In this Chapter, two tests are considered for SP estimation, both based on contingency tables. Here, *G*_{m,λm} is a χ^{2} distribution with *C* degrees of freedom and noncentrality parameter λ_{m}, that is *G*_{m,λm} = χ^{2}_{C–1,λm}.

First, the test for comparing the outcomes of a categorical variable in two populations is studied. Then, the problem of comparing two proportions over a certain number of strata is considered, and the so-called Mantel-Haenszel test is adopted.

Let’s now focus on categorical data with *C* different categories. Consider the *m*_{1} sized sample from the first population (i.e. *X*_{1j}, *j* = 1,…, *m*_{1}) and the *m*_{2} sized one from the second population (i.e. *X*_{2j}, *j* = 1,…, *m*_{2}) falling into *C* different categories. For each group there is a specific probability to fall into the generic category *h*: *P*(*X*_{ij} = *h*) = π_{i,h} for each *j*, with *h* - 1,…, *C, i* = 1, 2. These probabilities are collected in two vectors π_{i} = (π_{i,1},…, π_{i,C}), *i* = 1, 2, whose sums are equal to 1 in each group, i.e. . In other words, the random variables *X*_{ij} have multinomial distribution _{t}*F*_{i} = π_{i}, *j* = 1,…, *m*_{i}, *i* = 1, 2.

The null hypothesis is that there is no difference between _{t}*F*_{1} and _{t}*F*_{2}, that is, between the two vectors of probabilities: *H*_{0} : π_{1} = π_{2}, whereas the alternative is that there are some differences: ...

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