Hi all,
I need to compare some population means, but my data are non-normally distributed and with unequal variance (even after transformation) so I can use neither parametric t-tests nor rank-based tests.
On the book "Experimental Design and Data Analysis for Biologists" by Quinn and Keough (2002) there is a mention to a so-called "Wilcox H test" that is good for unequal variance and skewed distributions, so it would suit my case. This test should be the one described in Wilcox (1989). But I can't find any software that performs this test.
In R I found a function called wilcox.test, but that's actually for the Wilcoxon tests and then the function WilcoxH, that is another test from the same author.
Does anyone know where I could perform this or an analogous test?
Thank you very much
Wilcox, R. R. (1989). Adjusting for unequal variances when comparing means in one-way and two-way fixed effects ANOVA models. Journal of Educational Statistics, 14(3), 269-278.
That's the one you use, the one in R that you found! Or Mann Whitney. There is no such thing as the 'Wilcox H' test
The R function wilcox.test will suit your statistics. The difference in the test is in the 'paired'-parameter. When TRUE, then your test is a Wilcoxon signed-rank test for one population but with difference in time or similar or two related populations. When FALSE, then your test is a Mann-Whitney U / Wilcoxon rank-sum test for two independent populations. The choice in the 'paired'-parameter will then depend on what your values represent and how these values were obtained.
If you consider these tests to be inaccurate for you situation you can see if you can download/adapt some source code for the Wilcox H test from these two links, see:
https://www.rdocumentation.org/packages/mixPHM/versions/0.7-2/topics/WilcoxH
https://rdrr.io/cran/mixPHM/src/R/WilcoxH.R
There appears to be some confusion. The wilcox.test function in R is named after a couple of tests by Frank Wilcoxon. The Wilcox H test appears to be one by Rand Wilcox. In the Quinn and Keough book, they mention it as a test based on M estimators and bootstrapped values. There's a paper by R. Wilcox called Tests of independence and zero correlations among P variables that pops up in search results, but doesn't seem related. I would look at books by Rand Wilcox about robust statistics. Also note that some tests from these books are implemented in supplemental R packages like WRS2, but also others are implemented in functions available from the author.
Thank you all for your answers.
I agree with Salvatore view. These are two different tests made by different authors (with similar surnames).
The paper where Wilcox describes that test is:
Wilcox, R. R. (1989). Adjusting for unequal variances when comparing means in one-way and two-way fixed effects ANOVA models. Journal of Educational Statistics, 14(3), 269-278.
But I can't find anything else...
It is a pity because a test robust to both non-normality and unequal variance would be very handy especially in ecology
If you're working in R, a few things to explore:
General Factorial Designs
https://cran.r-project.org/web/packages/GFD/index.html
Permutation Tests for Linear Models
https://cran.r-project.org/web/packages/lmPerm/index.html
Functions from Rand Wilcox
https://cran.r-project.org/web/packages/WRS2/index.html
As to what Quinn and Keough suggest, it sounds like just using robust M-estimators for the central tendency of the samples (See, Article An introduction to robust estimation with R functions
), and then calculating confidence intervals for these values. Presumably, one would consider groups with non-overlapping confidence intervals to be statistically different.I have some code below in R as an example to do this.
I am the author of the rcompanion package, so there's a caveat there. The groupwiseHuber function uses the DescTools::HuberM function to calculate the estimators and the bootstrapped confidence intervals for the estimators. With the "boot" option, by default it uses the percentile method for determining the confidence intervals, with 1000 iterations.
set.seed(1234)
A = rnorm(16, 10, 5)
B = rlnorm(16, 1, 1)
C = runif(16, 1, 20)
Response = c(A, B, C)
Factor = factor(c(rep("A", length(A)), rep("B", length(B)), rep("C", length(C))))
Data = data.frame(Factor, Response)
#####################
plot(Response ~ Factor, data=Data)
#####################
require(MASS)
huber(Response[Factor=="A"], k=1.345)
huber(Response[Factor=="B"], k=1.345)
huber(Response[Factor=="C"], k=1.345)
### $mu ### [1] 8.535279
### $mu ### [1] 2.010136
### $mu ### [1] 8.400159
require(rcompanion)
groupwiseHuber(Response ~ Factor, data=Data, conf.type="boot")
### Factor n M.Huber lower.ci upper.ci ### 1 A 16 8.535279 6.337090 10.733469 ### 2 B 16 2.010136 1.271110 2.749162 ### 3 C 16 8.400159 5.455877 11.344440
### So, here, A and C are rather similar in their central tendency, and B is different from them.
### Also note that the higher observations in B (c. 8 and 30), don't have much of an effect on the M-estimators or confidence intervals.
B[order(B)]
hist(B, col="darkgray")
Thank you Salvatore,
your answer was very helpful.
In the meanwhile, I've also found that Unequal variance t-test on ranked data should perform well in my case (Ruxton 2006).
Ruxton, G. D. (2006). The unequal variance t-test is an underused alternative to Student's t-test and the Mann–Whitney U test. Behavioral Ecology, 17(4), 688-690.
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