Note that the effect of the Wilcoxon-signed-ranks tests are semi-parametric in some sense since this test can only be applied to metric data where differences can be built.  There exist, however, methods and procedures for purly nonparametric effects for paired data and for longitudinal data. Please find a list of some papers below. 

If you send me your e-mail address then I can send you these papers. My e-mail address is:  [email protected]

Brunner, E. and Langer, F. (2000). Nonparametric Analysis of Ordered

Categorical Data in Designs with Longitudinal Observations and Small Sample

Sizes. Biometrical Journal 42, 663-675.

Brunner, E. and Puri, M.L. (2001). Point and interval estimators for

nonparametric treatment effects in designs with repeated measures. In: Data

Analysis from Statistical Foundations: Festschrift in Honor of Donald A.S.

Fraser. (Editor: A.K.E. Saleh) Nova Science Publishers, Inc., New York, 167–

178.

Konietschke, F., Bathke, A.C., Hothorn, L.A., and Brunner, E. (2010). Testing

and Estimation of purely nonparametric effects in repeated measures designs.

Computational Statistics & Data Analysis 53, 730-741.

Konietschke, F., Hothorn, L. A., Brunner, E. (2012). Rank-based multiple test

procedures and simultaneous confidence intervals. The Electronic Journal of

Statistics 6, 737-758.

Noguchi, K., Gel, Y., Brunner, E. , Konietschke, F. (2012). nparLD: An R

Software Package for the Nonparametric Analysis of Longitudinal Data in

Factorial Experiments. Journal of Statistical Software 50, Iss. 12.

Brunner, E., Konietschke, F., Pauly, M., and Puri, M.L. (2016). Rank-Based

Procedures in Factorial Designs: Hypotheses about Nonparametric Treatment

Effects. Journal of the Royal Statistical Society Series B, (accepted for

Publication).

Brunner, E., Domhof S. and Langer, F. (2002). Nonparametric Analysis of

Longitudinal Data in Factorial Designs. Wiley, New York.

Kind regards,

Edgar Brunner

>>>>>>>>>>>>>>>>>>>>>>

Dear scientists

Thank you for taking the time to answer me, it will be an interesting read. I am hoping to have the opportunity to discuss the information one of the next days (it's a lot of information).

Kind regards.

Hello Hugo, did you find the answer? Because I have the same question. I tried to review all the information suggested but I did not find it.

I would like to calculate the confidence interval of an effect size estimated using using rank-biserial correlation. Thanks in advance

@Beatriz, you might look at using bootstrap for confidence intervals. It can be used for a variety of statistics. The idea is that if you re-sample the data and calculate the statistic many times, then the empirical distribution should be similar to the distribution of the statistic.

This is easy in R, except that I'm not sure what rank-biserial correlation is, or how to do it in R.

If you're using Stata, there's an example with all code necessary in this paper

http://www.stata-journal.com/sjpdf.html?articlenum=st0253

Hi Beatriz

There is a formula in this paper:

Article To adjust or not adjust: Nonparametric effect sizes, confide...

I do not know how valid it is. I hope it helps.

PS: Thanks to the other scientists for your comments.

Kind regards.

This is an answer to the original post, with code in R.

There is an effect size used for Wilcoxon tests, called r. There are variants for one-sample, two-sample, and paired tests.

r is computed as the z value from the test divided by the square root of N.

Running the code will take a while. This is because the wilcoxonOneSampleR function uses a permutation test from the coin package to get the z value, and then the bootstrap procedure needs many replicates. So basically, it's a re-sampling procedure within a re-sampling procedure.

# # # # #

if(!require(boot)){install.packages("boot")}

if(!require(rcompanion)){install.packages("rcompanion")}

library(rcompanion)

library(boot)

Pre = c(1,2,3,4,5, 6, 7, 8, 9, 10)

Post = c(2,4,6,4,4,12,14,16, 8, 11)

Diff = Post - Pre

### Effect size r

wilcoxonOneSampleR(Diff, mu=0)

### r ### 0.65

### Bootstrap procedure

### Define function

Function = function(input, index){

Input = input[index]

Stat = wilcoxonOneSampleR(Input, mu=0)

return(Stat)}

### Test Function

n = length(Diff)

Function(Diff, 1:n)

### r

### 0.65

### Produce Stat estimate by bootstrap

Boot = boot(Diff,

Function,

R=1000)

mean(Boot$t[,1])

median(Boot$t[,1])

hist(Boot$t[,1])

### Produce confidence interval by bootstrap

boot.ci(Boot,

conf = 0.95,

type = "perc")

### Options: "norm", "basic", "stud", "perc", "bca", "all"

### Level Percentile ### 95% ( 0.149, 0.890 ) ### Calculations and Intervals on Original Scale

wilcox.test(Diff, mu=0)

Dear all,

Thanks for your useful answers. It is complicated to find the correct procedure when you are not a statistician : ))

Regards!!

Hey all, any follow up on this? I've found how to calculate effect sizes for non-paremetric data, however I haven't found how to compute confidence intervals for those effect sizes. Could anyone comment on this please?

Thanks.

Matheus

I had already answered this question here about two years ago. Here is some more recent literature:

For independent observations, you may read our book which will appear in a few weeks:

Brunner, E., Bathke, A.C., and Konietschke, F. (2018). Rank- and Pseudo-Rank

Procedures for Independent Observations in Factorial Designs – Using R and

SAS. Springer Series in Statistics, Springer, Heidelberg (in production). ISBN:

978-3-030-02912-8.

This book is in print and will appear soon. For paired samples and repeated measures, please consult the literature listed belwow. In case of any questions, just drop me an email to my regular email account: [email protected]

Brunner, E. and Puri, M.L. (2001). Point and interval estimators for

nonparametric treatment effects in designs with repeated measures. In: Data

Analysis from Statistical Foundations: Festschrift in Honor of Donald A.S.

Fraser. (Editor: A.K.E. Saleh) Nova Science Publishers, Inc., New York, 167–

178.

Konietschke, F., Bathke, A.C., Hothorn, L.A., and Brunner, E. (2010). Testing

and Estimation of purely nonparametric effects in repeated measures designs.

Computational Statistics & Data Analysis 53, 730-741.

Konietschke, F., Hothorn, L. A., Brunner, E. (2012). Rank-based multiple test

procedures and simultaneous confidence intervals. The Electronic Journal of

Statistics 6, 737-758.

Noguchi, K., Gel, Y., Brunner, E. , Konietschke, F. (2012). nparLD: An R

Software Package for the Nonparametric Analysis of Longitudinal Data in

Factorial Experiments. Journal of Statistical Software 50, Iss. 12.

Brunner, E., Konietschke, F., Pauly, M., and Puri, M.L. (2017). Rank-Based

Procedures in Factorial Designs: Hypotheses about Nonparametric Treatment

Effects. Journal of the Royal Statistical Society Series B, Journal of the Royal

Statistical Society Series B 79, 1463–1485.

Brunner, E., Domhof S. and Langer, F. (2002). Nonparametric Analysis of

Longitudinal Data in Factorial Designs. Wiley, New York.

Kind regards,

Edgar Brunner

Thanks for your detailed references, Edgar Brunner. I have also emailed you just to clarify a few things. Kind regards, Matheus

Follow

in case of a pure shift effect there exists a standard procedure. See, e.g., the textbook by Hollander, Wolfe and Chicken. If you want to consider a nonparametric effect, then you should use the paired ranks test. This is refered to in the review paper by Brunner and Puri: "Nonparametric Methods in Factorial Designs", (2001, Statistical Papers, p.1-52) or in the textbook by Brunner, Domhof and Langer: "Nonparametric Analysis of Longitudinal Data in Factorial Designs" (Wiley, 2002). See also: "A studentized permutation test for the nonparametric Behrens-Fisher problem in paired data": Konietschke and Pauly (Electronic Journal of Statistics, 2012).

If you are interested in more details papers then just drop me a mail on my regular email account: [email protected]