ?MASS::rnegbin (where it's defined as
mu + mu^2/theta) into Maxima and then solving for theta, I get:

solve(
[
p = mu / sigma^2,
r = mu^2/(sigma^2-mu),
sigma^2 = mu + mu^2/theta
],
[mu, sigma, theta]
);
[
mu = ((1-p)*r)/p,
sigma = sqrt(r-p*r)/p,
theta = r
]

That is, the theta from MASS seems to be equivalent to the number of
successes from the formulation in the Wikipedia article.

--
Best regards,
Ivan

```