Noncentral Hypergeometric Distributions - The Difference Between The Two Noncentral Hypergeometric Distributions

The Difference Between The Two Noncentral Hypergeometric Distributions

Wallenius’ and Fisher’s distributions are approximately equal when the odds ratio 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 (w = 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 = ½. 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 ≈ ½. 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 ≈ ½. 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 (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 (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.