For more than two populations, check the normality distribution first. If the data are normally distributed, you can use one way ANOVA. If the data is not normally distributed, you can use Kruskal-Wallis non-parametric test.

For multiple comparison, one way ANOVA is usually followed by post hoc test such as Tukey's , Benferroni's, etc.

Multiple comparison for non-parametric test can be done using Mann Whitney test for each pair of interest.

Mann-Whitney test (MWT). Possibly GraphPad (I dont have this software, so I am only guessing) might have the possibility to do all pairwise comparisons, even with controlling the FWER, after a Kruskal-Wallis test. In any case individual MWTs can be performed for any comparison you like.

(Skip this if you get confused: Another way doing a nonparametric test here was to do a t-test on the ranks of the data. This would be formally ok, but supposedly not what the reviewer might have had in mind...)

I agree with Jochen, adding that you can use Wilcoxon–Wilcox multiple-comparisons test instead of MWT. [Basic Statistics: Tales of Distributions, Ninth Edition, Chris Spatz, Chap 14]. Most of Softwares can do KW and MWT, but I'm afraid they can't do Wilcoxon–Wilcox multiple-comparisons.

The WWMC is not difficult to perform, as one can read in "Tales of Distributions".

Here would be a table of critical values: http://www.vodppl.upm.edu.my/uploads/docs/edu5950_1299314424.pdf

Hi,

I agree with Abdualrahman. For multiple comparison I suggest to use the R program.

The function TukeyHSD() computes the differences between your groups (if the data are normally distributed). View here: http://stat.ethz.ch/R-manual/R-patched/library/stats/html/TukeyHSD.html

Also, the function pairwise.wilcox.test() computes the differences if the data are not normally distributed. View here: http://stat.ethz.ch/R-manual/R-patched/library/stats/html/pairwise.wilcox.test.html

Moreover, with the R package agricolae (http://cran.r-project.org/web/packages/agricolae/index.html) you will be able to graph the results on a graph with letters indicating significant differences.

Good luck!

You have only 5 elements in each group. You can't check the normality of distribution, you have to use nonparametric test - in this case Kruskal-Wallis test with the post-hoc test. You can use e.g. the Statistica software (StatSoft).

Robert, it is not a logical argument that one has to do a non-parametric test when the assumptions of a test can't be checked on the observed data. Assumptions are assumptions. Either I have reasons for them or I don't. Sometimes I can check if my observations do strongly deviate from the assumptions, but even this does not automatically render otherwise reasonable assumptions obsolete.

Further, there are 5 groups with 5 values, so this is a total of 25 values. With so many values a residual analysis can show major deviations from the assumptions (if they exist).

I think what the reviewer is looking for is a non-parametric kruskal-wallis test.

This test is applicable here since you only have one factor with five levels,

That should work fine and is available in basically every stats software.

If you are familiar with R, here a basic example for titrations from three blood types A,B and C.

###############################################################

# create data set

Titration = c(1.991, 1.342, 1.857, 2.330, 1.826, 1.079, 1.114, 1.663, 1.580, 1.690, 1.590, 1.470, 1.935, 1.079, 1.690, 1.519, 1.857)

Blood.type = c(rep("A",5),rep("B",7),rep("C",5))

dat = data.frame(Titration,Blood.type)

# Run non-parametric Kruskal-Wallis test

kruskal.test(Titration ~ Blood.type, data= dat)

################################################################

In this case, there is no significant difference between blood types

Good luck

Jochen, of course you're right, but if you do not have a basis for assumptions (as in this case - I think) you should not use parametric tests. I think it was more practical (not theoretical) question.

I see your point, Robert, and I agree.

However ... I think it is (or should be) a rare case where one has no assumptions. 2 reasons:

1) the response (its distribution) is known from experience or literature. Or it is a response very similar to known responses. For instance, concentrations of cellular compounds (like proteins or transcripts) are almost always lognormal distributed. The same is true for fold-changes. Many continuous medical parameter like blood glucose, hematocrit and so on that are tightly regulated are very similar to a normal distribution. Body sized, weights, and even BMIs are known to have distributions very similar to a normal distribution. All the above mentioned responses can actually be described by gamma distributions. Further, proportions (like oxygen saturation, saturation in general, percentages of some totals) are known to be beta distributed.

2) When there is no experience, one can often derive reasonable assumptions about the distribution when one tries to understand the process the response is generated and measured and how this is influenced by disturbances. And actually this should be the aim anyway. One should aim to understand how the response is generated. Simply noting that something has some influence on something else ("a treatment changes the blood pressure", for instance) is really only the very beginning of science. An understanding takes much more than this, and for an understanding, not the test-decision (significant or not, yes or no) is relevant but the effect size and the way *how* things are depending on each other. But before we have no reasonably good model about what is (supposed to) going on, where is the benefit of doing hypothesis tests?

I still think that pretty much of non-parametric analyses are an escape from (possibly very difficult) *scientific* questions. Noteably, most of these analyses are applied only to get p-values from hypothesis tests (I only know the rank-correlation as an exception - but there may be more I am not aware of), and most of such tests are scientifically sensless (by their logic, they do not allow us to learn from a single case - what would be neccesary in research! This is often misunderstood). It is a warm and cosy place, where reviewers request just p-values (instead of a reasonable model) and any user-friendly software will provide p-values for any kind of data by simply pressing a button. This leads to a world where most of the brains is used to interpret along the p-values to get a consistent story, instead of thinking harder about how exactly things are realted, and how much we now abouth these relations, and how (un)certain we are about things. It's a long story...

Jochen, you are right all the time.

But I live in a mundane world where medics think "how to publish", and reviewers want to see the p-value only... And no one wants to talk how this should be done in order to be "solidly" ...

With regard to non-parametric methods, I really appreciate them. Maybe because I too often meet with abuse of parametric methods. I know people who do not believe in the existence of normality :-) and do not use parametric methods (see http://en.wikipedia.org/wiki/El%C5%BCbieta_Pleszczy%C5%84ska ). It's a very extreme point of view, but prof. Pleszczynska gave me a lot to think about, especially when he caused a "sharp" discussion after my presentation at the biostatistical conference :-)

You're right, that is a long story... It's a bit like outlook, we can have opposing views and both be right :-)

Thank you very much for all of your valuable suggestions that helped me doing my analyses. I did by one-way analysis of variance (ANOVA) followed by the Kruskal-Wallis non-parametric test for between-group differences and Dunnett’s correction of the significance level for multiple comparisons, wherever applicable.

Dear Robert, you made the point saying:

"But I live in a mundane world where medics think "how to publish", and reviewers want to see the p-value only... And no one wants to talk how this should be done in order to be "solidly" ... "

But then I wonder why anyone should bother with the kind of test or anything. I mean, if you really ask editors, reviewers, authors and readers of such publications, what the meaning of a p-value is, you will get more than 99% wrong answers. So anyway nobody knows what it is, but everybody wants to have it and most people argue around p-values or significances. Since this is anyway wrong, why should it make a difference if the values - that are wrongly interpreted anyway - are solidly justified or given based on assumptions or methods that are more or lass renadomly selected or selected by "looking how others did it"?

I do not see the point wasting time on making something very correctly to get a result which meaning is not understood and largly misinterpreted.

The need of low p-values (smaller 0.05, say) enforced the bad habit of digging, re-experimenting and re-analyzing so long until it happens that a p0.05 and they then do not see how the could play auround with the analysis or repeat the experiment in order to finally get some p

A simple approach could be:

1) Kruskal - Wallis non-parametric test and if there is statistical significance then proceed to

2) Mann-Whitney test for between-group comparisons with Bonferroni correction for multpiple comparisons