[R] Non-parametric four-way interactions?

[Frank E Harrell Jr]

Paul Smith wrote:
> 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

Tony Lachenbruch has written some about this, also see my book (it's web 
page is biostat.mc.vanderbilt.edu/rms).  I don't know about the power of 
interaction tests, but for main effect tests in the absence of 
interaction, the Wilcoxon test (a special case of PO model) has 
efficiency of 3/pi compared to the t-test if normality holds.

Other approaches: Cox PH model, avas, ace.  My book covers these too.

Frank

-- 
Frank E Harrell Jr   Professor and Chair           School of Medicine
                      Department of Biostatistics   Vanderbilt University