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Article

A Novel Sequential Three-Way Decision Model for Medical Diagnosis

School of Business, Central South University, Changsha 410083, China
*
Author to whom correspondence should be addressed.
Submission received: 6 April 2022 / Revised: 3 May 2022 / Accepted: 13 May 2022 / Published: 15 May 2022
(This article belongs to the Special Issue Soft Computing and MCDA Methods for Support Decision Making)

Abstract

:
In the sequential three-way decision model (S3WD), conditional probability and decision threshold pair are two key elements affecting the classification results. The classical model calculates the conditional probability based on the strict equivalence relationship, which limits its application in reality. In addition, little research has studied the relationship between the threshold change and its cause at different granularity levels. To deal with these deficiencies, we propose a novel sequential three-way decision model and apply it to medical diagnosis. Firstly, we propose two methods of calculating conditional probability based on similarity relation, which satisfies the property of symmetry. Then, we construct an S3WD model for a medical information system and use three different kinds of cost functions as the basis for modifying the threshold pair at each level. Subsequently, the rule of the decision threshold pair change is explored. Furthermore, two algorithms used for implementing the proposed S3WD model are introduced. Finally, extensive experiments are carried out to validate the feasibility and effectiveness of the proposed model, and the results show that the model can achieve better classification performance.

1. Introduction

In the course of patient treatment, obtaining the correct diagnostic results and the appropriate treatment plan are the most important requirements for patients. However, if the information about the patients’ condition is vague and imprecise or the doctor lacks clinical experience, the doctors are likely to make the wrong judgment and delay the patient’s treatment. Therefore, knowing how to assist doctors in making the right diagnosis under uncertain scenarios is an urgent problem that needs to be solved.
With the advent of the big-data era, new technologies are constantly emerging to provide decision support for medical diagnosis, including data mining [1,2], deep learning [3,4,5], pattern recognition, etc. However, as the task of collecting complete information is time-consuming and expensive, patient information is usually inaccurate and uncertain [6]. Consequently, these classification methods dependent on data-driven methodologies may lead to the risk of diagnosing pending medical cases [7]. The reliability and certainty of information are critical factors in medical decision making and reasoning [8]. As an effective method to reduce the impact of uncertainty, the three-way decision (3WD) has been considered to delay the decision making in the absence of sufficient evidence. This approach is feasible for medical diagnosis.
The 3WD model, as proposed by Yao [9], was derived from a decision-theoretic rough set. It divides a universe into three disjoint regions by setting a threshold pair, and then different decision-making strategies are adopted [10]. Compared with the two-way decision making with a single threshold or Pawlak rough set [11], the 3WD model has a higher fault tolerance rate, which reduces the risk and cost of decision making [12]. Nowadays, many researchers pay much attention to the extended 3WD model in which the uncertain information decision-making method is integrated [13,14,15,16,17,18,19], whilst other researchers have combined 3WD with machine learning [20,21], utility theory [22] or prospect theory [23]. In addition, some scholars have applied 3WDs to practical applications, including face recognition [24], e-mail spam filtering [25,26], software defect prediction [27], credit scoring [28], attribute reduction [29,30], medical diagnosis [6,31,32,33], knowledge harnessing [34], and others.
In reality, collecting information is a gradual process; additionally, it is expensive and time-consuming. When the effective information of an object is incomplete and insufficient, 3WDs may lead to wrong judgment. To overcome this shortcoming, Yao [35,36,37] proposed a cost-effective, sequential, multi-step 3WD method based on the idea of granular computing, also called sequential three-way decisions (S3WDs). In each step, when the available resources are not enough for the decision maker to offer a judgment, the decision will be delayed to the next stage, and new information will be added until the evidence is sufficient. The boundary region will decrease with the progressive decision. To avoid decision revision and achieve higher accuracy at the lower level, the threshold pairs at each level need to be interpreted and calculated.
At present, the main solution to the problem mentioned above is to construct a cost loss function based on Bayesian decision theory; 3WD is generally considered an optimization problem, and the threshold pair is calculated by minimizing the decision risk. Deng and Yao [38] introduced an information-theoretic rough set model and determined a pair of thresholds to minimize the overall uncertainty of three regions. Zhang [39] used the Gini coefficient of rough set regions to obtain effective probabilistic thresholds. Azam and Yao [40] used a game-theoretic rough set model for calculation. Li et al. [30] combined three types of criteria, including the positive region, decision cost, and mutual information. However, the above method with high complexity is not directly applicable to the sequential three-way decision. Meanwhile, there is a lack of reasonable analysis and explanation of the change of decision thresholds at different granularity levels.
A patient visit is a gradual process of obtaining various examination indicators; thus, using the S3WD model for medical diagnosis is reasonable. In order to make the model achieve a highly accurate diagnosis, we must deal with two crucial challenges. The first challenge is the processing of medical data. Many kinds of data can be used to describe a patient’s condition [31] (e.g., set-valued data, categorical data, Boolean data, real-valued data, and missing data), indicating that the method of finding an equivalence class based on an equivalence relation is unsuitable. As such, on the basis of previous research [6,41,42], this study uses the similarity relation to replace the equivalence relation. In addition, for different types of data, various equations need to be constructed to measure similarity. On the basis of the abovementioned concepts, two methods of calculating conditional probability are proposed. The second challenge is the calculation of a threshold pair for each layer. The calculation of threshold pairs with the minimum decision risk as the goal is closely related to the decision cost matrix. As the decision progresses, the threshold pair changes with increased cost. However, precisely estimating the decision cost is difficult. Inspired by the penalty function [43] and other formations of the cost matrix [16,44] and in consideration of the actual situation, we use three different kinds of cost functions as the basis for modifying the cost matrix at each level. The rule of the threshold pair change is also determined.
The main contributions of this paper are summarized as follows: (1) Different similarity measures are proposed for dealing with different types of data. (2) Based on the similarity relationship, two new calculation methods of conditional probability are defined. (3) According to the threshold pairs of different granularity layers, three types of modified cost functions and reasonable semantic interpretation are given.
The rest of this paper is structured as follows. In Section 2, we briefly review the related work of the decision-theoretic rough set model and S3WD. In Section 3, we propose a novel S3WD model and two algorithms based on the concept of similarity. In Section 4, we verify the model by consolidating a case study and make a detailed analysis of the results. Section 5 summarizes the research and elaborates on future studies.

2. Preliminaries

In this section, we summarize the basic concept of the decision-theoretic rough set model, the 3WD model, and the S3WD model.

2.1. Decision-Theoretic Rough Set Model

Definition 1
([45]). Let a decision information table have the discrete value attributes given by S = ( U , A t = C D , V = { V a | a A t } , I = { I a | a A t } ) , where U represents a non-empty finite set of objects, A t represents a non-empty finite set of attributes, C is a set of condition attributes with discrete values, D is a set of decision attributes, V a is the set of values for each attribute a A t and I a is a complete information function that maps an object of U to exactly one value in V a .
In the decision-theoretic rough set model, an equivalence relation E A is a special binary relation, which can be described using a subset of attributes A A t as follows:
x E A y = { ( x , y ) U × U | a A ( I a ( x ) = I a ( y ) ) } ,
where I a ( x ) and I a ( y ) represent the value of objects x and y on the attribute a , respectively. The binary relation satisfies the following properties:
(1)
Reflexivity: x U , it exits ( x , x ) E A .
(2)
Symmetry: x , y U , if ( x , y ) E A , then ( y , x ) E A .
(3)
Transitivity: x , y , z U , if ( x , y ) E A , ( x , z ) E A , then ( y , z ) E A .
According to the equivalence relation E A , we can obtain the division of the universe U , i.e., U / E A = { [ x ] A | x U } . The equivalence class of object x is given by
[ x ] A = { y U | a A ( I a ( x ) = I a ( y ) ) } .
Definition 2
([45]). Suppose X U is a set of objects belonging to the same concept. On the basis of the abovementioned decision information table, the lower and upper approximations of X with respect to A are given as follows:
A * ( X ) = { x U | [ x ] A X } , A * ( X ) = { x U | [ x ] A X } .
Definition 3
([45]). According to the lower and upper approximations, the subset X can be divided into three disjoint regions, which are expressed by
P O S A ( X ) = A * ( X ) = { x | x U , [ x ] A X } , B N D A ( X ) = A * ( X ) A * ( X ) = { x | x U , [ x ] A X , [ x ] A X } , N E G A ( X ) = U A * ( X ) = { x | x U , [ x ] A X = } .
In addition, we can describe them in the form of a probability formula as follows:
P O S A ( X ) = { x | x U , P ( X | [ x ] A ) = 1 } , B N D A ( X ) = { x | x U , 0 < P ( X | [ x ] A ) < 1 } , N E G A ( X ) = { x | x U , P ( X | [ x ] A ) = 0 } ,
where P ( X | [ x ] A ) = | X [ x ] A | / | [ x ] A | is the conditional probability, | X [ x ] A | represents the number of an equivalence class of object x belonging to subset X , and | [ x ] A | represents the number of an equivalence class of object x .

2.2. Three-Way Decision Model

Definition 4
([9]). X is the target subset. Given a conditional probability and a pair of thresholds ( α , β ) , where 0 β < α 1 , we can obtain the new positive, boundary, and negative regions with respect to X as follows:
P O S ( X ) = { x | x U , P ( X | [ x ] A ) α } , B N D ( X ) = { x | x U , β < P ( X | [ x ] A ) < α } , N E G ( X ) = { x | x U , P ( X | [ x ] A ) β } .
The threshold pair can be calculated using the risk function. Given a decision cost matrix (Table 1), λ P P , λ B P , λ N P represents the incurred cost when action a P , a B , a N is taken as the object belonging to X . Furthermore, λ P N , λ B N , λ N N represents the cost when the same action is taken as the object that does not belong to X . The expected risk cost for each of the actions can be described as follows:
E ( a B | [ x ] A ) = λ B P P ( X | [ x ] A ) + λ B N P ( X ¬ | [ x ] A ) , E ( a N | [ x ] A ) = λ N P P ( X | [ x ] A ) + λ N N P ( X ¬ | [ x ] A ) , E ( a P | [ x ] A ) = λ P P P ( X | [ x ] A ) + λ P N P ( X ¬ | [ x ] A ) .
On the basis of the Bayesian decision procedure, the minimum cost decision rules can be expressed as follows:
  • If E ( a P | [ x ] A ) E ( a B | [ x ] A ) and E ( a P | [ x ] A ) E ( a N | [ x ] A ) , then x P O S ( X ) ,
  • If E ( a B | [ x ] A ) E ( a P | [ x ] A ) and E ( a B | [ x ] A ) E ( a N | [ x ] A ) , then x B N D ( X ) ,
  • If E ( a N | [ x ] A ) E ( a P | [ x ] A ) and E ( a N | [ x ] A ) E ( a B | [ x ] A ) , then x N E G ( X ) .
In normal cases, assuming λ P P λ B P λ N P , λ N N λ B N λ P N , we can simplify the decision rules as
  • If P ( X | [ x ] A ) α , then x P O S ( X ) ,
  • If β < P ( X | [ x ] A ) < α , then x B N D ( X ) ,
  • If P ( X | [ x ] A ) β , then x N E G ( X ) ,
where the parameters of α , β are calculated as
α = ( λ P N λ B N ) / { ( λ P N λ B N ) + ( λ B P λ P P ) } , β = ( λ B N λ N N ) / { ( λ B N λ N N ) + ( λ N P λ B P ) } .

2.3. Sequential Three-Way Decision Model

Definition 5
([35,37]). Let A t = ( A t 1 , A t 2 , , A t ( k 1 ) , A t k ) is a multilevel granular structure, and A t 1 A t 2 A t k A t . Each layer i has a pair of different thresholds ( α i , β i ) depending on the cost matrix, which satisfies the conditions.
0 β i < α i 1 , 1 i k , β 1 β 2 β k α k α 2 α 1 .
At level i , additional attributes in A t i A t i 1 are added, and P i ( X | [ x ] A ) is recalculated. Then, B N D i 1 is divided into three new regions as follows:
P O S i ( α i , β i ) = { x | x B N D i 1 , P ( X | [ x ] A ) α i } , B N D i ( α i , β i ) = { x | x B N D i 1 , β i < P ( X | [ x ] A ) < α i } , N E G i ( α i , β i ) = { x | x B N D i 1 , P ( X | [ x ] A ) β i } .
In addition, the real positive, boundary, and negative regions at the level i can be expressed as follows:
P O S i = U j = 1 i P O S j ( α j , β j ) , B N D i = B N D i ( α i , β i ) , N E G i = U j = 1 i N E G j ( α j , β j ) .
Thus, the S3WD model can be regarded as a multi-step 3WD, and it is more flexible and efficient.

3. Sequential Three-Way Decisions in Medical Information System

A medical information system can be considered a container with various data types, such as real-, integer-, and Boolean-type data, amongst others. As such, calculating the conditional probability by using the equivalence relation is unreasonable. In this section, we focus on discussing the S3WD model in the medical information table. Firstly, we define a new similarity relation to the replace equivalence relation. Secondly, we interpret the decision cost at each level and then propose an S3WD model based on the medical information system. Finally, we describe the model and construct the algorithm.

3.1. Medical Information System and the Similarity Measure

The traditional S3WD model can only deal with nominal data, which limits its practical application. In order to handle various data types, we give the following definitions.
Definition 6.
Given the formula M I S = ( U , A t = C D , V = { V a | a A t } , I = { I a | a A t } ) , where U stands for a non-empty finite set of objects, let A t denote a non-empty set of attributes with various types of values, V as the set of values for each attribute and I : U × A V as an information function, and D represents the decision attribute set. For a binary classification problem, the set of states U = { X , X ¬ } denotes that an object is in X and not in X .
Definition 7.
Suppose MIS has n objects, k conditional attributes, and one decision attribute. U = { x 1 , x 2 , , x n } , and C = { a 1 , a 2 , , a k } . For x i , x j U , a m C , the similarity degree between x i and x j on a m can be expressed as the following four cases:
(1)
Consider the value type of a m as a finite ordered discrete, and the value size represents degree, e.g., V a m = { 1 , 2 , , 5 } . Then,
S i m ( a m ) ( x i , x j ) = { 1 V a m ( x i ) = V a m ( x j ) 0 V a m ( x i ) V a m ( x j ) .
(2)
Consider the value type of a m as continuous, e.g., V a m = { 0.25 , 0.36 , ...0.98 } . Then,
S i m ( a m ) ( x i , x j ) = { 1 | V a m ( x i ) V a m ( x j ) | / | max ( V a m ( x i ) , V a m ( x j ) ) | , V a m ( x i ) , V a m ( x j ) V a m ( x i ) = V a m ( x j ) 0 1 , V a m ( x i ) = V a m ( x j ) = 0 .
(3)
Consider the value type of a m as Boolean, e.g., V a m = { y e s , n o } . Then,
S i m ( a m ) ( x i , x j ) = { 1 V a m ( x i ) = V a m ( x j ) 0 V a m ( x i ) V a m ( x j ) .
(4)
Consider the value type of a m as one that can be described in language, and the value of a m denotes the degree, e.g., V a m = { L o w , N o r m a l , H i g h } . Then, we turn these expressions into ordered discrete and calculate the similarity according to the first method.
On the basis of the above discussions, the similarity degree between the two objects x i and x j can be defined as
S i m ( x i , x j ) = ( 1 / k ) i = 1 k S i m ( a m ) ( x i , x j ) .
Definition 8.
Given M I S = ( U , A t = C D , V = { V a | a A t } , I = { I a | a A t } ) , and θ is a given threshold that satisfies 0 θ 1 . We can obtain a new similarity relation as follows:
S i m θ ( x i , x j ) = { ( x i , x j ) | S i m ( x i , x j ) θ } .
On the basis of the value of θ , we can define the similarity class of object x as follows:
[ x i ] s i m θ = { x j | x j U , S i m ( x i , x j ) θ } ,
Thus, θ is a parameter to ensure the accuracy of the classification. If the value of θ is extremely high, then the number of similar objects will be less. If the value of θ is extremely low, then the objects belonging to different categories will likely be classified into the similarity class of x . Subsequently, P ( X | [ x ] s i m θ ) can be calculated by
P ( X | x ) = | [ x ] s i m θ X | / | [ x ] s i m θ | .
When the setting of parameter θ is unreasonable, it causes [ x i ] s i m θ to be an empty set, and the conditional probability cannot be calculated. Inspired by K-nearest neighbor, we give another calculation method of conditional probability.
Definition 9.
Given M I S = ( U , A t = C D , V = { V a | a A t } , I = { I a | a A t } ) , and k is a given integer value that satisfies 1 k . We can obtain the top k objects with the highest similarity to the object x , which is described as [ x ] s i m k . Thus, the conditional probability of x belonging to X can be described as follows:
P ( X | x ) = | [ x ] s i m k X | / k ,
where | [ x ] s i m k X | is the number of objects belonging to X in k .
Example 1.
Given a medical information system (Table 2) with X = { x 3 , x 4 , x 6 } , an object to be classified can be described as x = { a 1 = 7.8 , a 2 = 1 , a 3 = 3 , a 4 = H i g h } .
According to Definition 7, we can calculate the similarity between object x and each object in the medical information system. We set the parameter θ = 0.65 , and k = 3 .
(1)
On level 1, the similar class of x based on Definition 8 is [ x ] s i m θ = { x 1 , x 2 , x 3 , x 4 , x 6 } , and the top three variables most similar to x based on Definition 9 are [ x ] s i m k = { x 1 , x 2 , x 6 } . Then, the conditional probability can be expressed as
P ( X | x ) = | [ x ] s i m θ X | / | [ x ] s i m θ | = 3 / 5 , P ( X | x ) = | [ x ] s i m k X | / k = 1 / 3 .
Given a threshold pair ( α = 0.9 , β = 0.1 ) , x is divided into the boundary region.
(2)
On level 2, the similar class of x based on Definition 8 is [ x ] s i m θ = { x 3 , x 4 , x 6 } , and the top three variables most similar to x based on Definition 9 are [ x ] s i m k = { x 3 , x 4 , x 6 } . Then, the conditional probability can be expressed as
P ( X | x ) = | [ x ] s i m θ X | / | [ x ] s i m θ | = 1 , P ( X | x ) = | [ x ] s i m k X | / k = 1 .
Given a threshold pair ( α = 0.9 , β = 0.1 ) , x is divided into the positive region.

3.2. Sequential Three-Way Decision-Based Medical Information System

With respect to the S3WD model, the key factor in the S3WD-based medical information system is the interpretation of a threshold pair at each level. According to the Bayesian decision procedure, we can deduce that the value of the threshold pair is closely related to the decision cost. Furthermore, with the increase in available attributes, the objects may be classified correctly. Therefore, once an incorrect classification occurs, the corresponding decision-making cost should also increase. In view of achieving high classification accuracy, it is reasonable to change the decision cost with the layer.
Definition 10.
Suppose A t = ( A t 1 , A t 2 , , A t ( k 1 ) , A t k ) is a k -level granule structure composed of an attribute set. At level i ( 1 i k ) , A t i = ( a 1 , a 2 , , a i ) , we define a cost matrix λ i , including six decision costs ( λ P P i , λ B P i , λ N P i , λ N N i , λ B N i , λ P N i ) , which satisfies the following condition: λ P P i λ B P i λ N P i , λ N N i λ B N i λ P N i .
The expected cost associated with the determination of the three different actions for the objects in [ x ] can be expressed as follows:
E i ( a P | [ x ] i ) = λ P P i P ( X | [ x ] i ) + λ P N i P ( X ¬ | [ x ] i ) , E i ( a B | [ x ] i ) = λ B P i P ( X | [ x ] i ) + λ B N i P ( X ¬ | [ x ] i ) , E i ( a N | [ x ] i ) = λ N P i P ( X | [ x ] i ) + λ N N i P ( X ¬ | [ x ] i ) ,
where [ x ] i represents a similar class of an object x at level i , and P ( X | [ x ] i ) is the conditional probability that is defined by Definitions 8 and 9.
According to the Bayesian decision procedure, the decision rules with the minimum cost criterion can be described as follows:
(1)
If E i ( a P | [ x ] i ) E i ( a B | [ x ] i ) and E i ( a P | [ x ] i ) E i ( a N | [ x ] i ) , then x P O S ( X ) ,
(2)
If E i ( a B | [ x ] i ) E i ( a P | [ x ] i ) and E i ( a B | [ x ] i ) E i ( a N | [ x ] i ) , then x B N D ( X ) ,
(3)
If E i ( a N | [ x ] i ) E i ( a P | [ x ] i ) and E i ( a N | [ x ] i ) E i ( a B | [ x ] i ) , then x N E G ( X ) .
The decision rules can be simplified in the following forms:
(1)
If P ( X | [ x ] i ) α i , then decide x P O S ( X ) ,
(2)
If β i < P ( X | [ x ] i ) < α i , then decide x B N D ( X ) ,
(3)
If P ( X | [ x ] i ) β i , then decide x N E G ( X ) ,
where the threshold pair can be computed as follows:
α i = ( λ P N i λ B N i ) / { ( λ P N i λ B N i ) + ( λ B P i λ P P i ) } , β i = ( λ B N i λ N N i ) / { ( λ B N i λ N N i ) + ( λ N P i λ B P i ) } .
On the basis of Definition 5, we can deduce the relationship between the thresholds at different levels. In general, the cost of judging correctly whether a patient is ill is set as zero. Thus, at any level i , λ P P i = λ N N i = 0 . Furthermore, in SW3D, when a clear answer cannot be derived regarding the patient’s condition, we need to obtain more evidence, and then, the corresponding cost will increase. At the same time, delay in decision making will bring about the risk of worsening the disease or aggravating the patient’s psychological burden. Motivated by the penalty function [43], we propose a novel method of temporarily increasing the value of λ B P and λ B N from level i to level i + 1 . In particular, we define the relations between λ B P i and λ B P i + 1 and between λ B N i and λ B N i + 1 as follows:
λ B P i = { λ B P 1 i = 1 λ B P 1 + φ ( i ) i > 1 , λ B N i = { λ B N 1 i = 1 λ B N 1 + φ ( i ) i > 1 .
where φ ( i ) represents the cost function, and we use three different forms of function to describe:
φ ( i ) = i c , φ ( i ) = i 2 c , φ ( i ) = i c ,
where c is a constant. Thus, we have i , 1 i k , 0 < λ B P i λ B P i + 1 , 0 < λ B N i λ B N i + 1 .
In addition, misdiagnosing a healthy person as ill leads to the wastage of medical resources and loss of money. Moreover, diagnosing a sick person as not ill can delay the patient’s condition and even endanger his or her life. These completely misclassified situations are worse than not giving any results. Therefore, we assume that the loss of making a wrong diagnosis at any level is equally enormous. Then, the cost parameters satisfy i , 1 i k , λ N P i = λ N P i + 1 , λ P N i = λ P N i + 1 , and λ B P k λ N P i , λ B N k λ P N i .
On the basis of the above discussion, we can derive the following theorem:
Theorem 1.
When λ B P i and λ B N i of level increase, then the threshold α i + 1 of level i + 1 decreases, and the threshold β i + 1 of level i + 1 increases.
Proof. 
For α i = ( λ P N i λ B N i ) / ( ( λ P N i λ B N i ) + ( λ B P i λ P P i ) ) , where λ P P i = 0 , λ P N i is a fixed value, and the parameter can be reduced to α i = ( λ P N i λ B N 1 φ ( i ) ) / ( λ P N i λ B N 1 + λ B P 1 ) . Let α i + 1 = ( λ P N i λ B N 1 φ ( i + 1 ) ) / ( λ P N i λ B N 1 + λ B P 1 ) . Then,
α i + 1 α i = ( λ P N i λ B N 1 φ ( i + 1 ) ) / ( λ P N i λ B N 1 + λ B P 1 ) ( λ P N i λ B N 1 φ ( i ) ) / ( λ P N i λ B N 1 + λ B P 1 ) = ( φ ( i ) φ ( i + 1 ) ) / ( λ P N i λ B N 1 + λ B P 1 ) ,
and φ ( i ) φ ( i + 1 ) 0 . Consequently, a i + 1 < a i can be obtained. The threshold α decreases.
For β i = ( λ B N i λ N N i ) / [ ( λ B N i λ N N i ) + ( λ N P i λ B P i ) ] , where λ N N i = 0 , λ N P i is a fixed value, and the parameter can be reduced to β i = λ B N 1 + φ ( i ) / ( λ B N 1 + λ N P i λ B P 1 ) . Let β i + 1 = λ B N 1 + φ ( i + 1 ) / ( λ B N 1 + λ N P i λ B P 1 ) . Then,
β i + 1 β i = ( λ B N 1 + φ ( i + 1 ) ) / ( λ B N 1 + λ N P i λ B P 1 ) ( λ B N 1 + φ ( i ) ) / ( λ B N 1 + λ N P i λ B P 1 ) = ( φ ( i + 1 ) φ ( i ) ) / ( λ B N 1 + λ N P i λ B P 1 ) ,
and φ ( i + 1 ) φ ( i ) 0 . Consequently, β i + 1 > β i can be obtained. The threshold β increases. □

3.3. Algorithm to Construct the Sequential Three-Way in Medical Information System

In this section, we briefly describe the steps of the sequential three-way model in medical information system and propose a basic algorithm. Firstly, the medical data are divided into a training set and a test set. Secondly, for each object in the training set, find its similar class in the test set and calculate the conditional probability. Thirdly, at each granularity level, the relationship between the conditional probability of the object and the calculated threshold pair is compared. Finally, the objects are divided into corresponding regions according to the decision rules. Repeat the above steps until all objects are divided into corresponding positive and negative domains. The model process framework is shown in Figure 1.
According to the aforementioned two different methods of calculating the conditional probability, we extend the basic algorithm (see Algorithm 1) to Algorithm A and Algorithm B in the process of experimental verification. Specifically, Algorithm A needs to add an input parameter θ to set the threshold of similar classes. The objects whose similarity is greater than the parameter θ in the training set are selected, and their categories are counted. Subsequently, we can calculate P ( X | x ) according to the formula in Definition 8. In the same manner, Algorithm B needs to add the input parameter k to obtain the top k objects with the highest similarity. After calculating the conditional probability, the basic algorithm process is adopted to obtain the positive and negative domains.
Algorithm 1. A basic algorithm for sequential three-way decision in medical information system
Input: A medical information system, λ P P , λ P N , λ B P , λ B N , λ N P , λ N N .
Output: POS and NEG
Split training set and test set
1.while U i 0 and i < = s u m ( a t )  do
2.     P ( X | x ) , α , β ;
3.     if P ( X | x ) α then
4.     P O S P O S x ;
5.     end if
6.     if β < P ( X | x ) < α then
7.     B N D B N D x ;
8.     end if
9.     if P ( X | x ) β  then
10.      N E G N E G x ;
11.     end if
12.     U i + 1 = B N D ( X ) ;
13.     i + = 1 .
14.     λ B P = λ B P 1 + φ ( i )
15.     λ B N = λ B N 1 + φ ( i )
16.else
17.     for each x U i do
18.       P ( X | x ) ;
19.      if P ( X | x ) 1 / 2 then
20.      P O S P O S x ;
21.      end if
22.      if P ( X | x ) 1 / 2 then
23.       N E G N E G x ;
24.      end if
25.    end for
26.    else
27.       P O S P O S x
28.return POS, NEG

4. Experiment and Analysis

In this section, we verify the two algorithms by means of experimental analysis. Algorithm A is used to calculate the conditional probability according to Definition 8, whilst Algorithm B is used to calculate the conditional probability according to Definition 9.

4.1. Experimental Setting

To illustrate the validity and advantages of the two algorithms, we select four medical datasets from the UCI database (http://archive.ics.uci.edu/ml, accessed on 1 April 2022) and pre-process these datasets and delete the data with missing values. The relevant details of the datasets are provided in Table 3.
To ensure that the experimental results are highly intuitive and credible, we use the three indexes of precision rate, recall rate, and accuracy rate to evaluate the performance of the different models, which can be described as follows:
p r e c i s i o n = | T P | / | T P + F P |
r e c a l l = | T P | / | T P + F N |
a c c u r a c y = | T P + T N | / | T P + T N + F P + F N |
F 1 = 2 precision r e c a l l / ( p r e c i s i o n + r e c a l l )
where T P , T N represent the number of objects correctly divided into P O S ( X ) and N E G ( X ) respectively, and F P , F N represent the number of objects incorrectly divided into P O S ( X ) and N E G ( X ) respectively.

4.2. Comparison of Performance between Models

In this section, we compare the two algorithms proposed in this study with K-nearest neighbor (KNN) and decision tree. As the dataset does not include the cost of obtaining the relevant attributes, we default to the expression λ N P = 4 λ P N , φ ( λ B P 1 ) = i λ B P 1 . We use ten-fold cross-validation to ensure that the results are highly reliable. The performance of the four algorithms on the four datasets is shown in Figure 2.
As shown in Figure 2, although the two algorithms proposed in this study do not always yield the highest results in terms of precision rate, they have high F1 on the premise of high classification accuracy, especially based on Algorithm A. In particular, for the Chronic Kidney Disease dataset, all the four evaluation indicators of Algorithm A are 100%. The reason is that the change in the cost of delaying the decision is reasonable. In addition, Figure 3 indicates Algorithm B has different evaluation values when using different cost functions in Equation (23). For the Statlog (Heart) dataset, compared with the other two cost functions, the precision rate and accuracy rate of Function 2 is lower, mainly because the threshold α drops too much, while β rises. Therefore, in order to achieve high accuracy, it is necessary to select the appropriate cost function for different datasets.

4.3. Sensitivity Analysis

The classification accuracy of S3WD is greatly affected by a threshold value. Thus, the algorithm’s stability should be tested by means of sensitivity analysis. In addition, the influence of parameter θ in Algorithm A and parameter k in Algorithm B on the result should be determined.
Firstly, we compare the results of the different thresholds. As the threshold value depends on the cost matrix, we change the multiple relations between λ N P and λ P N . The specific results are shown in Table 4, Table 5, Table 6 and Table 7.
When the multiple between cost parameters increases, the classification accuracy and precision of the two algorithms show a downward trend, whereas recall shows an upward trend. This trend can be explained by reduced loss caused by misclassification; that is, the higher the multiple loss, the more objects will be divided into the positive domain. Furthermore, the two algorithms proposed in this study have their own advantages for the different datasets. For the Statlog (Heart) dataset, when the loss ratio is small, the recall and F1 of Algorithm B are better than that of Algorithm A, but for the Chronic Kidney Disease dataset, Algorithm A outperforms Algorithm B at any time. Algorithm B limits the number of similar classes, whereas Algorithm A is more flexible in selecting similar classes. In reality, the decision maker can choose the algorithm according to his or her own preference.
Secondly, the results under different parameters of θ of Algorithm A are calculated. As shown in Figure 4, for most of the datasets, both higher and lower parameter values of θ may lead to lower accuracy and precision rates. A reason is that an extremely low parameter value causes the objects of the different categories to be classified into similar classes, whereas an extremely high parameter value makes it difficult for the objects to find similar classes such that they can only be divided into the positive domain by default in the last stage. In addition, different datasets vary in their optimal parameters.
Thirdly, we change the parameter k in Algorithm B to observe the changes in the three indexes. As shown in Figure 5 for the datasets of Breast Cancer Wisconsin and Chronic Kidney Disease, the results do not change regardless of the value of parameter k between 4 and 10. For the other datasets, a change in k has a negligible effect on the evaluation indicators. All of these trends indicate that Algorithm B has high stability.
Through the above analysis, we believe that this model has a better classification accuracy. It is helpful to help doctors realize efficient consultation, reduce the occurrence of misdiagnosis, and improve the satisfaction of patients in practice.

5. Conclusions

The main difficulties in studying the S3WD model are the calculation of conditional probability and the determination of threshold at different levels. To solve these problems, we propose the improvements as follows. Firstly, in the case of various types of medical data, different similarity measures are presented. On this basis, we introduce two ways to calculate conditional probability. Among them, the second method, which draws on the idea of the K-nearest neighbor, makes up for the first method when the value of θ is unreasonable. Secondly, we modify the cost parameters in the next stage in light of the cost of acquiring attributes and explain the variation rule of the threshold values between the different levels. It fills the gap in the analysis and interpretation of the decision thresholds change between different granularity levels. Subsequently, two algorithms are applied to the medical diagnosis. We conduct corresponding experiments to prove the reliability of the two algorithms. The experimental results demonstrate that the model proposed in this study can help determine the patients correctly, which can effectively reduce the waste of medical resources and the increase in loss caused by misdiagnosis in reality. Furthermore, we perform sensitivity analysis by changing the different parameters. The findings have verified the validity and stability of our model. We hope that the model proposed can promote the further development of the sequential three-way decision, especially combined with data-mining methods such as machine learning.
In this paper, we take the order of attributes in the medical information system as the basis for adding attributes at each granularity level by default, which may lead to decision errors in the coarse granularity layer. In fact, the number of attributes added to each layer and the order in which they are added will affect the classification accuracy to a certain extent. In our future study, we will try to solve the problem of attribute addition order and optimal granularity selection.

Author Contributions

Conceptualization, J.H. and W.C.; methodology, W.C.; software, W.C. and P.L.; validation, J.H., W.C. and P.L.; formal analysis, W.C.; investigation, W.C.; resources, J.H.; data curation, W.C.; writing—original draft preparation, W.C.; writing—review and editing, J.H. and P.L.; visualization, P.L.; supervision, J.H.; project administration, P.L.; funding acquisition, J.H. All authors have read and agreed to the published version of the manuscript.

Funding

This work is supported partially by the National Natural Science Foundation of China under grant number 71871229 and the Hunan Provincial Natural Science Foundation of China under grant number 2021JJ30031.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Available online: http://archive.ics.uci.edu/ml, accessed on 1 April 2022.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Figure 1. The framework of the S3WD in medical information system.
Figure 1. The framework of the S3WD in medical information system.
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Figure 2. Comparisons of precision, recall, accuracy, and F1: (a) Breast Cancer Wisconsin; (b) Chronic Kidney Disease; (c) SPECT Heart; (d) Statlog (Heart).
Figure 2. Comparisons of precision, recall, accuracy, and F1: (a) Breast Cancer Wisconsin; (b) Chronic Kidney Disease; (c) SPECT Heart; (d) Statlog (Heart).
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Figure 3. Results of Algorithm B with different cost functions: (a) Breast Cancer Wisconsin; (b) Chronic Kidney Disease; (c) SPECT Heart; (d) Statlog (Heart).
Figure 3. Results of Algorithm B with different cost functions: (a) Breast Cancer Wisconsin; (b) Chronic Kidney Disease; (c) SPECT Heart; (d) Statlog (Heart).
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Figure 4. Results of different parameters of θ: (a) Breast Cancer Wisconsin; (b) Chronic Kidney Disease; (c) SPECT Heart; (d) Statlog (Heart).
Figure 4. Results of different parameters of θ: (a) Breast Cancer Wisconsin; (b) Chronic Kidney Disease; (c) SPECT Heart; (d) Statlog (Heart).
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Figure 5. The results of different parameter k: (a) Breast Cancer Wisconsin; (b) Chronic Kidney Disease; (c) SPECT Heart; (d) Statlog (Heart).
Figure 5. The results of different parameter k: (a) Breast Cancer Wisconsin; (b) Chronic Kidney Disease; (c) SPECT Heart; (d) Statlog (Heart).
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Table 1. Cost matrix.
Table 1. Cost matrix.
X ( P ) X C ( N )
a P λ P P λ P N
a B λ B P λ B N
a N λ N P λ N N
Table 2. Medical information system.
Table 2. Medical information system.
U a 1 a 2 a 3 a 4 D
x 1 5.803Low0
x 2 6.301Normal0
x 3 1215High1
x 4 11.515High1
x 5 4.706Normal0
x 6 6.515Low1
Table 3. Description of datasets.
Table 3. Description of datasets.
NameInstancesConditional Attributes
Breast Cancer Wisconsin6839
Chronic Kidney Disease15624
SPECT Heart26722
Statlog (Heart)27013
Table 4. Results of two algorithms based on the different cost multiple relations for Breast Cancer Wisconsin.
Table 4. Results of two algorithms based on the different cost multiple relations for Breast Cancer Wisconsin.
MetricsAlgorithm λ N P = λ P N λ N P = 2 λ P N λ N P = 3 λ P N λ N P = 4 λ P N λ N P = 5 λ P N
PrecisionAlgorithm A95.70%95.70%95.70%95.70%95.71%
Algorithm B95.64%95.64%95.66%95.69%95.73%
RecallAlgorithm A95.38%95.38%95.38%95.38%95.67%
Algorithm B93.83%93.83%94.29%94.59%95.64%
AccuracyAlgorithm A96.47%96.47%96.47%96.47%96.62%
Algorithm B95.59%95.59%95.74%95.88%96.32%
F1Algorithm A95.54%95.54%95.54%95.54%95.69%
Algorithm B94.73%94.73%94.97%95.14%95.68%
Table 5. Results of two algorithms based on the different cost multiple relations for Chronic Kidney Disease.
Table 5. Results of two algorithms based on the different cost multiple relations for Chronic Kidney Disease.
MetricsAlgorithm λ N P = λ P N λ N P = 2 λ P N λ N P = 3 λ P N λ N P = 4 λ P N λ N P = 5 λ P N
PrecisionAlgorithm A100.00%100.00%100.00%100.00%100.00%
Algorithm B100.00%100.00%100.00%100.00%100.00%
RecallAlgorithm A100.00%100.00%100.00%100.00%100.00%
Algorithm B82.22%82.22%84.44%84.44%84.44%
AccuracyAlgorithm A100.00%100.00%100.00%100.00%100.00%
Algorithm B94.67%94.67%95.33%95.33%95.33%
F1Algorithm A100.00%100.00%100.00%100.00%100.00%
Algorithm B90.24%90.24%91.57%91.57%91.57%
Table 6. Results of two algorithms based on the different cost multiple relations for SPECT Heart.
Table 6. Results of two algorithms based on the different cost multiple relations for SPECT Heart.
MetricsAlgorithm λ N P = λ P N λ N P = 2 λ P N λ N P = 3 λ P N λ N P = 4 λ P N λ N P = 5 λ P N
PrecisionAlgorithm A89.81%89.85%89.01%88.73%81.50%
Algorithm B87.84%80.78%79.62%79.62%79.62%
RecallAlgorithm A87.94%88.39%88.39%90.33%96.18%
Algorithm B91.18%96.68%100.00%100.00%100.00%
AccuracyAlgorithm A82.31%82.69%81.92%83.08%79.23%
Algorithm B83.08%78.85%79.62%79.62%79.62%
F1Algorithm A88.87%89.11%88.70%89.52%88.23%
Algorithm B89.48%88.02%88.65%88.65%88.65%
Table 7. Results of two algorithms based on the different cost multiple relations for Statlog (Heart).
Table 7. Results of two algorithms based on the different cost multiple relations for Statlog (Heart).
MetricsAlgorithm λ N P = λ P N λ N P = 2 λ P N λ N P = 3 λ P N λ N P = 4 λ P N λ N P = 5 λ P N
PrecisionAlgorithm A89.70%89.70%89.70%89.70%77.83%
Algorithm B85.29%85.29%78.69%72.52%72.29%
RecallAlgorithm A75.36%75.36%75.36%75.36%87.67%
Algorithm B80.33%81.05%87.01%89.64%90.41%
AccuracyAlgorithm A84.81%84.81%84.81%84.81%82.22%
Algorithm B84.44%84.81%82.59%78.52%78.52%
F1Algorithm A81.91%81.91%81.91%81.91%82.46%
Algorithm B82.74%83.11%82.64%80.17%80.34%
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Hu, J.; Cao, W.; Liang, P. A Novel Sequential Three-Way Decision Model for Medical Diagnosis. Symmetry 2022, 14, 1004. https://0-doi-org.brum.beds.ac.uk/10.3390/sym14051004

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Hu J, Cao W, Liang P. A Novel Sequential Three-Way Decision Model for Medical Diagnosis. Symmetry. 2022; 14(5):1004. https://0-doi-org.brum.beds.ac.uk/10.3390/sym14051004

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Hu, Junhua, Wanying Cao, and Pei Liang. 2022. "A Novel Sequential Three-Way Decision Model for Medical Diagnosis" Symmetry 14, no. 5: 1004. https://0-doi-org.brum.beds.ac.uk/10.3390/sym14051004

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