Tables
For each of c populations, independent SRSs of sizes
are drawn. Each individual in a sample is
classified according to a categorical outcome variable with r
possible values. For the jth population the probability that an
individual will fall into category i is 
.
The null hypothesis is that the distributions of the outcome variable
are the same in all c populations. Letting denote the
proportion of population j in category i, the null hypothesis is
The alternative hypothesis is 
: at least one of the equalities in
does not hold.
The samples sizes from each of the populations are the column totals
in the sample count table. Call these sample sizes 
. In the
first model, the are random variables. The total samples size
n is set by the researcher, and the column sums are known only
after the data are analyzed. For the second model, the column sums
are the sample sizes selected at the design phase of the research.
The null hypothesis in both models says that there is no relationship
between the column variable and the row variable. Although the
hypothesis is expressed differently, the test of the hypothesis in
each case is the same.