Parallel Boxplots After Tail Transformation For a Block Design
For an IxJ block design, a boxplot of column differences after transformation. See the help file for ttrans() for information about the transformation.
boxplotTTBlockDesign(bd, grp, p = -1, qu = 0.95, tcks = NULL, internal = TRUE, ylab = "", xlab = "", main = "", cex.axis = .8, cex.lab = .8, cex.main = .8, symZero=TRUE)
bd: An I x J matrix or data.frame of outcomes with no missing outcomes.grp: A vector of length J = number of columns of bd. If grp is 1:J, then the plot contains choose(J,2) boxplots for the differences of choose(J,2) pairs of columns of bd, and these are labeled 1-2, 1-3, 2-3, .... If grp contains J distinct short names, the same boxplots appear but labeled name1-name2, name1-name3,.... One-letter names look best. If J=4 and grp is (1,2,2,2) and internal=FALSE, then one boxplot appears, comparing column 1 to each of the columns labeled 2, using column 1 three times in comparison with columns 2, 3 and 4, so an Ix4 design produces 3I differences. If J=4 and grp is (1,2,2,2) and internal=TRUE, then two boxplots appear. The first boxplot is the same as for internal=FALSE. The second boxplot displays 2I x choose(J-1,2) differences, or 2I x 3 differences if grp is (1,2,2,2), namely bd[,2]-bd[,3], bd[,3]-bd[,2], bd[,2]-bd[,4], bd[,4]-bd[,2], bd[,3]-bd[,4], bd[,4]-bd[,3], that is, all pair differences among distinct columns labeled 2. See details. An error will result if all J entries in grp are the same.p: The power of the transformation. See the help file for the ttrans() function.qu: One number strictly between 0 and 1, commonly 0.9 or 0.95. Define beta to be the qu quantile of abs(as.vector(y)). Then, values between -beta and beta are not transformed.tcks: A vector of untransformed values to become tick marks for the y axis after transformation. Although the y axis was transformed, it is labeled with corresponding untransformed values given by tcks. If is.null(tcks), then the y axis has no tick marks.internal: If the J entries in grp are all different, say grp=(1,2,3,4), then changing internal has no effect. If one or more entries in grp are duplicated, as in grp=(1,2,3,3), and if internal=TRUE, then three boxplots compare 1-2, 1-3 and 2-3, and a fourth boxplot displays the 2I symmetrized differences among controls labeled 3, namely c(bd[,3]-bd[,4], bd[,4]-bd[,3]). If grp=(1,2,3,3) and internal=FALSE, then the fourth boxplot does not appear. See details.ylab: Label for the y-axis.xlab: Label for the x-axis.main: Title for the plot. If main="", then no title appears, and the sample size appears above each boxplot. If you prefer no title and no sample sizes, set main=" ". Note that the sample size refers to the number of differences; see details.cex.lab: Sets the label size for the boxplot using the standard graphics parameter cex.lab.cex.main: Sets the title size for the boxplot using the standard graphics parameter cex.main.cex.axis: Sets the axis size for the boxplot using the standard graphics parameter cex.axis.symZero: If TRUE, makes the y-axis symmetric about zero. If FALSE, the maximum and minimum values in ylist determine the y-axis.See the help file for ttrans() for an explanation of the transformation.
If the columns of the I x J block design bd represent different groups, say grp=(1,2,...,J), then the choose(J,2) pairwise differences among pairs of columns are plotted after tail transformation.
Suppose that two or more columns represent the same group, so J>length(unique(grp)), as in grp(1,2,3,3,4,4) with a block size of J=6 but just 4 groups. If internal=FALSE, then there is a boxplot plot if I differences comparing groups 1 and 2, bd[,1]-bd[,2]. There is a boxplot of 2I differences comparing groups 1 and 3, namely c(bd[,1]-bd[,3], bd[,1]-bd[,4]). There is a boxplot of 4I differences comparing groups 3 and 4.
In the same situation, but with internal=TRUE, the two columns for group 3 are compared, and so are the two columns for group 4. Both columns 5 and 6 each represent I individuals from group 4. A boxplot shows the variation in group 4 within blocks by plotting the 2I symmetrized differences, c(bd[,5]-bd[,6], bd[,6]-bd[,5]). That boxplot is symmetric about zero, and shows how people in the same block differ by chance when they are in the same group. That boxplot is a benchmark for comparing two different groups.
For discussion and examples of symmetrized boxplots, see Ye et al. (2022, Figure 1) and Rosenbaum (2022, Figure 5).
Parallel boxplots after tail transformation.
Rosenbaum, P. R. (2022) doi:10.1080/00031305.2022.2063944 A new transformation of treated-control matched-pair differences for graphical display. American Statistician, 76, 346-352.
Ye, T., Small, D. S. and Rosenbaum, P. R. (2022) doi:10.1214/22-AOAS1611 Dimensions, power and factors in an observational study of behavioral problems after physical abuse of children. Annals of Applied Statistics, 16, 2732-2754.
Paul R. Rosenbaum
ttrans
# Makes Figure 5(ii) in Rosenbaum (2022). data(aHDL) y<-t(matrix(aHDL$hdl,4,406)) y<-y[,c(1,3,2,4)] grp<-c("D","N","R","B") # Figure 3 in Rosenbaum (2022) boxplotTTBlockDesign(y,grp=grp,tcks=c(-200,-50,-20,0,20,50,200), ylab="HDL Difference",xlab="Group Comparisons") # Same figure, different transformation, p=-2 boxplotTTBlockDesign(y,p=-2,grp=grp,tcks=c(-200,-50,-20,0,20,50,200), ylab="HDL Difference",xlab="Group Comparisons") # Figure 5 in Rosenbaum (2022). The three control groups have been merged. # Note that the C-C boxplot is perfectly symmetric about zero, and less # dispersed than the T-C boxplot. grp<-c("D","C","C","C") boxplotTTBlockDesign(y,grp=grp,tcks=c(-200,-50,-20,0,20,50,200), ylab="HDL Difference",xlab="Group Comparisons") # The same figure can be produced explicitly using boxplotTTlist. TC<-c(y[,1]-y[,2],y[,1]-y[,3],y[,1]-y[,4]) CC<-c(y[,2]-y[,3],y[,2]-y[,4],y[,3]-y[,4]) CC<-c(CC,-CC) ylist<-list(TC=TC,CC=CC) boxplotTTlist(ylist,tcks=c(-200,-50,-20,0,20,50,200),ylab="HDL Difference") # More variations: Keep group B separate grp<-c("D","C","C","B") boxplotTTBlockDesign(y,grp=grp,tcks=c(-50,-20,0,20,50), symZero=FALSE,main=" ",p=1/2) boxplotTTBlockDesign(y,grp=grp,tcks=c(-200,-50,-20,0,20,50,200), main="",p=-1,internal=FALSE)
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