logcondens (version 2.1.8)

logConROC: Compute ROC curve based on log-concave estimates for the constituent distributions

Description

The receiver operating characteristic (ROC) curve for two constituent distributions \(F\) and \(G\) is defined as

$$R(t; F, G) = 1 - G(F^{-1}(1 - t))$$

for \(t \in [0, 1]\). It is typically used to assess the performance of a diagnostic test used to discriminate between healthy and diseased individuals based on a continuous variable.

Usage

logConROC(cases, controls, grid, smooth = TRUE)

Value

m

Number of control measurements.

n

Number of case measurements.

fROC

Estimated ROC curve based on the log-concave density estimate.

fROC.smooth

Estimated ROC curve based on the smoothed log-concave density estimate.

res0

dlc object as a result of a call to logConDens for the data of the controls.

res1

dlc object as a result of a call to logConDens for the data of the cases.

Arguments

cases

A vector of measurements for the cases.

controls

A vector of measurements for the controls.

grid

A vector specifying the grid where the ROC curve is computed on.

smooth

Logical, indicating whether ROC curve and AUC should also be computed based on the smoothed log-concave density estimator.

Author

Kaspar Rufibach, ,
http://www.kasparrufibach.ch

Details

In Rufibach (2011) it was shown that the ROC curve based on log-concave density estimates exhibit nice properties for finite sample sizes as well as asymptotically. Its performance is typically much better than that of the empirical ROC curve and only, if at all, sligthly worse compared to the binormal model when in fact the underlying densities are normal. However, log-concavity encompasses many parametric densities, so this new model is much more flexible than the binormal one, at little efficiency sacrifice.

References

Duembgen, L. and Rufibach, K. (2009). Maximum likelihood estimation of a log--concave density and its distribution function: basic properties and uniform consistency. Bernoulli, 15(1), 40--68.

Duembgen, L. and Rufibach, K. (2011). logcondens: Computations Related to Univariate Log-Concave Density Estimation. Journal of Statistical Software, 39(6), 1--28. tools:::Rd_expr_doi("https://doi.org/10.18637/jss.v039.i06")

Rufibach, K. (2012). A smooth ROC curve estimator based on log-concave density estimates. Int. J. Biostat., 8(1), 1--29.

See Also

Confidence intervals at given false-positive fractions for the ROC curve based on log-concave densities can be computed using confIntBootLogConROC_t0. For the computation of the AUC the function ROCx is used. In the example below we analyze the pancreas data.

Examples

## ROC curve for pancreas data 
data(pancreas)
status <- factor(pancreas[, "status"], levels = 0:1, 
    labels = c("healthy", "diseased"))
var <- log(pancreas[, "ca199"])
cases <- var[status == "diseased"]
controls <- var[status == "healthy"]

## compute and plot empirical ROC curve
## code modified from https://stat.ethz.ch/pipermail/r-help/2008-October/178531.html
xx <- c(-Inf, sort(unique(c(cases, controls))), Inf)
sens <- sapply(xx, function(x){mean(cases >= x)})  
spec <- sapply(xx, function(x){mean(controls < x)})

## compute log-concave ROC curve
grid <- seq(0, 1, by = 1 / 500)
roc.logcon <- logConROC(cases, controls, grid)        

## plot
plot(0, 0, xlim = c(0, 1), ylim = c(0, 1), type = 'l', 
    main = "ROC curves for pancreas data", xlab = "1 - specificity", 
    ylab = "sensitivity", pty = 's')
legend("bottomright", c("empirical ROC", "log-concave ROC", "smooth log-concave ROC"), 
    lty = c(1, 1, 2), lwd = 2, col = 2:4, bty = "n")
segments(0, 0, 1, 1, col = 1)
lines(1 - spec, sens, type = 'l', col = 2, lwd = 2)
lines(grid, roc.logcon$fROC, col = 3, lwd = 2)
lines(grid, roc.logcon$fROC.smooth, col = 4, lwd = 2, lty = 2)

if (FALSE) {
## bootstrap confidence intervals at 1 - specificity = 0.2 and 0.8:
res <- confIntBootLogConROC_t0(controls, cases, grid = c(0.2, 0.8), conf.level = 0.95, 
    M = 1000, smooth = TRUE, output = TRUE)
res
}