# auc

##### Compute the area under the ROC curve

This function computes the numeric value of area under the ROC curve
(AUC) with the trapezoidal rule. Two syntaxes are possible: one object of class “roc”, or either
two vectors (response, predictor) or a formula (response~predictor) as
in the `roc`

function.
By default, the total AUC is computed, but a portion of the ROC curve
can be specified with `partial.auc`

.

- Keywords
- utilities, nonparametric, univar, ROC

##### Usage

```
auc(...)
# S3 method for roc
auc(roc, partial.auc=FALSE, partial.auc.focus=c("specificity",
"sensitivity"), partial.auc.correct=FALSE,
allow.invalid.partial.auc.correct = FALSE, ...)
# S3 method for smooth.roc
auc(smooth.roc, ...)
# S3 method for multiclass.roc
auc(multiclass.roc, ...)
# S3 method for formula
auc(formula, data, ...)
# S3 method for default
auc(response, predictor, ...)
```

##### Arguments

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

`roc`

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

function, or a “multiclass.roc” or “mv.multiclass.roc” from the`multiclass.roc`

function.- response, predictor
arguments for the

`roc`

function.- formula, data
a formula (and possibly a data object) of type response~predictor for the

`roc`

function.- partial.auc
either

`FALSE`

(default: consider total area) or a numeric vector of length 2: boundaries of the AUC to consider in [0,1] (or [0,100] if percent is`TRUE`

).- partial.auc.focus
if

`partial.auc`

is not`FALSE`

and a partial AUC is computed, specifies if`partial.auc`

specifies the bounds in terms of specificity (default) or sensitivity. Can be shortened to spec/sens or even sp/se. Ignored if`partial.auc=FALSE`

.- partial.auc.correct
logical indicating if the correction of AUC must be applied in order to have a maximal AUC of 1.0 and a non-discriminant AUC of 0.5 whatever the

`partial.auc`

defined. Ignored if`partial.auc=FALSE`

. Default:`FALSE`

.- allow.invalid.partial.auc.correct
logical indicating if the correction must return

`NA`

(with a`warning`

) when attempting to correct a pAUC below the diagonal. Set to`TRUE`

to return a (probably invalid) corrected AUC. This is useful especially to avoid introducing a bias against low pAUCs in bootstrap operations.- …
further arguments passed to or from other methods, especially arguments for

`roc`

when calling`auc.default`

,`auc.formula`

,`auc.smooth.roc`

. Note that the`auc`

argument of`roc`

is not allowed. Unused in`auc.roc`

.

##### Details

This function is typically called from `roc`

when `auc=TRUE`

(default). It is also used by `ci`

. When it is called with
two vectors (response, predictor) or a formula (response~predictor)
arguments, the `roc`

function is called and only the AUC is
returned.

By default the total area under the curve is computed, but a partial AUC (pAUC)
can be
specified with the `partial.auc`

argument. It specifies the bounds of
specificity or sensitivity (depending on `partial.auc.focus`

) between
which the AUC will be computed. As it specifies specificities or
sensitivities, you must adapt it in relation to the 'percent'
specification (see details in `roc`

).

`partial.auc.focus`

is ignored if
`partial.auc=FALSE`

(default). If a partial AUC is computed,
`partial.auc.focus`

specifies if the bounds specified in
`partial.auc`

must be interpreted as sensitivity or
specificity. Any other value will produce an error. It is recommended to
`plot`

the ROC curve with `auc.polygon=TRUE`

in order to
make sure the specification is correct.

If a pAUC is defined, it can be standardized (corrected). This correction is
controled by the `partial.auc.correct`

argument. If `partial.auc.correct=TRUE`

,
the correction by McClish will be applied:

$$\frac{1+\frac{auc-min}{max-min}}{2}$$

where auc is the uncorrected pAUC computed in the region defined by `partial.auc`

,
min is the value of the non-discriminant AUC (with an AUC of 0.5 or 50
in the region and max is the maximum possible AUC in the region. With this correction, the AUC
will be 0.5 if non discriminant and 1.0 if maximal, whatever the region
defined. This correction is fully compatible with `percent`

.

Note that this correction is undefined for curves below the diagonal (auc < min). Attempting
to correct such an AUC will return `NA`

with a warning.

##### Value

The numeric AUC value, of class `c("auc", "numeric")`

(or
`c("multiclass.auc", "numeric")`

or `c("mv.multiclass.auc", "numeric")`

if a “multiclass.roc” was supplied), in
fraction of the area or in percent if `percent=TRUE`

, with the
following attributes:

if the AUC is full (FALSE) or partial (and in this case the bounds), as defined in argument.

only for a partial AUC, if the bound specifies the sensitivity or specificity, as defined in argument.

only for a partial AUC, was it corrected? As defined in argument.

whether the AUC is given in percent or fraction.

the original ROC curve, as a “roc”, “smooth.roc” or “multiclass.roc” object.

##### Smoothed ROC curves

There is no difference in the computation of the area under a smoothed
ROC curve, except for curves smoothed with `method="binomial"`

. In this case
and only if a full AUC is requested, the classical binormal AUC formula is applied:

$$auc=\phi\frac{a}{\sqrt{1 + b^2}}.$$

If the ROC curve is smoothed with any other `method`

or if a partial AUC
is requested, the empirical AUC described in the previous section is applied.

##### Multi-class AUCs

With an object of class “multiclass.roc”, a multi-class AUC is computed as an average AUC as defined by Hand and Till (equation 7).

$$auc=\frac{2}{c(c-1)}\sum{aucs}$$

with aucs all the pairwise roc curves.

##### References

Tom Fawcett (2006) ``An introduction to ROC analysis''. *Pattern
Recognition Letters* **27**, 861--874. DOI:
10.1016/j.patrec.2005.10.010.

David J. Hand and Robert J. Till (2001). A Simple Generalisation of
the Area Under the ROC Curve for Multiple Class Classification
Problems. *Machine Learning* **45**(2), p. 171--186. DOI:
10.1023/A:1010920819831.

Donna Katzman McClish (1989) ``Analyzing a Portion of the ROC
Curve''. *Medical Decision Making* **9**(3), 190--195. DOI:
10.1177/0272989X8900900307.

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.

##### See Also

##### Examples

```
# NOT RUN {
# Create a ROC curve:
data(aSAH)
roc.s100b <- roc(aSAH$outcome, aSAH$s100b)
# Get the full AUC
auc(roc.s100b)
# Get the partial AUC:
auc(roc.s100b, partial.auc=c(1, .8), partial.auc.focus="se", partial.auc.correct=TRUE)
# }
```

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