Logistic Kernel: A Sensitive Biomarker for Kidney Cancer by ROC

Curve

JAVARIA AHMAD KHAN, ATIF AKBAR

Department of Statistics

Bahauddin Zakariya University, Multan,

PAKISTAN

Abstract: The receiver operating characteristic (ROC) curve is a well-known graphical method to

describe the accuracy of a diagnostic test. In this paper, Logistic kernel is proposed with its optimal

Key-Words: nonparametric ROC curve, AUC, symmetrical kernel, Logistic Kernel, Hemoglobin

Level, Fibrinogen Concentration

Received: June 9, 2022. Revised: August 22, 2023. Accepted: September 16, 2023. Published: October 16, 2023.

1 Introduction

In diagnostic medicine, it is important to assess the

accuracy of a diagnostic test in discriminating

diseased patients from healthy ones. For this

purpose, the receiver operating characteristic

sensitivity and specificity. When the response of

the test is continuous (i.e. blood pressure provides

continuous measurements), its accuracy is best

measured by the receiver operating characteristic

(ROC) curve, which is a plot of sensitivity versus

1- specificity [4]. However, applications of ROC

curve are recently extended to many other fields

like economics and data mining. More

comprehensive review of the literature about the

ROC curves and their possible applications can be

greater than c and negative otherwise. Let F and G

be completely unknown distribution functions of

the random variables X and Y, respectively. The

sensitivity of the test is defined as the SE(t) = 1 −

G(t), which is the probability that a truly diseased

individual has a positive test result. Similarly, the

specificity of the test is given by SP(t) = F(t) and

describes the probability that a truly non diseased

individual has a negative test result. The receiver

operating characteristic (ROC) curve is defined as

Although the empirical ROC curve is very simple

and popular, but estimator has some drawbacks; its

obvious weakness is being a step function. These

methods are sensitive to the assumptions and can

only provide a limited range of distributional

forms. Moreover, such methods may suffer from

large variability, particularly for small sample sizes

[13–15]. To overcome these problems

nonparametric methods are proposed [13]. The

commonly used nonparametric estimator is the

where 

empirical quantile function and the empirical

cumulative distribution function of the samples

respectively [16,17]. Asymptotic properties of this

estimator were studied by [16], they showed that,

under some basic assumptions for distribution

R(t) is to use the kernel smoothing method

proposed by [18]. [13], using kernel estimates

directly for F and G, obtained a smooth ROC curve

estimator given by

[15] extended the idea of [19] and constructed a

continuous and easily invertible estimator of the

distribution function by using order statistics. They

claimed that this idea leads to obtain a continuous

󰇛󰇜

 , 󰇟󰇠

(4)

where h > 0 is a bandwidth parameter and 

denotes the empirical distribution function of the

sample . [21] adopted Bernstein polynomials to

construct the ROC curve estimator and studied the

󰇟󰇠 (5)

In literature, further work has been done to

examine the effect of smoothing parameter, with

symmetric kernels (usually Bi-weight [18] and

Epanechnikov kernel [4,20]. In this paper, we will

examine the performance of asymmetrical kernel

(Logistic kernel) against symmetrical kernel

(Gaussian kernel), with different bandwidths.

The rest of this paper is organized as following. In

kernel. In Section 5 the performance is compared

on basis of a real data set and section 6 concludes.

2 Development of Logistic kernel

estimator

Let  be a random sample from a

distribution with an unknown probability density

function f which has support on -∞, ∞.

Representation of pdf of Logistic 󰇛󰇜 is