pooledROC.emp {ROCnReg}

Empirical estimation of the pooled ROC curve.

Description

This function estimates the pooled ROC curve using the empirical estimator proposed by Hsieh and Turnbull (1996).

Usage

pooledROC.emp(marker, group, tag.h, data, 
	p = seq(0, 1, l = 101), B = 1000, ci.level = 0.95, 
    method = c("ncoutcome", "coutcome"), pauc = pauccontrol(),
  	parallel = c("no", "multicore", "snow"), ncpus = 1, cl = NULL)

Arguments

marker

A character string with the name of the diagnostic test variable.

group

A character string with the name of the variable that distinguishes healthy from diseased individuals.

tag.h

The value codifying healthy individuals in the variable group.

data

Data frame representing the data and containing all needed variables.

p

Set of false positive fractions (FPF) at which to estimate the pooled ROC curve.

B

An integer value specifying the number of bootstrap resamples for the construction of the confidence intervals. The default is 1000.

ci.level

An integer value (between 0 and 1) specifying the confidence level. The default is 0.95.

method

A character string specifying if bootstrap resampling (for the confidence intervals) should be done with or without regard to the disease status (“coutcome” or “noutcome”). In both cases, a naive bootstrap is used. By default, the resampling is done conditionally on the disease status.

pauc

A list of control values to replace the default values returned by the function pauccontrol. This argument is used to indicate whether the partial area under the pooled ROC curve should be computed, and in case it is computed, whether the focus should be placed on restricted false positive fractions (FPFs) or on restricted true positive fractions (TPFs), and the upper bound for the FPF (if focus is FPF) or the lower bound for the TPF (if focus is TPF).

parallel

A characters string with the type of parallel operation: either "no" (default), "multicore" (not available on Windows) or "snow".

ncpus

An integer with the number of processes to be used in parallel operation. Defaults to 1.

cl

An object inheriting from class cluster (from the parallel package), specifying an optional parallel or snow cluster if parallel = "snow". If not supplied, a cluster on the local machine is created for the duration of the call.

Details

Estimates the pooled ROC curve (ROC) defined as

ROC(p) = 1 - F_{D}\{F_{\bar{D}}^{-1}(1-p)\},

where

F_{D}(y) = Pr(Y_{D} \leq y),

F_{\bar{D}}(y) = Pr(Y_{\bar{D}} \leq y).

The method implemented in this function estimates F_{D}(\cdot) and F_{\bar{D}}(\cdot) by means of the empirical dsitributions. More precisely, and letting \{y_{\bar{D}i}\}_{i=1}^{n_{\bar{D}}} and \{y_{Dj}\}_{j=1}^{n_{D}} be two independent random samples from the nondiseased and diseased populations, respectively, the distribution functions in each group take the form

\widehat{F}_{D}(y)=\frac{1}{n_D}\sum_{j=1}^{n_D}I(y_{Dj}\leq y),

\widehat{F}_{\bar{D}}(y)=\frac{1}{n_{\bar{D}}}\sum_{i=1}^{n_{\bar{D}}}I(y_{\bar{D}i}\leq y).

The area under the curve is

AUC=\int_{0}^{1}ROC(p)dp

and is estimated empirically by means of the Mann-Whitney U-statistic. With regard to the partial area under the curve, when focus = "FPF" and assuming an upper bound u_1 for the FPF, what it is computed is

pAUC_{FPF}(u_1)=\int_0^{u_1} ROC(p)dp,

where again is estimated empirically. The returned value is the normalised pAUC, pAUC_{FPF}(u_1)/u_1 so that it ranges from u_1/2 (useless test) to 1 (perfect marker). Conversely, when focus = "TPF", and assuming a lower bound for the TPF of u_2, the partial area corresponding to TPFs lying in the interval (u_2,1) is computed as

pAUC_{TPF}(u_2)=\int_{u_2}^{1}ROC_{TNF}(p)dp,

where ROC_{TNF}(p) is a 270^\circ rotation of the ROC curve, and it can be expressed as ROC_{TNF}(p) = F_{\bar{D}}\{F_{D}^{-1}(1-p)\}. Again, ROC_{TNF}(p) is estimated empirically. The returned value is the normalised pAUC, pAUC_{TPF}(u_2)/(1-u_2), so that it ranges from (1-u_2)/2 (useless test) to 1 (perfect test).

Value

As a result, the function provides a list with the following components:

call

The matched call.

marker

A list with the diagnostic test outcomes in the healthy (h) and diseased (d) groups.

missing.ind

A logical value indicating whether missing values occur.

p

Set of false positive fractions (FPF) at which the pooled ROC curve has been estimated.

ci.level

The value of the argument ci.level used in the call.

ROC

Estimated pooled ROC curve, and corresponding ci.level*100% pointwise confidence band (if computed).

AUC

Estimated pooled AUC, and corresponding ci.level*100% confidence interval (if computed).

pAUC

If computed, estimated partial area under the pooled ROC curve along with its ci.level*100% confidence interval (if B greater than zero). Note that the returned values are normalised, so that the maximum value is one (see more on Details).

References

Hsieh, F., and Turnbull, B.W. (1996). Nonparametric and semiparametric estimation of the receiver operating characteristic curve, The Annals of Statistics, 24, 25–40.

See Also

AROC.bnp, AROC.sp, AROC.kernel, pooledROC.BB, pooledROC.emp, pooledROC.kernel, pooledROC.dpm, cROC.bnp, cROC.sp or AROC.kernel.

Examples

library(ROCnReg)
data(psa)
# Select the last measurement
newpsa <- psa[!duplicated(psa$id, fromLast = TRUE),]

# Log-transform the biomarker
newpsa$l_marker1 <- log(newpsa$marker1)

m0_emp <- pooledROC.emp(marker = "l_marker1", group = "status",
tag.h = 0, data = newpsa, p = seq(0,1,l=101), B = 10,
method = "coutcome", pauc = pauccontrol(compute = TRUE, value = 0.5, focus = "FPF"))

summary(m0_emp)

plot(m0_emp)