Non-Null Ranking Models. I

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. Kendall (1950) has remarked that the major outstanding problem in ranking theory is the specification of a suitable population of ranks in non-null cases. Much attention has been concentrated on situations which Daniels (1950, ? 5) calls of type (i):'The sample is regarded as having been randomly chosen from a bivariate population of ranks' the underlying population being either finite or infinite, e.g. bivariate Normal. Rather less work has been done on Daniels's type (ii):'There is a fixed set of individuals being assessed by a population of judges, or by the same judge in repeated trials, on a particular attribute whose ranking is known a priori. The random element is uncertainty of preference, the correlation being the result of real differences between the individuals, and the population is one of rankings conditional on a given objective order.' Daniels (1950), following Babington Smith, Thurstone and Mann, treats this problem as one of regression. The present approach is by way of paired-comparison theory. The judge is assumed to arrive at a ranking of n objects U1, U2, ..., U. by first making all the nC2 comparisons between pairs independently, but then only accepting the resultg if they are consistent with a ranking of the n objects.Various non-null models are proposed. The general model (eqn. (1)) depends on 2 parameters-this number is then reduced to n-1 by using the Bradley-Terry (1952) paired-comparison model (eqn. (4)). An alternative method of simplification is proposed which makes the probability of putting (in paired-comparisons) any two objects Ui and U, in the correct order equal to i + i tanh (k log 0 + log s),where 0 and qS are parameters, and k = ji is the difference between the true ranks of the two objects. Thus this probability is a simple monotonic function of a quantity which is composed of a term increasing linearly with k, and a constant term. The null hypothesis corresponds to a = 0 = 1. It is found that 0 is associated with Spearman's coefficient r8, and 0 with Kendall's tk (eqn. (9)). Each of these parameters has a further interpretation. Thus 0 may be regarded as assigning weights to the n objects, these weights being in geometric progression. The paired-comparisons are then made in such a way that the probability of ranking one object higher than another is a simple function of the weights assigned to these two objects. The parameter qS has the following further interpretation: having obtained a ranking of length n-I and wishing to introduce a further object, 0 specifies the probabilities of this object being ranked in the various possible position...

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