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What is a Page test?
The Page test is an interesting alternative to the Friedman test when one wants to use an alternative hypothesis that ranks the treatments rather than just consider the case of a difference between at least two treatments. This test, when it makes sense, allows to extract more information from the data.
If t1, t2, …, tk correspond to the k treatments, the null and alternative hypotheses used in the Page test are:
- H0 : The k treatments are not significantly different.
- Ha : t1 ≤. t2 ≤ … . ≤ tk
- Ha : t1 ≥. t2 ≥ … . ≥ tk
Where, for the alternative hypotheses, at least one inequality is strict.
Dataset for running a Page test in Excel
A study was conducted on 20 students who were assessed three times during the year to measure their knowledge. Ratings are on a 1 to 5 scale. The three tests correspond here to what is commonly referred to as treatment.
We now want to know if students' scores have not changed, or if we can assume a change in a given direction (here we can expect an increase). The design of experiment corresponds here to a complete block design. Nevertheless, XLSTAT can also calculate the test if missing data are present.
An Excel sheet with both the data and the results can be downloaded by clicking here.
Setting up a Page test in Excel using XLSTAT
Once XLSTAT-Pro is activated, select the XLSTAT / Nonparametric tests / Page test command, or click on the corresponding button of the Nonparametric test menu (see below).
Once you've clicked the button, the dialog box appears. You can then select the data on the Excel sheet.
In the Options tab we choose to compute the asymptotic p-value to proceed as most software does.
XLSTAT can also use Monte Carlo simulations to get a better estimate of the p-value.
After you have clicked on the OK button, the results are displayed on a new Excel sheet (because the Sheet option has been selected for outputs).
Interpreting the results of a Page test
We see that the null hypothesis is rejected. As a consequence we can consider that the students improved during the year.
Multiple pairwise comparisons
As the null hypothesis is being rejected, it is legitimate to ask oneself which treatment(s) is/are responsible for the rejection of H0. To investigate, we can use the multiple comparisons procedure of Cabilio and Peng (2008). The p-value used to determine if the difference is significant or not, we use the normal approximation (option selected in the dialog box).
We see here that the three tests are well ordered with each a strict inequality T1<T2<T3.