[R] Weighted Mann-Whitney-Wilcoxon-Test

[Alexander Sommer]

Hi fellow R-users,

well, say I got two groups, A and B. Nested within each group are subgroups and in each subgroup are objects with values x and y to a certain attribute. So, I can compute the portion of x-objects for each subgroup as #x/(#x?+?#y).

Artificial example with 12 subgroups in group A and 8 subgroups in group B:

set.seed(123)
count.x <- NULL
count.y <- NULL
j <- 1
for (i in sample(x = 10:20, size = 20, replace = TRUE)){
count.x[j] <- sample(x = 0:i, size = 1)
count.y[j] <- i - count.x[j]
j <- j + 1
}
data <- data.frame(x.portion = (count.x/(count.x + count.y)),
x.portion = (count.y/(count.x + count.y)),
group = c(rep("A", 12), rep("B", 8),
weight = (count.x + count.y)
)

I am now interested in whether or not there is a difference in the portions of x-objects between group A and B and consider it a good idea ? as seen in the example above ? to weight for the total number of objects in each subgroup. Given data that is not considered a realization of some normal distribution, thinking of a test that uses ranks still does not look like a natural solution to this problem. But I guess it is possible. Though, Xie & Priebe (2002)* are not exactly aiming at this, their paper might give an idea how weighting may look like in the special case of the Mann/Whitney/Wilcoxon statistic. (Despite the hint by John & Priebe (2007)** that this ?is not a candidate for the practitioner?s toolbox?.)

Anyway, trying to apply the Wilcoxon rank sum test to weighted data, I was first tempted to replicate each portion by its weight. (Bad idea: data bloat, ties and probably a number of problems even worse.) Function wilcox_test in package coin has got a weight argument, but

library(coin)
wilcox_test(formula = x.portion ~ group, data = data, weight = ~ weight)

leads to warning ?Rank transformation doesn?t take weights into account?. Though, results differ from

wilcox_test(formula = x.portion ~ group, data = data)

and

wilcox.test(formula = x.portion ~ group, data = data)

The code in wilcox_test() and the functions it depends on looks a little bit interlaced to me, but I guess it is not what I am after.

tl;dr: I am looking for a nonparametric alternative to wtd.t.test in package weights.

Is anyone aware of an(other) implementation in R?

Cheers,

Alex


PS: For real, it is a little bit trickier as there are more than two values and maybe even more than two groups. So Kruskal-Wallis test might be of interest, but I thought I keep it simple for the moment.

* Jingdong Xie & Carey E. Priebe: A weighted generalization of the Mann?Whitney?Wilcoxon statistic. In: Journal of Statistical Planning and Inference 102 (2), 2002-04-01, pages 441?466. (http://dx.doi.org/10.1016/S0378-3758(01)00111-2.)
** Majnu John & Carey E. Priebe: A data-adaptive methodology for finding an optimal weighted generalized Mann-Whitney-Wilcoxon statistic. In: Computational Statistics & Data Analysis 51 (9), 2007-05-15, pages 4337?4353. (http://dx.doi.org/10.1016/j.csda.2006.06.003.)
--
Alexander Sommer
wissenschaftlicher Mitarbeiter

Technische Universit?t Dortmund
Fakult?t Erziehungswissenschaft, Psychologie und Soziologie
Forschungsverbund Deutsches Jugendinstitut/Technische Universit?t Dortmund
Vogelpothsweg 78
44227 Dortmund

Telefon: [phone]
Fax: [phone]
E-Mail: Alexander.Sommer at tu-dortmund.de
WWW: http://www.forschungsverbund.tu-dortmund.de/

[Thomas Lumley]

]On Wed, Aug 20, 2014 at 2:01 AM, Alexander Sommer

Post by Alexander Sommer
tl;dr: I am looking for a nonparametric alternative to wtd.t.test in package weights.
Is anyone aware of an(other) implementation in R?

survey::svyranktest

T. Lumley and A.J. Scott (2013). Two-sample rank tests under complex
sampling. Biometrika, 100, 831-842.


-thomas

[Alexander Sommer]

Post by Thomas Lumley

Post by Alexander Sommer
Is anyone aware of an(other) implementation in R?

survey::svyranktest

Oh, that easy. Thanks a lot.

Actually, in my case the weights do not derive from some selection probabilities, but your test works anyway.


For completeness (and this time with a working example):

set.seed(seed = 123)
count.x <- NULL
count.y <- NULL
j <- 1
for (i in sample(x = 10:20, size = 20, replace = TRUE)){
count.x[j] <- sample(x = 0:i, size = 1)
count.y[j] <- i - count.x[j]
j <- j + 1
}
data <- data.frame(x.portion = (count.x/(count.x + count.y)),
y.portion = (count.y/(count.x + count.y)),
group = c(rep("A", 12), rep("B", 8)),
weight = (count.x + count.y)
)

I first considered the unweighted case with

library(package = survey)
design <- svydesign(ids = ~0, data = data)
svyranktest(formula = x.portion ~ group, design = design)

and compared it to the default Wilcoxon test by

wilcox.test(formula = x.portion ~ group, data = data, exact = FALSE, correct = FALSE)

Or, if you prefer

library(package = coin)
wilcox_test(formula = x.portion ~ group, data = data)

The resulting p-values differ, as I understood due to an approximation in package /survey/.

Now, finally, the weighted case:

design <- svydesign(ids = ~0, data = data, weights = ~weight)
svyranktest(formula = x.portion ~ group, design)

And, by the way, package /survey/ seems also to be the preferable way, if you want to go for a parametric test. Once again, the unweighted case first:

design <- svydesign(ids = ~0, data = data)
svyttest(formula = x.portion ~ group, design)

And, yet again, the results differ from the default t test

t.test(formula = x.portion ~ group, data = data)

This time, I guess, it is due to the way standard errors are computed.

Finally (this time for real), the weighted case:

design <- svydesign(ids = ~0, data = data, weights = ~weight)
svyttest(x.portion ~ group, design)

Note that function /wtd.t.test/ from package /weights/ depends on the scale of the weights, /svyttest/ not.


Thomas, one more time: thank you for your help.

Cheers,

Alex
--
Alexander Sommer
wissenschaftlicher Mitarbeiter

Technische Universit?t Dortmund
Fakult?t Erziehungswissenschaft, Psychologie und Soziologie
Forschungsverbund Deutsches Jugendinstitut/Technische Universit?t Dortmund
Vogelpothsweg 78
44227 Dortmund

Telefon: [phone]
Fax: [phone]
E-Mail: Alexander.Sommer at tu-dortmund.de
WWW: http://www.forschungsverbund.tu-dortmund.de/