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

Ben Bolker bbo|ker @end|ng |rom gm@||@com
Fri Sep 15 16:02:39 CEST 2023

```    Yes, theta is the 'size' or overdispersion parameter.  Sometimes
also denoted as k. Wikipedia discusses this parameterization in the
paragraph starting "In negative binomial regression ..." (but they call
this parameter r rather than theta or k).

You can also see this in MASS on google books:

This parameterization was added to R in version 1.3.0 ...

On 2023-09-15 2:27 a.m., Ivan Krylov wrote:
> 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.
>

--
Dr. Benjamin Bolker
Professor, Mathematics & Statistics and Biology, McMaster University
Director, School of Computational Science and Engineering
(Acting) Graduate chair, Mathematics & Statistics
> E-mail is sent at my convenience; I don't expect replies outside of
working hours.

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