pROC (version 1.8)

roc.test: Compare the AUC of two ROC curves

Description

This function compares the AUC or partial AUC of two correlated (or paired) or uncorrelated (unpaired) ROC curves. Several syntaxes are available: two object of class roc (which can be AUC or smoothed ROC), or either three vectors (response, predictor1, predictor2) or a response vector and a matrix or data.frame with two columns (predictors).

Usage

# roc.test(...)
"roc.test"(roc1, roc2, method=c("delong", "bootstrap",
"venkatraman", "sensitivity", "specificity"), sensitivity = NULL,
specificity = NULL, alternative = c("two.sided", "less", "greater"),
paired=NULL, reuse.auc=TRUE, boot.n=2000, boot.stratified=TRUE,
ties.method="first", progress=getOption("pROCProgress")$name,
parallel=FALSE, ...)
"roc.test"(roc1, roc2, ...)
"roc.test"(roc1, roc2, ...)
"roc.test"(formula, data, ...)
"roc.test"(response, predictor1, predictor2=NULL,
na.rm=TRUE, method=NULL, ...)

Arguments

roc1, roc2

the two ROC curves to compare. Either “roc”, “auc” or “smooth.roc” objects (types can be mixed).

response

a vector or factor, as for the roc function.

predictor1

a numeric or ordered vector as for the roc function, or a matrix or data.frame with predictors two colums.

predictor2

only if predictor1 was a vector, the second predictor as a numeric vector.

formula

a formula of the type response~predictor1+predictor2.

data

a matrix or data.frame containing the variables in the formula. See model.frame for more details.

na.rm

if TRUE, the observations with NA values will be removed.

method

the method to use, either “delong”, “bootstrap” or “venkatraman”. The first letter is sufficient. If omitted, the appropriate method is selected as explained in details.

sensitivity, specificity

if method="sensitivity" or method="specificity", the respective level where the test must be assessed as a numeric of length 1.

alternative

specifies the alternative hypothesis. Either of “two.sided”, “less” or “greater”. The first letter is sufficient. Default: “two.sided”. Only “two.sided” is available with method="venkatraman".

paired

a logical indicating whether you want a paired roc.test. If NULL, the paired status will be auto-detected by are.paired. If TRUE but the paired status cannot be assessed by are.paired will produce an error.

reuse.auc

if TRUE (default) and the “roc” objects contain an “auc” field, re-use these specifications for the test. See the AUC specification section for more details.

boot.n

for method="bootstrap" and method="venkatraman" only: the number of bootstrap replicates or permutations. Default: 2000.

boot.stratified

for method="bootstrap" only: should the bootstrap be stratified (same number of cases/controls in each replicate than in the original sample) or not. Ignored with method="venkatraman". Default: TRUE.

ties.method

for method="venkatraman" only: argument for rank specifying how ties are handled. Defaults to “first” as described in the paper.

progress

the name of progress bar to display. Typically “none”, “win”, “tk” or “text” (see the name argument to create_progress_bar for more information), but a list as returned by create_progress_bar is also accepted. See also the “Progress bars” section of this package's documentation.

parallel

if TRUE, the bootstrap is processed in parallel, using parallel backend provided by plyr (foreach).

...

further arguments passed to or from other methods, especially arguments for roc and roc.test.roc when calling roc.test.default or roc.test.formula. Arguments for auc, and txtProgressBar (only char and style) if applicable.

Value

A list of class "htest" with following content:

p.value

the p-value of the test.

statistic

the value of the Z (method="delong") or D (method="bootstrap") statistics.

alternative

the alternative hypothesis.

method

the character string “DeLong's test for two correlated ROC curves” (if method="delong") or “Bootstrap test for two correlated ROC curves” (if method="bootstrap").

null.value

the expected value of the statistic under the null hypothesis, that is 0.

estimate

the AUC in the two ROC curves.

data.name

the names of the data that was used.

parameter

for method="bootstrap" only: the values of the boot.n and boot.stratified arguments.

AUC specification

The comparison of the AUC of the ROC curves needs a specification of the AUC. The specification is defined by:

1. the “auc” field in the “roc” objects if reuse.auc is set to TRUE (default)
2. passing the specification to auc with ... (arguments partial.auc, partial.auc.correct and partial.auc.focus). In this case, you must ensure either that the roc object do not contain an auc field (if you called roc with auc=FALSE), or set reuse.auc=FALSE.

If reuse.auc=FALSE the auc function will always be called with ... to determine the specification, even if the “roc” objects do contain an auc field. As well if the “roc” objects do not contain an auc field, the auc function will always be called with ... to determine the specification. The AUC specification is ignored in the Venkatraman test. Warning: if the roc object passed to roc.test contains an auc field and reuse.auc=TRUE, auc is not called and arguments such as partial.auc are silently ignored.

Computation details

With method="bootstrap", the processing is done as follow:

1. boot.n bootstrap replicates are drawn from the data. If boot.stratified is TRUE, each replicate contains exactly the same number of controls and cases than the original sample, otherwise if FALSE the numbers can vary.
2. for each bootstrap replicate, the AUC of the two ROC curves are computed and the difference is stored.
3. The following formula is used: $$D=\frac{AUC1-AUC2}{s}$$ where s is the standard deviation of the bootstrap differences and AUC1 and AUC2 the AUC of the two (original) ROC curves.
4. D is then compared to the normal distribution, according to the value of alternative.

See also the Bootstrap section in this package's documentation. With method="delong", the processing is done as described in DeLong et al. (1988) for paired ROC curves. Only comparison of two ROC curves is implemented. The method has been extended for unpaired ROC curves where the p-value is computed with an unpaired t-test with unequal sample size and unequal variance. With method="venkatraman", the processing is done as described in Venkatraman and Begg (1996) (for paired ROC curves) and Venkatraman (2000) (for unpaired ROC curves) with boot.n permutation of sample ranks (with ties breaking). For consistency reasons, the same argument boot.n as in bootstrap defines the number of permutations to execute, even though no bootstrap is performed. For method="specificity", the test assesses if the sensitivity of the ROC curves are different at the level of specificity given by the specificity argument, which must be a numeric of length 1. Bootstrap is employed as with method="bootstrap" and boot.n and boot.stratified are available. This is identical to the test proposed by Pepe et al. (2009). The method="sensitivity" is very similar, but assesses if the specificity of the ROC curves are different at the level of sensitivity given by the sensitivity argument.

Warnings

If “auc” specifications are different in both roc objects, the warning “Different AUC specifications in the ROC curves. Enforcing the inconsistency, but unexpected results may be produced.” is issued. Unexpected results may be produced. If one or both ROC curves are “smooth.roc” objects with different smoothing specifications, the warning “Different smoothing parameters in the ROC curves. Enforcing the inconsistency, but unexpected results may be produced.” is issued. This warning can be benign, especially if ROC curves were generated with roc(..., smooth=TRUE) with different arguments to other functions (such as plot), or if you really want to compare two ROC curves smoothed differently. If method="delong" and the AUC specification specifies a partial AUC, the warning “Using DeLong's test for partial AUC is not supported. Using bootstrap test instead.” is issued. The method argument is ignored and “bootstrap” is used instead. If method="delong" and the ROC curve is smoothed, the warning “Using DeLong's test for smoothed ROCs is not supported. Using bootstrap test instead.” is issued. The method argument is ignored and “bootstrap” is used instead. If method="venkatraman", and the AUC specification specifies a partial AUC, the AUC specification is ignored with the warning “Partial AUC is ignored in Venkatraman's test.”. If method="venkatraman", and alternative is “less” or “greater”, the warning “Only two-sided tests are available for Venkatraman. Performing two-sided test instead.” is produced and a two tailed test is performed. Both DeLong and Venkatraman's test ignores the direction of the ROC curve so that if two ROC curves have a different differ in the value of direction, the warning “(DeLong|Venkatraman)'s test should not be applied to ROC curves with different directions.” is printed. However, the spurious test is enforced. If boot.stratified=FALSE and the sample has a large imbalance between cases and controls, it could happen that one or more of the replicates contains no case or control observation, or that there are not enough points for smoothing, producing a NA area. The warning “NA value(s) produced during bootstrap were ignored.” will be issued and the observation will be ignored. If you have a large imbalance in your sample, it could be safer to keep boot.stratified=TRUE. When both ROC curves have an auc of 1 (or 100%), their variances and covariance will always be null, and therefore the p-value will always be 1. This is true for both “delong”, “bootstrap” and “venkatraman” methods. This result is misleading, as the variances and covariance are of course not null. A warning will be displayed to inform of this condition, and of the misleading output.

Errors

An error will also occur if you give a predictor2 when predictor1 is a matrix or a data.frame, if predictor1 has more than two columns, or if you do not give a predictor2 when predictor1 is a vector. If density.cases and density.controls were provided for smoothing, the error “Cannot compute the statistic on ROC curves smoothed with density.controls and density.cases.” is issued. If method="venkatraman" and one of the ROC curves is smoothed, the error “Using Venkatraman's test for smoothed ROCs is not supported.” is produced. With method="specificity", the error “Argument 'specificity' must be numeric of length 1 for a specificity test.” is given unless the specificity argument is specified as a numeric of length 1. The “Argument 'sensitivity' must be numeric of length 1 for a sensitivity test.” message is given for method="sensitivity" under similar conditions.

Acknowledgements

We would like to thank E. S. Venkatraman and Colin B. Begg for their support in the implementation of their test.

Details

This function compares two ROC curves. It is typically called with the two roc objects to compare. roc.test.default is provided as a convenience method and creates two roc objects before calling roc.test.roc.

Three methods are available: “delong”, “bootstrap” and “venkatraman” (see “Computational details” section below). “delong” and “bootstrap” are tests over the AUC whereas “venkatraman” compares the the ROC curves themselves.

Default is to use “delong” method except for comparison of partial AUC, smoothed curves and curves with different direction, where bootstrap is used. Using “delong” for partial AUC and smoothed ROCs is not supported in pROC (a warning is produced and “bootstrap” is employed instead). It is spurious to use “delong” for roc with different direction (a warning is issued but the spurious comparison is enforced). “venkatraman”'s test cannot be employed to compare smoothed ROC curves. Additionally, partial AUC specifications are ignored (with a warning), and comparison of ROC curves with different direction should be used with care (a warning is produced as well).

If alternative="two.sided", a two-sided test for difference in AUC is performed. If alternative="less", the alternative is that the AUC of roc1 is smaller than the AUC of roc2. For method="venkatraman", only “two.sided” test is available. If the paired argument is not provided, the are.paired function is employed to detect the paired status of the ROC curves. It will test if the original response is identical between the two ROC curves (this is always the case if the call is made with roc.test.default). This detection is unlikely to raise false positives, but this possibility cannot be excluded entierly. It would require equal sample sizes and response values and order in both ROC curves. If it happens to you, use paired=FALSE. If you know the ROC curves are paired you can pass paired=TRUE. However this is useless as it will be tested anyway.

For smoothed ROC curves, smoothing is performed again at each bootstrap replicate with the parameters originally provided. If a density smoothing was performed with user-provided density.cases or density.controls the bootstrap cannot be performed and an error is issued.

References

Elisabeth R. DeLong, David M. DeLong and Daniel L. Clarke-Pearson (1988) ``Comparing the areas under two or more correlated receiver operating characteristic curves: a nonparametric approach''. Biometrics 44, 837--845. James A. Hanley and Barbara J. McNeil (1982) ``The meaning and use of the area under a receiver operating characteristic (ROC) curve''. Radiology 143, 29--36.

Margaret Pepe, Gary Longton and Holly Janes (2009) ``Estimation and Comparison of Receiver Operating Characteristic Curves''. The Stata journal 9, 1. Xavier Robin, Natacha Turck, Jean-Charles Sanchez and Markus Müller (2009) ``Combination of protein biomarkers''. useR! 2009, Rennes. http://www.r-project.org/nosvn/conferences/useR-2009/abstracts/user_author.html Xavier Robin, Natacha Turck, Alexandre Hainard, et al. (2011) ``pROC: an open-source package for R and S+ to analyze and compare ROC curves''. BMC Bioinformatics, 7, 77. DOI: 10.1186/1471-2105-12-77.

E. S. Venkatraman and Colin B. Begg (1996) ``A distribution-free procedure for comparing receiver operating characteristic curves from a paired experiment''. Biometrika 83, 835--848. DOI: 10.1093/biomet/83.4.835.

E. S. Venkatraman (2000) ``A Permutation Test to Compare Receiver Operating Characteristic Curves''. Biometrics 56, 1134--1138. DOI: 10.1111/j.0006-341X.2000.01134.x. Hadley Wickham (2011) ``The Split-Apply-Combine Strategy for Data Analysis''. Journal of Statistical Software, 40, 1--29. URL: www.jstatsoft.org/v40/i01.

See Also

roc, power.roc.test CRAN package plyr, employed in this function.

Examples

data(aSAH)

# Basic example with 2 roc objects
roc1 <- roc(aSAH$outcome, aSAH$s100b)
roc2 <- roc(aSAH$outcome, aSAH$wfns)
roc.test(roc1, roc2)

## Not run: 
# # The latter used Delong's test. To use bootstrap test:
# roc.test(roc1, roc2, method="bootstrap")
# # Increase boot.n for a more precise p-value:
# roc.test(roc1, roc2, method="bootstrap", boot.n=10000)
# ## End(Not run)

# Alternative syntaxes
roc.test(aSAH$outcome, aSAH$s100b, aSAH$wfns)
roc.test(aSAH$outcome, data.frame(aSAH$s100b, aSAH$wfns))

# If we had a good a priori reason to think that wfns gives a
# better classification than s100b (in other words, AUC of roc1
# should be lower than AUC of roc2):
roc.test(roc1, roc2, alternative="less")

## Not run: 
# # Comparison can be done on smoothed ROCs
# # Smoothing is re-done at each iteration, and execution is slow
# roc.test(smooth(roc1), smooth(roc2))
# # or:
# roc.test(aSAH$outcome, aSAH$s100b, aSAH$wfns, smooth=TRUE, boot.n=100)
# ## End(Not run)
# or from an AUC (no smoothing)
roc.test(auc(roc1), roc2)

## Not run: 
# # Comparison of partial AUC:
# roc3 <- roc(aSAH$outcome, aSAH$s100b, partial.auc=c(1, 0.8), partial.auc.focus="se")
# roc4 <- roc(aSAH$outcome, aSAH$wfns, partial.auc=c(1, 0.8), partial.auc.focus="se")
# roc.test(roc3, roc4)
# # This is strictly equivalent to:
# roc.test(roc3, roc4, method="bootstrap")
# 
# # Alternatively, we could re-use roc1 and roc2 to get the same result:
# roc.test(roc1, roc2, reuse.auc=FALSE, partial.auc=c(1, 0.8), partial.auc.focus="se")
# 
# # Comparison on specificity and sensitivity
# roc.test(roc1, roc2, method="specificity", specificity=0.9)
# roc.test(roc1, roc2, method="sensitivity", sensitivity=0.9)
# ## End(Not run)

# Spurious use of DeLong's test with different direction:
roc5 <- roc(aSAH$outcome, aSAH$s100b, direction="<")
roc6 <- roc(aSAH$outcome, aSAH$s100b, direction=">")
roc.test(roc5, roc6, method="delong")

## Not run: 
# # Comparisons of the ROC curves
# roc.test(roc1, roc2, method="venkatraman")
# ## End(Not run)

# Unpaired tests
roc7 <- roc(aSAH$outcome, aSAH$s100b)
# artificially create an roc8 unpaired with roc7
roc8 <- roc(aSAH$outcome[1:100], aSAH$s100b[1:100])
## Not run: 
# roc.test(roc7, roc8, paired=FALSE, method="delong")
# roc.test(roc7, roc8, paired=FALSE, method="bootstrap")
# roc.test(roc7, roc8, paired=FALSE, method="venkatraman")
# roc.test(roc7, roc8, paired=FALSE, method="specificity", specificity=0.9)
# ## End(Not run)