[R] Non-parametric four-way interactions?

Frank E Harrell Jr f.harrell at vanderbilt.edu
Thu Jul 27 05:42:58 CEST 2006

Paul Smith wrote:
> Dear All
> I am trying to study four-way interactions in an ANOVA problem.
> However, qqnorm+qqline result
> (at http://phhs80.googlepages.com/qqnorm.png)
> is not promising regarding the normality of data (960 observations).
> The result of Shapiro-Wilk test is also not encouraging:
> W = 0.9174, p-value < 2.2e-16
> (I am aware of the fact that normality tests tend to reject normality
> for large samples.)
> By the way, the histogram is at:
> http://phhs80.googlepages.com/hist.png
> To circumvent the problem, I looked for non-parametric tests, but I
> found nothing, but the article:
> http://www.pgia.ac.lk/socs/asasl/journal_papers/PDFformat/g.bakeerathanpaper-2.pdf
> Finally, my question is: has R got implemented functions to use
> non-parametric tests to avoid the fulfillment of the normality
> assumption required to study four-way interactions?
> Thanks in advance,
> Paul

Yes, although I seldom want to look at 4th order interactions.  You can 
fit a proportional odds model for an ordinal response which is a 
generalization of the Wilcoxon/Kruskal-Wallis approach, and allows one 
to have N-1 intercepts in the model when there are N data points (i.e., 
it works even with no ties in the data).  However if N is large the 
matrix operations will be prohibitive and you might reduce Y to 100-tile 
groups.  The PO model uses only the ranks of Y so is monotonic 
transformation invariant.

library(Design)  # also requires library(Hmisc)
f <- lrm(y ~ a*b*c*d)

Also see the polr function in VR
Frank E Harrell Jr   Professor and Chair           School of Medicine
                      Department of Biostatistics   Vanderbilt University

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