Part IX
Tests on Contingency Tables
Contingency tables are frequently used to present the outcome of a sample of categorical random variables. These variables can also be the result of categorizing the output of continuous random variables. Of interest are, for example, homogeneity or independence between the variables. We focus on two-dimensional tables, where the categories of one variable define the rows and the categories of another variable the columns. Each cell then contains the frequency of occurrence of the row/column combination in the sample. The simplest case is a 
table:
Here we have two binary random variables 
and 
with marginal binomial distribution, or two random variables which are dichotomized into two outcome groups, with labels 
and 
. Usually the absolute counts are listed, so 
is the count of outcome 
of random variable 
and outcome 
of random variable 
, whereas 
usually denotes the (marginal) sum of the counts of the first column. Instead of absolute counts in a contingency table sometimes relative counts are reported. If not stated otherwise, we work with absolute counts.
Extending this notation to 
and 
possible outcomes of 
and 
, respectively, we get:
While setting up tests we formulate a test statistic as a function of the random sample to be observed. For this purpose we further denote the random variable withoutput 
by 
, 
, 
. Concerning distributional assumptions for contingency tables commonly three different sampling distributions are distinguished for the 
's, depending on the employed sampling scheme. If the sample size is not known beforehand, for example, if observations are taken over a specific period of time, it is assumed that each 
follows an independent Poisson distribution. For fixed sample sizes 
we get a multinomial distribution characterized by 
and the cell probabilities 
. In experimental studies the total number of individuals in each group is also often fixed and the resulting sample distribution is a product of independent multinomial distributions. Throughout Chapter 14 we use the above notation.