## Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 35, Number 1 (1964), 122-137.

### Rank Tests for Paired-Comparison Experiments Involving Several Treatments

#### Abstract

Two nonparametric tests presently available for testing the equality of several treatments (varieties, objects etc.) on the basis of paired-comparisons are: (a) the Bradley-Terry [4] likelihood ratio test; (b) Durbin's [7] test (which, in fact, covers the general balanced incomplete block design). These tests, however, were proposed for the case when no meaningful measurements on the quality of treatments are possible. Instead, one can merely decide for each individual comparison which treatment to prefer. Both these tests are, thus, instances where only signs of the comparison differences are involved, and as such can be regarded as generalizations of the Sign-test. The large sample properties of these tests have been discussed by Bradley [3] and by Van Elteren and Noether [17] respectively. As shown in the latter paper, both these tests have asymptotic (Pitman) efficiency equal to $2/\pi$ relative to the $F$-test (under normality). One can reasonably hope to improve this efficiency by taking into consideration the magnitudes of the observed comparison differences, when they are available. A test of this nature, based on a generalization of the Wilcoxon-one-sample ranking procedure, is proposed and investigated below for the case when all comparisons are performed under the same experimental conditions (Section 2). The asymptotic distribution of the test statistic proposed is obtained using the results of Godwin and Zaremba [8] and Konijn [12] (Section 3). It is shown that the asymptotic efficiencies of this test relative to the Durbin and the Bradley-Terry tests and the corresponding $F$-test are independent of the number of treatments involved. These results are also extended to the case of non-uniformity of experimental conditions (Section 4). (The attention of the reader is also drawn to a rank procedure suggested by Hodges and Lehmann [11] for the general incomplete block design which, in particular, is also applicable to the present problem. However, the efficiency of this procedure has so far not been fully investigated.)

#### Article information

**Source**

Ann. Math. Statist., Volume 35, Number 1 (1964), 122-137.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177703734

**Digital Object Identifier**

doi:10.1214/aoms/1177703734

**Mathematical Reviews number (MathSciNet)**

MR193717

**Zentralblatt MATH identifier**

0131.36102

**JSTOR**

links.jstor.org

#### Citation

Mehra, K. L. Rank Tests for Paired-Comparison Experiments Involving Several Treatments. Ann. Math. Statist. 35 (1964), no. 1, 122--137. doi:10.1214/aoms/1177703734. https://projecteuclid.org/euclid.aoms/1177703734