[R] Non-parametric four-way interactions?

[Paul Smith]

On 7/27/06, Frank E Harrell Jr <f.harrell at vanderbilt.edu> wrote:
> > 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?
>
> 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)
> f
> anova(f)
>
> Also see the polr function in VR

Thanks, Frank. It is very encouraging to learn that, even without
normality, I can still study my four-way interactions. I am also aware
of transformations that may work in some non-normal cases, and I have
tried some of them, but with no success.

I am not familiar with the solutions that you suggest, and I would
like to learn how they work theoretically, in some book or on the
Internet. In particular, I would like to see, regarding power, how the
non-parametric suggested approach compares with the classical ANOVA
approach. Could you please indicate some references to help me with
that?

Again, thanks in advance.

Paul