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Table of Contents Bivariate Data Correlation Ordinal Association | |
See also: Pearson's Correlation Coefficient |
Gamma defines perfect association as weak monotonicity (see discussion in the section on association). Under statistical independence, gamma will be 0, but it can be 0 at other times as well (whenever concordant minus discordant pairs are 0).
tau-b = (P - Q)/ SQRT[((P + Q + Y0)(P + Q + X0))]
There is no well-defined intuitive meaning for tau-b, which is the surplus of concordant over discordant pairs as a percentage of concordant, discordant, and approximately one-half of tied pairs. The rationale for this is that if the direction of causation is unknown, then the surplus of concordant over discordant pairs should be compared with the total of all relevant pairs, where those relevant are the concordant pairs, the discordant pairs, plus either the X-ties or Y-ties but not both, and since direction is not known, the geometric mean is used as an estimate of relevant tied pairs.
Tau-b defines perfect association as strict monotonicity, as discussed in the section on association. Although it requires strict monotonicity to reach 1.0, it does not penalize ties as much as some other measures. It defines null relationship as statistical independence.
tau-c = (P - Q)*[2m/(n2(m-1))]
where m is the number of rows or columns, whichever is smaller, and n is sample size. Correlation coefficients Suppose we rank a group of eight people by height and by weight:
Person A B C D E F G H Rank by Height 1 2 3 4 5 6 7 8 Rank by Weight 3 4 1 2 5 7 8 6
We can see that there is some correlation between the two rankings but that the correlation is far from perfect, and we would like some way of objectively measuring the degree of correspondence. In the 1940s Maurice Kendall developed a coefficient, t, for this purpose that has the following properties:
It is defined by
where n is the number of items, and P is a quantity derived from the rankings as follows: In the Weight ranking above, the first entry, 3, has five higher ranks to the right of it; the contribution to P of this entry is 5. Moving to the second entry, 4, we see that there are four higher ranks to the right of it and the contribution to P is 4. Continuing this way, we find that P = 5 + 4 + 5 + 4 + 3 + 1 + 0 + 0 = 22. Thus . This result indicates that there is strong agreement between the rankings, as expected.
Last Update: 2006-Jän-18