Correlation Analysis of Paired Ranking Data

Abstract

A ranking represents the order of preference one has with respect to a set of t objects. If we label the objects by the integers 1 to t, a ranking can then be thought of as a permutation of the integers $(1,2,\ldots,t)$. We may denote such a permutation by μ = (μ(1), μ(2), …, μ(t))′ which may also be conceptualized as a point in t-dimensional space. It is natural to measure the spread between two individual permutations μ, ν by means of a distance function.

Chapter Notes

In this chapter, the traditional rank correlation has been extended to include incomplete rankings. This was made possible using the notion of compatibility which was developed by Alvo and Cabilio in a series of papers. Cabilio and Tilley (1999) report the results of a simulation study where they considered linear, quadratic, and square root trends. They observed that when there were no missing observations, the Spearman statistic was more powerful than Kendall’s. In the incomplete case, however, the new Kendall statistic has superior power for more patterns.

The calculation of the exact variance of $\mathcal{A}_{K}^{{\ast}}$ under H ₂, in Theorem 3.4, is more involved, and the reader is referred to Alvo and Cabilio (1992), where it is shown that

$$\displaystyle{V ar\left (\mathcal{A}_{K}^{{\ast}}\right ) = \frac{\kappa _{1}\kappa _{2}} {9t\left (t - 1\right )}\left [\frac{\left (2t + k_{1} + 3\right )\left (2t + k_{2} + 3\right )} {2} + \frac{\left (t^{2} - k_{1} - 2\right )\left (t^{2} - k_{2} - 2\right )} {\left (t - 2\right )} \right ].}$$

An important application of the results presented above which do not discard missing data is in tests of trend where k ₂ = t and k ₁ < t. It is seen that in this context, the superiority of the extended Spearman statistic is established through the calculation of its asymptotic relative efficiency relative to the “naive” statistic. (Alvo and Cabilio 1994) applied these methods to test for trend in precipitation data for St John and Fredericton (NB) and showed that the extended statistic based on Spearman distance is more sensitive in detecting trends than the statistic which ignores the missing observations. Tables of selected critical values of $\mathcal{A}_{S}^{{\ast}}$ and $\mathcal{A}_{K}^{{\ast}}$ for the trend case when k ≥ t∕2 have been developed for both hypotheses (Alvo and Cabilio 1993). The results of this section have been extended to the case of ties (Yu et al. 2002) and applied to deal with tests of independence in opinion surveys. A further extension to assess trend in proportions appears in Chap. 7.

Alvo and Smrz (2005) proposed an arc model which serves as a good approximation to Kendall distance.

Although not considered in this book, Alvo and Park (2002) were concerned with multivariate tests of trend when the data are partially incomplete. Such is the case in environmental studies when pH data for one or more lakes are often recorded over regular time intervals and examined for monotone increasing or decreasing trends in order to test for trend in acidification. In monitoring recovering patients, one looks for trends in their vital signs which are often multivariate data in nature. There may be as many as 20–30 blood constituents measured weekly over a period of several months or years. In those case, the use of separate tests on each constituent is inefficient.