Noncentral hypergeometric distributions

Wallenius' distribution can be explained as follows. Assume that an urn contains ${\displaystyle m_{1}}$ red balls and ${\displaystyle m_{2}}$ white balls, totalling ${\displaystyle N=m_{1}+m_{2}}$ balls. ${\displaystyle n}$ balls are drawn at random from the urn one by one without replacement. Each red ball has the weight ${\displaystyle \omega _{1}}$, and each white ball has the weight ${\displaystyle \omega _{2}}$. We assume that the probability of taking a particular ball is proportional to its weight. The physical property that determines the odds may be something else than weight, such as size or slipperiness or some other factor, but it is convenient to use the word weight for the odds parameter.

The probability that the first ball picked is red is equal to the weight fraction of red balls:

${\displaystyle p_{1}={\frac {m_{1}\omega _{1}}{m_{1}\omega _{1}+m_{2}\omega _{2}}}.}$

The probability that the second ball picked is red depends on whether the first ball was red or white. If the first ball was red then the above formula is used with ${\displaystyle m_{1}}$ reduced by one. If the first ball was white then the above formula is used with ${\displaystyle m_{2}}$ reduced by one.

The important fact that distinguishes Wallenius' distribution is that there is competition between the balls. The probability that a particular ball is taken in a particular draw depends not only on its own weight, but also on the total weight of the competing balls that remain in the urn at that moment. And the weight of the competing balls depends on the outcomes of all preceding draws.

A multivariate version of Wallenius' distribution is used if there are more than two different colors.

The distribution of the balls that are not drawn is a complementary Wallenius' noncentral hypergeometric distribution.

In the Fisher model, the fates of the balls are independent and there is no dependence between draws. One may as well take all n balls at the same time. Each ball has no "knowledge" of what happens to the other balls. For the same reason, it is impossible to know the value of n before the experiment. If we tried to fix the value of n then we would have no way of preventing ball number n + 1 from being taken without violating the principle of independence between balls. n is therefore a random variable, and the Fisher distribution is a conditional distribution which can only be determined after the experiment when n is observed. The unconditional distribution is two independent binomials, one for each color.

Fisher's distribution can simply be defined as the conditional distribution of two or more independent binomial variates dependent upon their sum. A multivariate version of the Fisher's distribution is used if there are more than two colors of balls.

Comparison of distributions with same odds:
Blue: Wallenius ω = 0.5
Red: Fisher ω = 0.5
Green: Central hypergeometric ω = 1.
m₁ = 80, m₂ = 60, n = 100

Comparison of distributions with same mean:
Blue: Wallenius ω = 0.5
Red: Fisher ω = 0.28
Green: Central hypergeometric ω = 1.
m₁ = 80, m₂ = 60, n = 100

Wallenius' and Fisher's distributions are approximately equal when the odds ratio ${\displaystyle \omega =\omega _{1}/\omega _{2}}$ is near 1, and n is low compared to the total number of balls, N. The difference between the two distributions becomes higher when the odds ratio is far from one and n is near N. The two distributions approximate each other better when they have the same mean than when they have the same odds (ω = 1) (see figures above).

Both distributions degenerate into the hypergeometric distribution when the odds ratio is 1, or to the binomial distribution when n = 1.

To understand why the two distributions are different, we may consider the following extreme example: An urn contains one red ball with the weight 1000, and a thousand white balls each with the weight 1. We want to calculate the probability that the red ball is not taken.

First we consider the Wallenius model. The probability that the red ball is not taken in the first draw is 1000/2000 = 1⁄2. The probability that the red ball is not taken in the second draw, under the condition that it was not taken in the first draw, is 999/1999 ≈ 1⁄2. The probability that the red ball is not taken in the third draw, under the condition that it was not taken in the first two draws, is 998/1998 ≈ 1⁄2. Continuing in this way, we can calculate that the probability of not taking the red ball in n draws is approximately 2⁻ⁿ as long as n is small compared to N. In other words, the probability of not taking a very heavy ball in n draws falls almost exponentially with n in Wallenius' model. The exponential function arises because the probabilities for each draw are all multiplied together.

This is not the case in Fisher's model, where balls are taken independently, and possibly simultaneously. Here the draws are independent and the probabilities are therefore not multiplied together. The probability of not taking the heavy red ball in Fisher's model is approximately 1/(n + 1). The two distributions are therefore very different in this extreme case, even though they are quite similar in less extreme cases.

The following conditions must be fulfilled for Wallenius' distribution to be applicable:

• Items are taken randomly from a finite source containing different kinds of items without replacement.
• Items are drawn one by one.
• The probability of taking a particular item at a particular draw is equal to its fraction of the total "weight" of all items that have not yet been taken at that moment. The weight of an item depends only on its kind (e.g., color).
• The total number n of items to take is fixed and independent of which items happen to be taken first.

The following conditions must be fulfilled for Fisher's distribution to be applicable:

• Items are taken randomly from a finite source containing different kinds of items without replacement.
• Items are taken independently of each other. Whether one item is taken is independent of whether another item is taken. Whether one item is taken before, after, or simultaneously with another item is irrelevant.
• The probability of taking a particular item is proportional to its "weight". The weight of an item depends only on its kind (e.g., color).
• The total number n of items that will be taken is not known before the experiment.
• n is determined after the experiment and the conditional distribution for n known is desired.

The following examples illustrate which distribution applies in different situations.

Conclusion to the examples

These examples show that the distribution of the types of fish caught depends on the way they are caught. Many situations will give a distribution that lies somewhere between Wallenius' and Fisher's noncentral hypergeometric distributions.

A consequence of the difference between these two distributions is that one will catch more of the heavy fish, on average, by catching n fish one by one than by catching all n at the same time. In general, we can say that, in biased sampling, the odds parameter has a stronger effect in Wallenius' distribution than in Fisher's distribution, especially when n/N is high.

Probability mass function for Wallenius' noncentral hypergeometric distribution for different values of the odds ratio ω.
m₁ = 80, m₂ = 60, n = 100, ω = 0.1 ... 20

Probability mass function for Fisher's noncentral hypergeometric distribution for different values of the odds ratio ω.
m₁ = 80, m₂ = 60, n = 100, ω = 0.01 ... 1000

• Wallenius' noncentral hypergeometric distribution
• Fisher's noncentral hypergeometric distribution
• Hypergeometric distribution
• Urn problem
• Bias
• Biased sample

Johnson, N. L.; Kemp, A. W.; Kotz, S. (2005), Univariate Discrete Distributions, Hoboken, New Jersey: Wiley and Sons.

McCullagh, P.; Nelder, J. A. (1983), Generalized Linear Models, London: Chapman and Hall.

Fog, Agner (2007), Random number theory.

Fog, Agner (2008), "Calculation Methods for Wallenius' Noncentral Hypergeometric Distribution", Communications in Statistics - Simulation and Computation, vol. 37, no. 2, pp. 258–273, doi:10.1080/03610910701790269, S2CID 9040568.