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```
library(shiny)
# Define UI for application that draws a histogram
ui <- fluidPage(
# Application title
titlePanel("Medical Decision Methods: Receiver Operating Characteristics (ROC) for Binormal Distributions"),
# Sidebar with a slider input for number of bins
sidebarLayout(
sidebarPanel(
p("The point of intersection is equal to the optimal dichtomous threshold at which the sum of Sensitivity and Specificity (Se + Sp) is maximized,
and the sum of errors (FNR + FPR) is minimized. You manipulate the percentages of errors with the sliders."),
sliderInput("FNR",
"False Negative Rate: Percentage of true patients with test scores below the intersection (1 - Se):",
min = 1,
max = 50,
value = 15),
sliderInput("FPR",
"False Positive Rate: Percentage of true non-patients with test scores above the intersection (1 - Sp):",
min = 1,
max = 50,
value = 15),
checkboxInput("combineSliders", "Combine patients slider with non-patient slider", TRUE),
),
# Show a plot of the generated distribution
mainPanel(
plotOutput("distPlot"),
h1('Demonstration of the Receiver Operating Characteristic curve (ROC)'),
h2("Hands-on Demonstration of ROC"),
p("1. Check the checkbox to combine the two sliders. Move the upper
slider to the left or right, simulating a better or
worse test. ROC curves closer to the top-left corner indicate
better performance (AUC). A curve close to the diagonal indicates
a worthless test (AUC = .5). The ROC curves shows us the relative
trade-off between true positives (benefits) and false positives (costs).
The ROC curve does not show us the actual cut-off scores."),
p("2. Uncheck the checkbox to release the lower slider. Move the
upper slider to 30, and simulate different tests with the lower slider.
The two tests have now different variances. Observe that the ROC
curve sometimes crosses the daigonal. In the left plot you can see
that this is caused by a secondary point of intersection. This is
undesirable, because the test scores around the second intersection
have a problematic and inconsistent interpretation: they no longer
allow us to say that only the higher scores indicate the targeted disease. "),
p("3. Create different tests and observe the values of AUC. AUC is
abbreviation of 'Area under the Curve'. If you were wondering 'Which curve?',
well, it is the Area under the ROC curve, and AUROCC would be a better name.
As we do not want the curve
below the diagonal, AUC varies in practice between .5
(complete overlap) and 1.0 (no overlap)."),
h2('Using the dash board'),
p('The grey panel offers a dash board where the user can create many
different tests. The tests differ in their overlap of the scores
for the two groups. The overlap is chosen with two sliders: the
upper slider sets the percentage of true patients with test scores
below the point of intersection (that is the blue dotted line in the left graph). The
lower slider sets the percentage of patients which are known to not
have the disease that have test scores above the intersection.
The true presence or absence of the
disease is known and is determined with superior means, called a "gold standard".'),
p('A checkbox allows the combinations of the two sliders. When the two sliders are combined, the variance remains equal for
the two distributions. Unchecking makes it possible to use the two
sliders seperately, allowing the two distributions to have different variance.'),
p('The total overlap is here defined as the sum of these two percentages. In this way,
a large amount of tests of varying strenths can be simulated.
The strength of the test is directly determined by the overlap of
the distributions of test scores: a test is stronger when the
overlap is smaller. For convenience, the',
span("AUC statistic", style="color:blue"), 'is presented, which is
also an estimate of the strength of the test. '),
h2("Background Information"),
p("When a test intended for comfirming the presence or absence of a
disease, the test is evaluated using two groups: a group of true patients
who have the disease and a group of patients who truly do not have
the disease (shortly called non-patients). The true status of each
patient is determined with a 'gold standard'. The left plot above shows
the two densities of the two groups. The right plot shows the ROC,
and shows the trade-off between Sensitivity and Specificity. Clearly,
Sensitivity increases while Specificity decreases and vice versa."),
h2('Two normal densities and ROC'),
p("The bi-normal distributions shown in the left plot show the densities
of the obtained simulated test scores. The test scores of the
non-patients are always standard normal distributed, with mean of 0
and standard deviation of 1: N(0, 1). The distribution of the true
patients can vary widely. The difference of the two densities
indicates for a given test score from which of the two groups of patients
the test score is most likely drawn. When the sliders are combined,
both distributions have a standard deviation of 1 and only the means
differs. The application starts with a distribution of test scores
with a mean of 2.07 and a standard deviation of 1 (N(2.07, 1))."),
h2("Trichotomization versus dichotomization"),
p("The classic techniques such as the ROC curve for evaluating tests for medical decision
make use of dichotomization of the test scores. All test scores are
considered as equally useful for a positive or
negative classification of each patient concerning the disease that
is the target of the test. Trichotomization methods say that some test
scores offer an insufficient distinction between the two groups of patients and
try to identify test scores that are insufficiently valid and are
better not used for classification. "),
h2('In conclusion'),
p('The ROC method is useful for showing the trade-off between
Sensitivity and Specificity. It is also useful to compare different
tests: a test is better when the curve is more drawn to the upperleft
corner. This is directly related to AUC: the Area Under the curve.
Furthermore, it allows us to identify short-comings of the test, as
it shows a crossing of the diagonal when the test has an undesirable
secondary point of intersection.'),
p("The ROC curve shows us the trade-off between Sensitivity and
Specificity. These statistics are however group statistics.
Sensitivity gives us the percentage patients with a correct positive
classification and concerns the patients with test scores higher
than the cut-off score. Specificity gives us the percentage patients
with a correct negative classification and concerns the patients
with test scores higher than the cut-off score. Individual patients
do not have a true test score that is equal or higher than the cut-off
score, but their true score lies around the received test score.
For such a small interval of true scores around the received test
score, the left plot is easier
to interpret: simply look at the difference between the two density
plots. A large difference indicates good discrimination for these
test scores, a small difference indicates test scores that perhaps
are better not used for classification.
"),
)
)
)
# Define server logic required to draw a histogram
server <- function(input, output, session) {
output$distPlot <- renderPlot({
# generate pdf's based on input$bins from ui.R
if (input$combineSliders) updateSliderInput(session, "FPR", value = input$FNR)
FPR = input$FPR / 100
FNR = input$FNR / 100
acc = ifelse(input$acc==1, .9, .95)
m0=0; sd0=1;
is = qnorm(1-FPR, 0, 1) # x value for 15% FP = intersection
Z = qnorm(FNR) # Z value = 15% FN; Z = (x-mean) / sd
dens = dnorm(is, 0, 1) # density at the point of intersection
sd1 = (1/(dens*sqrt(2*pi)))*exp(-.5 * Z^2)
m1 = is-Z*sd1
x <- seq(-4, 4+m1, length=200)
y0 <- dnorm(x, m0, sd0)
y1 = dnorm(x, m1, sd1)
par(mfrow=c(1,2))
plot(x, y0, type="l", col='black', xlab='Test score', ylab='Density',
main='Probability Density Functions', ylim=c(0,.5))
lines(x, y1, type="l", col='red')
lines(x=c(is,is), y=c(0,.5),col='blue', lty=3)
a = (m1-m0)/sd1
b = sd0/sd1
AUC = round(unname(pnorm(a/sqrt(1+b^2))), 4)
legend('topleft', c(paste('Non-Patients = N(', m0,', ', sd0,')', sep=''),
paste('True Patients = N(', round(m1,2),', ', round(sd1,2),')', sep='')),
text.col= c('black','red'))
legend('topright', c('intersection',paste('AUC = ', AUC, sep='')), lty=c(3, 0),col=c('blue', 'black'))
Se = 1-pnorm(x, m1, sd1)
Sp = pnorm(x, 0, 1)
plot(1-Sp, Se, type='l', col='red', xlab='1 - Specificity',
ylab='Sensitivity', main="ROC curve of 1-Specificity and Sensitivity")
lines(x=c(0,1), y=c(0,1),col='black') # diagonal
par(mfrow=c(1,1))
})
}
# Run the application
shinyApp(ui = ui, server = server)
```

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