# var.roc

##### Variance of a ROC curve

These functions compute the variance of the AUC of a ROC curve.

- Keywords
- utilities, nonparametric, univar, ROC

##### Usage

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

##### Arguments

- roc, smooth.roc, auc
a “roc” object from the

`roc`

function, a “smooth.roc” object from the`smooth`

function or an “auc” object from the`auc`

function.- 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`.- 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

`var.roc`

when calling`var`

,`var.auc`

and`var.smooth.roc`

. Arguments for`auc`

(if`reuse.auc=FALSE`

) and`txtProgressBar`

(only`char`

and`style`

) if applicable.

##### Details

The `var`

function computes the variance of the AUC of a ROC
curve. It is typically called with the roc object of
interest. Two methods are available: “delong” and
“bootstrap” (see “Computational
details” section below).

The default is to use “delong” method except for 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.

`var.default`

forces the usage of the
`var`

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

should continue to function
normally.

##### Value

The numeric value of the variance.

##### AUC specification

`var`

needs a specification of the AUC to compute
the variance of the AUC of the ROC curve.
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 ROC curve is computed and stored.

the variance of the resampled AUCs are computed and returned.

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 4, p. 1530--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)) var(roc(response, norm.x, ...), ...)

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

##### Warnings

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 test instead.” is
issued. The `method`

argument is ignored and “bootstrap”
is used instead.

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 the ROC curve has an `auc`

of 1 (or 100%), the variance will always be null.
This is true for both “delong” and “bootstrap” methods that can
not properly assess the variance in this case. This result is misleading, as the variance is of course not null.
A `warning`

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

##### 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.

##### 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
roc1 <- roc(aSAH$outcome, aSAH$s100b)
roc2 <- roc(aSAH$outcome, aSAH$wfns)
var(roc1)
var(roc2)
# We could also write it in one line:
var(roc(aSAH$outcome, aSAH$s100b))
# }
# NOT RUN {
# The latter used Delong. To use bootstrap:
var(roc1, method="bootstrap")
# Decrease boot.n for a faster execution
var(roc1,method="bootstrap", boot.n=1000)
# }
# NOT RUN {
# To use obuchowski:
var(roc1, method="obuchowski")
# }
# NOT RUN {
# Variance of smoothed ROCs:
# Smoothing is re-done at each iteration, and execution is slow
var(smooth(roc1))
# }
# NOT RUN {
# or from an AUC (no smoothing)
var(auc(roc1))
## 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)
roc3 <- roc(disease.present, field.strength.1)
roc4 <- roc(disease.present, field.strength.2)
# Assess the variance:
var(roc3)
var(roc4)
# }
# NOT RUN {
# With bootstrap:
var(roc3, method="bootstrap")
var(roc4, method="bootstrap")
# }
# NOT RUN {
# }
```

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