In statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the Page test for multiple comparisons between ordered correlated variables is the counterpart of Spearman's rank correlation coefficient

Spearman's rank correlation coefficient

In statistics, Spearman's rank correlation coefficient or Spearman's rho, named after Charles Spearman and often denoted by the Greek letter \rho or as r_s, is a non-parametric measure of statistical dependence between two variables. It assesses how well the relationship between two variables can...

 which summarizes the association of continuous variables. It is also known as Page's trend test or Page's L test. It is a repeated measure trend test.

The Page test is useful where:

• there are three or more conditions,
• a number of subjects (or other randomly sampled entities) are all observed in each of them, and
• 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

Null hypothesis

The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...

 that, for n conditions, where m_i is a measure of the central tendency

Central tendency

In statistics, the term central tendency relates to the way in which quantitative data is clustered around some value. A measure of central tendency is a way of specifying - central value...

 of the ith condition,


against the alternative hypothesis that


It has more statistical power

Statistical power

The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false . The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis...

 than the Friedman test

Friedman test

The Friedman test is a non-parametric statistical test developed by the U.S. economist Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves ranking each row together, then...

 against the alternative that there is a difference in trend. Friedman's test considers the alternative hypothesis that the central tendencies of the observations under the n conditions are different without specifying their order.

Procedure for the Page test, with k subjects each exposed to n conditions:

• Arrange the n conditions in the order implied by the alternative hypothesis, and assign each of them a rank Y_i.
• 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 X_i.
• Multiply X_i by Y_i 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

may be compared with values of the chi-squared distribution with one degree of freedom

Degrees of freedom (statistics)

In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the...

. This gives a two-tailed test

Two-tailed test

The two-tailed test is a statistical test used in inference, in which a given statistical hypothesis, H0 , will be rejected when the value of the test statistic is either sufficiently small or sufficiently large...

. 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 0.05 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 was developed by Jonckheere (1954).