In statistics, the Page test for multiple comparisons between ordered correlated variables is the counterpart of Spearman's rank correlation coefficient 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 Ellis Batten Page (1963). More formally, the test considers the

null hypothesis In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is d ...

that, for ''n'' conditions, where ''m''_''i'' is a measure of the

central tendency In statistics, a central tendency (or measure of central tendency) is a central or typical value for a probability distribution.Weisberg H.F (1992) ''Central Tendency and Variability'', Sage University Paper Series on Quantitative Applications in ...

of the ''i''th condition, :

m_1 = m_2 = m_3 = \cdots = m_n\,

against the

alternative hypothesis In statistical hypothesis testing, the alternative hypothesis is one of the proposed proposition in the hypothesis test. In general the goal of hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting ...

that :

m_1 > m_2 > m_3 > \cdots > m_n.\,

It has more

statistical power In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true. It is commonly denoted by 1-\beta, and represents the chances ...

than the

Friedman test The Friedman test is a non-parametric statistical test developed by Milton Friedman. Similar to the parametric repeated measures ANOVA, it is used to detect differences in treatments across multiple test attempts. The procedure involves rank ...

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 In probability theory and statistics, the chi-squared distribution (also chi-square or \chi^2-distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables. The chi-square ...

with one

degree of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...

. 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 = 12''L''/''k''(''n''³ − ''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 Jonckheere was a Belgian motor coach and bus builder, founded in 1881 by Henri Jonckheere in Roeselare. History In 1881, Henri Jonckheere began to build horse-drawn carriages in the village of Beveren, near Roeselare in the province of West F ...

(1954).

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Nonparametric Tests of Trend
Statistical tests

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