Bi-power Lindley ROC curve analysis: Application to Malignant Tumors Diagnosis

Abstract

This paper conducts Receiver Operating Characteristic (ROC) curve analysis for measurementsthat follow the recently proposed power Lindley distribution. A new index isintroduced to determine a cut-off point, providing an alternative to the well-known YoudenJ index. Firstly, several point estimators are examined for the cut-off points and the AreaUnder the Curve (AUC) measure in the bi-power Lindley (bi-PL) ROC model. Secondly,confidence intervals for these measures are discussed. The point and interval estimates arecompared by using a simulation study. Two numerical examples are presented, one involvingreal data and the other utilizing simulated data, for illustrative purposes. An R library calledPLindleyROC is being developed to enable practitioners to utilize the methodology proposedin this paper.

Full text

Bi-power Lindley ROC curve analysis: Application to
Malignant Tumors Diagnosis
Ertan AKGENÇ
Selçuk University
Coşkun KUŞ
Selçuk University
Research Article
Keywords: AUC, Bootstrap, Maximum likelihood estimation, power Lindley distribution, ROC curve,
Youden’s J index
Posted Date: June 17th, 2024
DOI: https://doi.org/10.21203/rs.3.rs-4507738/v1
License:   This work is licensed under a Creative Commons Attribution 4.0 International License. 
Read Full License
Additional Declarations: No competing interests reported.

Bi-power Lindley ROC curve analysis: Application to
Malignant Tumors Diagnosis
Ertan Akgenç1and Coşkun Kuş1
1Department of Statistics, Faculty of Science, Selçuk University, 42250, Konya, Turkey
Abstract
This paper conducts Receiver Operating Characteristic (ROC) curve analysis for mea-
surements that follow the recently proposed power Lindley distribution. A new index is
introduced to determine a cut-off point, providing an alternative to the well-known Youden
J index. Firstly, several point estimators are examined for the cut-off points and the Area
Under the Curve (AUC) measure in the bi-power Lindley (bi-PL) ROC model. Secondly,
confidence intervals for these measures are discussed. The point and interval estimates are
compared by using a simulation study. Two numerical examples are presented, one involving
real data and the other utilizing simulated data, for illustrative purposes. An R library called
PLindleyROC is being developed to enable practitioners to utilize the methodology proposed
in this paper.
Keywords: AUC, Bootstrap, Maximum likelihood estimation, power Lindley distribution,
ROC curve, Youden’s J index.
1 Introduction
The ROC curve analysis has broad applications in modern medicine, engineering, and various cross-
industry fields. Especially in the medical field, it is highly recommended for accurately diagnosing
individuals’ diseases through diagnostic tests. The diagnosis of diseases using biomarkers involves
creating the ROC graph (Attwood et al. (2022)).
In ROC curve analysis, several key concepts are elucidated as follows: Diagnostic tests provide
binary results, indicating "negative" (-) and "positive" (+) outcomes for diseases (Kumar and
Indrayan (2011)). The True Negative Rate (TNR) represents the probability of correctly diagnosing
the absence of a disease in individuals with negative results. The False Negative Rate (FNR)
denotes the probability of incorrectly diagnosing the absence of a disease in individuals with positive
results. The True Positive Rate (TPR) is the probability of correctly diagnosing the presence of a
disease in individuals with positive results. The False Positive Rate (FPR) signifies the probability
1

of incorrectly diagnosing the presence of a disease in individuals with negative results (Fawcett
(2006); Adler and Lausen (2009); Calì and Longobardi (2015)). It is worth mentioning that TPR
and TNR are also known as sensitivity and specificity. The ROC is a two-dimensional graph that
originates at the point (0,0) and is delimited by the point (1,1). The y-axis represents sensitivity,
while the x-axis represents 1-specificity in the ROC graph (Han (2022)).
Kannan and Vardhan (2022) studied the mix Gamma ROC model when the data is non-
normally distributed. Noma et al. (2021) introduced a bootstrap algorithm designed to calculate
the bootstrap confidence interval for the AUC of the summary ROC curve. Additionally, they
developed a user-friendly software package specifically tailored for diagnostic test accuracy meta-
analysis. Pundir and Amala (2014) discussed the ROC analysis under the bi-Weibull distribution.
Bertail et al. (2008) discussed confidence bands for the ROC curve and considered a re-sampling
procedure based on a smoothed version of the empirical distribution. Recently, with regard to
ROC curve analysis, we can refer to the following works: Bianco and Boente (2023), Kannan and
Vardhan (2022), Tafiadis et al. (2022), Ying et al. (2022), Bantis et al. (2021), Mosier and Bantis
(2021), Hu et al. (2021), Yin et al. (2021), Jokiel-Rokita and Topolnicki (2020), Annen et al. (2019)
and Yu and Hwang (2019).
Recently, Attwood et al. (2022) discussed the ROC model based on the skew exponential power
(SEP) distribution, utilizing maximum likelihood methodology for parameter estimation. They
compared SEP to bi-normal, non-parametric, and kernel smoothing in terms of AUC, Youden’s J
index, and corresponding sensitivity and specificity. Their results highlighted the importance of the
distributional assumption in determining the AUC and optimal cut-off points. The determination
of the cut-off point is a critical outcome in ROC curve analysis, and various methods are suggested
to minimize different objective functions for this purpose. For instance, Youden (1950) proposed
a cut-off point that maximizes the sum of specificity and sensitivity. Other current cut-off point
indices can be found in Liu (2012), Nahm (2022), and Perkins and Schisterman (2006).
In our study, we focus on the power Lindley (PL) distribution proposed by Ghitany et al.
(2013), a recently popular lifetime distribution known for its flexibility in modeling lifetime data.
This distribution has attracted significant attention in the literature, as evidenced by references
such as Ghitany et al. (2015), Valiollahi et al. (2018), Joukar et al. (2020), and Çetinkaya (2021).
Our emphasis is on estimation methods in the ROC curve model based on the PL distribution.
We recognize the importance of not only model selection but also the choice of the parameter
estimation method. Distinct from Attwood et al. (2022), we explore various estimators beyond
the Maximum Likelihood Estimation (MLE) method in ROC curve analysis. Furthermore, we
introduce novel methods for establishing the cut-off point and provide a comprehensive analysis
of their strengths and weaknesses in comparison to existing methodologies.
The remainder of the paper is organized as follows: In Section 3, we describe the bi-PL ROC
model. Section 3 introduce the novel cut-off point in ROC curve analysis. In Sections 4-5, we
2

discuss various estimators and confidence intervals (CIs) for ROC parameters, respectively. A
simulation study is conducted to observe the performance of these estimators and CIs in Section
6. In Section 7, two numerical examples are provided for illustrative purposes. Section 8 concludes
the paper with concluding remarks.
2 Bi-power Lindley ROC Model
In this section, we discuss the ROC curve under PL distribution. The PL distribution, proposed
by Ghitany et al. (2013), is defined by the probability density function and cumulative distribution
function (CDF) as follows:
f(x;θ) = αβ2
β+ 1 (1 + xα)xα−1exp (−βxα), x > 0(1)
and
F(x;θ) = 1 −1 + β
β+ 1xαexp (−βxα),(2)
where θ=(α, β), α > 0is a shape parameter, and β > 0is a scale parameter. Using CDF (2),
the quantile function (QF) of PL distribution is provided by
Q(u;θ) = F−1(u;θ) = −W((1 + β) (−1 + u) exp (−(1 + β))) + 1 + β
β1
α
,(3)
where 0< u < 1and W(·)is Lambert Wfunction (Zeghdoudi and Bensid (2017)). We noticed
that PL reduces to Lindley distribution with α= 1 (Barco et al. (2017)). In the next section, we
discuss the ROC model based on PL distribution. It is noted that, we denote the random variable
Xhaving CDF (2) by X∼P L (θ).
In the context of examining the PL ROC model for healthy and diseased populations, where
X1∼P L (θ1)and X2∼P L (θ2)with θi= (αi, βi), i = 1,2represent the measurements for healthy
and diseased populations, respectively. Likewise, F(x1;θ1)and F(x2;θ2)are the corresponding
CDFs. Then, 1-specificity= 1 −Spc(FPR) for a given cut-off value ccan be expressed as
1−Spc(θ) = 1 −F(c;θ1)
=1 + β1
β1+ 1cα1exp (−β1cα1).(4)
On the other hand, we can write the cut-off value cas
c=Q(specificity;θ1),(5)
3

where Qis a QF given in (3). In a similar way, sensitivity=Sec(TPR) is given by
Sec(θ) = 1 −F(c;θ2)
= 1 −F(Q(specificity;θ1) ; θ2)
The ROC curve is plotted with sensitivity on the y-axis versus 1-specificity on the x-axis, both
of which are confined to the range of 0 to 1 (Attwood et al. (2022); Gonçalves et al. (2014)).
Therefore, we can express the bi-PL ROC model as follows:
ROC (θ) = {r, 1−F(Q(1 −r;θ1) ; θ2), r ∈(0,1)}
= 1 + β2
β2+ 1 −W((1 + β1) (−r) exp (−(1 + β1))) + 1 + β1
β1α2
α1!
×exp −β2−W((1 + β1) (−r) exp (−(1 + β1))) + 1 + β1
β1α2
α1!,(6)
where W(·)is a Lambert Wfunction, Qis a quantile function given in (3), θ= (α1, β1, α2, β2) =
(θ1,θ2)and r∈(0,1).
The AUC is a measure of the accuracy of the test or the performance of the ROC model and
it is defined as the area under the ROC curve. Then the AUC for bi-PL ROC model is defined as
AUC (θ) =
1
Z0
{1−F(Q(1 −r;θ1) ; θ2)}dr, (7)
where Fand Qare defined as in (2) and (3), respectively. There is a simple way to calculate the
integral given in equation 7. Using the fact AUC =P(X1< X2)then the AUC can be expressed
more compact form as
AUC (θ) = P(X1< X2)
=
∞
Z0α2β2
2
β2+ 1 (1 + xα2)xα2−1exp (−β2xα2)
×1−1 + β1
β1+ 1xα1exp (−β1xα1)dx, (8)
where X1∼F(x;θ1)and X2∼F(x;θ2).
Recently, Ghitany et al. (2013) derived the stress-strength reliability for PL distribution for
α1=α2.. Utilizing their result and the well-known relationship AUC =P(X1< X2),we have
AUC (θ) = β2
1
β1+ 1 1
β1+β2
+2β2+ 1
(β2+ 1) (β1+β2)2+2β2
(β2+ 1) (β1+β2)3
4

for α1=α2.Let us discuss the determination of the cut-off point in the next section.
3 Current and proposed cut-off points
In this section, the current criteria are summarized for determining the cut-off points used in ROC
curve analysis. In addition, a new index for cut-off point is introduced. One of the well-known
tools to determine the cut-off point is Youden’s J index (J)in ROC curve analysis. Youden (1950)
first employed it, and it has been studied in the context of diagnostic tests in medicine.
Let F(x1;θ1)and F(x2;θ2)be the CDFs for healthy and diseased populations, respectively.
Then the Youden’s J index is given by
c1(θ) = arg max
c
{Sec(θ)−(1 −S pc(θ))}
= arg max
c
{F(c;θ1)−F(c;θ2)}.
It is noted that the aim of the Youden’s J index is to determine the cut-off point which
maximizes the difference between sensitivity and 1-specificity. Another index is the closest to
(0,1) criteria (ER), also known as the Euclidean distance (Nahm (2022)). This index is defined as
the value that minimizes the root of the sum of errors 1-sensitivity and 1-specificity. It is given by
c2(θ) = arg min
cq(1 −Spc(θ))2+ (1 −S ec(θ))2
= arg min
cq(1 −F(c;θ1))2+ (F(c;θ2))2.
The current third index is the concordance probability method (CZ) is defined by (Liu (2012)).
It is obtained by maximizing the product of the sensitivity and specificity. The CZ index is written
as
c3(θ) = arg max
c
{Sec(θ)×S pc(θ)}
= arg max
c
{(1 −F(c;θ2)) ×F(c;θ1)}.
In this paper, our primary aim is to propose a new index (NI) to determine a cut-off point
by ROC curve analysis. The motivation of the NI index is illustrated in Figure 1 for better
comprehension. To understand the definition of the proposed index, readers should first direct
their attention to the areas defined in Figure 1. In this figure, Area 1is (1 −Sp)×(1 −Se), Area
2is Sp ×(1 −S e), Area 3is (1 −Sp)×Se and Area 4is Sp ×Se. Then, our index is defined as
the value that maximizes the cnew =Area 4−Area 3on the ROC curves.
5

1
0 1
Area 3 Area 4
Area 1 Area 2
Sensitivity
1−Specificity
0
{
{
{
{
1-Se
Se
1-Sp Sp
Figure 1: Scheme for motivation of the new index
That is, the NI is defined as
cnew (θ) = arg max
c
{(Spc(θ)×S ec(θ)) −((1 −Spc(θ)) ×Sec(θ))}
= arg max
c
{F(c;θ1)×(1 −F(c;θ2)) −((1 −F(c;θ1)) ×(1 −F(c;θ2)))}.
It is recalled that the CZ index is obtained by maximizing Area 4. Our new index, the min-
imization of Area 3reduces 1-specificity while the maximization of Area 4increases sensitivity.
Considering the minimization of Area 3alongside the maximization of Area 4, it makes sense to
use cnew, which maximizes Area 4−Area 3. Given the structure of the new index, it is clear that
its 1-specificity is lower compared to the CZ index.
In the following section, we discuss the estimation methods for AUC and cut-off point.
4 Point Estimation
Recently, Ghitany et al. (2013) discussed the maximum likelihood (ML) estimation for PL distri-
bution. In this section, we start ML estimation in ROC curve analysis for PL distribution. Let
X1, X2, . . . , Xnbe a random sample from P L (α1, β1)with size nand Y1, Y2, . . . , Ymbe a random
6

sample from P L (α2, β2)with size m. Then the log-likelihood function is written by,
ℓ(θ|x, y) = nlog (α1) + 2nlog (β1)−nlog (1 + β1)
+
n
X
i=1
log (1 + xα1
i) + (α1−1)
n
X
i=1
log (xi)−β1
n
X
i=1
xi
+mlog (α2) + 2mlog (β2)−mlog (1 + β2)
+
m
X
j=1
log 1 + yα2
j+ (α2−1)
m
X
j=1
log (yj)−β2
m
X
j=1
yj.(9)
Then ML estimates b
β1and b
β2of parameters β1and β2are given, respectively, by
b
β1=1
2
n−
n
P
i=1
x
bα1
i+sn2+ 6nn
P
i=1
x
bα1
i+n
P
i=1
x
bα1
i2
n
P
i=1
x
bα1
i
,(10)
b
β2=1
2
m−
m
P
j=1
y
bα2
j+v
u
u
tm2+ 6m m
P
j=1
y
bα2
j!+ m
P
j=1
y
bα2
j!2
m
P
j=1
y
bα2
j
,(11)
where bα1and bα2are the ML estimates of α1and α2and they should be obtained numerically by
solving the equations
n
bα1
+
n
X
i=1
x
bα1
ilog (xi)
1 + x
bα1
i
+
n
X
i=1
log (xi)−b
β1
n
X
i=1
x
bα1
ilog (xi) = 0,
m
bα2
+
m
X
j=1
y
bα2
jlog (yj)
1 + y
bα2
j
+
m
X
j=1
log (yj)−b
β2
m
X
j=1
y
bα2
jlog (yj) = 0,(12)
respectively. We provide fixed-point iteration to obtain ML estimates of α1and α2as follows
bα(h+1)
1=n

b
β(h)
1
n
X
i=1
x
bα(h)
1
ilog (xi)−
n
X
i=1
x
bα(h)
1
ilog (xi)
1 + x
bα(h)
1
i
−
n
X
i=1
log (xi)


−1
,
bα(h+1)
2=m

b
β(h)
2
m
X
j=1
y
bα(h)
2
jlog (yj)−
m
X
j=1
y
bα(h)
2
jlog (yj)
1 + y
bα(h)
2
j
−
m
X
j=1
log (yj)


−1
,
for h= 1,2,..., bα(0)
1and bα(0)
2are initial values for the iteration, and b
β(h)
1and b
β(h)
2are the substi-
tutions of bα1and bα2with bα(h)
1and bα(h)
2in Eqs. (10) and (11), respectively.
According to the invariance principle in ML methodology, the ML estimate of AUC and ROC
7

curve can be estimated by AUC b
θand ROC b
θ,where b
θ1=bα1,b
β1,b
θ2=bα2,b
β2and
b
θ=bα1,b
β1,bα2,b
β2. In the following, b
θ1,b
θ2and b
θstand for the other estimates of parameters
vector θ1,θ2and θ.
Let X(i): 1 ≤i≤nand Y(j): 1 ≤j≤mbe the order statistics based on the healthy and
diseased populations, respectively. The least squares (LS) estimates of θ1and θ2are given by
(Swain et al. (1988)),
b
θ1= arg min
θ1(n
X
i=1 Fx(i);θ1−i
n+ 12),
and
b
θ2= arg min
θ2(m
X
j=1 Fy(j);θ2−j
m+ 12),
respectively. The weighted least squares (WLS) estimates of θ1and θ2are
b
θ1= arg min
θ1(n
X
i=1
(n+ 1)2(n+ 2)
i(n−i+ 1) Fx(i);θ1−i
n+ 12),
and
b
θ2= arg min
θ2(m
X
j=1
(m+ 1)2(m+ 2)
j(m−j+ 1) Fy(j);θ2−j
m+ 12).
The Anderson-Darling (AD) estimation, which is related to the goodness-of-fit statistics has
been proposed by T. W. Anderson Jr. and D. A. Darling (Anderson and Darling (1952)). AD
estimates of θ1and θ2are are presented as
b
θ1= arg min
θ1(−n−1
n
n
X
i=1
(2i−1) log Fx(i);θ1+ log 1−Fx(n−i+1:n);θ1),
and
b
θ2= arg min
θ2

−m−1
m
m
X
j=1
(2j−1) log Fy(j);θ2+ log 1−Fy(m−j+1:m);θ2

,
The Cramer-von Mises (CvM) estimates of θ1and θ2are by (Macdonald (1971)),
b
θ1= arg min
θ1(1
12n+
n
X
i=1 Fx(i);θ1−2i−1
2n2),
b
θ2= arg min
θ2(1
12m+
m
X
j=1 Fy(j);θ2−2j−1
2m2).
Utilizing the invariance principle of ML estimation, the estimates for AUC (θ), ROC (θ),ci(θ)
8

(i= 1,2,3), and cnew (θ)are expressed as AUC b
θ, ROC b
θ,cib
θ, and cnew b
θ, where
b
θ=b
θ1,b
θ2any estimates of θ1and θ2discussed above.
An approximation of ML estimate for the AUC can be expressed as
[
AUCBi−P L =1
h"1
2+
h−1
X
i=1
1−FQi
h;b
θ1;b
θ2#,(13)
where his the number of sub-ranges of [0,1],b
θ1and b
θ2are ML estimates of θ1and θ2,Fand Q
are the CDF and QF given in (2) and (3), respectively (Attwood et al. (2022)). In a similar way,
the non-parametric estimate of the AUC is written by
[
AUC =1
h"1
2+
h−1
X
i=1
1−b
F2b
Q1i
h#,(14)
b
F2and b
Q1are the empirical CDF based on the second sample in disease population and sample
quantile function based on the first sample in healthy populations, respectively. We close this
section here and discuss interval estimation in ROC curve analysis in the next section.
5 Intervals Estimation
In this section, we discuss ROC characteristics in terms of ML-based and bootstrap CIs. First, we
address ML-based CIs. Let g(θ)stands for AUC (θ), ROC(θ),ci(θ) (i= 1,2,3) and cnew (θ).
The asymptotic confidence interval of g(θ)is given by,
gb
θ−z1−α
2×se gb
θ, g b
θ+z1−α
2×se gb
θ,(15)
where zbis bth standard normal quantile and se gb
θis the asymptotic standard error of the
gb
θ.Using Delta rule, we can write
se gb
θ=∇Tgb
θI−1b
θ∇gb
θ,
where Ib
θis the observed Fisher information matrix (Attwood et al. (2022)). It can be easily
be obtained by using R function optim.
The bootstrap (B) is a resampling technique widely used to construct CIs without relying on
explicit theoretical results. According to (Gu et al. (2008); Pundir and Amala (2014); Ouyang
et al. (2023)), we give the steps of the algorithm to get B CIs.
Algorithm for parametric bootsrap
9

•Step 1: Let X1, X2, . . . , Xnbe a random sample from P L (α1, β1)with size nand
Y1, Y2, . . . , Ymbe a random sample from P L (α2, β2)with size m.
•Step 2: Generate the bootstrap samples from first and second sample, respectively, and
obtain ML estimates (say b
θb) of parameter θ.
•Step 3: Calculate ξb=gb
θb.
The bootstrap sample is obtained by repeating Btimes Steps 2-3. We denote bootstrap sample
as ξ1, ξ2, . . . , ξB.
At this point, we discuss the four methods to construct the bootstrap CIs which are the normal
bootstrap (NB), basic bootstrap (BB), percentile bootstrap (PB) and adjusted bootstrap percentile
(BCa) (Ugarte et al. (2008)). The NB confidence interval of g(θ)is given by
gb
θ−b
β−z1−α/2×bσ, g b
θ−b
β+z1−α/2×bσ,
where ztis the tth quantile of the standard normal distribution,
b
β=1
B
B
X
b=1
ξb−gb
θ,
and
bσ=
1
B−1
B
X
b=1 ξb−1
B
B
X
b=1
ξb!2

1
2
.
The BB and PB confidence intervals of g(θ)are given, respectively, by
2gb
θ−ξ(B+1)×(1−α/2),2gb
θ−ξ(B+1)×α/2
and ξ(B+1)×α/2, ξ(B+1)×(1−α/2),
where ξtis the tth sample quantile based on sample ξ1, ξ2, . . . , ξB.The BCa bootstrap confidence
interval of g(θ)is given by ξ(B+1)×a1, ξ(B+1)×a2,
where
a1= Φ "zB+zB+zα/2
1−azB+zα/2#, a2= Φ "zB+zB+z1−α/2
1−azB+z1−α/2#.
In addition, the bias factor and skewness correction factors are
zB= Φ−1"1
B
B
X
b=1
Inξb< g b
θo#
10

and
a=
n
P
i=1 1
n
n
P
j=1
ξ(−j)
b−ξ(−i)
b!3
6


n
P
i=1 1
n
n
P
j=1
ξ(−j)
b−ξ(−i)
b!2


3
2
,
respectively, where ξ(−t)
b=gb
θ(−t), t = 1,2, . . . , n is the value of gb
θwhen the tth value is
deleted from the original sample, Iis the indicator function, and Φis CDF of the standard normal
distribution.
6 Simulation Study
In this section, a simulation study is conducted to assess the performance of point estimates and
CIs for the AUC and indices (cut-off points) across 5000 trials, maintaining a fixed 0.95 confidence
level for CIs. In the simulation, the sample sizes for the two groups are selected to be equal,
with n1=n2(30, 50, 100, 250). The re-sampling size is set to B=500, and the nominal level
for CIs is fixed at 0.95. To compare the CIs, a simulation is carried out in a scenario where X1
denotes the healthy population following a PL distribution with parameters α1= 2 and β1= 5,
and X2represents the diseased population following a PL distribution with parameters α2= 5
and β2= 5. Additionaly, the second scenario where X1denotes the healthy population following
a PL distribution with parameters α1= 2 and β1= 5, and X2represents the diseased population
following a PL distribution with parameters α2= 6 and β2= 1. Table 1 and 7 indicate that the
coverage probabilities of all intervals for AUC and cut-off points approach 0.95 with increasing
sample size. Additionally, the mean lengths decrease to zero as the sample size increases for all
CIs. In terms of coverage probability, the most reliable confidence interval is the BCa confidence
interval.
The corresponding simulated bias, root mean squared error (RMSE), mean squared error
(MSE), sensitivity, 1-specificity, and sensitivity+specificity values are also provided in Tables 2-6
and Tables 8-12. Upon analyzing the results, it is evident that MSE values tend to approach
zero with increasing sample size. According to the MSE criterion, the MLE outperforms other
estimators, particularly in scenarios involving small sample sizes. As the sample size increases, all
estimators exhibit comparable performance. From Table 6 and 12, it is observed that the specificity
based on a new index is higher than that of the others.
11

Table 1: CPs and Bandwidth of CIs for AUC and ROC Model Indices at θ= (2,5,5,5)
Performance Coverage Probabilities Band Width
Assesments n1 = n2MLE NB BB PB BCA MLE NB BB PB BCA
[
AUC
30 0.8993 0.9000 0.8720 0.9111 0.9212 0.1994 0.1964 0.1963 0.1963 0.2010
50 0.9285 0.9240 0.9225 0.9321 0.9321 0.1560 0.1550 0.1559 0.1559 0.1579
100 0.9622 0.9552 0.9490 0.9550 0.9654 0.1118 0.1111 0.1119 0.1119 0.1128
250 0.9484 0.9454 0.9410 0.9472 0.9490 0.0714 0.0706 0.0710 0.0710 0.0713
b
J
30 0.9220 0.9193 0.9175 0.9210 0.9180 0.1094 0.1199 0.1115 0.1115 0.1119
50 0.9290 0.9230 0.9280 0.9153 0.9173 0.0851 0.0846 0.0853 0.0853 0.0855
100 0.9422 0.9413 0.9400 0.9380 0.9380 0.0607 0.0604 0.0608 0.0608 0.0610
250 0.9513 0.9540 0.9535 0.9555 0.9562 0.0384 0.0384 0.0387 0.0387 0.0388
c
ER
30 0.9364 0.9383 0.9370 0.9382 0.9392 0.1017 0.1033 0.1041 0.1041 0.1043
50 0.9364 0.9380 0.9373 0.9343 0.9380 0.0793 0.0798 0.0805 0.0805 0.0806
100 0.9400 0.9450 0.9410 0.9492 0.9471 0.0566 0.0567 0.0572 0.0572 0.0573
250 0.9520 0.9522 0.9513 0.9560 0.9535 0.0359 0.0360 0.0363 0.0363 0.0363
c
CZ
30 0.9283 0.9262 0.9315 0.9300 0.9310 0.1007 0.1015 0.1022 0.1022 0.1021
50 0.9274 0.9230 0.9252 0.9271 0.9292 0.0786 0.0787 0.0793 0.0793 0.0794
100 0.9423 0.9453 0.9460 0.9400 0.9430 0.0562 0.0561 0.0566 0.0566 0.0566
250 0.9533 0.9520 0.9523 0.9552 0.9573 0.0356 0.0357 0.0360 0.0360 0.0360
c
NI
30 0.9292 0.9820 0.9460 0.9623 0.9654 0.1074 0.2774 0.2441 0.2441 0.2457
50 0.9370 0.9610 0.9400 0.9480 0.9390 0.0841 0.1317 0.1116 0.1116 0.1144
100 0.9515 0.9495 0.9533 0.9512 0.9491 0.0601 0.0635 0.0617 0.0617 0.0622
250 0.9540 0.9530 0.9535 0.9530 0.9521 0.0381 0.0382 0.0385 0.0385 0.0385
12

Table 2: Evaluation Measures for AUC Estimation at θ= (2,5,5,5)
Method n1 = n2Mean [
AUC Bias [
AUC Standard Error [
AUC RMSE [
AUC MSE [
AUC
MLE
30 0.8311 0.00177 0.0519 0.0518 0.0027
50 0.8312 0.00199 0.0407 0.0407 0.0017
100 0.8298 0.00060 0.0291 0.0291 0.0009
250 0.8299 0.00069 0.0184 0.0185 0.0003
ADE
30 0.8235 -0.00582 0.0528 0.0531 0.0028
50 0.8262 -0.00305 0.0416 0.0417 0.0017
100 0.8271 -0.00216 0.0299 0.0300 0.0009
250 0.8289 -0.00033 0.0190 0.0190 0.0004
CvME
30 0.8298 0.00054 0.0567 0.0567 0.0032
50 0.8302 0.00094 0.0444 0.0444 0.0020
100 0.8290 -0.00025 0.0318 0.0318 0.0010
250 0.8297 0.00046 0.0203 0.0203 0.0004
LSE
30 0.8175 -0.01180 0.0561 0.0573 0.0033
50 0.8226 -0.00666 0.0441 0.0446 0.0020
100 0.8251 -0.00410 0.0317 0.0320 0.0010
250 0.8281 -0.00109 0.0202 0.0203 0.0004
WLSE
30 0.8211 -0.00820 0.0543 0.0549 0.0030
50 0.8253 -0.00390 0.0421 0.0422 0.0018
100 0.8270 -0.00224 0.0301 0.0302 0.0009
250 0.8291 -0.00019 0.0191 0.0191 0.0004
Table 3: Evaluation Measures for J Estimation at θ= (2,5,5,5)
Method n1 = n2Mean Bias Standard Error RMSE MSE Mean Mean Mean Mean
bcbcbcbcbcSensitivity Specificity 1-Specificity Sensitivity+Specificity
MLE
30 0.5335 0.00358 0.0291 0.0293 0.0009 0.8382 0.7071 0.2929 1.5453
50 0.5327 0.00279 0.0217 0.0219 0.0005 0.8397 0.7038 0.2962 1.5435
100 0.5311 0.00111 0.0157 0.0157 0.0003 0.8395 0.6998 0.3002 1.5393
250 0.5307 0.00072 0.0100 0.0100 0.0001 0.8395 0.6988 0.3012 1.5383
ADE
30 0.5292 -0.00077 0.0302 0.0302 0.0009 0.8367 0.6977 0.3023 1.5344
50 0.5299 -0.00007 0.0226 0.0226 0.0005 0.8388 0.6975 0.3025 1.5364
100 0.5295 -0.00043 0.0163 0.0163 0.0003 0.8392 0.6963 0.3037 1.5356
250 0.5300 0.00004 0.0103 0.0103 0.0001 0.8394 0.6975 0.3025 1.5370
CvME
30 0.5327 0.00275 0.0317 0.0318 0.0010 0.8374 0.7089 0.2911 1.5463
50 0.5321 0.00214 0.0239 0.0240 0.0006 0.8394 0.7042 0.2958 1.5436
100 0.5306 0.00069 0.0170 0.0170 0.0003 0.8395 0.6995 0.3005 1.5390
250 0.5304 0.00047 0.0109 0.0109 0.0001 0.8395 0.6989 0.3011 1.5383
LSE
30 0.5268 -0.00320 0.0325 0.0326 0.0011 0.8351 0.6922 0.3078 1.5272
50 0.5284 -0.00154 0.0242 0.0243 0.0006 0.8381 0.6940 0.3060 1.5321
100 0.5288 -0.00118 0.0172 0.0172 0.0003 0.8389 0.6944 0.3056 1.5332
250 0.5297 -0.00028 0.0109 0.0109 0.0001 0.8392 0.6968 0.3032 1.5360
WLSE
30 0.5282 -0.00171 0.0311 0.0311 0.0010 0.8359 0.6956 0.3044 1.5315
50 0.5296 -0.00032 0.0230 0.0230 0.0005 0.8386 0.6968 0.3033 1.5354
100 0.5296 -0.00037 0.0164 0.0164 0.0003 0.8391 0.6964 0.3036 1.5355
250 0.5301 0.00015 0.0104 0.0104 0.0001 0.8394 0.6977 0.3023 1.5372
13

Table 4: Evaluation Measures for ER Estimation at θ= (2,5,5,5)
Method n1 = n2Mean Bias Standard Error RMSE MSE Mean Mean Mean Mean
bcbcbcbcbcSensitivity Specificity 1-Specificity Sensitivity+Specificity
MLE
30 0.5605 0.00072 0.0268 0.0268 0.0007 0.7969 0.7428 0.2572 1.5397
50 0.5609 0.00112 0.0207 0.0208 0.0004 0.7968 0.7412 0.2589 1.5379
100 0.5601 0.00032 0.0145 0.0145 0.0002 0.7954 0.7385 0.2615 1.5339
250 0.5600 0.00027 0.0093 0.0093 0.0001 0.7951 0.7379 0.2621 1.5330
ADE
30 0.5583 -0.00145 0.0273 0.0273 0.0008 0.7928 0.7354 0.2646 1.5281
50 0.5595 -0.00024 0.0210 0.0210 0.0004 0.7942 0.7362 0.2638 1.5304
100 0.5594 -0.00036 0.0147 0.0147 0.0002 0.7941 0.7358 0.2642 1.5299
250 0.5597 -0.00008 0.0094 0.0094 0.0001 0.7946 0.7369 0.2631 1.5315
CvME
30 0.5592 -0.00051 0.0286 0.0286 0.0008 0.7970 0.7432 0.2568 1.5402
50 0.5602 0.00046 0.0218 0.0218 0.0005 0.7969 0.7409 0.2591 1.5378
100 0.5598 0.00002 0.0151 0.0151 0.0002 0.7953 0.7380 0.2620 1.5334
250 0.5598 0.00005 0.0097 0.0097 0.0001 0.7951 0.7379 0.2622 1.5329
LSE
30 0.5571 -0.00261 0.0284 0.0285 0.0008 0.7900 0.7303 0.2697 1.5203
50 0.5589 -0.00084 0.0217 0.0218 0.0005 0.7926 0.7331 0.2669 1.5257
100 0.5591 -0.00065 0.0151 0.0151 0.0002 0.7932 0.7341 0.2659 1.5273
250 0.5595 -0.00022 0.0097 0.0097 0.0001 0.7942 0.7363 0.2637 1.5305
WLSE
30 0.5578 -0.00193 0.0277 0.0277 0.0008 0.7915 0.7334 0.2666 1.5250
50 0.5594 -0.00030 0.0212 0.0212 0.0005 0.7938 0.7355 0.2645 1.5293
100 0.5594 -0.00032 0.0147 0.0147 0.0002 0.7940 0.7358 0.2642 1.5298
250 0.5597 -0.00002 0.0094 0.0094 0.0001 0.7947 0.7371 0.2629 1.5318
Table 5: Evaluation Measures for CZ Estimation at θ= (2,5,5,5)
Method n1 = n2Mean Bias Standard Error RMSE MSE Mean Mean Mean Mean
bcbcbcbcbcSensitivity Specificity 1-Specificity Sensitivity+Specificity
MLE
30 0.5476 0.00227 0.0268 0.0268 0.0007 0.8173 0.7264 0.2736 1.5437
50 0.5473 0.00201 0.0204 0.0205 0.0004 0.8182 0.7238 0.2763 1.5420
100 0.5461 0.00077 0.0145 0.0145 0.0002 0.8174 0.7204 0.2796 1.5378
250 0.5458 0.00050 0.0092 0.0093 0.0001 0.8175 0.7194 0.2806 1.5369
ADE
30 0.5447 -0.00065 0.0273 0.0273 0.0007 0.8141 0.7185 0.2816 1.5326
50 0.5455 0.00013 0.0208 0.0208 0.0004 0.8162 0.7184 0.2816 1.5347
100 0.5451 -0.00021 0.0147 0.0147 0.0002 0.8166 0.7175 0.2826 1.5340
250 0.5454 0.00003 0.0094 0.0094 0.0001 0.8171 0.7184 0.2816 1.5355
CvME
30 0.5466 0.00126 0.0284 0.0284 0.0008 0.8170 0.7275 0.2725 1.5446
50 0.5467 0.00138 0.0216 0.0216 0.0005 0.8182 0.7239 0.2761 1.5420
100 0.5458 0.00044 0.0152 0.0152 0.0002 0.8174 0.7200 0.2800 1.5375
250 0.5456 0.00027 0.0097 0.0097 0.0001 0.8174 0.7195 0.2806 1.5369
LSE
30 0.5432 -0.00217 0.0285 0.0285 0.0008 0.8117 0.7135 0.2865 1.5252
50 0.5446 -0.00074 0.0216 0.0216 0.0005 0.8150 0.7153 0.2847 1.5303
100 0.5447 -0.00064 0.0152 0.0152 0.0002 0.8158 0.7157 0.2843 1.5316
250 0.5452 -0.00017 0.0097 0.0097 0.0001 0.8168 0.7177 0.2823 1.5345
WLSE
30 0.5441 -0.00123 0.0277 0.0277 0.0008 0.8130 0.7166 0.2834 1.5296
50 0.5453 0.00001 0.0210 0.0210 0.0004 0.8159 0.7177 0.2823 1.5336
100 0.5452 -0.00015 0.0148 0.0148 0.0002 0.8165 0.7175 0.2825 1.5339
250 0.5454 0.00011 0.0094 0.0094 0.0001 0.8172 0.7186 0.2815 1.5357
14

Table 6: Evaluation Measures for NI Estimation at θ= (2,5,5,5)
Method n1 = n2Mean Bias Standard Error RMSE MSE Mean Mean Mean Mean
bcbcbcbcbcSensitivity Specificity 1-Specificity Sensitivity+Specificity
MLE
30 0.6296 0.00312 0.0538 0.0539 0.0029 0.6642 0.8220 0.1780 1.4861
50 0.6264 -0.00007 0.0242 0.0242 0.0006 0.6704 0.8179 0.1821 1.4883
100 0.6262 -0.00030 0.0154 0.0154 0.0002 0.6681 0.8154 0.1846 1.4835
250 0.6265 0.00001 0.0099 0.0099 0.0001 0.6677 0.8147 0.1853 1.4824
ADE
30 0.6306 0.00411 0.0540 0.0541 0.0029 0.6560 0.8165 0.1835 1.4725
50 0.6272 0.00069 0.0254 0.0254 0.0007 0.6650 0.8142 0.1858 1.4793
100 0.6266 0.00007 0.0157 0.0156 0.0002 0.6654 0.8133 0.1867 1.4787
250 0.6266 0.00008 0.0100 0.0100 0.0001 0.6668 0.8140 0.1861 1.4807
CvME
30 0.6347 0.00818 0.0746 0.0750 0.0056 0.6551 0.8245 0.1755 1.4797
50 0.6275 0.00095 0.0355 0.0356 0.0013 0.6684 0.8183 0.1817 1.4867
100 0.6263 -0.00026 0.0175 0.0175 0.0003 0.6678 0.8151 0.1849 1.4828
250 0.6264 -0.00009 0.0103 0.0103 0.0001 0.6678 0.8147 0.1853 1.4824
LSE
30 0.6331 0.00660 0.0591 0.0594 0.0035 0.6481 0.8133 0.1868 1.4614
50 0.6284 0.00188 0.0293 0.0294 0.0009 0.6611 0.8121 0.1879 1.4733
100 0.6271 0.00053 0.0162 0.0162 0.0003 0.6636 0.8121 0.1879 1.4757
250 0.6268 0.00026 0.0103 0.0103 0.0001 0.6660 0.8135 0.1865 1.4795
WLSE
30 0.6335 0.00698 0.0630 0.0634 0.0040 0.6497 0.8160 0.1840 1.4656
50 0.6280 0.00145 0.0296 0.0296 0.0009 0.6635 0.8139 0.1861 1.4774
100 0.6267 0.00015 0.0157 0.0157 0.0003 0.6653 0.8134 0.1867 1.4787
250 0.6266 0.00008 0.0100 0.0100 0.0001 0.6669 0.8141 0.1859 1.4810
Table 7: CPs and Bandwidth of CIs for AUC and ROC Model Indices at θ= (2,5,6,1)
Performance Coverage Probabilities Band Width
Assesments n1 = n2MLE NB BB PB BCA MLE NB BB PB BCA
[
AUC
30 0.9240 0.8790 0.8030 0.9180 0.9290 0.0764 0.0666 0.0653 0.0653 0.0711
50 0.9069 0.8843 0.8229 0.9232 0.9334 0.0573 0.0516 0.0511 0.0511 0.0541
100 0.9350 0.9180 0.8800 0.9360 0.9350 0.0392 0.0380 0.0378 0.0378 0.0391
250 0.9309 0.9281 0.9127 0.9347 0.9407 0.0253 0.0245 0.0246 0.0246 0.0250
b
J
30 0.9470 0.9490 0.9500 0.9510 0.9520 0.1296 0.1346 0.1356 0.1356 0.1355
50 0.9411 0.9431 0.9442 0.9426 0.9416 0.1005 0.1026 0.1035 0.1035 0.1035
100 0.9470 0.9480 0.9490 0.9490 0.9470 0.0713 0.0719 0.0726 0.0726 0.0726
250 0.9380 0.9374 0.9391 0.9402 0.9391 0.0451 0.0452 0.0456 0.0456 0.0456
c
ER
30 0.9470 0.9430 0.9430 0.9430 0.9440 0.1453 0.1484 0.1499 0.1499 0.1497
50 0.9406 0.9380 0.9447 0.9416 0.9401 0.1131 0.1142 0.1153 0.1153 0.1152
100 0.9420 0.9410 0.9420 0.9430 0.9400 0.0803 0.0806 0.0813 0.0813 0.0813
250 0.9347 0.9341 0.9341 0.9330 0.9341 0.0509 0.0509 0.0512 0.0512 0.0513
c
CZ
30 0.9450 0.9500 0.9480 0.9480 0.9500 0.1315 0.1358 0.1371 0.1371 0.1370
50 0.9427 0.9437 0.9452 0.9411 0.9416 0.1021 0.1039 0.1048 0.1048 0.1048
100 0.9470 0.9470 0.9470 0.9480 0.9470 0.0725 0.0730 0.0737 0.0737 0.0737
250 0.9364 0.9363 0.9380 0.9363 0.9358 0.0459 0.0460 0.0463 0.0463 0.0463
c
NI
30 0.9460 0.9450 0.9520 0.9370 0.9430 0.1368 0.1412 0.1429 0.1429 0.1417
50 0.9355 0.9421 0.9442 0.9365 0.9380 0.1063 0.1081 0.1091 0.1091 0.1085
100 0.9460 0.9460 0.9510 0.9490 0.9440 0.0754 0.0760 0.0768 0.0768 0.0766
250 0.9298 0.9297 0.9325 0.9253 0.9264 0.0477 0.0479 0.0482 0.0482 0.0482
15

Table 8: Evaluation Measures for AUC Estimation at θ= (2,5,6,1)
Method n1 = n2Mean [
AUC Bias [
AUC Standard Error [
AUC RMSE [
AUC MSE [
AUC
MLE
30 0.9685 -0.00028 0.0177 0.0177 0.0003
50 0.9689 0.00006 0.0139 0.0139 0.0002
100 0.9687 -0.00011 0.0101 0.0101 0.0001
250 0.9688 0.00001 0.0064 0.0064 0.0000
ADE
30 0.9637 -0.00511 0.0203 0.0209 0.0004
50 0.9657 -0.00307 0.0156 0.0159 0.0003
100 0.9671 -0.00172 0.0111 0.0112 0.0001
250 0.9681 -0.00067 0.0069 0.0069 0.0001
CvME
30 0.9666 -0.00219 0.0213 0.0214 0.0005
50 0.9673 -0.00152 0.0168 0.0169 0.0003
100 0.9679 -0.00091 0.0121 0.0121 0.0002
250 0.9685 -0.00036 0.0076 0.0076 0.0001
LSE
30 0.9594 -0.00943 0.0237 0.0255 0.0007
50 0.9630 -0.00583 0.0180 0.0189 0.0004
100 0.9658 -0.00306 0.0126 0.0129 0.0002
250 0.9676 -0.00121 0.0077 0.0078 0.0001
WLSE
30 0.9619 -0.00694 0.0214 0.0225 0.0005
50 0.9651 -0.00376 0.0161 0.0166 0.0003
100 0.9670 -0.00180 0.0112 0.0113 0.0001
250 0.9682 -0.00061 0.0069 0.0069 0.0001
Table 9: Evaluation Measures for J Estimation at θ= (2,5,6,1)
Method n1 = n2Mean Bias Standard Error RMSE MSE Mean Mean Mean Mean
bcbcbcbcbcSensitivity Specificity 1-Specificity Sensitivity+Specificity
MLE
30 0.7338 -0.00080 0.0344 0.0344 0.0012 0.9245 0.9076 0.0924 1.8322
50 0.7343 -0.00027 0.0262 0.0261 0.0007 0.9237 0.9061 0.0939 1.8298
100 0.7345 -0.00008 0.0188 0.0188 0.0004 0.9227 0.9040 0.0960 1.8267
250 0.7345 -0.00007 0.0117 0.0117 0.0001 0.9220 0.9032 0.0968 1.8252
ADE
30 0.7314 -0.00321 0.0347 0.0349 0.0012 0.9205 0.8989 0.1011 1.8194
50 0.7327 -0.00189 0.0266 0.0267 0.0007 0.9211 0.9004 0.0996 1.8215
100 0.7336 -0.00097 0.0190 0.0191 0.0004 0.9215 0.9011 0.0990 1.8225
250 0.7342 -0.00043 0.0119 0.0119 0.0001 0.9215 0.9019 0.0981 1.8235
CvME
30 0.7314 -0.00317 0.0376 0.0377 0.0014 0.9246 0.9062 0.0938 1.8307
50 0.7328 -0.00184 0.0285 0.0285 0.0008 0.9234 0.9045 0.0955 1.8279
100 0.7337 -0.00094 0.0200 0.0200 0.0004 0.9227 0.9032 0.0968 1.8258
250 0.7342 -0.00039 0.0125 0.0125 0.0002 0.9220 0.9028 0.0972 1.8248
LSE
30 0.7296 -0.00503 0.0367 0.0370 0.0014 0.9180 0.8924 0.1076 1.8104
50 0.7316 -0.00301 0.0280 0.0282 0.0008 0.9195 0.8961 0.1039 1.8156
100 0.7330 -0.00156 0.0198 0.0199 0.0004 0.9207 0.8990 0.1011 1.8196
250 0.7340 -0.00064 0.0125 0.0125 0.0002 0.9212 0.9011 0.0989 1.8223
WLSE
30 0.7308 -0.00382 0.0357 0.0359 0.0013 0.9194 0.8960 0.1040 1.8154
50 0.7325 -0.00206 0.0270 0.0271 0.0007 0.9207 0.8993 0.1007 1.8200
100 0.7337 -0.00090 0.0192 0.0192 0.0004 0.9215 0.9009 0.0991 1.8224
250 0.7342 -0.00038 0.0120 0.0120 0.0001 0.9216 0.9021 0.0979 1.8237
16

Table 10: Evaluation Measures for ER Estimation at θ= (2,5,6,1)
Method n1 = n2Mean Bias Standard Error RMSE MSE Mean Mean Mean Mean
bcbcbcbcbcSensitivity Specificity 1-Specificity Sensitivity+Specificity
MLE
30 0.7409 -0.00215 0.0386 0.0386 0.0015 0.9197 0.9120 0.0881 1.8316
50 0.7418 -0.00121 0.0296 0.0296 0.0009 0.9186 0.9107 0.0893 1.8294
100 0.7425 -0.00048 0.0211 0.0211 0.0005 0.9174 0.9090 0.0910 1.8264
250 0.7427 -0.00035 0.0132 0.0132 0.0002 0.9167 0.9082 0.0918 1.8249
ADE
30 0.7403 -0.00270 0.0393 0.0394 0.0016 0.9142 0.9044 0.0956 1.8186
50 0.7414 -0.00159 0.0305 0.0306 0.0009 0.9152 0.9058 0.0942 1.8210
100 0.7423 -0.00068 0.0217 0.0218 0.0005 0.9156 0.9065 0.0936 1.8221
250 0.7426 -0.00039 0.0136 0.0137 0.0002 0.9160 0.9072 0.0928 1.8231
CvME
30 0.7389 -0.00407 0.0436 0.0438 0.0019 0.9192 0.9107 0.0893 1.8299
50 0.7407 -0.00234 0.0335 0.0336 0.0011 0.9180 0.9093 0.0907 1.8273
100 0.7419 -0.00108 0.0234 0.0235 0.0006 0.9171 0.9083 0.0917 1.8254
250 0.7425 -0.00052 0.0147 0.0147 0.0002 0.9165 0.9079 0.0921 1.8244
LSE
30 0.7400 -0.00304 0.0426 0.0427 0.0018 0.9106 0.8988 0.1012 1.8094
50 0.7412 -0.00176 0.0330 0.0331 0.0011 0.9128 0.9021 0.0979 1.8149
100 0.7422 -0.00080 0.0233 0.0233 0.0005 0.9145 0.9047 0.0954 1.8191
250 0.7426 -0.00041 0.0147 0.0147 0.0002 0.9155 0.9064 0.0936 1.8219
WLSE
30 0.7404 -0.00263 0.0409 0.0409 0.0017 0.9126 0.9019 0.0981 1.8145
50 0.7415 -0.00152 0.0313 0.0313 0.0010 0.9146 0.9049 0.0951 1.8194
100 0.7424 -0.00057 0.0220 0.0220 0.0005 0.9156 0.9064 0.0936 1.8220
250 0.7426 -0.00038 0.0137 0.0137 0.0002 0.9160 0.9073 0.0927 1.8233
Table 11: Evaluation Measures for CZ Estimation at θ= (2,5,6,1)
Method n1 = n2Mean Bias Standard Error RMSE MSE Mean Mean Mean Mean
bcbcbcbcbcSensitivity Specificity 1-Specificity Sensitivity+Specificity
MLE
30 0.7353 -0.00089 0.0349 0.0349 0.0012 0.9235 0.9086 0.0914 1.8321
50 0.7359 -0.00034 0.0266 0.0266 0.0007 0.9226 0.9071 0.0929 1.8298
100 0.7361 -0.00010 0.0191 0.0191 0.0004 0.9216 0.9051 0.0949 1.8267
250 0.7361 -0.00010 0.0119 0.0119 0.0001 0.9210 0.9042 0.0958 1.8252
ADE
30 0.7335 -0.00277 0.0353 0.0354 0.0013 0.9191 0.9003 0.0998 1.8193
50 0.7346 -0.00161 0.0271 0.0272 0.0007 0.9198 0.9016 0.0984 1.8215
100 0.7354 -0.00080 0.0194 0.0194 0.0004 0.9203 0.9022 0.0978 1.8225
250 0.7358 -0.00037 0.0122 0.0122 0.0002 0.9204 0.9030 0.0970 1.8234
CvME
30 0.7332 -0.00301 0.0383 0.0384 0.0015 0.9234 0.9073 0.0927 1.8307
50 0.7345 -0.00171 0.0291 0.0292 0.0009 0.9223 0.9056 0.0944 1.8279
100 0.7354 -0.00085 0.0204 0.0205 0.0004 0.9215 0.9043 0.0957 1.8258
250 0.7359 -0.00036 0.0128 0.0128 0.0002 0.9209 0.9038 0.0962 1.8248
LSE
30 0.7321 -0.00407 0.0374 0.0376 0.0014 0.9162 0.8941 0.1059 1.8104
50 0.7338 -0.00241 0.0287 0.0288 0.0008 0.9180 0.8976 0.1024 1.8156
100 0.7350 -0.00122 0.0203 0.0203 0.0004 0.9194 0.9002 0.0998 1.8196
250 0.7357 -0.00052 0.0128 0.0128 0.0002 0.9201 0.9022 0.0978 1.8223
WLSE
30 0.7331 -0.00316 0.0363 0.0364 0.0013 0.9178 0.8975 0.1025 1.8153
50 0.7345 -0.00170 0.0276 0.0276 0.0008 0.9194 0.9006 0.0994 1.8200
100 0.7355 -0.00071 0.0196 0.0196 0.0004 0.9203 0.9021 0.0979 1.8224
250 0.7359 -0.00033 0.0122 0.0122 0.0002 0.9205 0.9031 0.0969 1.8237
17

Table 12: Evaluation Measures for NI Estimation at θ= (2,5,6,1)
Method n1 = n2Mean Bias Standard Error RMSE MSE Mean Mean Mean Mean
bcbcbcbcbcSensitivity Specificity 1-Specificity Sensitivity+Specificity
MLE
30 0.7908 -0.00379 0.0364 0.0366 0.0013 0.8805 0.9386 0.0615 1.8191
50 0.7923 -0.00231 0.0279 0.0280 0.0008 0.8790 0.9376 0.0624 1.8165
100 0.7936 -0.00102 0.0198 0.0199 0.0004 0.8770 0.9361 0.0639 1.8132
250 0.7941 -0.00053 0.0124 0.0124 0.0002 0.8759 0.9356 0.0644 1.8115
ADE
30 0.7922 -0.00239 0.0369 0.0370 0.0014 0.8724 0.9325 0.0675 1.8049
50 0.7931 -0.00148 0.0285 0.0285 0.0008 0.8738 0.9336 0.0664 1.8074
100 0.7940 -0.00063 0.0203 0.0203 0.0004 0.8744 0.9341 0.0659 1.8085
250 0.7943 -0.00035 0.0128 0.0128 0.0002 0.8749 0.9347 0.0653 1.8096
CvME
30 0.7890 -0.00566 0.0412 0.0416 0.0017 0.8801 0.9373 0.0627 1.8174
50 0.7914 -0.00326 0.0313 0.0315 0.0010 0.8781 0.9363 0.0637 1.8144
100 0.7931 -0.00154 0.0218 0.0219 0.0005 0.8767 0.9355 0.0645 1.8121
250 0.7939 -0.00067 0.0137 0.0137 0.0002 0.8758 0.9353 0.0647 1.8110
LSE
30 0.7932 -0.00144 0.0398 0.0398 0.0016 0.8670 0.9280 0.0721 1.7949
50 0.7938 -0.00082 0.0307 0.0307 0.0009 0.8702 0.9306 0.0694 1.8008
100 0.7943 -0.00034 0.0216 0.0216 0.0005 0.8727 0.9326 0.0674 1.8053
250 0.7944 -0.00020 0.0136 0.0136 0.0002 0.8742 0.9341 0.0659 1.8083
WLSE
30 0.7928 -0.00180 0.0383 0.0383 0.0015 0.8700 0.9305 0.0695 1.8005
50 0.7934 -0.00121 0.0291 0.0292 0.0009 0.8729 0.9329 0.0672 1.8057
100 0.7941 -0.00052 0.0205 0.0205 0.0004 0.8744 0.9340 0.0660 1.8084
250 0.7943 -0.00036 0.0128 0.0128 0.0002 0.8750 0.9348 0.0652 1.8098
7 Illustrative Examples
To illustrate the methodologies under investigation, we generated data using the first bi-PL ROC
model with P L (α1= 2, β1= 5) and P L (α2= 5, β2= 5), and the second bi-PL ROC model with
P L (α1= 2, β1= 5) and P L (α2= 6, β2= 1) for healthy and diseased cases, respectively. The
simulated data are presented in Table 13 and 17, and the fitted (for the bi-PL ROC model) and
empirical ROC curves are depicted in Fig. 2 and 3. True values and their estimates for AUC and
indices are presented in Tables 14-15-18-19, respectively, while the corresponding CIs can be found
in Table 16 and 20. According to the results, the current cut-off points are closely situated, but
the proposed index is higher than the others. This suggests that the proposed cut-off point can be
employed in situations where high specificity is required.
18

Table 13: The simulated data for the first model
X10.4874 0.4037 0.2948 0.8115 0.5274 0.1609
0.2495 0.6011 0.1811 0.4738 0.7255 0.2681
0.2465 0.8394 0.5177 0.4229 0.5555 0.1991
0.8718 0.4053 0.6389 0.1420 0.3653 0.3859
0.5690 0.6850 0.3423 0.3151 0.8610 0.1914
X20.8051 0.8334 0.5174 0.6444 0.4186 0.5291
0.9105 0.7046 0.7933 0.6487 0.7381 0.8583
0.8343 0.6237 0.5394 0.6994 0.5441 0.6610
0.6468 0.7151 0.5671 0.6993 0.7705 0.6470
0.8556 0.5152 0.9569 0.6325 0.7389 0.5019
Figure 2: Empirical and Fitted ROC curve based on simulated data for the first model
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
1−Specificity
Sensitivity
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
1−Specificity
Sensitivity
Empirical ROC Curve
Fitted ROC Curve
Table 14: Estimates of AUC based on simulated data for the first model
Method α1β1α2β2AUC
TRUE 2 5 5 5 0.8292
MLE 2.2407 5.0882 5.7037 6.3677 0.8108
ADE 2.0570 4.5916 5.4281 6.0162 0.7955
CvME 2.0094 4.4936 5.3748 6.0255 0.7920
LSE 1.9127 4.1972 5.1153 5.5328 0.7801
WLSE 2.0103 4.4360 5.2669 5.6711 0.7898
19

Table 15: ROC curve analysis results based on simulated data for the first model
Method Index Cut-off Point Sensitivity Specificity 1-Specificitiy Sensitivity+Specificity
TRUE
J 0.5299 0.8397 0.6970 0.3030 1.5366
ER 0.5597 0.7946 0.7367 0.2633 1.5313
CZ 0.5453 0.8173 0.7179 0.2821 1.5352
NI 0.6265 0.6668 0.8136 0.1864 1.4803
MLE
J 0.5451 0.8410 0.6712 0.3288 1.5121
ER 0.5769 0.7870 0.7179 0.2821 1.5049
CZ 0.5623 0.8131 0.6969 0.3031 1.5100
NI 0.6395 0.6493 0.7983 0.2017 1.4476
ADE
J 0.5314 0.8460 0.6497 0.3503 1.4957
ER 0.5695 0.7837 0.7027 0.2973 1.4863
CZ 0.5526 0.8133 0.6796 0.3204 1.4929
NI 0.6370 0.6385 0.7844 0.2156 1.4230
CvME
J 0.5265 0.8481 0.6447 0.3553 1.4928
ER 0.5662 0.7836 0.6992 0.3008 1.4828
CZ 0.5486 0.8142 0.6756 0.3244 1.4898
NI 0.6346 0.6370 0.7809 0.2191 1.4178
LSE
J 0.5197 0.8478 0.6293 0.3707 1.4771
ER 0.5637 0.7783 0.6876 0.3124 1.4658
CZ 0.5448 0.8102 0.6632 0.3368 1.4735
NI 0.6374 0.6241 0.7724 0.2276 1.3964
WLSE
J 0.5290 0.8446 0.6425 0.3575 1.4871
ER 0.5689 0.7802 0.6971 0.3029 1.4774
CZ 0.5515 0.8102 0.6739 0.3261 1.4841
NI 0.6394 0.6312 0.7809 0.2191 1.4121
Table 16: 95% CIs based on simulated data for the first model
MLE NB BB PB BCA
AUC (0.7025, 0.9190) (0.7016, 0.9074) (0.7102, 0.9150) (0.7065, 0.9112) (0.6969, 0.9002)
J (0.4932, 0.5971) (0.4986, 0.5874) (0.4974, 0.5856) (0.5046, 0.5929) (0.4994, 0.5914)
ER (0.5297, 0.6241) (0.5320, 0.6220) (0.5313, 0.6209) (0.5329, 0.6225) (0.5336, 0.6225)
CZ (0.5149, 0.6098) (0.5175, 0.6052) (0.5167, 0.6053) (0.5193, 0.6079) (0.5190, 0.6075)
NI (0.5890, 0.6900) (0.4595, 0.7943) (0.1790, 0.6937) (0.5853, 0.6899) (0.5857, 0.6973)
20

Table 17: The simulated data for the second model
X10.3961 0.3752 0.2753 0.6561 0.4112 0.3617
0.7038 0.3787 0.4661 0.5040 0.3410 0.6658
0.1575 0.1842 0.5105 0.2311 0.1608 0.5201
0.2941 0.9310 0.5508 0.2789 0.7456 0.6878
0.9055 0.3325 0.5557 0.5371 0.2326 0.5790
X20.8058 0.6635 1.0959 1.1787 0.6854 0.8475
1.0052 1.0943 1.0735 0.9272 1.1369 0.9855
1.1373 1.0752 1.0993 0.7118 0.9897 1.1678
0.6297 0.9824 0.8767 1.2050 0.8959 1.0263
1.0806 0.7792 1.0608 1.1717 1.2998 0.7567
Figure 3: Empirical and Fitted ROC curve based on simulated data for the second model
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
1−Specificity
Sensitivity
Empirical ROC Curve
Fitted ROC Curve
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
1−Specificity
Sensitivity
Table 18: Estimates of AUC based on simulated data for the second model
Method α1β1α2β2AUC
TRUE 2 5 6 1 0.9688
MLE 2.4181 5.5050 6.1679 1.1060 0.9667
ADE 2.3089 5.2323 5.7061 1.1097 0.9605
CvME 2.2782 5.1746 5.6739 1.0860 0.9608
LSE 2.1688 4.7996 5.3858 1.0942 0.9525
WLSE 2.2237 4.9320 5.5257 1.1157 0.9545
21

Table 19: ROC curve analysis results based on simulated data for the second model
Method Index Cut-off Point Sensitivity Specificity 1-Specificitiy Sensitivity+Specificity
TRUE
J 0.7346 0.9217 0.9024 0.0976 1.8241
ER 0.7430 0.9162 0.9076 0.0924 1.8239
CZ 0.7362 0.9207 0.9034 0.0966 1.8241
NI 0.7946 0.8753 0.9351 0.0649 1.8104
MLE
J 0.7431 0.9081 0.9036 0.0964 1.8117
ER 0.7448 0.9068 0.9049 0.0951 1.8117
CZ 0.7434 0.9078 0.9039 0.0961 1.8117
NI 0.7969 0.8599 0.9381 0.0619 1.7980
ADE
J 0.7369 0.8992 0.8934 0.1066 1.7926
ER 0.7390 0.8976 0.8950 0.1050 1.7926
CZ 0.7374 0.8988 0.8937 0.1063 1.7926
NI 0.7957 0.8456 0.9317 0.0683 1.7773
CvME
J 0.7382 0.9003 0.8936 0.1064 1.7939
ER 0.7407 0.8984 0.8955 0.1045 1.7939
CZ 0.7387 0.8998 0.8941 0.1059 1.7939
NI 0.7976 0.8469 0.9318 0.0682 1.7787
LSE
J 0.7356 0.8922 0.8789 0.1211 1.7712
ER 0.7404 0.8884 0.8827 0.1173 1.7711
CZ 0.7368 0.8913 0.8799 0.1201 1.7711
NI 0.8012 0.8314 0.9223 0.0777 1.7537
WLSE
J 0.7355 0.8939 0.8824 0.1176 1.7763
ER 0.7397 0.8906 0.8856 0.1144 1.7762
CZ 0.7365 0.8931 0.8832 0.1168 1.7763
NI 0.7988 0.8348 0.9245 0.0755 1.7594
Table 20: 95% CIs based on simulated data for the second model
MLE NB BB PB BCA
AUC (0.9308, 0.9988) (0.93012, 0.9998) (0.9413, 0.9993) (0.9151, 0.9999) (0.9130, 0.9917)
J (0.6836, 0.8026) (0.6718, 0.8119) (0.6714, 0.8068) (0.6793, 0.8147) (0.6794, 0.8161)
ER (0.6780, 0.8116) (0.6659, 0.8211) (0.6670, 0.8233) (0.6662, 0.8225) (0.6657, 0.8218)
CZ (0.6828, 0.8041) (0.6710, 0.8131) (0.6711, 0.8089) (0.6779, 0.8158) (0.6786, 0.8171)
NI (0.7342, 0.8596) (0.7235, 0.8706) (0.7268, 0.8769) (0.7168, 0.8669) (0.7171, 0.8680)
The real dataset contains variables used for the diagnosis of benign and malignant tumors
in breast cancer patients (Street et al. (1993)). The data can be downloaded from Kaggle web
platform. The cut-off point for benign and malignant tumor diagnosis is examined using the
compactness variable. Compactness measures the similarity relationship between the shape, area
and circle of the breast tumor. When the compactness value approaches 1, the probability of a
22

malignant tumor decreases. The compactness (C) formula is given by,
C=A
4πL2,(16)
where Ais area of the tumor and Lis the perimeter of the breast tumor circle (Wei et al. (2020)).
The samples are gathered from patients with benign and malignant tumors, respectively. Figure
4 displays both the empirical and fitted ROC curves for the actual data, revealing the excellent
fit of the bi-PL ROC model to the dataset. Examination of the ROC curve analysis results in
Table 23 shows that the specificity of the NI is higher than that of the other indices. This suggests
that the recommended cut-off point can be effectively employed in situations where achieving high
specificity is paramount.
Table 21: Estimates of AUC based on real data
Method α1β1α2β2AUC
MLE 2.1064 28.5649 2.3107 8.0981 0.8371
ADE 2.1915 34.0245 2.4689 9.7021 0.8493
CvME 2.2601 39.0527 2.6425 11.7541 0.8606
LSE 2.2501 38.3909 2.6222 11.5281 0.8591
WLSE 2.2734 39.1087 2.6347 11.4230 0.8604
Table 22: 95% CIs based on real data
MLE NB BB PB BCA
AUC (0.8034, 0.8710) (0.8083, 0.8672) (0.8100, 0.8699) (0.8043, 0.8643) (0.8037, 0.8628)
J (0.2678, 0.2916) (0.2672, 0.2918) (0.2669, 0.2918) (0.2676, 0.2925) (0.2672, 0.2912)
ER (0.2458, 0.2679) (0.2463, 0.2667) (0.2463, 0.2668) (0.2468, 0.2673) (0.2462, 0.2669)
CZ (0.2559, 0.2787) (0.2561, 0.2778) (0.2567, 0.2782) (0.2564, 0.2778) (0.2559, 0.2772)
NI (0.3004, 0.3241) (0.2992, 0.3250) (0.2989, 0.3257) (0.2987, 0.3256) (0.2995, 0.3272)
23

Table 23: ROC curve analysis results based on the real data
Method Index Cut-off Point Sensitivity Specificity 1-Specificitiy Sensitivity+Specificity
MLE
J 0.2673 0.6834 0.8486 0.1514 1.5320
ER 0.2797 0.7317 0.7933 0.2067 1.5250
CZ 0.2568 0.7099 0.8200 0.1800 1.5300
NI 0.3122 0.6117 0.9075 0.0924 1.5193
ADE
J 0.2728 0.7000 0.8533 0.1467 1.5534
ER 0.2526 0.7445 0.8025 0.1975 1.5471
CZ 0.2624 0.7234 0.8284 0.1716 1.5517
NI 0.3032 0.6289 0.9111 0.0888 1.5401
CvME
J 0.2676 0.7168 0.8558 0.1442 1.5726
ER 0.2501 0.7570 0.8101 0.1899 1.5672
CZ 0.2589 0.7370 0.8343 0.1657 1.5713
NI 0.2967 0.6454 0.9134 0.0865 1.5588
LSE
J 0.2679 0.7148 0.8552 0.1448 1.5700
ER 0.2501 0.7555 0.8090 0.1910 1.5645
CZ 0.2590 0.7354 0.8333 0.1667 1.5686
NI 0.2971 0.6432 0.9130 0.0869 1.5562
WLSE
J 0.2702 0.7156 0.8574 0.1426 1.5731
ER 0.2522 0.7566 0.8109 0.1891 1.5674
CZ 0.2613 0.7361 0.8356 0.1644 1.5717
NI 0.2992 0.6452 0.9142 0.0857 1.5595
Figure 4: Empirical and Fitted ROC curve based on real data
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
1−Specificity
Sensitivity
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.2 0.4 0.6 0.8 1.0
1−Specificity
Sensitivity
Empirical ROC Curve
Fitted ROC Curve
24

8 Conclusion
In this study, we explore the bi-PL ROC model and introduce a novel cut-off point index. Point
estimators and CIs are proposed for AUC and cut-off point index. The stability of the proposed
five estimators and five CIs in functioning effectively for AUC and index is demonstrated through a
Monte Carlo simulation study. The proposed new index has potential greater specificity compared
to CZ and it can be particularly useful when a lower 1-specificity is desired. To empower practition-
ers in applying the methodology suggested in this paper, a specialized R library "PLindleyROC"
in CRAN is currently under development. Future research could involve proposing and examining
a new index through combinations of existing indices. Likelihood ratio confidence intervals can
be employed in ROC curve analysis, and estimators and CIs can be modified for censored sample
situations.
References
Adler, W. and Lausen, B. (2009). Bootstrap estimated true and false positive rates and roc curve.
Computational statistics & data analysis, 53(3):718–729.
Anderson, T. W. and Darling, D. A. (1952). Asymptotic theory of certain" goodness of fit" criteria
based on stochastic processes. The annals of mathematical statistics, 23(2):193–212.
Annen, J., Filippini, M. M., Bonin, E., Cassol, H., Aubinet, C., Carrière, M., Gosseries, O.,
Thibaut, A., Barra, A., Wolff, A., Sanz, L. R. D., Martial, C., Laureys, S., and Chatelle, C.
(2019). Diagnostic accuracy of the crs-r index in patients with disorders of consciousness. Brain
Injury, 33(11):1409–1412.
Attwood, K., Hou, S., and Hutson, A. (2022). Application of the skew exponential power distri-
bution to roc curves. Journal of Applied Statistics, 50(8):1–16.
Bantis, L. E., Nakas, C. T., and Reiser, B. (2021). Statistical inference for the difference be-
tween two maximized youden indices obtained from correlated biomarkers. Biometrical journal,
63(6):1241–1253.
Barco, K. V. P., Mazucheli, J., and Janeiro, V. (2017). The inverse power lindley distribution.
Communications in Statistics-Simulation and Computation, 46(8):6308–6323.
Bertail, P., Clémençon, S., and Vayatis, N. (2008). On constructing accurate confidence bands for
roc curves through smooth resampling. Hyper Articles en Ligne.
Bianco, A. M. and Boente, G. (2023). Addressing robust estimation in covariate–specific roc curves.
Econometrics and Statistics, 17(6):1–18.
25

Calì, C. and Longobardi, M. (2015). Some mathematical properties of the roc curve and their
applications. Ricerche di Matematica, 64:391–402.
Fawcett, T. (2006). An introduction to roc analysis. Pattern recognition letters, 27(8):861–874.
Ghitany, M., Al-Mutairi, D. K., and Aboukhamseen, S. (2015). Estimation of the reliability
of a stress-strength system from power lindley distributions. Communications in Statistics-
Simulation and Computation, 44(1):118–136.
Ghitany, M., Al-Mutairi, D. K., Balakrishnan, N., and Al-Enezi, L. (2013). Power lindley distri-
bution and associated inference. Computational Statistics & Data Analysis, 64:20–33.
Gonçalves, L., Subtil, A., Oliveira, M. R., and de Zea Bermudez, P. (2014). Roc curve estimation:
An overview. REVSTAT-Statistical journal, 12(1):1–20.
Gu, J., Ghosal, S., and Roy, A. (2008). Bayesian bootstrap estimation of roc curve. Statistics in
medicine, 27(26):5407–5420.
Han, H. (2022). The utility of receiver operating characteristic curve in educational assessment:
Performance prediction. Mathematics, 10(9):1493.
Hu, X., Li, C., Chen, J., and Qin, G. (2021). Confidence intervals for the youden index and
its optimal cut-off point in the presence of covariates. Journal of Biopharmaceutical Statistics,
31(3):251–272.
Jokiel-Rokita, A. and Topolnicki, R. (2020). Estimation of the roc curve from the lehmann family.
Computational Statistics & Data Analysis, 142:106820.
Joukar, A., Ramezani, M., and MirMostafaee, S. (2020). Estimation of p (x> y) for the power
lindley distribution based on progressively type ii right censored samples. Journal of Statistical
Computation and Simulation, 90(2):355–389.
Kannan, A. S. and Vardhan, R. V. (2022). Estimation of area under the roc curve in the framework
of gamma mixtures. Communications in Statistics: Case Studies, Data Analysis and Applica-
tions, 8(4):714–727.
Kumar, R. and Indrayan, A. (2011). Receiver operating characteristic (roc) curve for medical
researchers. Indian pediatrics, 48:277–287.
Liu, X. (2012). Classification accuracy and cut point selection. Statistics in medicine, 31(23):2676–
2686.
26

Macdonald, P. (1971). Comments and queries comment on “an estimation procedure for mix-
tures of distributions” by choi and bulgren. Journal of the Royal Statistical Society: Series B
(Methodological), 33(2):326–329.
Mosier, B. R. and Bantis, L. E. (2021). Estimation and construction of confidence intervals
for biomarker cutoff-points under the shortest euclidean distance from the roc surface to the
perfection corner. Statistics in medicine, 40(20):4522–4539.
Nahm, F. S. (2022). Receiver operating characteristic curve: overview and practical use for clini-
cians. Korean journal of anesthesiology, 75(1):25–36.
Noma, H., Matsushima, Y., and Ishii, R. (2021). Confidence interval for the auc of sroc curve
and some related methods using bootstrap for meta-analysis of diagnostic accuracy studies.
Communications in Statistics: Case Studies, Data Analysis and Applications, 7(3):344–358.
Ouyang, L., Dey, S., Byun, J.-H., and Park, C. (2023). Confidence intervals of the process capability
index c pc c_pc revisited via modified bootstrap technique and roc curves. Quality and Reliability
Engineering International, 39:2162–2184.
Perkins, N. J. and Schisterman, E. F. (2006). The inconsistency of “optimal” cutpoints obtained
using two criteria based on the receiver operating characteristic curve. American journal of
epidemiology, 163(7):670–675.
Pundir, S. and Amala, R. (2014). Evaluation of area under the constant shape bi-weibull roc curve.
Journal of Modern Applied Statistical Methods, 13(1):305–328.
Street, W. N., Wolberg, W. H., and Mangasarian, O. L. (1993). Nuclear feature extraction for
breast tumor diagnosis. Biomedical image processing and biomedical visualization, 1905:861–870.
Swain, J. J., Venkatraman, S., and Wilson, J. R. (1988). Least-squares estimation of distribution
functions in johnson’s translation system. Journal of Statistical Computation and Simulation,
29(4):271–297.
Tafiadis, D., Helidoni, M. E., Chronopoulos, S. K., Kosma, E. I., Alexandropoulou, A., Velegrakis,
S., Konitsiotis, S., and Ziavra, N. (2022). Roc analysis cut-off points of hellenic voice hand-
icap index for neurogenic voice disorders patients: An exploratory study. Journal of Voice,
36(6):875.e25–875.e33.
Ugarte, M. D., Militino, A. F., and Arnholt, A. T. (2008). Probability and Statistics with R. CRC
press.
27

Valiollahi, R., Raqab, M. Z., Asgharzadeh, A., and Alqallaf, F. (2018). Estimation and prediction
for power lindley distribution under progressively type ii right censored samples. Mathematics
and Computers in Simulation, 149:32–47.
Wei, M., Du, Y., Wu, X., Su, Q., Zhu, J., Zheng, L., Lv, G., and Zhuang, J. (2020). A benign and
malignant breast tumor classification method via efficiently combining texture and morphological
features on ultrasound images. Computational and Mathematical Methods in Medicine, 2020.
Yin, J., Mutiso, F., and Tian, L. (2021). Joint hypothesis testing of the area under the receiver
operating characteristic curve and the youden index. Pharmaceutical statistics, 20(3):657–674.
Ying, G.-S., Maguire, M. G., Glynn, R. J., and Rosner, B. (2022). Tutorial on biostatistics:
Receiver-operating characteristic (roc) analysis for correlated eye data. Ophthalmic Epidemiol-
ogy, 29(2):117–127.
Youden, W. J. (1950). Index for rating diagnostic tests. Cancer, 3(1):32–35.
Yu, D. and Hwang, W.-T. (2019). Optimal cutoffs for continuous biomarkers for survival data under
competing risks. Communications in Statistics-Simulation and Computation, 48(5):1330–1345.
Zeghdoudi, H. and Bensid, A. (2017). On the lindley family distribution: quantile function, limiting
distributions of sample minima and maxima and entropies. Balkan Journal of Geometry and Its
Applications, 24:1–18.
Çetinkaya, Ç. (2021). Estimation in step-stress partially accelerated life tests for the power lindley
distribution under progressive censoring. Gazi University Journal of Science, 34(2):579–590.
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