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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)
kk + 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
0
1
2
3
4
5
6
7
8
No. hosts
No. parasites
164
147
92
43
25
11
3
0
1
486
641
162.4
149.6
91.4
46.5
21.2
9.0
3.7
1.4
0.5
 
 

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