Success Probability Estimation with Applications to Clinical Trials

CHAPTER 8

SP ESTIMATION FOR CHI-SQUARE STATISTICAL TESTS

In this Chapter, two tests are considered for SP estimation, both based on contingency tables. Here, G_m,λm is a χ² distribution with C degrees of freedom and noncentrality parameter λ_m, that is G_m,λm = χ²_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.

8.1 Test for two multinomial distributions: 2 × C comparative trial

Let’s now focus on categorical data with C different categories. Consider the m₁ sized sample from the first population (i.e. X_1j, j = 1,…, m₁) and the m₂ sized one from the second population (i.e. X_2j, j = 1,…, m₂) 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 _tF_i = π_i, j = 1,…, m_i, i = 1, 2.

The null hypothesis is that there is no difference between _tF₁ and _tF₂, that is, between the two vectors of probabilities: H₀ : π₁ = π₂, whereas the alternative is that there are some differences: ...