Fisher's noncentral hypergeometric distribution

The following symmetry relations apply:

${\displaystyle \operatorname {fnchypg} (x;n,m_{1},N,\omega )=\operatorname {fnchypg} (n-x;n,m_{2},N,1/\omega )\,.}$

${\displaystyle \operatorname {fnchypg} (x;n,m_{1},N,\omega )=\operatorname {fnchypg} (x;m_{1},n,N,\omega )\,.}$

${\displaystyle \operatorname {fnchypg} (x;n,m_{1},N,\omega )=\operatorname {fnchypg} (m_{1}-x;N-n,m_{1},N,1/\omega )\,.}$

Recurrence relation:

${\displaystyle \operatorname {fnchypg} (x;n,m_{1},N,\omega )=\operatorname {fnchypg} (x-1;n,m_{1},N,\omega ){\frac {(m_{1}-x+1)(n-x+1)}{x(m_{2}-n+x)}}\omega \,.}$

The distribution is affectionately called "finchy-pig," based on the abbreviation convention above.

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,

${\displaystyle X\sim \operatorname {Bin} (m_{X},\pi _{X}),\quad Y\sim \operatorname {Bin} (m_{Y},\pi _{Y})\,}$.

Their odds ratio is given as

${\displaystyle \omega ={\frac {\omega _{X}}{\omega _{Y}}}={\frac {\pi _{X}/(1-\pi _{X})}{\pi _{Y}/(1-\pi _{Y})}}}$.

The responder prevalence ${\displaystyle \pi _{i}}$ is fully defined in terms of the odds ${\displaystyle \omega _{i}}$, ${\displaystyle i\in \{X,Y\}}$, which correspond to the sampling bias in the urn scheme above, i.e.

${\displaystyle \pi _{i}={\frac {\omega _{i}}{1+\omega _{i}}}}$.

The trial can be summarized and analyzed in terms of the following contingency table.

In the table, ${\displaystyle n=x+y}$ 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, ${\displaystyle Pr(X=x\;|\;X+Y=n,m_{X},m_{Y},\omega _{X},\omega _{Y})}$, is noncentral hypergeometric:

${\displaystyle {\begin{aligned}F(X,\omega ):&=Pr(X=x\;|\;X+Y=n,m_{X},m_{Y},\omega _{X},\omega _{Y})\\&={\frac {Pr(X=x,X+Y=n\;|\;m_{X},m_{Y},\omega _{X},\omega _{Y})}{Pr(X+Y=n\;|\;m_{X},m_{Y},\omega _{X},\omega _{Y})}}\\&={\frac {Pr(X=x\;|\;m_{X},\omega _{X})Pr(Y=n-x\;|\;m_{Y},\omega _{Y},X=x)}{Pr(X+Y=n\;|\;m_{X},m_{Y},\omega _{X},\omega _{Y})}}\\&={\frac {{\binom {m_{X}}{x}}\pi _{X}^{x}(1-\pi _{X})^{m_{X}-x}{\binom {m_{Y}}{n-x}}\pi _{Y}^{n-x}(1-\pi _{Y})^{m_{Y}-(n-x)}}{Pr(X+Y=n\;|\;m_{X},m_{Y},\omega _{X},\omega _{Y})}}\\&={\frac {{\binom {m_{X}}{x}}\omega _{X}^{x}(1-\pi _{X})^{m_{X}}{\binom {m_{Y}}{n-x}}\omega _{Y}^{n-x}(1-\pi _{Y})^{m_{Y}}}{Pr(X+Y=n\;|\;m_{X},m_{Y},\omega _{X},\omega _{Y})}}\\&={\frac {{\binom {m_{X}}{x}}{\binom {m_{Y}}{n-x}}\omega ^{x}(1-\pi _{X})^{m_{X}}\omega _{Y}^{n}(1-\pi _{Y})^{m_{Y}}}{(1-\pi _{X})^{m_{X}}\omega _{Y}^{n}(1-\pi _{Y})^{m_{Y}}\sum _{u=\max(0,n-m_{Y})}^{\min(m_{X},n)}{\binom {m_{X}}{u}}{\binom {m_{Y}}{n-u}}\omega ^{u}}}\\&={\frac {{\binom {m_{X}}{x}}{\binom {m_{Y}}{n-x}}\omega ^{x}}{\sum _{u=\max(0,n-m_{Y})}^{\min(m_{X},n)}{\binom {m_{X}}{u}}{\binom {m_{Y}}{n-u}}\omega ^{u}}}\end{aligned}}}$

Note that the denominator is essentially just the numerator, summed over all events of the joint sample space ${\displaystyle (X,Y)}$ for which it holds that ${\displaystyle X+Y=n}$. Terms independent of X can be factored out of the sum and cancel out with the numerator.

The order of the colors is arbitrary so that any colors can be swapped.

The weights can be arbitrarily scaled:

${\displaystyle \operatorname {mfnchypg} (\mathbf {x} ;n,\mathbf {m} ,{\boldsymbol {\omega }})=\operatorname {mfnchypg} (\mathbf {x} ;n,\mathbf {m} ,r{\boldsymbol {\omega }})\,\,}$ for all ${\displaystyle r\in \mathbb {R} _{+}.}$

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:

${\displaystyle {\begin{aligned}&{}\operatorname {mfnchypg} \left(\mathbf {x} ;n,\mathbf {m} ,(\omega _{1},\ldots ,\omega _{c-1},\omega _{c-1})\right)\\&{}=\operatorname {mfnchypg} \left((x_{1},\ldots ,x_{c-1}+x_{c});n,(m_{1},\ldots ,m_{c-1}+m_{c}),(\omega _{1},\ldots ,\omega _{c-1})\right)\,\cdot \\&\qquad \operatorname {hypg} (x_{c};x_{c-1}+x_{c},m_{c},m_{c-1}+m_{c})\end{aligned}}}$

where ${\displaystyle \operatorname {hypg} (x;n,m,N)}$ is the (univariate, central) hypergeometric distribution probability.