Page's trend test
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In statistics, the Page test for multiple comparisons between ordered alternatives is a generalisation of the test of the statistical significance of a correlation performed using Spearman's rank correlation coefficient. It is also known as Page's trend test or Page's L test.
The Page test is useful in the situation where:
- there are three or more conditions,
- a number of subjects (or other randomly sampled entities) are all observed in each of them
- we predict that the observations will have a particular order.
For example, a number of subjects might each be given three trials at the same task, and we predict that performance will improve from trial to trial. A test of the significance of the trend between conditions in this situation was developed by Page (1963). More formally, the test considers the null hypothesis that, for n conditions, where mi is a measure of the central tendency of the ith condition,
m1 = m2 = m3 = ... = mn
against the alternative hypothesis that
m1 > m2 > m3 > ... > mn
As such it is more powerful than a test such as the Friedman test that uses the data in similar ways, but tests for the alternative hypothesis that the central tendencies of the observations under the n conditions are different, without specifying their order.
The procedure for carrying out the Page test, when there are k subjects each exposed to n conditions, is as follows:
- Arrange the n conditions in the order implied by the alternative hypothesis, and assign each of them a rank Yi
- For each of the k subjects separately, rank the n observations from 1 to n.
- Add the ranks for each condition to give a total Xi.
- Multiply Xi by Yi and add all the products together; this sum is called L.
- To test whether there is a significant trend, values of L can be compared with those tabulated by Page (1963).
- Alternatively, the quantity
(12L - 3kn(n+1)2)2/(kn2(n2-1)(n+1))
- may be compared with values of chi-squared with one degree of freedom. This gives a two-tailed test. The approximation is reliable for more than 20 subjects with any number of conditions, for more than 12 subjects when there are 4 or more conditions, and for any number of subjects when there are 9 or more conditions.
- If a measure of the overall correlation between the conditions and the data is required, it can be calculated as
ρ = 12L/k(n3-n) - 3(n+1)/(n-1)
- if k=1, this reduces to the familiar Spearman coefficient.
The Page test is most often used with fairly small numbers of conditions and subjects. The minimum values of L for significance at the .05 level, one-tailed, with three conditions, are 56 for 4 subjects (the lowest number that is capable of giving a significant result at this level), 54 for 5 subjects, 91 for 7 subjects, 128 for 10 subjects, 190 for 15 subjects and 251 for 20 subjects.
A corresponding extension of Kendall's tau correlation coefficient was developed by Jonckheere (1954).
References
- Jonckheere, A. R., (1954). A test of significance for the relation between m rankings and k ranked categories. British Journal of Statistical Psychology, 7, 93-100.
- Page, E. B. (1963). Ordered hypotheses for multiple treatments: A significance test for linear ranks. Journal of the American Statistical Association, 58, 216-230.