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Cochran's Q test for differences between related-sample percentages or proportions has generally been incorrectly presented in secondary sources. The most common mistake results from failure to recognize that rows containing only 1's or only O's, i.e., only successes or only failures, do not affect the value of Q. The F test, however, is affected by such rows. The probabilities from the x[sup 2] and F approximations are compared with the exact probabilities in three sets of data. A rule of thumb, based on extensive study of the distribution of Q in small samples, is given as an aid in judging when the x[sup 2] approximation is satisfactory for practical purposes. [ABSTRACT FROM AUTHOR]

When the hypothesis H[sub 0]: theta = theta[sub 0] is tested against the alternative H[sub alpha]: theta is not equal to theta[sub 0], theta a real-valued parameter, it would often be more appropriate to make simultaneous tests of the two hypotheses [sub 1]H[sub 0]: theta less than or equal to theta[sub 0] (with alternative [sub 1]H[sub alpha]: theta > theta[sub 0]) and [sub 2]H[sub 0]: theta is greater than or equal to theta[sub 0] (with alternative : [sub 2]H[sub 0]: theta < theta[sub 0]). The criterion of unbiasedness in choosing an optimal test of Ho is inapplicable for choosing an optimal simultaneous procedure for testing [sub 1]H[sub 0] and [sub 2]H[sub 0]. A bidirectional unbiased procedure is defined and its properties enumerated. In many important applications such a procedure for testing [sub 1]H[sub 0] and [sub 2]H[sub 0] is essentially equivalent to an "equal tails" test of H[sub 0]. [ABSTRACT FROM AUTHOR]

The article comments on a weakened principle of conditionality. The idea that the principle of conditionality ought to be weakened to because all the data of a! experiment beyond the minimal sufficient statistic are superfluous seems to me like the plausible and frustrating idea of the geometry pupil who, since lines of construction cannot affect the truth of a theorem, concludes that they ought not be referred to in its demonstration. How common and right it is to say that sampling with, and sampling without, replacement are practically equivalent when the finite population is large compared to the sample size. Solid though seems to me, it represents an assumption and must be judged by the reader himself, as must a somewhat less familiar continuity assumption to be introduced a little later. Incidentally some seem to find the notion of a function, Ev, the value of which is the evidential meaning of the outcome x to the experiment B so disturbing as to impede further discussion. They, like me, might prefer to begin with the notion that sometimes the outcome x of experiment B cannot justifiably provoke any different statistical reaction from those provoked by x' and B'. Such a pair might be called evidentially equivalent, though just which pairs are equivalent is still under discussion.

The conditionality (ancillarity) concept is discussed with particular reference to Durbin's proposed reformulation (C') of Birnbaum's formulation (C) and Fisher's original formulations [6, 7]. It is pointed out that (C') is not in general suitable for applications in the usual sense, since it fails to eliminate all ambiguities of choice of the conditional frame of reference in which standard statistical methods may be applied. The alternative formulations of conditionality are discussed in relation to sufficiency, likelihood, and related methodological problems. [ABSTRACT FROM AUTHOR]