Page test

What is the Page non parametric test

The goal of the test proposed by Page (1963) is to allow analyzing rigorously the results of a study carried out within the framework of a complete design, to verify if a series of several treatments should be considered as not different, or if alternatively a ranking of the treatment makes sense. The Page test is a nonparametric method, thus not making any assumption on the distribution of the measurements. This test differs from the Friedman test by the fact that the alternative hypothesis is a ranking of the treatments and not only a difference. This test has been extended to the case of incomplete blocks by Alvo and Cabilio (2005).

If t1, t2, …, tk correspond to the k treatments, the null and alternative hypotheses used in the test are:
•    H0 : The k treatments are not significantly different.
•    Ha : t1 ≤. t2 ≤ … . ≤ tk
or
•    Ha : t1 ≥. t2 ≥ … . ≥ tk
Where, for the alternative hypotheses, at least one inequality is strict.

Computation of the p-values

To compute the p-values corresponding to the various statistics, XLSTAT offers two alternative methods:

• Asymptotic method: The p-value is obtained using the asymptotic approximation of the distribution of the z statistics. The reliability of the approximation depends on the number of treatments and on the number of blocks.
• Monte Carlo method: The computation of the p-value is based on random resamplings. The user must set the number of resamplings. A confidence interval on the p-value is provided. The more resamplings are performed, the better the estimation of the p-value.

In order to avoid freezing Excel because of too long computations, it is possible with the two latter methods to set the maximum time that should be spent computing the p-value.

Multiple pairwise comparisons

If the p-value is such that the H0 hypothesis has to be rejected, then at least one treatment is different from another. To identify which treatment(s) is/are responsible for rejecting H0, a multiple comparison procedure can be used, XLSTAT allows using the procedure suggested by Cabilio and Peng (2008), with two alternative ways to compute the p-value of the paired comparisons. It can either use the normal approximation of a Monte Carlo based -pvalue.