LINEAR HYPOTHESES IN 2XALPHA FREQUENCY TABLES

An approximate method, with decision‐based error‐rates, for testing linear statistical hypotheses (0κi) about a independent binomial parameters (φj) is given which is more general than the usual methods for partitioning Xm2 from a 2 × a table. Parameters in Table 2 may be used to choose sample‐size (n) for stated alternatives to 0κi and power (or expected proportion of rejections). The procedure may be used for planned tests (v1 = 1) or post‐hoc tests (v1 = a − 1 or v1 = a) and in the latter case the probability of rejecting r of the 0κi (r = 1, 2, …, v1) can be directly calculated for any statement of what is true in the populations. The results from sampling experiments are used to examine the adequacy of the approximation and compare the method proposed (Rv1) with two others (Gv1 and Zv1) Although the latter methods have a power advantage under some circumstances, their experimental results for n ≤ 30 are generally less close to ‘theory’ than those of Rv1, and they are always arithmetically and conceptually more complex. The results from these experiments suggest that the Rv1 approximation is adequate when n ≥ 10 and the row of smaller expected frequencies (nφ) is greater than or equal to 1. When the φj differ and the true φj approach 0 or 1, “end‐effects” appear but these diminish as n increases. These methods can be used in a 2×a generalization of the 2 × 2 median test. If this is interpreted not as a test of population medians, but as a test of proportions of population distributions to the right of Me (the value of the common sample median), which is φj in population j, it is distribution‐free. 1969 The British Psychological Society