R: Smooth a ROC curve

smooth {pROC} | R Documentation

## Smooth a ROC curve

### Description

This function smoothes a ROC curve of numeric predictor. By default, a binormal smoothing is performed, but density or custom smoothings are supported.

### Usage

    smooth(...)
    ## Default S3 method:
    smooth(...)
    ## S3 method for class 'roc'
    smooth(roc,
    method=c("binormal", "density", "fitdistr", "logcondens",
    "logcondens.smooth"), n=512, bw = "nrd0", density=NULL,
    density.controls=density, density.cases=density,
    start=NULL, start.controls=start, start.cases=start,
    reuse.auc=TRUE, reuse.ci=FALSE, ...)
    ## S3 method for class 'smooth.roc'
    smooth(smooth.roc, ...)

### Arguments

`roc, smooth.roc` | a “roc” object from the `roc` function, or a “smooth.roc” object from the `smooth` function.
`method` | “binormal”, “density”, “fitdistr”, “logcondens”, “"logcondens.smooth"”, or a function returning a list of smoothed sensitivities and specificities.
`n` | the number of equally spaced points where the smoothed curve will be calculated.
`bw` | if `method="density"` and `density.controls` and `density.cases` are not provided, `bw` is passed to `density` to determine the bandwidth of the density Can be a character string (“nrd0”, “nrd”, “ucv”, “bcv” or “SJ”, but any name matching a function prefixed with “bw.” is supported) or a numeric value, as described in `density`. Defaults to “nrd0”.
`density, density.controls, density.cases` | if `method="density"`, a numeric value of density (over the y axis) or a function returning a density (such as `density`. If `method="fitdistr"`, a `densfun` argument for `fitdistr`. If the value is different for control and case observations, `density.controls` and `density.cases` can be employed instead, otherwise `density` will be propagated to both `density.controls` and `density.cases`.
`start, start.controls, start.cases` | if `method="fitdistr"`, optionnal `start` arguments for . `start.controls` and `start.cases` allows to specify different distributions for controls and cases.
`reuse.auc, reuse.ci` | if `TRUE` (default for reuse.auc) and the “roc” objects contain “auc” or “ci” fields, re-use these specifications to regenerate `auc` or `ci` on the smoothed ROC curve with the original parameters. If `FALSE`, the object returned will not contain “auc” or “ci” fields. It is currently not possible to redefine `auc` and `ci` options directly: you need to call `auc` or `ci` later for that.
`...` | further arguments passed to or from other methods, and especially to `density` (only `cut`, `adjust`, and `kernel`, plus `window` for compatibility with S+) and `fitdistr`. Also passed to to `method` if it is a function.

### Details

If `method="binormal"`, a linear model is fitted to the quantiles of the sensitivities and specificities. Smoothed sensitivities and specificities are then generated from this model on `n` points. This simple approach was found to work well for most ROC curves, but it may produce hooked smooths in some situations (see in Hanley (1988)).

With `method="density"`, the `density` function is employed to generate a smooth kernel density of the control and case observations as described by Zhou et al. (1997), unless `density.controls` or `density.cases` are provided directly. `bw` can be given to specify a bandwidth to use with `density`. It can be a numeric value or a character string (“nrd0”, “nrd”, “ucv”, “bcv” or “SJ”, but any name matching a function prefixed with “bw.” is supported). In the case of a character string, the whole predictor data is employed to determine the numeric value to use on both controls and cases. Depending on your data, it might be a good idea to specify the `kernel` argument for `density`. By default, “gaussian” is used, but “epanechnikov”, “rectangular”, “triangular”, “biweight”, “cosine” and “optcosine” are supported. As all the kernels are symetrical, 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))
        smooth(roc(response, norm.x, ...), ...)

Additionally, `density` can be a function which must return either a numeric vector of densities over the y axis or a list with a “y” item like the `density` function. It must accept the following input:

        density.fun(x, n, from, to, bw, kernel, ...)

It is important to honour `n`, `from` and `to` in order to have the densities evaluated on the same points for controls and cases. Failing to do so and returning densities of different length will produce an error. It is also a good idea to use a constant smoothing parameter (such as `bw`) especially when controls and cases have a different number of observations, to avoid producing smoother or rougher densities.

If `method="fitdistr"`, the `fitdistr` function from the MASS package is employed to fit parameters for the density function `density` with optionnal start parameters `start`. The density function are fitted separately in control (`density.controls`, `start.controls`) and case observations (`density.cases`, `start.cases`). `density` can be one of the character values allowed by `fitdistr` or a density function (such as `dnorm`, `dweibull`, ...).

The `method="logcondens"` and `method="logcondens.smooth"` use the logcondens package to generate a non smoothed or smoothed (respectively) log-concave density estimate of of the control and case observation with the logConROC function.

Finally, `method` can also be a function. It must return a list with exactly 2 elements named “sensitivities” and “specificities”, which must be numeric vectors between 0 and 1 or 100 (depending on the `percent` argument to `roc`). It is passed all the arguments to the `smooth` function.

`smooth.default` forces the usage of the `smooth` function in the stats package, so that other code relying on `smooth` should continue to function normally.

Smoothed ROC curves can be passed to smooth again. In this case, the smoothing is not re-applied on the smoothed ROC curve but the original “roc” object will be re-used.

Note that a `smooth.roc` curve has no threshold.

### Value

A list of class “smooth.roc” with the following fields:

`sensitivities` | the smoothed sensitivities defining the ROC curve.
`specificities` | the smoothed specificities defining the ROC curve.
`percent` | if the sensitivities, specificities and AUC are reported in percent, as defined in argument.
`direction` | the direction of the comparison, as defined in argument.
`call` | how the function was called. See `match.call` for more details.
`smoothing.args` | a list of the arguments used for the smoothing. Will serve to apply the smoothing again in further bootstrap operations.
`auc` | if the original ROC curve contained an AUC, it is computed again on the smoothed ROC.
`ci` | if the original ROC curve contained a CI, it is computed again on the smoothed ROC.
`fit.controls, fit.cases` | with `method="fitdistr"` only: the result of MASS's `fitdistr` function for controls and cases, with an additional “densfun” item indicating the density function, if possible as character.
`logcondens` | with `method="logcondens"` and `method="logcondens.smooth"` only: the result of logcondens's logConROC function.
`model` | with `method="binormal"` only: the linear model from `lm` used to smooth the ROC curve.

#### Attributes

Additionally, the original `roc` object is stored as a “roc” attribute.

### Errors

If `method` is a function, the return values will be checked thoroughly for validity (list with two numeric elements of the same length named “sensitivities” and “specificities” with values in the range of possible values for sensitivities and specificities).

The message “The 'density' function must return a numeric vector or a list with a 'y' item.” will be displayed if the `density` function did not return a valid output. The message “Length of 'density.controls' and 'density.cases' differ.” will be displayed if the returned value differ in length.

Binormal smoothing cannot smooth ROC curve defined by only one point. Any such attempt will fail with the error “ROC curve not smoothable (not enough points).”.

If the smooth ROC curve was generated by `roc` with `density.controls` and `density.cases` numeric arguments, it cannot be smoothed and the error “Cannot smooth a ROC curve generated directly with numeric 'density.controls' and 'density.cases'.” is produced.

`fitdistr` and `density` smoothing methods require a numeric `predictor`. If the ROC curve to smooth was generated with an ordered factor only binormal smoothing can be applied and the message “ROC curves of ordered predictors can be smoothed only with binormal smoothing.” is displayed otherwise.

`fitdistr`, `logcondens` and `logcondens.smooth` methods require additional packages. If not available, the following message will be displayed with the required command to install the package: “Package ? not available, required with method='?'. Please install it with 'install.packages("?")'. ”

### References

James E. Hanley (1988) “The robustness of the “binormal” assumptions used in fitting ROC curves”. Medical Decision Making 8, 197–203.

Lutz Duembgen, Kaspar Rufibach (2011) “logcondens: Computations Related to Univariate Log-Concave Density Estimation”. Journal of Statistical Software, 39, 1–28. URL: jstatsoft.org/v39/i06.

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.

Kaspar Rufibach (2011) “A Smooth ROC Curve Estimator Based on Log-Concave Density Estimates”. The International Journal of Biostatistics, 8, accepted. DOI: 10.1515/1557-4679.1378. arXiv: 1103.1787.

William N. Venables, Brian D. Ripley (2002). “Modern Applied Statistics with S”. New York, Springer. Google books.

Kelly H. Zou, W. J. Hall and David E. Shapiro (1997) “Smooth non-parametric receiver operating characteristic (ROC) curves for continuous diagnostic tests”. Statistics in Medicine 18, 2143–2156. DOI: 10.1002/(SICI)1097-0258(19971015)16:19<2143::AID-SIM655>3.0.CO;2-3.

### See Also

`roc`

CRAN packages MASS and logcondens employed in this function.

### Examples

    data(aSAH)

    ##  Basic example
    rocobj <- roc(aSAH$outcome, aSAH$s100b)
    smooth(rocobj)
    # or directly with roc()
    roc(aSAH$outcome, aSAH$s100b, smooth=TRUE)

    # plotting
    plot(rocobj)
    rs <- smooth(rocobj, method="binormal")
    plot(rs, add=TRUE, col="green")
    rs2 <- smooth(rocobj, method="density")
    plot(rs2, add=TRUE, col="blue")
    rs3 <- smooth(rocobj, method="fitdistr", density="lognormal")
    plot(rs3, add=TRUE, col="magenta")
    rs4 <- smooth(rocobj, method="logcondens")
    plot(rs4, add=TRUE, col="brown")
    rs5 <- smooth(rocobj, method="logcondens.smooth")
    plot(rs5, add=TRUE, col="orange")
    legend("bottomright", legend=c("Empirical", "Binormal", "Density", "Log-normal",
                                   "Log-concave density", "Smoothed log-concave density"),
           col=c("black", "green", "blue", "magenta", "brown", "orange"), lwd=2)

    ## Advanced smoothing

    # if we know the distributions are normal with sd=0.1 and an unknown mean:
    smooth(rocobj, method="fitdistr", density=dnorm, start=list(mean=1), sd=.1)
    # different distibutions for controls and cases:
    smooth(rocobj, method="fitdistr", density.controls="normal", density.cases="lognormal")

    # with densities
    bw <- bw.nrd0(rocobj$predictor)
    density.controls <- density(rocobj$controls, from=min(rocobj$predictor) - 3 * bw,
                                to=max(rocobj$predictor) + 3*bw, bw=bw, kernel="gaussian")
    density.cases <- density(rocobj$cases, from=min(rocobj$predictor) - 3 * bw,
                                to=max(rocobj$predictor) + 3*bw, bw=bw, kernel="gaussian")
    smooth(rocobj, method="density", density.controls=density.controls$y,
           density.cases=density.cases$y)
    # which is roughly what is done by a simple:
    smooth(rocobj, method="density")

    ## Not run:
    ## Smoothing artificial ROC curves

    rand.unif <- runif(1000, -1, 1)
    rand.exp <- rexp(1000)
    rand.norm <-
    rnorm(1000)

    # two normals
    roc.norm <- roc(controls=rnorm(1000), cases=rnorm(1000)+1, plot=TRUE)
    plot(smooth(roc.norm), col="green", lwd=1, add=TRUE)
    plot(smooth(roc.norm, method="density"), col="red", lwd=1, add=TRUE)
    plot(smooth(roc.norm, method="fitdistr"), col="blue", lwd=1, add=TRUE)
    plot(smooth(roc.norm, method="logcondens"), col="brown", lwd=1, add=TRUE)
    plot(smooth(roc.norm, method="logcondens.smooth"), col="orange", lwd=1, add=TRUE)
    legend("bottomright", legend=c("empirical", "binormal", "density", "fitdistr",
                                   "logcondens", "logcondens.smooth"),
           col=c(par("fg"), "green", "red", "blue", "brown", "orange"), lwd=c(2, 1, 1, 1))

    # deviation from the normality
    roc.norm.exp <- roc(controls=rnorm(1000), cases=rexp(1000), plot=TRUE)
    plot(smooth(roc.norm.exp), col="green", lwd=1, add=TRUE)
    plot(smooth(roc.norm.exp, method="density"), col="red", lwd=1, add=TRUE)
    # Wrong fitdistr: normality assumed by default
    plot(smooth(roc.norm.exp, method="fitdistr"), col="blue", lwd=1, add=TRUE)
    # Correct fitdistr
    plot(smooth(roc.norm.exp, method="fitdistr", density.controls="normal",
                density.cases="exponential"), col="purple", lwd=1, add=TRUE)
    plot(smooth(roc.norm.exp, method="logcondens"), col="brown", lwd=1, add=TRUE)
    plot(smooth(roc.norm.exp, method="logcondens.smooth"), col="orange", lwd=1, add=TRUE)
    legend("bottomright", legend=c("empirical", "binormal", "density",
                                   "wrong fitdistr", "correct fitdistr",
                                   "logcondens", "logcondens.smooth"),
           col=c(par("fg"), "green", "red", "blue", "purple", "brown", "orange"), lwd=c(2, 1, 1, 1, 1))

    # large deviation from the normality
    roc.unif.exp <- roc(controls=runif(1000, 2, 3), cases=rexp(1000)+2, plot=TRUE)
    plot(smooth(roc.unif.exp), col="green", lwd=1, add=TRUE)
    plot(smooth(roc.unif.exp, method="density"), col="red", lwd=1, add=TRUE)
    plot(smooth(roc.unif.exp, method="density", bw="ucv"), col="magenta", lwd=1, add=TRUE)
    # Wrong fitdistr: normality assumed by default (uniform distributions not handled)
    plot(smooth(roc.unif.exp, method="fitdistr"), col="blue", lwd=1, add=TRUE)
    plot(smooth(roc.unif.exp, method="logcondens"), col="brown", lwd=1, add=TRUE)
    plot(smooth(roc.unif.exp, method="logcondens.smooth"), col="orange", lwd=1, add=TRUE)
    legend("bottomright", legend=c("empirical", "binormal", "density",
                                   "density ucv", "wrong fitdistr",
                                   "logcondens", "logcondens.smooth"),
           col=c(par("fg"), "green", "red", "magenta", "blue", "brown", "orange"), lwd=c(2, 1, 1, 1, 1))

    ## End(Not run)

    # 2 uniform distributions with a custom density function
    unif.density <- function(x, n, from, to, bw, kernel, ...) {
      smooth.x <- seq(from=from, to=to, length.out=n)
      smooth.y <- dunif(smooth.x, min=min(x), max=max(x))
      return(smooth.y)
    }
    roc.unif <- roc(controls=runif(1000, -1, 1), cases=runif(1000, 0, 2), plot=TRUE)
    s <- smooth(roc.unif, method="density", density=unif.density)
    plot(roc.unif)
    plot(s, add=TRUE, col="grey")

    ## Not run:
    # you can bootstrap a ROC curve smoothed with a density function:
    ci(s, boot.n=100)

    ## End(Not run)

[Package pROC version 1.16.0 Index]