# Two-way Contingency Tables

• Erling B. Andersen

## Abstract

A two-way contingency is a number of observed counts set up in a matrix with I rows and J columns. Data are thus given as a matrix
$$X\,\, = \,\,\left[ {\begin{array}{*{20}{c}} {{x_{11}} \ldots {x_{1J}}}\\ { \vdots \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}\\ {{x_{11}} \ldots {x_{IJ}}} \end{array}} \right]$$
The statistical model for such data depends on the way the data are collected. A great variety of such tables can, however, be treated by three closely connected statistical models. Let the random variables corresponding to the contingency table be X11,...,XIJ. Then in the first model the Xij’s are assumed to be independent with
$${x_{ij}}\, \sim \,Ps\left( {{\lambda _{ij}}} \right),$$
i.e. Xij is Poisson distributed with parameter λij. The likelihood function for this model is
$$f\left( {{x_{11}},...,{x_{IJ}}|{\lambda _{11}},...,{\lambda _{IJ}}} \right)\,\, = \,\,\mathop {II}\limits_{i\, = \,1}^I \,\mathop {II}\limits_{j\, = \,1}^J \,\,\frac{{{\lambda _{ij}}^{{x_{ij}}}}}{{{x_{ij}}!}}\,{e^{ - {\lambda _{ij}}}}.$$
(4.1)

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