Column-wise Area Under ROC Curve (AUC)
Calculate Area Under the ROC Curve (AUC) for every column of a matrix. Also, can be used to plot the ROC curves.
colAUC(X, y, plotROC=FALSE, alg=c("Wilcoxon","ROC"))X |
A matrix or data frame. Rows contain samples and columns contain features/variables. |
y |
Class labels for the |
plotROC |
Plot ROC curves. Use only for small number of features.
If |
alg |
Algorithm to use: "ROC" integrates ROC curves, while "Wilcoxon" uses Wilcoxon Rank Sum Test to get the same results. Default "Wilcoxon" is faster. This argument is mostly provided for verification. |
AUC is a very useful measure of similarity between two classes measuring area
under "Receiver Operating Characteristic" or ROC curve.
In case of data with no ties all sections of ROC curve are either horizontal
or vertical, in case of data with ties diagonal
sections can also occur. Area under the ROC curve is calculated using
trapz function. AUC is always in between 0.5
(two classes are statistically identical) and 1.0 (there is a threshold value
that can achieve a perfect separation between the classes).
Area under ROC Curve (AUC) measure is very similar to Wilcoxon Rank Sum Test
(see wilcox.test) and Mann-Whitney U Test.
There are numerous other functions for calculating AUC in other packages. Unfortunately none of them had all the properties that were needed for classification preprocessing, to lower the dimensionality of the data (from tens of thousands to hundreds) before applying standard classification algorithms.
The main properties of this code are:
Ability to work with multi-dimensional data (X can have many
columns).
Ability to work with multi-class datasets (y can have more
than 2 different values).
Speed - this code was written to calculate AUC's of large number of features, fast.
Returned AUC is always bigger than 0.5, which is equivalent of
testing for each feature colAUC(x,y) and colAUC(-x,y) and
returning the value of the bigger one.
If those properties do not fit your problem, see "See Also" and "Examples" sections for AUC functions in other packages that might be a better fit for your needs.
An output is a single matrix with the same number of columns as X and
"n choose 2" ( n!/((n-2)! 2!) = n(n-1)/2 )
number of rows,
where n is number of unique labels in y list. For example, if y
contains only two unique class labels ( length(unique(lab))==2 ) than
output
matrix will have a single row containing AUC of each column. If more than
two unique labels are present than AUC is calculated for every possible
pairing of classes ("n choose 2" of them).
For multi-class AUC "Total AUC" as defined by Hand & Till (2001) can be
calculated by colMeans(auc).
Jarek Tuszynski (SAIC) jaroslaw.w.tuszynski@saic.com
Mason, S.J. and Graham, N.E. (1982) Areas beneath the relative operating characteristics (ROC) and relative operating levels (ROL) curves: Statistical significance and interpretation, Q. J. R. Meteorol. Soc. textbf30 291-303.
Fawcett, Tom (2004) ROC Graphs: Notes and Practical Considerations for Researchers
Hand, David and Till, Robert (2001) A Simple Generalization of the Area Under the ROC Curve for Multiple Class Classification Problems; Machine Learning 45(2): 171-186
See http://www.medicine.mcgill.ca/epidemiology/hanley/software/ to find articles below:
Hanley and McNeil (1982), The Meaning and Use of the Area under a Receiver Operating Characteristic (ROC) Curve, Radiology 143: 29-36.
Hanley and McNeil (1983), A Method of Comparing the Areas under ROC curves derived from same cases, Radiology 148: 839-843.
McNeil and Hanley (1984), Statistical Approaches to the Analysis of ROC curves, Medical Decision Making 4(2): 136-149.
wilcox.test and pwilcox
wilcox.exact from exactRankTests package
wilcox_test from coin package
performance from ROCR package
ROC from Epi package
roc.area from verification package
rcorr.cens from Hmisc package
# Load MASS library with "cats" data set that have following columns: sex, body
# weight, hart weight. Calculate how good weights are in predicting sex of cats.
# 2 classes; 2 features; 144 samples
library(MASS); data(cats);
colAUC(cats[,2:3], cats[,1], plotROC=TRUE)
# Load rpart library with "kyphosis" data set that records if kyphosis
# deformation was present after corrective surgery. Calculate how good age,
# number and position of vertebrae are in predicting successful operation.
# 2 classes; 3 features; 81 samples
library(rpart); data(kyphosis);
colAUC(kyphosis[,2:4], kyphosis[,1], plotROC=TRUE)
# Example of 3-class 4-feature 150-sample iris data
data(iris)
colAUC(iris[,-5], iris[,5], plotROC=TRUE)
cat("Total AUC: \n");
colMeans(colAUC(iris[,-5], iris[,5]))
# Test plots in case of data without column names
Iris = as.matrix(iris[,-5])
dim(Iris) = c(600,1)
dim(Iris) = c(150,4)
colAUC(Iris, iris[,5], plotROC=TRUE)
# Compare calAUC with other functions designed for similar purpose
auc = matrix(NA,12,3)
rownames(auc) = c("colAUC(alg='ROC')", "colAUC(alg='Wilcox')", "sum(rank)",
"wilcox.test", "wilcox_test", "wilcox.exact", "roc.area", "AUC",
"performance", "ROC", "auROC", "rcorr.cens")
colnames(auc) = c("AUC(x)", "AUC(-x)", "AUC(x+noise)")
X = cbind(cats[,2], -cats[,2], cats[,2]+rnorm(nrow(cats)) )
y = ifelse(cats[,1]=='F',0,1)
for (i in 1:3) {
x = X[,i]
x1 = x[y==1]; n1 = length(x1); # prepare input data ...
x2 = x[y==0]; n2 = length(x2); # ... into required format
data = data.frame(x=x,y=factor(y))
auc[1,i] = colAUC(x, y, alg="ROC")
auc[2,i] = colAUC(x, y, alg="Wilcox")
r = rank(c(x1,x2))
auc[3,i] = (sum(r[1:n1]) - n1*(n1+1)/2) / (n1*n2)
auc[4,i] = wilcox.test(x1, x2, exact=0)$statistic / (n1*n2)
## Not run:
if (require("coin"))
auc[5,i] = statistic(wilcox_test(x~y, data=data)) / (n1*n2)
if (require("exactRankTests"))
auc[6,i] = wilcox.exact(x, y, exact=0)$statistic / (n1*n2)
if (require("verification"))
auc[7,i] = roc.area(y, x)$A.tilda
if (require("ROC"))
auc[8,i] = AUC(rocdemo.sca(y, x, dxrule.sca))
if (require("ROCR"))
auc[9,i] = performance(prediction( x, y),"auc")@y.values[[1]]
if (require("Epi")) auc[10,i] = ROC(x,y,grid=0)$AUC
if (require("limma")) auc[11,i] = auROC(y, x)
if (require("Hmisc")) auc[12,i] = rcorr.cens(x, y)[1]
## End(Not run)
}
print(auc)
stopifnot(auc[1, ]==auc[2, ]) # results of 2 alg's in colAUC must be the same
stopifnot(auc[1,1]==auc[3,1]) # compare with wilcox.test results
# time trials
x = matrix(runif(100*1000),100,1000)
y = (runif(100)>0.5)
system.time(colAUC(x,y,alg="ROC" ))
system.time(colAUC(x,y,alg="Wilcox"))Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.