ks (version 1.8.13)

kroc: Kernel receiver operating characteristic (ROC) curve

Description

Kernel receiver operating characteristic (ROC) curve for 1- to 3-dimensional data.

Usage

kroc(x1, x2, H1, h1, hy, gridsize, gridtype, xmin, xmax, supp=3.7, eval.points,
   binned=FALSE, bgridsize, positive=FALSE, adj.positive, w, verbose=FALSE,
   approx.ref="pmvnorm")
## S3 method for class 'kroc':
summary(object, ...)

Arguments

x1,x2

vector/matrix of data values

H1,h1,hy

bandwidth matrix/scalar bandwidths. If these are missing, Hpi.kcde, hpi.kcde is called by default.

gridsize

vector of number of grid points

gridtype

not yet implemented

xmin,xmax

vector of minimum/maximum values for grid

supp

effective support for standard normal

eval.points

not yet implemented

binned

flag for binned estimation. Default is FALSE.

bgridsize

vector of binning grid sizes

positive

flag if 1-d data are positive. Default is FALSE.

adj.positive

adjustment applied to positive 1-d data

w

vector of weights. Default is a vector of all ones.

verbose

flag to print out progress information. Default is FALSE.

approx.ref

use "kcde" or "pmvnorm" (default) to compute reference ROC curve.

object

object of class kroc, output from kroc

...

other summary parameters

Value

• A kernel ROC curve is an object of class kroc which is a list with fields:
• x
  list of data values x1, x2 - same as input
• eval.points
  points at which the estimate is evaluated
• estimate
  ROC curve estimate at eval.points
• gridtype
  "linear"
• gridded
  flag for estimation on a grid
• binned
  flag for binned estimation
• names
  variable names
• w
  weights
• tail
  "lower.tail"
• h1
  scalar bandwidth for first sample (1-d only)
• H1
  bandwidth matrix for first sample
• hy
  scalar bandwidth for ROC curve
• indices
  summary indices of ROC curve.

Details

The usual method for computing 1-d ROC curves is based on $R(z) = F_2 (\bar{F}_1^{-1}(z))$ where $\bar{F}_1, F_2$ are the respective survival and cumulative distribution function, and $\bar{F}_1^{-1}(z)$ is the $z$-th quantile of $\bar{F}_1$, is not valid for multivariate data. Duong (2013), based on Lloyd (1998), develops an alternative formulation $R(z) = F_Y(z)$ as the cumulative distribution function of $Y = \bar{F}_1(\bold{X}_2)$.

In this set-up, the values in the first sample x1 should be larger in general that those in the second sample x2.

If the bandwidth H1 is missing from kroc, then the default bandwidth is the binned 2-stage plug-in selector Hpi.kcde(, nstage=2, binned=TRUE). Likewise for missing h1,hy. A bandwidth matrix H1 is required for x1 for d>1, but the second bandwidth hy is always a scalar since Y is a 1-d variable. These bandwidth selectors are optimal for cumulative distribution/survival functions, see Duong (2013).

To compute the reference ROC curve, either kcde or pmvnorm (from the mvtnorm package) is used. The former is faster because it's a binned kcde computation, whereas the latter is a more exact computation of the multivariate normal CDF and thus can give a smoother ROC curve. Binning/exact estimation and positive 1-d data behaviour is the same as for kde. --The summary method for kroc objects prints out the summary indices of the ROC curve, as contained in the indices field, namely the AUC (area under the curve) and Youden index, see Duong (2013).

References

Duong, T. (2013) Non-parametric kernel estimation of multivariate cumulative distribution functions and receiver operating characteristic curves. Submitted.

Lloyd, C. (1998) Using smoothed receiver operating curves to summarize and compare diagnostic systems. Journal of the American Statistical Association, 93, 1356--1364.

See Also

kcde

Examples

samp <- 1000
x <- rnorm.mixt(n=samp, mus=0, sigmas=1, props=1)
y <- rnorm.mixt(n=samp, mus=0.5, sigmas=1, props=1)
Rhat <- kroc(x1=x, x2=y)  
summary(Rhat)