# auc: Compute the area under the ROC curve In pROC: Display and Analyze ROC Curves

## Description

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`.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```auc(...) ## S3 method for class '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 class 'smooth.roc' auc(smooth.roc, ...) ## S3 method for class 'multiclass.roc' auc(multiclass.roc, ...) ## S3 method for class 'formula' auc(formula, data, ...) ## Default S3 method: 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:

(1+(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:

 `partial.auc` if the AUC is full (FALSE) or partial (and in this case the bounds), as defined in argument. `partial.auc.focus` only for a partial AUC, if the bound specifies the sensitivity or specificity, as defined in argument. `partial.auc.correct` only for a partial AUC, was it corrected? As defined in argument. `percent` whether the AUC is given in percent or fraction. `roc` 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:

pnorm(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).

2/(count * (count - 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.

`roc`, `ci.auc`
 ```1 2 3 4 5 6 7 8 9``` ```# 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) ```