Area under the ROC curve estimation based on ranked set sampling via genetic algorithm

1. Introduction

The receiver operating characteristic (ROC) curve is a common statistical method used to evaluate the performance or accuracy of classification models in many fields of science, e.g., medicine, psychology, engineering, economics, etc. It was firstly developed to analyze radar signals during the second world war with the aim of identifying enemy objects in battlefields and has been widely used in signal detection theory since then (Marcum, 1960). In 1960, Lusted employed the ROC curve in roentgen diagnosis as one of the very first applications in the context of medical research. Then, ROC analysis has gradually found its way into the field of medical diagnostic accuracy studies, see, for example, Aegerter et al. (1994), Cantor and Kattan (2000), Zhang et al. (2021), Yang et al. (2024), etc. It is now a popular device used to measure diagnostic tests’ discrimination ability between diseased and non-diseased patients. Many diagnostic tests use continuous biomarker values as test scores, such as prostate-specific antigen (PSA) for cancer, bilirubin for liver disease, creatinine for kidney malfunction, etc. Individuals whose test scores are above (or below) a predetermined threshold value are grouped as diseased (or non-diseased). A ROC curve graphically describes the relation between the true positive rate (TPR) and false positive rate (FPR) computed over the entire range of threshold values. The diagnostic information revealed by the ROC curve is summarized by the area under the ROC curve (AUC), which is the most widely applied diagnostic accuracy index in the literature, see Bamber (1975) for the details of AUC.

AUC is defined as the probability $P(X>Y)$ where $X$ and $Y$ are the test scores of randomly selected diseased and non-diseased patients, respectively. It represents the probability that a randomly chosen diseased patient has a higher test score than a randomly chosen non-diseased patient. In conventional ROC analysis, $X$ and $Y$ are assumed to follow a normal distribution; however, in real-life problems data often deviate from normality. Therefore, there exist many studies estimating AUC under the assumption of non-normally distributed test scores. For example, Campbell and Ratnaparkhi (1993) derived the ROC curve under the assumption of the Lomax distribution and used ML estimators of the related parameters to calculate AUC. Faraggi and Reiser (2002) compared two non-parametric and two parametric methods to estimate AUC by using Box-Cox transformation for non-normal cases in the parametric approaches. Bandos et al. (2006) discussed the resampling methods such as bootstrap, jackknife, and permutations in the context of AUC. Molodianovitch et al. (2006) extended the approach by Wieand et al. (1989) to non-normal data by applying a Box-Cox transformation to compare the areas under the two correlated ROC curves. Vardhan et al. (2012) focused on AUC estimation using normal, exponential and Weibull distributions and compared the performances of proposed methodologies via a simulation study. Ch et al. (2022) reported the results corresponding to AUC of the bi-generalized exponential ROC model.

In all of the above-mentioned studies about AUC estimation, simple random sampling (SRS) which is a well-accepted and practical sampling method based on the assumption that each unit in the population has an equal probability of being selected is used. However, reducing sample size, cost, time and effort can be considerably important in clinical trials due to expensive medical equipment, expensive laboratory analyses and lack of expert users to perform these analyses, etc. Ranked set sampling (RSS), on the other hand, is a cost-effective alternative to SRS when measurement of the variable of interest is either difficult or costly, but ranking can be performed easily, see McIntyre (1952). In this ranking process, exact measurement of the units is not performed. However, they are ranked by using visual assessment, subjective judgment, prior information, or an auxiliary variable that can readily be observed and correlated with the variable of interest, etc. In recent years, RSS has gained attention and become a desirable sampling scheme in the context of AUC estimation, especially in the field of medicine, since it yields highly efficient estimators at low costs. For example, Mahdizadeh and Zamanzade (2021) developed a non-parametric estimator of AUC using kernel density estimation based on multistage RSS. Mahdizadeh and Zamanzade (2022) developed three estimators for AUC based on RSS, one of which is under the normality assumption and the others are obtained through a Box-Cox transformation on data. Moon et al. (2022) proposed an empirical likelihood method to construct confidence intervals for AUC based on RSS. Abdallah (2023) proposed concomitant-based estimators for AUC based on paired RSS. Akbari Ghamsari et al. (2024) proposed four different approaches for AUC estimation based on nomination sampling which is regarded as a variation of RSS.

In this study, we have three motivations for estimating AUC under non-normality. Our primary motivation is to use the original data rather than the transformed data to avoid information loss in the data set and also to avoid difficulties in interpreting the results. Therefore, AUC is estimated under the assumption that the distributions of the test scores $X$ and $Y$ are generalized logistic (GL). The reason for assuming GL distribution for the test scores is that it is a flexible distribution that can be used in modeling negatively skewed, positively skewed and symmetric data sets. Also, GL distribution reduces to Logistic distribution when $b=1$, which is a plausible alternative to the widely used normal distribution with its symmetric shape and heavier tails. Our second motivation is to use the RSS sampling method as an efficient alternative to the commonly used SRS sampling method for estimating the AUC. The efficiencies of AUC estimators are then compared with respect to the mentioned sampling methods. Our final motivation in this study is the use of the genetic algorithm (GA), a population-based metaheuristic method, in solving likelihood equations that include nonlinear expressions of distribution parameters with an aim to avoid the drawbacks of traditionally used derivative-based numeric methods such as getting stuck in a local optimum, non-convergence and mathematical difficulties. The performance of GA relative to traditional numeric methods in obtaining ML estimates of AUC is also evaluated.

The subsequent sections of this paper are arranged as follows. In Section 2, the related material and methods are briefly introduced. In Section 3, the steps of GA based ML estimation of AUC under RSS are explained in detail. Section 4 is reserved to the Monte Carlo simulation study that examines the performance of the proposed estimator. Section 5 investigates the performances of RSS based estimators under imperfect ranking conditions. Section 6 presents an application of the proposed methodology to a real data set. Finally, in Section 7, the study is summarized by highlighting its remarkable points.

2. Materials and Methods

In this section, brief descriptions of GL distribution, ROC curve, RSS method and GA method are given.

2.1 GL distribution

Let $X$ denote a random variable following GL distribution with location parameter $\mu$, scale parameter $\sigma$and shape parameter $b$. Its probability density function (PDF) is expressed as given in Eq. (1)

(1)

$\begin{array}{c}{f}_X\left(x\right)=\frac{b}{\sigma }\frac{e^{-\left(\frac{x-\mu }{\sigma }\right)}}{{\left(1+e^{-\left(\frac{x-\mu }{\sigma }\right)}\right)}^{\left(b+1\right)}};-\infty <x<\infty ;-\infty <\mu <\infty ;\sigma >0\end{array}$

and the corresponding cumulative distribution function (cdf) is given in Eq. (2)

(2)

$\begin{array}{c}{F}_X\left(x\right)={\left(1+e^{-\left(\frac{x-\mu }{\sigma }\right)}\right)}^{-b}.\end{array}$

Table 1 gives the skewness $\left(\sqrt{{\beta }_1}\right)$and kurtosis $\left({\beta }_2\right)$ values of GL distribution for various representative values of the shape parameter $b$.

Table 1. Skewness and kurtosis values of GL distribution.

$b$	0.5	1	2	4	6
$\sqrt{{\beta }_1}$	-0.86	0	0.33	0.75	0.92
${\beta }_2$	5.40	4.20	4.33	4.76	4.95

Table 1 clearly shows that GL is a highly flexible distribution representing negatively skewed, positively skewed and symmetric distributions depending on whether its shape parameter $b < 1$, $b > 1$ or $b = 1$, respectively. Also, its kurtosis is always greater than 3 (leptokurtic) as seen from Table 1. For $b = 1$, it reduces to the logistic distribution, which is widely used as a long-tailed alternative to the well-known and widely used normal distribution, see, for example, Berkson (1951).

Fig. 1 shows the PDF plots of the standard GL distribution for the values of the shape parameter $b$ given in Table 1.

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Fig. 1.

The PDF plots of the $GL\left(\mu =0, \sigma =1, b\right)$ distribution.

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3. GA based ML Estimation

This section is reserved for the details of the estimation procedure of AUC. Here, test scores of diseased $\left(X\right)$ and non-diseased $\left(Y\right)$ individuals are assumed to follow a GL distribution with the assumption of known shape parameters. First, the unknown distribution parameters of the GL distribution are estimated via ML methodology based on RSS in subsection (3.1), then AUC is estimated based on these estimates in subsection (3.2).

3.2 Estimation of AUC

AUC is the most frequently employed index summarizing a diagnostic test’s discriminatory accuracy. It is calculated by integrating the area under the ROC curve as shown in Eq. (15)

(15)

$\begin{array}{c}\text{AUC}=P\left(X>Y\right)={\int }_0^1\left(1-F_X\left(c\right)\right)d\left(1-F_Y\left(c\right)\right)\end{array}$

or equivalently as given in Eq. (16),

(16)

$\begin{array}{c}\text{AUC}={\int }_0^{1}1-F_X\left(F_Y^{-1}\left(1-t\right)\right)dt,\end{array}$

see subsection (2.2) for the details of ROC curve. Notice that larger values of AUC indicate higher diagnostic accuracy. Fig. 2 illustrates three hypothetical cases: $\left(i\right)$ An ideal test with $\text{AUC}=1$ (line A), $\left(ii\right)$ A typical ROC curve with $0.50<\text{AUC}<1$ (curve B) and $\left(iii\right)$ An uninformative test with $\text{AUC}=0.50$ (line C).

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Fig. 2.

Three hypothetical cases illustrated by three ROC curves: $\left(i\right)$ An ideal test with $\text{AUC}=1$ (line A), $\left(ii\right)$ A typical ROC curve with $0.50<\text{AUC}<1$ (curve B) and $\left(iii\right)$ An uninformative test with $\text{AUC}=0.50$ (line C).

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In our case, the cdfs $F_X\left(x\right)$ and $F_Y\left(y\right)$ corresponding to $GL\left({\mu }_x,{\sigma }_x,b_x\right)$ and $GL\left({\mu }_y,{\sigma }_y,b_y\right)$ distributions, respectively, are incorporated into (16), and AUC is calculated as given in Eq. (17)

(17)

$\begin{array}{c}\text{AUC}={\int }_0^1\left(1-{\left(1+exp\left(\frac{-\left(u-{\mu }_x\right)}{{\sigma }_x}\right)\right)}^{-b_x}\right)dt,\end{array}$

where $u={\mu }_y+{\sigma }_y\ln \left(1/\left({\left(1-t\right)}^{\left(1/b_y\right)}-1\right)\right)$, see Eq. (2) for the cdf of GL distribution. In computing AUC, GA based ML estimates of ${\mu }_x,{\sigma }_x,$ ${\mu }_y$ and ${\sigma }_y$ under RSS are plugged into Eq. (17) and the resulting estimator of AUC is denoted by $A\hat{U}{C}_{ML-GA}$. SRS counterpart of this estimator is denoted by $A\hat{U}{C}_{ML-GA}^{*}$ and proposed for the first time in this study. In its computation, firstly MML estimators of the location and scale parameters of GL distribution under SRS are taken from Şenoğlu and Tiku (2001) and then, confidence intervals based on them are constructed to define the search space for GA. Here, we resort to numeric integration methods in order to solve the integral in Eq. (17) since it cannot be solved explicitly.

4. Monte Carlo Simulation Study

In this section, the efficiencies of $A\hat{U}{C}_{ML-GA}$ and $A\hat{U}{C}_{ML-GA}^{*}$ are compared with the corresponding ML estimators obtained using traditional numeric methods, i.e., $A\hat{U}{C}_{ML}$ and $A\hat{U}{C}_{ML}^{*}$, respectively. It should be noted here that in computing the ML estimates of AUC via the traditional numeric methods, optim() function with BFGS algorithm in R statistical software is utilized. In the comparison of these estimators,

criteria are used where $\hat{\theta }$ and $s$ are the estimators of the parameter $\theta$ and the number of Monte Carlo runs, respectively, see Eq. (18).

Table 3 provides some representative values of the location, scale and shape parameters of GL distribution corresponding to the diseased $\left(X\right)$ population, i.e., $GL\left({\mu }_x,{\sigma }_x,b_x\right)$, so as to produce AUC of 0.5, 0.7 and 0.9. The location and scale parameters corresponding to the non-diseased $\left(Y\right)$ population are taken as $({\mu }_y,{\sigma }_y)=(0,1)$for all cases. The shape parameter values for both diseased $\left(X\right)$ and non-diseased $\left(Y\right)$ populations are assumed to be equal in each scenario, i.e., $b_x=b_y=b$.

Also, the following cycle, set and sample sizes are considered throughout the simulation study.

The GA parameters are set in accordance with the values widely used in the literature as follows: population size = 250, probability of crossover = 0.9, probability of mutation = 0.1, elitism = 8 and selection is "roulette-wheel", see Gen and Cheng (1999), Haupt and Haupt (2004) and Eiben and Smith (2015). Note that the search space for the unknown parameters of GL distribution is given in Eqs. (12) and (13).

All computations in the Monte Carlo simulation study are conducted in R software. Simulated $bias$ and $MSE$ values for the estimators of AUC obtained from 10,000 Monte Carlo runs are given in Table S1. It is clear from the results that all AUC estimators have negligibly small bias values in each scenario. Efficiency comparisons according to sampling and estimation methods are given in detail as follows.

According to sampling methods: RSS based AUC estimators perform better than their SRS analogs for all cases. The superiority of the proposed method becomes increasingly evident with larger set sizes in most of the scenarios. When the variances of test scores of diseased and non-diseased populations are different, it is observed that the efficiencies of AUC estimators obtained using RSS and SRS sampling methods increase when the sample sizes are determined directly proportional to the variances. It should also be realized for these cases that the efficiency gain under RSS is greater than the efficiency gain under SRS in the supplementary materials (Table S1).

According to estimation methods: To provide fair and consistent comparisons for the efficiencies of the AUC estimators according to the estimation methods, line plots are used (Figs. 3 and 4).

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Fig. 3.

MSE values vs. set sizes $\left(m_x,m_y\right)$ for the RSS based AUC estimators $A\hat{U}{C}_{ML}$ and $A\hat{U}{C}_{ML-GA}$ under $\left({\mu }_y,{\sigma }_y\right)=\left(0,1\right)$ with $b_x=b_y=b$.

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Fig. 4.

MSE values vs. sample sizes $\left(n_x,n_y\right)$ for the SRS based AUC estimators $A\hat{U}{C}_{ML}^{*}$ and $A\hat{U}{C}_{ML-GA}^{*}$ under $\left({\mu }_y,{\sigma }_y\right)=\left(0,1\right)$ with $b_x=b_y=b$.

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Efficiency comparisons according to estimation methods are done separately for RSS and SRS as follows:

$\left(i\right)$ Under RSS scheme, the performance of $A\hat{U}{C}_{ML-GA}$is similar to that of $A\hat{U}{C}_{ML}$ and for $\text{AUC}=0.9$ however, there are still instances where $A\hat{U}{C}_{ML-GA}$ performs better than $A\hat{U}{C}_{ML}$. As the distributions of the diseased and non-diseased populations move closer and the AUC becomes 0.7, $A\hat{U}{C}_{ML-GA}$ is more efficient than $A\hat{U}{C}_{ML}$ in many of the cases especially when $b=1.$ When the distributions of the diseased and non-diseased populations fully overlap, i.e., $\text{AUC}=0.5$, $A\hat{U}{C}_{ML-GA}$ outperforms $A\hat{U}{C}_{ML}$ in most of the cases (Fig. 3).

$\left(ii\right)$ Under SRS scheme, when the overlapping area of distributions of the diseased and non-diseased populations is the smallest, i.e., $\text{AUC}=0.9$, $A\hat{U}{C}_{ML-GA}^{*}$ is more efficient than its rival in many of the scenarios, especially for the symmetric case $\left(b=1\right).$ When $\text{AUC}=0.7$, $A\hat{U}{C}_{ML-GA}^{*}$ is better than $A\hat{U}{C}_{ML}^{*}$ in each scenario for the symmetric case $\left(b=1\right).$ For non-symmetrical cases ($b=0.5$ and $4$), $A\hat{U}{C}_{ML-GA}^{*}$ still outperforms $A\hat{U}{C}_{ML}^{*}$ in many cases however, the performances of the two methodologies are much closer to each other than the previous case. When the distributions of the diseased and non-diseased populations are exactly the same, i.e., $\text{AUC}=0.5$, $A\hat{U}{C}_{ML-GA}^{*}$ is more efficient than its rival for almost all cases. Its superiority is more obvious for $b=0.5$ (Fig. 4).

It should be realized that when the distributions of the diseased and non-diseased populations become more similar, the discrimination ability between the two groups decreases, making AUC estimation more complex. Simulation results reveal that the GA performs relatively better in such challenging scenarios since it explores the parameter space more broadly than the conventional optimization methods and is therefore more likely to find better solutions.

In light of these results, it is recommended to prioritize RSS based GA over traditional numeric methods in obtaining the ML estimates of AUC.

5. Imperfect Ranking

The previous parts of the study rely on the assumption of perfect ranking. Nevertheless, there may be cases where there exist errors in the ranking process of RSS since the ranking is conducted without doing any exact measurement as mentioned earlier. Therefore, in this part of the study, how imperfect ranking affects the efficiencies of $A\hat{U}{C}_{ML-GA}$ and $A\hat{U}{C}_{ML}$ is investigated.

In this section, we also consider the conventional unbiased non-parametric AUC estimator, which is closely related to the Mann–Whitney U statistic, see Bamber (1975), Hanley and McNeil (1982), Faraggi and Reiser (2002), Qin and Zhou (2006). Under the RSS framework it is defined as given in Eq. (19)

where $I\left(.\right)$ denotes the indicator function. Sengupta and Mukhuti (2008) demonstrated that $A\hat{U}{C}_N$ remains unbiased and exhibits higher efficiency than its SRS counterpart, even when the ranking is imperfect. To make the simulations more comprehensive, this estimator is also included into our study.

When comparing the efficiencies of estimators under the presence of ranking error, the simulation procedure introduced by Dell and Clutter (1972) is used. In this approach, random ranking errors are introduced by adding independent noise terms to the true measurements. First, random samples $X_{tic}$, $t,i=1,\dots , m_x;$ $c=1,\dots ,r_x$ and $Y_{sjk}$, $s, j=1,\dots , m_y;$ $k=1,\dots ,r_y$ are generated from $GL\left({\mu }_x,{\sigma }_x,b_x\right)$ and $GL\left({\mu }_y,{\sigma }_y,b_y\right)$ distributions, respectively. Then, random errors $e_{tic}$ and $e_{sjk}$ corresponding to $X_{tic}$ and $Y_{sjk}$ are generated from $N\left(0,{\sigma }^2\right)$ and summed as shown in Eqs. (20) and (21):

Here, $X_{tic}^{*}$ and $Y_{sjk}^{*}$ are called as the concomitants of $X_{tic}$ and $Y_{sjk}$, respectively. They are ranked in ascending order and the corresponding $X_{tic}$ and $Y_{sjk}$ values are selected to construct the RSS sample, see subsection (3.2). Then repeat this process for each value of ${\sigma }^2=0, 0.3, 0.6$ and $1$. It is obvious that the parameter ${\sigma }^2$controls the magnitude of ranking error. When ${\sigma }^2=0$, there is no ranking error, and it represents the perfect ranking case. As ${\sigma }^2$ increases, the ranking error becomes more apparent, representing higher degrees of imperfect ranking. In other words, ${\sigma }^2$ is a direct measure of the ranking error, with larger values indicating greater imperfection in ranking.

In the simulation study, we use the same settings yielding $\text{AUC}=0.7$ and take the cycle size as 5 just for an illustration, see Section (4) in the context of the simulation setup for the case of interest. Table 5 shows the related $bias$ and $MSE$ values under imperfect ranking.

Table 5 clearly demonstrates that the RSS based estimators of AUC have much better performance than their SRS based competitors in all cases, (Table S1, supplementary material). The relative efficiencies of RSS based estimators with respect to SRS based estimators slightly decrease as the error in ranking increases as expected. However, even with severe errors in ranking, RSS based estimators still perform much better than their rivals.

It is also obvious that $A\hat{U}{C}_{ML-GA}$ outperforms $A\hat{U}{C}_N$ in all cases. Moreover, $A\hat{U}{C}_{ML-GA}$ shows better performance than $A\hat{U}{C}_{ML}$ in almost all scenarios for the symmetric case $(b=1)$. For the non-symmetric cases ($b=0.5$ and $b=4$), although their performances are closer overall, there are still many instances where $A\hat{U}{C}_{ML-GA}$ is better than $A\hat{U}{C}_{ML}$. Particularly in equal variance cases, the relative efficiency of $A\hat{U}{C}_{ML-GA}$ over $A\hat{U}{C}_{ML}$ becomes even more pronounced for ${\sigma }^2=1,$ where the error in ranking is the highest.

6. Application

Diabetes is a chronic metabolic disorder defined by persistently increased blood glucose levels that is often caused by the pancreas's failure to produce enough insulin, body cells' resistance to insulin or a combination of both. It constitutes one of the major causes of death and disabilities such as blindness, renal failure, lower limb amputation, nerve damage, etc. According to International Diabetes Federation, diabetes affects approximately 532 million adults worldwide, thus imposing a heavy cost to health care systems. Therefore, there exists a vast literature on prevention, early diagnosis, and treatment of diabetes, most of which reinforce the importance of understanding the risk factors. Many studies have shown that body mass index ($BMI$), a measure of weight adjusted for height, can be regarded as a biomarker for diabetes.

In this part of the study, a data set taken from the National Health and Nutrition Examination Survey (NHANES) is used to illustrate the implementation of the proposed methodology given in Section (3). The data for the periods 2009-2010 and 2011-2012 are provided in NHANES package in R statistical software. In this study, we focus on $BMI$ and $Weight$ variables from the NHANES data for the period 2011-2012 and aim to estimate $\text{AUC}=P(X>Y),$where $X$ and $Y$ denote the $BMI$ values of individuals with and without diabetes, respectively. Here, $Weight$ is used as the concomitant of $BMI$ since they are highly correlated, demonstrating that the use of RSS is particularly appropriate for this practical example. Mahdizadeh and Zamanzade (2022) also used the same data to estimate AUC based on RSS by using Box-Cox transformation to achieve normality. Different than their study, here we used the original data to avoid the disadvantages caused by the use of transformed data and showed that GL distribution provides good fit for both $X$ and $Y$ (Fig. 5).

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Fig. 5.

GL Q-Q plots for the BMI values of the diseased and non-diseased patients; $b_x=b_y=10.$ (a) Diseased patients (X), (b) Non-diseased patients (Y).

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The above mentioned NHANES data with a size of 779 for the diseased and 7817 for the non-diseased population is taken as the hypothetical population. In estimating AUC, RSS samples are drawn from this hypothetical population by taking set and cycle sizes as $m_x=m_y=5$ and $r_x=r_y=10$, respectively. SRS samples of sizes $n_x=n_y=50$ are drawn from the same population in order to make a fair comparison. Then, a Monte Carlo simulation study based on NHANES data is carried out. RSS and SRS based AUC estimates are computed based on 10,000 samples. In order to compare the performances of the proposed AUC estimators with the existing ones, $bias$ and $MSE$ values are calculated (Table 6). Here, it should be noted that the real AUC is 0.78, which is calculated based on the entire information from the hypothetical population.

All estimators have negligibly small $bias$ values, as obviously seen from Table 6. In view of efficiency, RSS based AUC estimators perform better than their SRS based counterparts. Moreover, $A\hat{U}{C}_{ML-GA}$ and $A\hat{U}{C}_{ML-GA}^{*}$ are more efficient than $A\hat{U}{C}_{ML}$ and $A\hat{U}{C}_{ML}^{*}$, respectively. These results are also consistent with the simulation outcomes.

Estimating the AUC is crucial in medical diagnostics, as it reflects the ability of a test to discriminate between diseased and non-diseased patients. The use of RSS instead of the conventional SRS in the context of AUC estimation provides higher efficiencies at lower costs especially when the test scores are more easily ranked than quantified, which is the case for the present diabetes application.

7. Conclusions

AUC estimation plays a significant role in medical diagnostic studies, which is generally based on the normality assumption. However, test scores in medical diagnostic applications commonly follow non-normal distributions. Therefore, in this study, it was assumed that the test scores of diseased and non-diseased individuals follow GL distribution due to its flexible tails. Moreover, RSS method is preferred in selecting the sampling units as an alternative to the traditional SRS method, since cost-effectiveness is highly critical in clinical trials. Here, GA based ML estimation is employed in estimating the AUC corresponding to GL distributed test scores under RSS. An extensive Monte Carlo simulation is conducted to assess the performance of the proposed estimator in terms of bias, efficiency and robustness criteria. In the simulation study, comparisons are made with respect to sampling methods and estimation methods. First, RSS based estimators are compared with the traditionally used SRS based estimators. Then, ML estimators obtained by using GA are compared with the corresponding ML estimators obtained by using the conventional numeric methods. Simulation results show that the use of RSS significantly improve efficiency compared to SRS, as expected. Also, GA demonstrates superior performance compared to traditional numeric methods in obtaining ML estimates of AUC in majority of the cases. Furthermore, robustness part of the simulation study shows that the proposed AUC estimator retains a high level of efficiency under plausible deviations from the underlying model. The efficiencies of RSS based estimators are also evaluated under imperfect ranking conditions and it is found that $A\hat{U}{C}_{ML-GA}$ outperforms its parametric and non-parametric counterparts in most of the cases. The application of proposed methodology to a diabetes data set further confirms its practical utility in real world scenarios. Overall, it is recommended to prioritize GA based ML for AUC estimation, as it generally outperforms its competitor in most of the cases.