colAUC {caTools}

Column-wise Area Under ROC Curve (AUC)

Description

Calculate Area Under the ROC Curve (AUC) for every column of a matrix. Also, can be used to plot the ROC curves.

Usage

  colAUC(X, y, plotROC=FALSE, alg=c("Wilcoxon","ROC"))

Arguments

X

A matrix or data frame. Rows contain samples and columns contain features/variables.

y

Class labels for the X data samples. A response vector with one label for each row/component of X. Can be either a factor, string or a numeric vector.

plotROC

Plot ROC curves. Use only for small number of features. If TRUE, will set alg to "ROC".

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.

Details

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.

Value

An output is a single matrix with the same number of columns as X and "n choose 2" ( \frac{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).

Author(s)

Jarek Tuszynski (SAIC) [e-mail]

References

• 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.

See Also

• 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

Examples

# 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"))