# cov.roc

##### Covariance of two paired ROC curves

This function computes the covariance between the AUC of two correlated (or paired) ROC curves.

- Keywords
- multivariate, utilities, nonparametric, ROC

##### Usage

```
cov(...)
# S3 method for default
cov(...)
# S3 method for auc
cov(roc1, roc2, ...)
# S3 method for smooth.roc
cov(roc1, roc2, ...)
# S3 method for roc
cov(roc1, roc2, method=c("delong", "bootstrap", "obuchowski"),
reuse.auc=TRUE, boot.n=2000, boot.stratified=TRUE, boot.return=FALSE,
progress=getOption("pROCProgress")$name, parallel=FALSE, ...)
```

##### Arguments

- roc1, roc2
the two ROC curves on which to compute the covariance. Either “roc”, “auc” or “smooth.roc” objects (types can be mixed as long as the original ROC curve are paired).

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

- reuse.auc
if

`TRUE`

(default) and the “roc” objects contain an “auc” field, re-use these specifications for the test. See details.- boot.n
for

`method="bootstrap"`

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. Default:`TRUE`.- boot.return
if

`TRUE`and`method="bootstrap"`

, also return the bootstrapped values. See the “Value” section for more details.- 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

`cov.roc`

when calling`cov`

,`cov.auc`

or`cov.smooth.roc`

. Arguments for`auc`

(if`reuse.auc=FALSE`

) and`txtProgressBar`

(only`char`

and`style`

) if applicable.

##### Details

This function computes the covariance between the AUC of two
correlated (or paired, according to the detection of `are.paired`

) ROC
curves. It is typically called with the two roc objects of
interest. Two methods are available: “delong” and
“bootstrap” (see “Computational
details” section below).

The default is to use “delong” method except with partial AUC and smoothed curves where “bootstrap” is employed. Using “delong” for partial AUC and smoothed ROCs is not supported.

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.

`cov.default`

forces the usage of the
`cov`

function in the stats package, so
that other code relying on `cov`

should continue to function
normally.

##### Value

The numeric value of the covariance.

If `boot.return=TRUE`

and `method="bootstrap"`

, an attribute
`resampled.values`

is set with the resampled (bootstrapped)
values. It contains a matrix with the columns representing the two ROC
curves, and the rows the `boot.n`

bootstrap replicates.

##### AUC specification

To compute the covariance of the AUC of the ROC curves, `cov`

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

the “auc” field in the “roc” objects if

`reuse.auc`

is set to`TRUE`

(default)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.

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:

`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.for each bootstrap replicate, the AUC of the two ROC curves are computed and stored.

the variance (as per

`var.roc`

) of the resampled AUCs and their covariance are assessed in a single bootstrap pass.The following formula is used to compute the final covariance: \(Var[AUC1] + Var[AUC2] - 2cov[AUC1,AUC2]\)

With `method="delong"`

, the processing is done as described in
Hanley and Hajian-Tilaki (1997) using the algorithm by Sun and Xu (2014).

With `method="obuchowski"`

, the processing is done as described
in Obuchowski and McClish (1997), Table 1 and Equation 5, p. 1531. The
computation of \(g\) for partial area under the ROC curve is
modified as:

$$expr1 * (2 * pi * expr2) ^ {(-1)} * (-expr4) - A * B * expr1 * (2 * pi * expr2^3) ^ {(-1/2)} * expr3$$.

##### Binormality assumption

The “obuchowski” method makes the assumption that the data is binormal.
If the data shows a deviation from this assumption, it might help to
normalize the data first (that is, before calling `roc`

),
for example with quantile normalization:

norm.x <- qnorm(rank(x)/(length(x)+1)) cov(roc(response, norm.x, ...), ...)

“delong” and “bootstrap” methods make no such assumption.

##### Errors

If `density.cases`

and `density.controls`

were provided
for smoothing, the error “Cannot compute the covariance on ROC
curves smoothed with density.controls and density.cases.” is
issued.

##### 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 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 for
smoothed ROCs is not supported. Using bootstrap instead.” is
issued. The `method`

argument is ignored and “bootstrap”
is used instead.

DeLong ignores the direction of the ROC curve so that if two
ROC curves have a different `direction`

, the warning
“"DeLong should not be applied to ROC curves with a different
direction."” is printed. However, the spurious computation 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 covariance will always be null.
This is true for both “delong” and “bootstrap” and methods. This result is misleading,
as the covariance is of course not null.
A `warning`

will be displayed to inform of this condition, and of the misleading output.

##### Messages

The covariance can only be computed on paired data. This
assumption is enforced by `are.paired`

. If the ROC curves
are not paired, the covariance is `0`

and the message “ROC
curves are unpaired.” is printed. If your ROC curves are paired, make
sure they fit `are.paired`

criteria.

##### 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 Karim O. Hajian-Tilaki (1997) ``Sampling
variability of nonparametric estimates of the areas under receiver
operating characteristic curves: An update''. *Academic
Radiology* **4**, 49--58. DOI:
10.1016/S1076-6332(97)80161-4.

Nancy A. Obuchowski, Donna K. McClish (1997). ``Sample size
determination for diagnostic accurary studies involving binormal ROC
curve indices''. *Statistics in Medicine*, **16**(13),
1529--1542. DOI: (SICI)1097-0258(19970715)16:13<1529::AID-SIM565>3.0.CO;2-H.

Xu Sun and Weichao Xu (2014) ``Fast Implementation of DeLongs Algorithm for Comparing
the Areas Under Correlated Receiver Operating Characteristic Curves''. *IEEE Signal
Processing Letters*, **21**, 1389--1393.
DOI: 10.1109/LSP.2014.2337313.

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

CRAN package plyr, employed in this function.

##### Examples

```
# NOT RUN {
data(aSAH)
# Basic example with 2 roc objects
roc1 <- roc(aSAH$outcome, aSAH$s100b)
roc2 <- roc(aSAH$outcome, aSAH$wfns)
cov(roc1, roc2)
# }
# NOT RUN {
# The latter used Delong. To use bootstrap:
cov(roc1, roc2, method="bootstrap")
# Decrease boot.n for a faster execution:
cov(roc1, roc2, method="bootstrap", boot.n=1000)
# }
# NOT RUN {
# To use Obuchowski:
cov(roc1, roc2, method="obuchowski")
# }
# NOT RUN {
# Comparison can be done on smoothed ROCs
# Smoothing is re-done at each iteration, and execution is slow
cov(smooth(roc1), smooth(roc2))
# }
# NOT RUN {
# or from an AUC (no smoothing)
cov(auc(roc1), roc2)
# }
# NOT RUN {
# With bootstrap and return.values, one can compute the variances of the
# ROC curves in one single bootstrap run:
cov.rocs <- cov(roc1, roc2, method="bootstrap", boot.return=TRUE)
# var(roc1):
var(attr(cov.rocs, "resampled.values")[,1])
# var(roc2):
var(attr(cov.rocs, "resampled.values")[,2])
# }
# NOT RUN {
# }
# NOT RUN {
# Covariance 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")
cov(roc3, roc4)
# This is strictly equivalent to:
cov(roc3, roc4, method="bootstrap")
# Alternatively, we could re-use roc1 and roc2 to get the same result:
cov(roc1, roc2, reuse.auc=FALSE, partial.auc=c(1, 0.8), partial.auc.focus="se")
# }
# 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=">")
cov(roc5, roc6, method="delong")
## Test data from Hanley and Hajian-Tilaki, 1997
disease.present <- c("Yes", "No", "Yes", "No", "No", "Yes", "Yes", "No",
"No", "Yes", "No", "No", "Yes", "No", "No")
field.strength.1 <- c(1, 2, 5, 1, 1, 1, 2, 1, 2, 2, 1, 1, 5, 1, 1)
field.strength.2 <- c(1, 1, 5, 1, 1, 1, 4, 1, 2, 2, 1, 1, 5, 1, 1)
roc7 <- roc(disease.present, field.strength.1)
roc8 <- roc(disease.present, field.strength.2)
# Assess the covariance:
cov(roc7, roc8)
# }
# NOT RUN {
# With bootstrap:
cov(roc7, roc8, method="bootstrap")
# }
# NOT RUN {
# }
```

*Documentation reproduced from package pROC, version 1.16.2, License: GPL (>= 3)*