# Non-parametric test for repeated measures and post-hoc single comparisons in R?

Some attribute $x$ of 17 individuals was recorded repeatedly on 6 time points using a Likert scale with 7 distractors. Which statistical test(s) can I apply to check whether the changes along the 6 time points were significant?

set.seed( 123 )
x <- matrix( sample( 1:7, 17*6, repl=T ),
nrow = 17, byrow = TRUE,
dimnames = list(1:17, paste( 'T', 1:6, sep='' ))
)


I found the Friedman test and the Quade test for testing the overall hypothesis.

friedman.test( x )
quade.test( x )


However, the R help files, my text books (Bortz, Lienert and Boehnke, 2008; Köhler, Schachtel and Voleske, 2007; both German), and the Wikipedia texts differ in what they propose as requirements for the tests. R says that data need to be unreplicated. I read 'unreplicated' as 'not-repeated', but is that right? If so, the example, in contrast, in friedman.test() appears to use indeed repeated measures. Yet, Wikipedia says the contrary that is to say the test is good especially if data represents repeated measures. The text books say either (in the same paragraph, which is very confusing). What is right?

In addition, what would be an appropriate test for post-hoc single comparisons for the indication which column differs from others significantly?

Bortz, Lienert, Boehnke (2008). Verteilungsfreie Methoden in der Biostatistik. Berlin: Springer Köhler, Schachtel, Voleske (2007). Biostatistik: Eine Einführung für Biologen und Agrarwissenschaftler. Berlin: Springer

• "Unreplicated" just means that each person is observed exactly once at each time point, but not twice of more often. So "unreplicated" $\neq$ "not repeated". For the post-hoc comparisons, this question or this question provide a start. Commented Feb 19, 2012 at 9:53

## 1 Answer

It may be worth looking at the Brunner-Munzel-Test, see e.g. Brunner, Munzel, Puri (1999) "Rank-score tests in factorial designs with repeated measures", Journal of Multivariate Analysis. You can use it even if you have any kind of ordinal scale and not too small sample sizes. Yet I'm not sure about your particular model and hypotheses. But the first author wrote some books about nonparametric statistics (some of them in German) with plenty of examples. There or in another of his publications you'll likely also find more about multiple comparisons.