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
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, > > PaulYes, 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 -- Frank E Harrell Jr Professor and Chair School of Medicine Department of Biostatistics Vanderbilt University
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 VRThanks, 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