BayesianFROC (version 0.4.0)

AFROC: AFROC curve (alternative free-response ROC curve)

Description

An AFROC curve is a plane curve whose area under the curve (AUC) indicates an observer performance ability. In the following, \(\Phi()\) denotes the cumulative distribution function on the standard Gaussian disribution.

The so-called AFROC curve is defined by

$$ (\xi(t),\eta(t) ) =(1-e^{-t}, \Phi( b\Phi^{-1}(\exp(-t) )- a ) )$$

for all \(t >0\) and some fixed real numbers \(a,b\).

Specifying two real numbers \(a\) and \(b\), we can plot an AFROC curve.

The are under the AFROC curve, or breafly AUC, is calculated as follows, whic are used to evaluate how physicians detect lesions in radiographs.

$$ AUC = \int \eta(t) d\xi(t) = \frac{ a }{ \sqrt{1+ b^2} }. $$

Note that the so-called FROC curve can be interpreted as the curve of expectations of data points. On the other hand, AFROC curve cannot be interpreted as the fitted curve, but its AUC is finite. Because AFROC can be obtained by modifying FROC curve, it reflects obeserver performance.

Usage

AFROC(t, a = 0.14, b = 0.19, x.coordinate.also = FALSE)

Arguments

t

A real number which moves in the domain of FROC curve

a, b

One of the parameter of model which characterize AFROC curve

x.coordinate.also

Logical, whether a vector of 1-exp(-t) is included in a return value.

Value

if x.coordinate.also =TRUE, then A list, contains two vectors as x,y cooridinates of the AFROC curve for drawing curves. if x.coordinate.also =FALSE, then return is a vector as y coodinates of the AFROC curve exclueded its x-coordinates. (x coodinates is omitted.)

Examples

# NOT RUN {
#========================================================================================
#             Plot AFROC curve
#========================================================================================

tt <- seq(0, 1, length.out = 111)
ttt <- stats::runif(1000,0.001,100)
t <- c(tt,ttt)
a <-  AFROC(t,x.coordinate.also=TRUE)

plot(a$x,a$y)

# We note that the x-coordinates of AFROC curve is not t but x = 1 - exp(-t).
# To emphasize that x-coordinates is not t, we prepare the another example

#========================================================================================
#             Plot AFROC curve
#========================================================================================

tt <- seq(0, 1, length.out = 111)
ttt <- stats::runif(1000,0.001,100)
t <- c(tt,ttt)
y <-  AFROC(t,x.coordinate.also=FALSE)

plot(1-exp(-t),y)



     Close_all_graphic_devices() # 2020 August
# }