Two-way Contingency Tables

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 $$X = \left[ \begin{array}{l} {x_{11}} \cdots {x_{1J}}\\ \vdots \\ {x_{I1}} \cdots {x_{IJ}} \end{array} \right]$$ The statistical model for such data depends on the way the data are collected. A great variety of tables can, however, be treated by three closely connected statistical models. Let the random variables corresponding to the contingency table be X₁₁,…,X_IJ. Then in the first model the X_ij’sare assumed to be independent with $${X_{ij}} \sim Ps({\lambda _{ij}}),$$ i.e. X_ij is Poisson distributed with parameter λ_ij. The likelihood function for this model is 4.1$$f({x_{11}},...,{x_{IJ}}|{\lambda _{11}},...,{\lambda _{IJ}}) = \mathop {II}\limits_{i = 1}^I \;\mathop {II}\limits_{j = 1}^J \frac{{\lambda _{ij}^{{x_{ij}}}}}{{x_{ij}^!}}{e^{ - {\lambda _{ij}}}}$$ The log-likelihood is accordingly given by 4.2$$\ln {\rm{L(}}{\lambda _{11}}{\rm{,}} \ldots ,{\lambda _{{\rm{IJ}}}}{\rm{) = }}\sum\limits_{\rm{i}} {\sum\limits_{\rm{j}} {{{\rm{x}}_{{\rm{ij}}}}\ln {\lambda _{{\rm{ij}}}} - \sum\limits_{\rm{i}} {\sum\limits_{\rm{j}} {{\rm{ln}}{{\rm{x}}_{{\rm{ij}}}}!} - \sum\limits_{\rm{i}} {\sum\limits_{\rm{j}} {{\lambda _{{\rm{ij}}}}.} } } } }$$ The model is thus a IJ-dimensional log-linear model with canonical parameters lnλ₁₁,...,1nλ_IJ and sufficient statistics T_ij=X_ij, i=1,...,I, j=1,...,J.