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# Negative binomial distribution  ### Estimating k for the negative binomial distribution

k can be estimated in a number of different ways, some approximate, some precise. We will give one approximate method with a worked example, and also the maximum likelihood method which would usually be done by computer.

#### Algebraically speaking -

We can rearrange the formula for the variance above to give the following formula for k:

 k = m2 (s2 − m)
where

• m is the mean number of individuals per sampling unit
• s2 is the variance of individuals per sampling unit.

This provides what is known as the moment estimate which is a reasonable approximation under these conditions:

• for small values of m when k/m > 6,
• for large values of m when k > 13,
• for intermediate values of m, when (k+m)(k+2)/m ≥15.

#### Algebraically speaking -

Alternatively you can use the maximum likelihood estimator, which is now available on many computer packages:

 N × loge ( 1 + ) = Σ ( Ar )  k k + r
Where
• Ar is the sum of the observed frequencies containing more than r individuals.
• k is the estimated value of k.

This must be solved iteratively and is best done on a computer. It assumes the frequency distribution is smooth and does not have any extremely large values.

#### Worked example

 Number of acanthocephalan parasites in ducks Number of parasites Number observed Number predicted 012345678 No. hostsNo. parasites 16414792432511301486641 162.4149.691.446.521.29.03.71.40.5

Variance = 1.863

Mean = 1.319

Approximate value of k = 1.3192 / (1.863 − 1.319) = 3.20

Maximum likelihood estimate of k = 3.054 ± SE 0.726 