
In
probability theory
Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, Fisher's noncentral hypergeometric distribution is a generalization of the
hypergeometric distribution
In probability theory and statistics, the hypergeometric distribution is a Probability distribution#Discrete probability distribution, discrete probability distribution that describes the probability of k successes (random draws for which the ...
where sampling probabilities are modified by weight factors. It can also be defined as the
conditional distribution
Conditional (if then) may refer to:
* Causal conditional, if X then Y, where X is a cause of Y
*Conditional probability, the probability of an event A given that another event B
* Conditional proof, in logic: a proof that asserts a conditional, ...
of two or more
binomially distributed variables dependent upon their fixed sum.
The distribution may be illustrated by the following
urn model. Assume, for example, that an urn contains ''m''
1 red balls and ''m''
2 white balls, totalling ''N'' = ''m''
1 + ''m''
2 balls. Each red ball has the weight ω
1 and each white ball has the weight ω
2. We will say that the odds ratio is ω = ω
1 / ω
2. Now we are taking balls randomly in such a way that the probability of taking a particular ball is proportional to its weight, but independent of what happens to the other balls. The number of balls taken of a particular color follows the
binomial distribution
In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
. If the total number ''n'' of balls taken is known then the conditional distribution of the number of taken red balls for given ''n'' is Fisher's noncentral hypergeometric distribution. To generate this distribution experimentally, we have to repeat the experiment until it happens to give ''n'' balls.
If we want to fix the value of ''n'' prior to the experiment then we have to take the balls one by one until we have ''n'' balls. The balls are therefore no longer independent. This gives a slightly different distribution known as
Wallenius' noncentral hypergeometric distribution. It is far from obvious why these two distributions are different. See the entry for
noncentral hypergeometric distributions for an explanation of the difference between these two distributions and a discussion of which distribution to use in various situations.
The two distributions are both equal to the (central)
hypergeometric distribution
In probability theory and statistics, the hypergeometric distribution is a Probability distribution#Discrete probability distribution, discrete probability distribution that describes the probability of k successes (random draws for which the ...
when the
odds ratio
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of event A taking place in the presence of B, and the odds of A in the absence of B ...
is 1.
Unfortunately, both distributions are known in the literature as "the" noncentral hypergeometric distribution. It is important to be specific about which distribution is meant when using this name.
Fisher's noncentral hypergeometric distribution was first given the name extended hypergeometric distribution (Harkness, 1965), and some authors still use this name today.
Univariate distribution
The probability function, mean and variance are given in the adjacent table.
An alternative expression of the distribution has both the number of balls taken of each color and the number of balls not taken as random variables, whereby the expression for the probability becomes symmetric.
The calculation time for the probability function can be high when the sum in ''P''
0 has many terms. The calculation time can be reduced by calculating the terms in the sum recursively relative to the term for ''y'' = ''x'' and ignoring negligible terms in the tails (Liao and Rosen, 2001).
The mean can be approximated by:
:
,
where
,
,
.
The variance can be approximated by:
:
.
Better approximations to the mean and variance are given by Levin (1984, 1990), McCullagh and Nelder (1989), Liao (1992), and Eisinga and Pelzer (2011). The saddlepoint methods to approximate the mean and the variance suggested Eisinga and Pelzer (2011) offer extremely accurate results.
Properties
The following symmetry relations apply:
:
:
:
Recurrence relation:
:
The distribution is affectionately called "finchy-pig," based on the abbreviation convention above.
Derivation
The univariate noncentral hypergeometric distribution may be derived alternatively as a conditional distribution in the context of two binomially distributed random variables, for example when considering the response to a particular treatment in two different groups of patients participating in a clinical trial. An important application of the noncentral hypergeometric distribution in this context is the computation of exact confidence intervals for the odds ratio comparing treatment response between the two groups.
Suppose ''X'' and ''Y'' are binomially distributed random variables counting the number of responders in two corresponding groups of size ''m''
X and ''m''
Y respectively,
:
.
Their odds ratio is given as
:
.
The responder prevalence
is fully defined in terms of the odds
,
, which correspond to the sampling bias in the urn scheme above, i.e.
:
.
The trial can be summarized and analyzed in terms of the following contingency table.
In the table,
corresponds to the total number of responders across groups, and ''N'' to the total number of patients recruited into the trial. The dots denote corresponding frequency counts of no further relevance.
The sampling distribution of responders in group X conditional upon the trial outcome and prevalences,
,
is noncentral hypergeometric:
Note that the denominator is essentially just the numerator, summed over all events of the joint sample space
for which it holds that
. Terms independent of ''X'' can be factored out of the sum and cancel out with the numerator.
Multivariate distribution
The distribution can be expanded to any number of colors ''c'' of balls in the urn. The multivariate distribution is used when there are more than two colors.
The probability function and a simple approximation to the mean are given to the right. Better approximations to the mean and variance are given by McCullagh and Nelder (1989).
Properties
The order of the colors is arbitrary so that any colors can be swapped.
The weights can be arbitrarily scaled:
:
for all
Colors with zero number (''m''
''i'' = 0) or zero weight (ω
''i'' = 0) can be omitted from the equations.
Colors with the same weight can be joined:
:
where
is the (univariate, central) hypergeometric distribution probability.
Applications
Fisher's noncentral hypergeometric distribution is useful for models of biased sampling or biased selection where the individual items are sampled independently of each other with no competition. The bias or odds can be estimated from an experimental value of the mean. Use
Wallenius' noncentral hypergeometric distribution instead if items are sampled one by one with competition.
Fisher's noncentral hypergeometric distribution is used mostly for tests in
contingency table
In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the multivariate frequency distribution of the variables. They are heavily used in survey research, business int ...
s where a conditional distribution for fixed margins is desired. This can be useful, for example, for testing or measuring the effect of a medicine. See McCullagh and Nelder (1989).
Software available
FisherHypergeometricDistributionin
Mathematica
Wolfram (previously known as Mathematica and Wolfram Mathematica) is a software system with built-in libraries for several areas of technical computing that allows machine learning, statistics, symbolic computation, data manipulation, network ...
.
* An implementation for the
R programming language
R is a programming language for statistical computing and data visualization. It has been widely adopted in the fields of data mining, bioinformatics, data analysis, and data science.
The core R language is extended by a large number of so ...
is available as the package name
BiasedUrn Includes univariate and multivariate probability mass functions, distribution functions,
quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities or dividing the observations in a sample in the same way. There is one fewer quantile t ...
s,
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
generating functions, mean and variance.
* The
R packag
MCMCpackincludes the univariate probability mass function and random variable generating function.
*
SAS System
SAS (previously "Statistical Analysis System") is a statistical software suite developed by SAS Institute for data management, advanced analytics, multivariate analysis, business intelligence, and predictive analytics.
SAS was developed at No ...
includes univariate probability mass function and distribution function.
* Implementation in
C++ is available fro
www.agner.org
* Calculation methods are described by Liao and Rosen (2001) and Fog (2008).
See also
*
Noncentral hypergeometric distributions
*
Wallenius' noncentral hypergeometric distribution
*
Hypergeometric distribution
In probability theory and statistics, the hypergeometric distribution is a Probability distribution#Discrete probability distribution, discrete probability distribution that describes the probability of k successes (random draws for which the ...
*
Urn models
*
Biased sample
*
Bias
Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is inaccurate, closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individ ...
*
Contingency table
In statistics, a contingency table (also known as a cross tabulation or crosstab) is a type of table in a matrix format that displays the multivariate frequency distribution of the variables. They are heavily used in survey research, business int ...
*
Fisher's exact test
Fisher's exact test (also Fisher-Irwin test) is a statistical significance test used in the analysis of contingency tables. Although in practice it is employed when sample sizes are small, it is valid for all sample sizes. The test assumes that a ...
References
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{{DEFAULTSORT:Fisher's Noncentral Hypergeometric Distribution
Discrete distributions
Ronald Fisher