It looks to me like what you are doing is trying to judge significance of differences by non-overlap of single-sample confidence intervals. While this is appealing, it's not quite right.
I just looked into my copy of Applied Nonparametric Statistics (second ed.) by Wayne W. Daniel (Duxbury, 1990) but that only deals with the situation where there is a single replicate per block-treatment combination (whereas you have 10 reps) and block-treatment interaction is assumed to be non-existent. The method that Daniel prescribes in this simple setting seems to be no more than applying the Bonferroni method of multiple comparisons. (Daniel does not say; his book is very much a cook-book.) So you might simply try Bonferroni --- i.e. do all k-choose-2 pairwise comparisons between treatments (using the appropriate 2 sample method for each comparison) doing each comparison at the alpha/k-choose-2 significance level. Where k = the number of treatments = 4 in your case. This method is not going to be super-powerful but it is sometimes surprizing how well Bonferroni stacks up against more ``sophisticated'' methods. Daniel gives a reference to ``Nonparametric Statistical Methods'' by Myles Hollander and Douglas A. Wolfe, New York, Wiley, 1973, for ``an alternative multiple comparisons formula''. I don't have this book, and don't know what direction Hollander and Wolfe ride off in, but it ***might*** be worth trying to get your hands on it and see. Finally --- in what way are the assumptions of Anova violated? The conventional wisdom is that Anova is actually quite robust to non-normality. Particularly when the sample size is large --- and 10 reps per treatment combination is pretty good. Heteroskedasticity is more of a worry, but it's not so much of a worry when the design is nicely balanced. As yours is. And finally-finally --- have you tried transforming your data to make them a bit more normal and/or homoskedastic? I hope this is some help. cheers, Rolf Turner [EMAIL PROTECTED] Marco Chiarandini wrote: > I am conducting a full factorial analysis. I have one factor > consisting in algorithms, which I consider my treatments, and another > factor made of the problems I want to solve. For each problem I > obtain a response variable which is stochastic. I replicate the > measure of this response value 10 times. > > When I apply ANOVA the assumptions do not hold, hence I must rely on > non parametric tests. > > By transforming the response data in ranks, the Friedman test tells > me that there is statistical significance in the difference of the > sum of ranks of at least one of the treatments. > > I would like now to produce a plot for the multiple comparisons > similar to the Least Significant Difference or the Tukey's Honest > Significant Difference used in ANOVA. Since I am in the non > parametric case I can not use these methods. > > Instead, I compare graphically individual treatments by plotting the > sum of ranks of each treatment togehter with the 95% confidence > interval. To compute the interval I use the Friedman test as > suggested by Conover in "Practical Nonparametric statistics". > > I obtain something like this: > > Treat. A |-+-| > Treat. B |-+-| > Treat. C |-+-| > Treat. D |-+-| > > The intervals have all the same spread because the number of > replications was the same for all experimental units. > > I would like to know if someone in the list had a similar experience > and if what I am doing is correct. In alternative also a reference to > another list which could better fit my request is welcome. ______________________________________________ [EMAIL PROTECTED] mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide! http://www.R-project.org/posting-guide.html