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

Eric Berger er|cjberger @end|ng |rom gm@||@com
Fri Sep 15 07:37:25 CEST 2023

```I asked your question to ChatGPT4

In statistics, what is the theta parameter of the negative binomial
distribution?

The response contained some math symbols but I am sending in plain
text mode so there may be some loss, hopefully not serious.

The negative binomial distribution is used to model the number of
successes before observing a fixed number of failures in a Bernoulli
process. It has two parameters, typically denoted r and p:

r - This is the number of failures until the experiment is stopped.
p - This is the probability of success on any given trial.

In some contexts, the negative binomial distribution is described in
terms of the number of failures before a fixed number of successes. In
this case:

r would be the number of successes.
p would still be the probability of success on any given trial.

In some literature and software implementations, the negative binomial
distribution's parameterization is provided in terms of r (or k) and θ
instead of r and p. Here:

r (or k) is the number of successes.
θ represents a "success-to-failure" ratio.

The relationship between p and θ is:  p = θ / (θ+1)

[i.e. p = theta / (theta + 1) ]

Conversely, θ = p / (1−p)

[i.e. theta = p / (1-p) ]

In summary, θ is a parameter that captures the success-to-failure
ratio in some parameterizations of the negative binomial distribution,
and it can be related back to the success probability p.

On Fri, Sep 15, 2023 at 5:06 AM Andrew Robinson via R-help
<r-help using r-project.org> wrote:
>
> Hi John,
>
> the negative binomial is a tricky one - there are several different parameterisations and therefore different interpretations of the parameters.  Joseph Hilbe wrote a whole book on it that might be wroth checking.
>
> Cheers,
>
> Andrew
>
>
> --
> Andrew Robinson
> Chief Executive Officer, CEBRA and Professor of Biosecurity,
> School/s of BioSciences and Mathematics & Statistics
> University of Melbourne, VIC 3010 Australia
> Tel: (+61) 0403 138 955
> Email: apro using unimelb.edu.au
> Website: https://researchers.ms.unimelb.edu.au/~apro@unimelb/
>
> I acknowledge the Traditional Owners of the land I inhabit, and pay my respects to their Elders.
> On 15 Sep 2023 at 11:52 AM +1000, Sorkin, John <jsorkin using som.umaryland.edu>, wrote:
> External email: Please exercise caution
>
> Colleagues,
>
> I want to use the power_NetativeBinomial function from the PASSED library. The function requires a value for a parameter theta. The meaning of theta is not given in the documentation (at least I can�t find it) of the function. Further the descriptions of the negative binomial distribution that I am familiar with do not mention theta as being a parameter of the distribution. I noticed that when one runs the glm.nb function to perform a negative binomial regression one obtains a value for theta. This leads to two questions
>
> 1. Is the theta required by the power_NetativeBinomial function the theta that is produced by the glm.nb function
> 2. What is theta, and how does it relate to the parameters of the negative binomial distribution?
>
> Thank you,
> John
>
> [[alternative HTML version deleted]]
>
>
>         [[alternative HTML version deleted]]
>
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```