R help - Sep 2023 - Theta from negative binomial regression and power_NegativeBinomiial from PASSED

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

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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


--
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School/s of BioSciences and Mathematics & Statistics
University of Melbourne, VIC 3010 Australia
Tel: (+61) 0403 138 955
Email: apro at unimelb.edu.au
Website: https://researchers.ms.unimelb.edu.au/~apro at unimelb/

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to their Elders.
On 15 Sep 2023 at 11:52 AM +1000, Sorkin, John <jsorkin at
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]]

On Fri, 15 Sep 2023 01:51:27 +0000
"Sorkin, John" <jsorkin at 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