# smooth

##### Smooth a ROC curve

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

- Keywords
- smooth, utilities, nonparametric, univar, ROC

##### Usage

```
smooth(...)
# S3 method for default
smooth(...)
# S3 method for 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 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:

the smoothed sensitivities defining the ROC curve.

the smoothed specificities defining the ROC curve.

if the sensitivities, specificities and AUC are reported in percent, as defined in argument.

the direction of the comparison, as defined in argument.

how the function was called. See `match.call`

for
more details.

a list of the arguments used for the smoothing. Will serve to apply the smoothing again in further bootstrap operations.

if the original ROC curve contained an AUC, it is computed again on the smoothed ROC.

if the original ROC curve contained a CI, it is computed again on the smoothed ROC.

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.

with `method="logcondens"`

and `method="logcondens.smooth"`

only:
the result of logcondens's logConROC function.

with `method="binormal"`

only:
the linear model from `lm`

used to smooth the ROC curve.

`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

CRAN packages MASS and logcondens employed in this function.

##### Examples

```
# NOT RUN {
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))
# }
# 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)
# }
```

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