2. p-values. Consider a family of tests of H : 8 = 80 (or 8 :5; 80 ) , with level-a rejection regions Sa such that

(a) PSo{ X E Sa} = a for all 0 < a < 1, and

(b) Sao = (Ia>aoSa for all 0 < ao < 1, which in particular implies Sa C Sa' for a < a' . (i) Then the p-value a is given by a = a(x) = inf{ a : x E Sa}. (ii) When 8 = 80 , the distribution of ais the uniform distribution over (0, 1). (iii) If the tests Sa are unbiased, the distribution of aunder any alternative 8 satisfies Ps{a :5; a} PSo{a :5; a} =

a, so that it is shifted toward the origin. If p-values are available from a number of independent experiments, they can be combined by (ii) and (iii) to provide an overall test* of the hypothesis. [a :5; a if and only if x E Sa' and hence Ps{a:5; a} = Ps{X E Sa} = Pa (8), which is a for 8 = 80 and a if 8 is an alternative to H.] "For discussions of such tests see for example Koziol and Perlman (1978), Berk and Cohen (1979), Mudholkar and George (1979), Scholz (1982), and the related work of Marden (1982). Associated confidence intervals are proposed by Littell and Louv (1981).