Significance Tests Which May be Applied to Samples from Any Populations

Abstract

1. THE object of this paper is to show how we can devise valid tests of significance which involve no assumptions about the forms of the populations sampled. It is also shown that precise fiducial limits can be determined for the difference of means of populations of the same form, no matter what the form of the populations may be. While only one test is discussed in this paper, the principle is applicable to all tests. The main idea is not new, it seems to be implicit in all Fisher's writings; * but perhaps the approach to the subject, frankly starting from the sample and working towards the population instead of the reverse, may be a bit of a novelty. Discordant, Concordant, and Neutral Separations.Suppose that we have m + n numbers (not necessarily all different), and that their mean is z. Numbers which are equal in value are supposed to be distinguishable from one another-we may think of the m + n numbers as painted on m + n different marbles.Consider a separation of the numbers into two different classes of m and n, (m ;;;;: n), U 1, U z . . . Um, with mean u,The number, N, of such separations is m+nCm, provided that when m = n the two classes, though equal in size, are regarded as different, so that two separations in which the classes are simply interchanged are regarded as different separations.We shall call Iii~1: ' 1 the spread of the separation. Since mu + nv= (m + n)z, an alternative expression for the spread isLet M be a fixed integer less than N. Consider any particular separation R. If the number of separations with a spread equal to or greater than that of R is less than or equal to M, we shall call R

Efficient calculation ofp-values in linear-statistic permutation significance tests

“…There exists a huge literature in this area, including canonical works by Fisher [1], Pitman [2], Efron [3], Edgington [4], Davison and Hinkley [5] and Good [6]. Unfortunately, even if we restrict ourselves to the simplest case, in which a set of m observations and a set of n observations are pooled and then redistributed in all possible ways, the number of possible permutations can be vast (unless m and n are small), and the impossibility of exhaustively enumerating these in a reasonable time has restricted the extent to which exact permutation tests have been used in practice.…”

Randomization analysis of covariance and change-over designs

“…by Eden and Yates (1933) and others, that the significance level that is obtained by this process is "close to" the significance level that is obtained by using the upper tail area of the associated normal distribution theory. Theoretical work on this was done by Pitman (1937), Welch (1937), Ogawa (1974) and Arnold (1964).…”

“…In Chapters 3 and 4 of this thesis we will examine the behavior under randomization of the usual Gauss-Markov normal theory statistics in the presence of concomitant 26 variation for these two designs. Pitman (1937) and Welch (1937) Ogawa (1974) and Arnold (1964). Pitman (1937) investigated the problem of testing the null hypothesis that the treatments are all equal in the randomized block design for univariate situations.…”

“…Pitman (1937) and Welch (1937) Ogawa (1974) and Arnold (1964). Pitman (1937) investigated the problem of testing the null hypothesis that the treatments are all equal in the randomized block design for univariate situations. In the usual additive model the observation associated with the j-th treatment in the i-th block is denoted by:…”

“…Hence the ratio has an F distribution with (t-1) and (r-l)(t-l) degrees of freedom. Instead of the F ratio Pitman (1937) (ii) Fit a beta distribution by the use of the first two moments of W (also investigated by Welch (1937)). This will mean a decrease in the number of degrees of freedom in the beta distribution which applies under normality.…”