R: Covariance of two paired ROC curves

cov.roc {pROC} | R Documentation

## Covariance of two paired ROC curves

### Description

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

### Usage

    cov(...)
    ## Default S3 method:
    cov(...)
    ## S3 method for class 'auc'
    cov(roc1, roc2, ...)
    ## S3 method for class 'smooth.roc'
    cov(roc1, roc2, ...)
    ## S3 method for class '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:

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.

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 stored.

3. the variance (as per `var.roc`) of the resampled AUCs and their covariance are assessed in a single bootstrap pass.

4. 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

`roc`, `var.roc`

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)
    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)

    ## End(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))

    ## End(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])

    ## End(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")

    ## 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=">")
    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")

    ## End(Not run)

[Package pROC version 1.16.0 Index]