6 Friedman Rank
Nonparametric version of two-way ANOVA. The collected data of an experimental design involves two factors, each at two or more levels, the treatment factor and blocking factor.
library(BSDA)## Loading required package: lattice##
## Attaching package: 'BSDA'## The following object is masked from 'package:datasets':
##
## Orangex=c(1.83,0.50,1.62,2.48,1.68,1.88,1.55,3.06,1.30);
y=c(0.878,0.647,0.598,2.05,1.06,1.29,1.06,3.14,1.29);
plot(x,y,las=1); abline(0,1)
xy=cbind(x,y); xy## x y
## [1,] 1.83 0.878
## [2,] 0.50 0.647
## [3,] 1.62 0.598
## [4,] 2.48 2.050
## [5,] 1.68 1.060
## [6,] 1.88 1.290
## [7,] 1.55 1.060
## [8,] 3.06 3.140
## [9,] 1.30 1.290axy=apply(xy,1,rank); axy## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9]
## x 2 1 2 2 2 2 2 1 2
## y 1 2 1 1 1 1 1 2 1(rowMeans(axy)-3/2)^2## x y
## 0.07716049 0.07716049S=sum((rowMeans(axy)-3/2)^2)*12*9/6
S## [1] 2.777778pchisq(S,1,lower.tail=F)## [1] 0.0955807n=length(x); da=cbind(z=c(x,y),gr=c(rep(1,n),rep(2,n)),blk=rep(1:n,2))
da## z gr blk
## [1,] 1.830 1 1
## [2,] 0.500 1 2
## [3,] 1.620 1 3
## [4,] 2.480 1 4
## [5,] 1.680 1 5
## [6,] 1.880 1 6
## [7,] 1.550 1 7
## [8,] 3.060 1 8
## [9,] 1.300 1 9
## [10,] 0.878 2 1
## [11,] 0.647 2 2
## [12,] 0.598 2 3
## [13,] 2.050 2 4
## [14,] 1.060 2 5
## [15,] 1.290 2 6
## [16,] 1.060 2 7
## [17,] 3.140 2 8
## [18,] 1.290 2 9friedman.test(z~gr|blk,data=da)##
## Friedman rank sum test
##
## data: z and gr and blk
## Friedman chi-squared = 2.7778, df = 1, p-value = 0.09558test.stat=(sum(x>y)-n/2)/sqrt(n/4); t2=test.stat^2
c(2*pnorm(test.stat,lower.tail=F),t2,pchisq(t2,1,lower.tail=F))## [1] 0.0955807 2.7777778 0.0955807wilcox.test(x,y,paired=T)##
## Wilcoxon signed rank exact test
##
## data: x and y
## V = 40, p-value = 0.03906
## alternative hypothesis: true location shift is not equal to 0