Likelihood based Inference using Signed Ranks for Matched Pairs

SUMMARY Exact and approximate inference based on the marginal likelihood which results from the signed ranks of the within pair differences of independent matched pairs is con- sidered. Inference is made using Bayesian ideas. A quickly computed approximate analysis is introduced and this is shown to be extremely good when the within pair differences are assumed to have a normal distribution. Numerical comparisons are also made when the differences have a logistic distribution, but for this case the approxi- mation is not as good. The approximations involve the Wilcoxon signed rank statistic and the half-normal scores signed rank statistic. The approximation is extended to consider regression models for matched pairs data. An application is given illustrating the ideas. In this paper we consider exact and approximate inference based on the marginal rank-sign likelihood which results from independent matched pairs data. It is assumed that the within pair differences are independent and symmetrically distributed. Inference is made using Bayesian ideas. Suppose that D1, . ., 1Dn are observable differences which result from a matched pairs experiment and that it is possible to transform the Dj 's, giving Yj = sign (Dj) h (IDj 1), where h(.) is an unknown increasing and differentiable function on (0, oo) and h(O) = 0. It is assumed that the Y1's are independent and symmetrically distribtuted about 0 with a known density. Under such circumstances the signs of the Dj's and the ranks of the absolute values of the Dj's remain invariant under the transformation, and these are given by the signs and ranks of the Y1's. In non- parametric statistics inference can be made for 0 using the signs of the differences-the sign-test- and also by utilizing between pair comparisons, the ranks of the absolute differences, giving the signed rank test of Wilcoxon or the half-normal scores test. Lehmann (1975, Chapter 5) gives an-account of non-parametric tests based on signed ranks using matched pairs data. In this paper we consider the marginal likelihood, p(r, s I 0), of the Dj's based on their signs, s, and the ranks r, of their absolute values. The idea of a marginal rank likelihood was proposed by Kalbfleisch and Prentice (1973) and the likelihood p(r, s I 0), for matched pairs data, was considered by Woolson and Lachenbruch (1980), who developed test statistics for the hypothesis H: 0 = 0 with censored data. We use the marginal likelihood, p(r, s I 0), to make inferences about 0 from a Bayesian point of view. In particular, when the prior density for 0 is chosen to be locally uniform, the posterior density of 0 is given by the standardized likelihood