# Page's nonparametric trend test for levels of a repeated measures factor

Siegal and Castellan (1981, pp.184-188) describe a nonparametric test, Page's L test, which is similar to a one-way repeated measures ANOVA contrast of a trend. The presence of a trend in the response suggests increasing scores with conditions, ordered apriori. The trend being tested, unlike the ANOVA contrast approach, is not of a particular form, and is, for example, not necessarily linear. The single response is measured repeatedly under different conditions on the same subject.

Page's L test can be performed using this spreadsheet inputting the scores for each test made by each subject. The output from the spreadsheet consists of a test statistic, L, which represents the degree of association between the ordered conditions and the response (the higher the value of L the stronger the association) and a chi-square statistic which Siegal and Castellan say can be used for statistical significance when there are more than 20 subjects for three conditions or more than 12 subjects for between 4 and 10 conditions or any number of subjects when there are more than 10 conditions. If this is not the case critical values of L contained in Table N of their book (also given here) should be used (pp. 354-355) which in turn are taken from the original paper of Page (1963). The L statistic is simply the product of the group rank (ordered between 1 and the number of groups) and the rank of each score where each subject score is ranked within subject, from 1 to the number of conditions, with average ranks given for tied scores.

Unlike Jonckheere's nonparametric trend test for between subjects groups and the ANOVA parametric contrast tests of trend (both between and within subjects) both described elsewhere on the statswiki, Page's L test is not available in SPSS (at least upto version 16!). There is a page.trend.test procedure in R but please note that this R procedure, unlike the above spreadsheet and Siegal and Castellan, does not rank the observations in any way but uses raw values.

References

Page EB (1963). Ordered hypotheses for multiple treatments: a significance test for linear ranks. *Journal of the American Statistical Association* **58** 216-230. For CBSUers:This book is available for borrowing from the CBSU library.

Siegel S and Castellan Jr. NJ (1981). Nonparametric statistics for the behavioral sciences. McGraw-Hill:New York.