mWNChygeodds

mWNChygeodds returns Wallenius' multivariate non-central hypergeometric odds given means

expand all in page

Syntax

  • odds=mWNChygeodds(mu, m, n)example

Description

This function calls function oddsMWNCHypergeo_from_mean which is a translation into MATLAB of the corresponding C++ function of Fog (2008).

The notation which is used in mWNChygeodds and the order of the arguments is the one of MATLAB hyge. The notation which is used inside oddsMWNCHypergeo_from_mean is the original one of Fog.

To illustrate the meaning of Wallenius and Fisher' function parameters, let's use the classical biased urn example, with $K$ red balls and $M-K$ white balls, totalling $M$ balls. $n$ balls are drawn at random from the urn without replacement. Each red ball has the weight $\omega_{1}$, and each white ball has the weight $\omega_{2}$; the probability ratio of red over white balls is then given by $odds = \omega_{1} / \omega_{2}$. Note that the odds are fixed once and for all during the drawings.

If the balls are taken one by one, the probability (say $p_1$) that the first ball picked is red is equal to the weight fraction of red balls:

\[ p_1= \frac{K w_1}{K w_1 + (M-K) w_2} \]

In the Wallenius distribution 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 $K$ reduced by one. If the first ball was not red then the above formula is used with $M-K$ reduced by one. The number of red balls that we get in this experiment is a random variable with Wallenius' noncentral hypergeometric distribution.

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. 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.

More formally, if the total number $n$ of balls taken is not known before the experiment (i.e n is determined just after the experiment), then the conditional distribution of the number of taken red balls for given $n$ is Fisher's noncentral hypergeometric distribution.

These two distributions have important applications in evolutionary biology and population genetics. If animals of a particular species are competing for a limited food resource so that individual animals are dying one by one until there is enough food for the remaining animals, and if there are different variants of animals with different chances of finding food, then we can expect the animals that die to follow a Wallenius noncentral hypergeometric distribution. Fisher’s noncentral hypergeometric distribution may be used instead of Wallenius distribution in cases where the fates of the individual animals are independent and the total number of survivors is known or controlled as part of a simulation experiment (Fog 2008).

Wallenius distribution is a general model of biased sampling. Fisher’s noncentral hypergeometric distribution is used for statistical tests on contingency tables where the marginals are fixed (McCullagh and Nelder, 1983).

The multivariate Fishers and Wallenius noncentral hypergeometric distribution, are referred to the case where the type of different balls is greater than 2 and each ball has a different probability.

The two distributions are both equal to the (multivariate) hypergeometric distribution when the odds ratio is 1.

example

odds =mWNChygeodds(mu, m, n) Problem description.

Examples

expand all

  • Problem description.

    • We have 70 balls in a urn of 3 different colors. 

    % Initially in the urn we have 20 red balls, 30 white and 20 black balls.
m=[20,30,20];
% The vector of mean values is 
mu=[15, 30, 5];
% The extracted sample is n=50
n=50;
disp('We have an urn with 3 groups of balls; 20 red, 30 white and 20 black.');
disp('In total the urn contains 70 balls.');
disp('What are the weight of getting the 3 balls (odds)?');
disp('given the above values of mu')
odds=mWNChygeodds(mu,m,n);
disp(odds);
    We have an urn with 3 groups of balls; 20 red, 30 white and 20 black.
In total the urn contains 70 balls.
What are the weight of getting the 3 balls (odds)?
given the above values of mu
Warning: Inf values in  odds 
    1.0000
       Inf
    0.2075

    Related Examples

    expand all

  • Find the odds given mu, m and n.

    • Vector of mu values 

    mu=[8 46 29 17]; 
% Number of balls in the urn for each color
m=[30,60,50,20];
n=sum(mu); % number of extracted balls
odds=mWNChygeodds(mu, m, n);
disp('Odds')
disp(odds)

  • An example in which one of the odds is Inf.

    • The vector of mean values is 

    mu=[15, 30, 5]; 
% The number of balls inside the urn for each color 
m=[20,30,20]; 
% The extracted sample is n=50 
n=sum(mu); 
disp('We have an urn with 3 groups of balls; 20 red, 30 white and 20 black.');
disp('In total the urn contains 70 balls.'); 
disp('What are the weight of getting the 3 balls (odds)?'); 
disp('given the above values of mu')
odds=mWNChygeodds(mu,m,n); 
disp(odds);

  • An example in which one of the odds is 0 m=[20,30,20]; The vector of mean values is mu=[6, 0, 5]; The extracted sample is n=10 n=10; disp('We have an urn with 3 groups of balls; 20 red, 30 white and 20 black.

    • m=[20,30,20];
% The vector of mean values is 
mu=[6, 0, 5];
% The extracted sample is n=10
n=10;
disp('We have an urn with 3 groups of balls; 20 red, 30 white and 20 black.');
disp('In total the urn contains 70 balls.');
disp('What are the weight of getting the 3 balls (odds)?');
disp('given the above values of mu')
odds=mWNChygeodds(mu,m,n);
disp(odds);
    We have an urn with 3 groups of balls; 20 red, 30 white and 20 black.
In total the urn contains 70 balls.
What are the weight of getting the 3 balls (odds)?
given the above values of mu
Warning: 0 values in  odds 
    1.2398
         0
    1.0000

  • An example where one odd is much bigger.

    • m=[200,300,200,10];
% The vector of mean values is 
mu=[6, 1, 5, 9];
% The extracted sample n
n=sum(mu);
odds=mWNChygeodds(mu,m,n);
disp(odds);

  • Comparison between Fisher and Wallenius odds.

    • An example where one odd is much greater than the others 

    m=[200,300,200,10];
% The vector of mean values is 
mu=[6, 1, 5, 9];
% The extracted sample n
n=sum(mu);
oddsF=mFNChygeodds(mu,m,n);
oddsW=mWNChygeodds(mu,m,n);
% Row names and column names
rn="odds"+(1:length(mu));
cn=["Fisher" "Wallenius"];
oddsT=array2table([oddsF, oddsW],"RowNames",rn,"VariableNames",cn);
disp(oddsT);
                 Fisher     Wallenius
             _______    _________

    odds1          1           1 
    odds2    0.10814     0.10962 
    odds3    0.82906      0.8312 
    odds4        291      75.596

    Input Arguments

    expand all

    mu — Measured mean for each color. Vector.

    Length of vector mu is equal to the number of colors.

    The sum of the elements of mu must be equal to n.

    Data Types: single|double

    m — Initial number of balls of each color in the urn. Vector.

    The length of vector M is the number of colors in the urn. M=sum(m) is the the total number of balls.

    If the number of colors is equal to 2, m(1)+m(2) corresponds to M and m(1) corresponds to K (number of red balls in the urn).

    Data Types: single|double

    n — Total number of balls sampled. Scalar.

    Scalar which defines the number of balls which are drawn from the urn.

    Note that sum(mu) must be equal n, however, following the original C++ implementation of Fog we allow abs(sum(mu)-n)/n<=0.1

    Data Types: single|double

    Output Arguments

    expand all

    odds —Wallenius' odds (weights) for each color. Vector

    Vector containing the estimated odds for each color given the input means. The reference color has odds=1.

    References

    Fog, A. (2008), Calculation Methods for Wallenius' Noncentral Hypergeometric Distribution, "Communications in Statistics - Simulation and Computation", Vol. 37, pp. 258-273.

    See Also

    | | | | | | |

    This page has been automatically generated by our routine publishFS