Contingency Table

Definition

$J\times K$ contingency table

	$B_1$	$B_2$	$\dots$	$B_K$	Total
$A_1$	$Y_{11}$	$Y_{12}$	$\dots$	$Y_{1K}$	$Y_{1.}$
$A_2$	$Y_{21}$	$Y_{22}$	$\dots$	$Y_{2K}$	$Y_{2.}$
$⋮$	$⋮$	$⋮$	$\ddots$	$⋮$	$⋮$
$A_J$	$Y_{J1}$	$Y_{J2}$	$\dots$	$Y_{JK}$	$Y_{J.}$
Total	$Y_{.1}$	$Y_{.2}$	$\dots$	$Y_{.K}$	$n=Y_{..}$

Contingency table is a table displays the multivariate frequency distribution of the variables.

Examples

Poisson and Multinomial Distribution Cases

2-Dimensional Case

Consider a $J\times K$ contingency table.

No Margin Fixed

If $Y_{jk}$‘s are independent and $Y_{jk}\sim \mathrm{Pois}({\lambda }_{jk})$, then the joint PDF becomes $f(\mathbf{y};\mathbf{\lambda })={\prod }_{j=1}^J{\prod }_{k=1}^K\frac{{\lambda }_{jk}^{y_{jk}}{e}^{-{\lambda }_{jk}}}{y_{jk}!}$

Fixed Total

Given the total $n=Y_{..}={\sum }_{j=1}^J{\sum }_{k=1}^K{Y}_{jk}$, the conditional distribution is Multinomial Distribution. $f(\mathbf{y}|Y_{..}=n)=n!{\prod }_{j=1}^J{\prod }_{k=1}^K\frac{{\theta }_{jk}^{y_{jk}}}{y_{jk}!}\sim \mathrm{Mult}(n,{\theta }_{11},{\theta }_{12},\dots ,{\theta }_{JK})$ where ${\theta }_{jk}^{y_{jk}}=\frac{{\lambda }_{jk}}{{\sum }_{j=1}^J{\sum }_{k=1}^K{\lambda }_{jk}}$ and ${\sum }_{j=1}^J{\sum }_{k=1}^K{\theta }_{jk}=1$

One Margin Fixed

Given one fixed margin (in this case, row) $n_j=Y_{j.}$, the conditional distribution of frequency of $j$-th row is Multinomial Distribution. $f(y_{j1},y_{j2},\dots ,y_{jK}|y_{j.})=y_{j.}!{\prod }_{k=1}^K\frac{{\theta }_{jk}^{y_{jk}}}{y_{jk}!}\sim \mathrm{Mult}(n,{\theta }_{j1},{\theta }_{j2},\dots ,{\theta }_{jK})$ And if each row is independent, the conditional distribution is product-Multinomial Distribution (joint multinomial distribution). $f(\mathbf{y}|y_{j.}=n_j,j=1,\dots ,J)={\prod }_{j=1}^J\left[y_{j.}!{\prod }_{k=1}^K\frac{{\theta }_{jk}^{y_{jk}}}{y_{jk}!}\right]$

3-Dimensional Case

Consider a $J\times K\times L$ contingency table.

No Margin Fixed

If $Y_{jkl}$‘s are independent and $Y_{jkl}\sim \mathrm{Pois}({\lambda }_{jkl})$, then the joint PDF becomes $f(\mathbf{y};\mathbf{\lambda })={\prod }_{j=1}^J{\prod }_{k=1}^K{\prod }_{l=1}^L\frac{{\lambda }_{jkl}^{y_{jkl}}{e}^{-{\lambda }_{jkl}}}{y_{jkl}!}$

Fixed Total

Given the total $n=Y_{...}={\sum }_{j=1}^J{\sum }_{k=1}^K{\sum }_{k=1}^L{Y}_{jkl}$, the conditional distribution is Multinomial Distribution. $f(\mathbf{y}|Y_{...}=n)=n!{\prod }_{j=1}^J{\prod }_{k=1}^K{\prod }_{l=1}^L\frac{{\theta }_{jkl}^{y_{jkl}}}{y_{jkl}!}\sim \mathrm{Mult}(n,{\theta }_{111},{\theta }_{112},\dots ,{\theta }_{JKL})$ where ${\theta }_{jkl}^{y_{jkl}}=\frac{{\lambda }_{jkl}}{{\sum }_{j=1}^J{\sum }_{k=1}^K{\sum }_{l=1}^L{\lambda }_{jkl}}$ and ${\sum }_{j=1}^J{\sum }_{k=1}^K{\sum }_{l=1}^L{\theta }_{jkl}=1$

One Margin Fixed

Given one fixed margin (in this case, $L$) $n_l=Y_{..l}$, and each $l$-th rows are independent, the conditional distribution is product-Multinomial Distribution (joint multinomial distribution). $f(\mathbf{y}|y_{..l}=n_l,l=1,\dots ,L)={\prod }_{l=1}^L\left[y_{..l}!{\prod }_{j=1}^J{\prod }_{k=1}^K\frac{{\theta }_{jkl}^{y_{jkl}}}{y_{jkl}!}\right]$

Two Margin Fixed

If two margins are fixed (in this case, $J$ and $L$) $n_{jl}=Y_{j.l}$, and each $j$-th and $l$-th rows are independent, the conditional distribution is product-Multinomial Distribution (joint multinomial distribution). $f(\mathbf{y}|y_{j.l}=n_{jl},j=1,\dots ,J;l=1,\dots ,L)={\prod }_{j=1}^J{\prod }_{l=1}^L\left[y_{j.l}!{\prod }_{k=1}^K\frac{{\theta }_{jkl}^{y_{jkl}}}{y_{jkl}!}\right]$