Leveraging human-centered AI for clinical decision-making: a transparent, accurate rule extractor using non-dominated sorting genetic algorithm

Abstract

Background

The advent of health technologies associated with artificial intelligence (AI) is deemed a transformative shift in the delivery of medical care within our lifetime. Nevertheless, there is a communication gap between intelligent models and clinical experts. Transitioning towards Human-Centered AI can serve as a means to bridge this gap.

Methods

This study introduces a human-centered rule extraction model based on the Non-Dominated Sorting Genetic Algorithm (NSGA-II), designed to enhance the interpretability and clinical utility of diagnostic tools in healthcare. This model autonomously generates diagnostic rules, adjusts threshold values for variables, and involves clinical experts in evaluating the rules, thereby ensuring the relevance and applicability of extracted rules in real-world settings.

Results

Experiments on the WBC, WDBC, and Pima datasets showed that the proposed model outperformed state-of-the-art rule extraction methods in the literature in terms of predictive value accuracy (PVA) and support. On the subset of data covered by the extracted rules, it achieved accuracy comparable to traditional black-box methods without sacrificing interpretability. The extracted rules were clinically evaluated by 13 domain physicians, with all approved rules achieving a content validity index (CVI) of at least 0.85. Additionally, the model provides multiple high-performance alternative diagnostic rules per class, giving clinicians practical flexibility.

Conclusions

Our approach emphasizes the importance of multidisciplinary collaboration between AI specialists and healthcare professionals, aiming to build trust in AI-driven diagnostic systems through transparency and clinical validation.

Supplementary Information

The online version contains supplementary material available at https://doi.org/10.1186/s12911-026-03674-z.

Background

The widespread adoption of artificial intelligence (AI) and machine learning (ML) techniques in software solutions has emerged as the central focus of modern technologies [1, 2]. The remarkable successes achieved in various fields have rendered these technologies integral to people’s lives [3]. Consequently, numerous studies are currently underway to explore the future of AI/ML and their impact on the evolving world and humanity [4].

Simultaneously, with the continuous expansion of diverse medical information resources and the resultant need for processing vast volumes of data, the utilization of these technologies in addressing healthcare challenges has garnered significant attention. Governments and medical societies worldwide are convinced that the advent of AI-related health technologies will represent “the most radical change in the way healthcare is delivered in our lifetime” [5–7]. The goals of employing these technologies in the healthcare field encompass improving diagnostic accuracy, enhancing efficiency, reducing time and costs, facilitating remote management, and ensuring continuity of care [8–10].

However, the translation of these technologies from research to implementation in a clinical setting has been accompanied by obstacles and requires addressing technical, ethical, and regulatory challenges [5, 11]. Concerns regarding the trustworthiness of the outputs of ML/AI-based diagnostic systems have posed a significant challenge to their acceptance in practical applications [12–18]. The challenges encompass validating the performance of AI/ML tools in clinical applications, evaluating real-world clinical scenarios, ensuring result reproducibility, addressing transparency, and overcoming interpretability issues (commonly referred to as black-box issues) [19–22]. Additionally, considerations of shared responsibility and the active involvement of healthcare professionals in the design and evaluation process of intelligent tools are among the critical challenges [23, 24]. Failure to address these challenges leads to a lack of trust among healthcare professionals, ultimately impacting the widespread acceptance and eroding patient trust in these tools [12, 25].

As a result, gaining the trust of end users and achieving clinical acceptance of intelligent tools greatly depends on their high predictive accuracy, clinical utility, reliability, and a clear explanation of their decision-making process [26]. Moreover, according to the World Health Organization (WHO) Guidance for the Ethics and Governance of Artificial Intelligence for Health [27], maintaining human autonomy (meaning “humans should remain in full control of health-care systems and medical decisions”), transparency and understanding the way artificial intelligence technology arrives at a decision should be one of the main design principles of these systems. Therefore, the design of these tools is a multi-objective issue.

In recent years, despite the superior computational performance of black-box methods, especially boosting ones, rule extraction methods known as white-box methods have gained the attention of researchers, which are formulated as multi-objective optimization problems with the aim of providing transparent and interpretable rules [14, 20, 28–31]. Among these, evolutionary algorithms (EAs), particularly the multi-objective variants, have gained popularity for extracting precise and interpretable rules [8, 26, 32, 33]. These algorithms are utilized for various purposes, including rule selection, pruning in tree-based algorithms, feature selection, optimizing black-box algorithms such as artificial neural networks (ANNs), or a combination of these applications.

In [8, 32], the non-dominated sorting genetic algorithm (NSGA-II) and multi-objective evolutionary algorithm (MOEA) have been employed to select rules generated from random forests. In [34], a genetic algorithm (GA) was used for feature selection in combination with a temporal fuzzy ant miner decision tree (TFAMT) for the diagnosis of type 2 diabetes. Additionally, in [35, 36], GA was combined with ANNs for compressing rule sets and selecting optimal rule sets. Furthermore, in [37], GA was utilized for optimizing the weights of ANNs and feature weighting in the detection of cardiac arrhythmia. In other works, including [33, 38–40], models based on EAs have been proposed for rule selection in diagnostics or classification tasks.

Since multi-objective problems yield a set of solutions, the decision-maker must select a more suitable one. Thus, defining an appropriate selection criterion presents another challenge in the development of intelligent diagnostic tools [41]. The accuracy of the classification model is typically based on metrics related to the number of samples that are correctly classified. However, interpretability is measured based on the complexity of the model and lacks a mathematical criterion. Some sources define it as the degree to which a human can predict the model’s outcomes or understand the reasoning behind its decisions [20, 42]. Quantitative metrics commonly used for interpretability evaluation in the literature include the number of rules, the number of features used in each rule, and the coverage of a rule (meaning the percentage of samples that satisfy all rule conditions) [8, 21, 32, 33, 36].

Despite the definition of interpretability assessment criteria and efforts to gain the trust of experts in the outputs of ML/AI-based diagnostic models, the clinical evaluation of rules by domain experts has been overlooked. It is critical that AI development incorporates clinical insight when designing and implementing AI systems, especially in healthcare. However, upon examining 73 original articles in the field of EAs in clinical diagnosis (see Appendix A), it was observed that none of the presented rules had been evaluated by clinical experts. Furthermore, among 383 systematic reviews and 71 meta-analyses in the field of AI/ML-based diagnostic rule extraction, regardless of the algorithm used, none had evaluated the results of these studies and their extracted rules from a clinical perspective (see Appendices B and C). Thus, a gap persists between interpretability in practice and the goal of transparency, leading to a significant divide between intelligent models and clinical experts as well [20, 43].

To address this issue, it is crucial to place the user at the center of the development of AI/ML-based tools, considering their needs and values rather than focusing solely on technical aspects. This approach aligns with the concept of human-centered AI (HCAI), which aims to actively engage users in decision-making and designing intelligent models [44, 45]. Therefore, this study aims to realize the HCAI approach in developing clinical diagnostic applications based on AI/ML. Moreover, considering the importance of timely diagnosis and ensuring its accuracy, this study presents a transparent and precise diagnostic model within the HCAI framework, capable of providing multiple distinct classification rules to enhance functional validation. This model is designed to be assessable by human users without the need for a computer.

More precisely, the proposed model is a rule extraction model based on the NSGA-II algorithm. It autonomously handles the process of rule generation, determines the threshold boundaries for variables, and conducts classification by introducing a novel chromosomal structure. The evaluation of clinical priorities and rules is carried out by a team of healthcare professionals. Therefore, the composition of the research team adheres to the principle of multidisciplinary development of practical clinical tools, involving both clinical experts and AI specialists. To the best of our knowledge, this study marks the first implementation of a hybrid approach that combines qualitative insights from domain experts with quantitative criteria used by developers of AI/ML-based rule extraction models. This represents the most significant innovation of the study.

The remainder of the paper is structured as follows. In the next section, the proposed rule extractor model based on HCAI is described in detail, comprising three major components: the problem space, intelligent space, and interaction space. In the results section, we first specify the context of use and provide information about the datasets used to assess the model’s efficacy. Subsequently, the performance of the proposed model on the datasets will be precisely reported. Finally, the paper will conclude with an analytical comparison of the proposed model with other rule extraction methods in the literature.

Methods

In this section, the details of the proposed method based on the HCAI approach are presented. The rationale for designing the suggested clinical diagnostic model based on HCAI is to protect human autonomy and the high sensitivity of clinical decisions, which often involve matters of life and death [12]. Therefore, as clinical experts bear responsibility for the final decision-making, the adoption of ML/AI-based tools into clinical practice is highly dependent on reassuring clinicians that these tools are reliable, safe, and trustworthy.

Achieving this goal requires leveraging the cognitive abilities of the user in solving the problem under consideration and actively involving them in decision-making, as introduced in the concept of “human-in-the-loop” by Balasubramanian in [46]. Consequently, in the development of intelligent tools in clinical applications, high levels of human control must coexist with high levels of computer automation. These two aspects serve as key indicators in the design of technologies aimed at enhancing human performance and gaining broader acceptance, as emphasized in the HCAI approach [47].

Furthermore, based on these principles, the proposed classification model has been developed with the aim of creating a diagnostic tool suitable for use in real clinical environments. Consequently, the proposed model adheres to ISO 9241–110 and follows the four main activities in the design of human-centered interactive systems: understanding and specifying the context of use, specifying the user requirements, producing design solutions, and evaluating the design [48]. Figure 1 illustrates the hierarchical relationship of the key components of the proposed model.

Fig. 1.

Hierarchical relationship between the components of the proposed HCAI model

Specifying the context of use, and user requirements (problem space)

Because of the critical role of early disease diagnosis in selecting effective treatments [49], reducing mortality [50], and improving the quality of patients’ lives [14], alongside the emergence of new diseases and the time-consuming process of knowledge acquisition from experts [21, 51], the development of a clinical diagnostic model is the focus of this study. To comprehensively examine user requirements and understand the desired criteria in clinical settings, the formation of a multidisciplinary team is a fundamental principle in the development of clinical practice tools, as emphasized by the World Health Organization (WHO) [52], and the Institute of Medicine (IOM) [53]. Therefore, our research team is multidisciplinary, consisting of both AI/ML specialists and clinical experts.

AI/ML-based model: computational performance (intelligent space)

Model construction

Recognizing the necessity for clinical experts to ensure the accuracy of the decision-making model through an understanding of its process, the proposed diagnostic model is a rule-extraction model. Generally, white-box methods are more transparent but exhibit lower diagnostic accuracy compared to the complex black-box ones [54]. The proposed approach aims to bridge the gap between the transparency capability of white-box algorithms and the efficiency of black-box models.

Considering that the problem of selecting k rules from n rules is an NP-complete problem, finding an optimal solution is not feasible. However, it is possible to reach a sufficiently good solution using optimization algorithms. Therefore, in this study, EAs are considered. Among these, given the popularity and satisfactory performance of the NSGA-II algorithm mentioned in the introduction section, the proposed model is based on this evolutionary algorithm. The distinguishing feature of the models presented based on this algorithm lies in their chromosomal structure and fitness function.

In such methods, each chromosome encodes a set of candidate solutions in a way that, alongside other chromosomes in the population, covers the entire search space. In our study, the search space consists of a set of diagnostic rules related to disease classes. Since a smaller search space increases the chances of finding an optimal solution, we employ the idea of divide-and-conquer by breaking down the search space into the sub-classes related to a disease under consideration, and each diagnostic class is solved separately. Consequently, by solving smaller sub-problems and then combining them, the chances of overcoming the challenge of finding an optimal solution, which intensifies exponentially with the growth of the search space, increase. Therefore, the proposed model is executed separately for each diagnostic class. As a result, the proposed chromosomal structure encodes the set of diagnostic rules related to one particular diagnostic class. For ease of implementing mutation and recombination operators, all gene values in the proposed structure are real numbers between 0 and 1. Figure 2 illustrates the structure of the proposed chromosome.

Fig. 2.

The proposed chromosome which codes the rule base of a particular diagnostic class

The size of the proposed chromosome is calculated as follows:

1

Where chroSize determines the number of genes in a chromosome, i.e., the chromosome size, and Nc is the number of conditions (variables) constituting the antecedent of a rule; in other words, it indicates the length of the rule. Nc remains constant for all rules in different generations of each GA run, but it can vary in different runs. The maximum and minimum values of Nc are adjustable by the development team. To specify each condition, four genes are used (as described in the next paragraph). Nr represents the maximum number of rules encoded in each chromosome, which is equal to the size of the rule base. The value of this variable is determined by the development team based on interpretability considerations. The additional 1 in Eq. 1 is due to the first gene of the chromosome. The first gene, X⁰, is a special gene that represents the number of rules encoded in the chromosome and is randomly determined subject to the value of Nr. This gene specifies the value of n.

The set of encoded rules in the proposed chromosome is in disjunctive normal form, as follows:

2

Where Ruleⁱ, i = 1, 2, …, n, is a rule in conjunctive form of variables/conditions for classifying a diagnostic class, indicated by the “Diagnostic-class label”. In Ruleⁱ, the variables involved in the formation of the antecedent part of the rule are unique, and there are no duplicate variables within the antecedent part of a rule. Therefore, the antecedent part of Ruleⁱ, which is a conjunctive combination of dataset variables, is specified as follows:

3

Each condition j in the antecedent of Rule i, denoted as Xⁱ_j, is specified by a quadruplet as follow:

4

The first component specifies the variable number. Equation 5 is used to map the real value of this gene to the variable number of the dataset:

5

Where Nv represents the number of independent variables in the dataset and does not include the dependent one. Therefore, varIndex will always be a numerical value between 1 and the number of independent variables in the dataset. Figure 3 illustrates how to map this normal, real-value component to a variable index from a hypothetical dataset with 31 independent variables numbered from 1 to 31.

Fig. 3.

Illustrating the decoding of the chromosome genes to the variable index contributed to the antecedent of the rule, which is part of the hypothetical data set containing 31 variables

The second and third components, respectively, determine the lower threshold (α) and the upper threshold (β) of the variable. These two components create five states: low, low or normal, normal, normal or high, and high. The fourth component specifies under which state(s) the condition becomes active. In other words, it determines which states of the variable are acceptable. The mapping of the real value of this gene to the five mentioned states is based on the following formula:

6

Where MNS represents the maximum number of considered states for each variable, which is set to five. Figure 4 illustrates the relationship between the mentioned components and the states.

Fig. 4.

Decoding the genes of the condition part of the chromosome: each variable can have 5 states according to its relative location to the threshold values

To better explain the decoding of genes in the first part of a condition in a rule, let us consider a hypothetical dataset containing 31 independent variables normalized between 0 and 1. Suppose the seventh variable in the dataset represents “blood pressure”. Figure 5 illustrates t the process of extracting a condition encoded in the antecedent part of a hypothetical rule. It should be noted that there are no duplicate rules among the n possible ones encoded in a chromosome.

Fig. 5.

Illustration of the decoding of a condition in an antecedent of a hypothetical rule

In addition to encoding a diagnostic rule base in a chromosome, the desired criteria for evaluating its performance should be embedded in the form of a fitness function. The criteria used are based on two metrics: support and positive predictive value (PPV). Support of a rule is defined as the percentage of examples that satisfy all the conditions of the rule [32]. In other words, it represents the frequency of examples covered by the rule that were correctly classified [55]. Since in this study, each diagnostic class (referred to as the target class) is examined separately, and each model run aims to detect only one class, the support based on examples of the class is used, which is defined as “the degree of coverage that the rule offers to examples of that class” [55]. This metric, referred to as support for simplicity, is calculated as follows:

7

Where NTR is the number of samples of the target class, and NCTRⁱ represents the number of samples of the target class that satisfy the conditions of rule i and are correctly covered by rule i. This metric is used to filter out less important rules; any rule whose support is lower than the threshold set by the development team will be considered as an unimportant rule and will be removed.

Since it is desirable for each diagnostic rule to cover the maximum number of samples of the target class and the minimum number of samples from other classes, two accuracy metrics, Predictive Value Accuracy (PVA) and Predictive Error (PE), are customized based on PPV. These two metrics are calculated as follows:

8

9

Where NCR is the number of records covered by the rule base, and NCORⁱ is the number of samples belonging to other diagnostic classes that satisfy the conditions of rule i but are mistakenly covered by rule i. Figure 6 illustrates how these metrics are calculated.

Fig. 6.

Illustration of the calculation of fitness function criteria

Accordingly, the objective functions used in forming the Pareto front in this study will be PVA and 1-PE; the proposed model seeks to optimize these two parameters. It should be noted that each model run focuses solely on one diagnostic class, i.e., the target class, and the other diagnostic classes are collectively considered as other diagnostic classes.

Model refinement

After training the model, there is a need to select an optimal subset of the extracted rules for external evaluation, as multiple solutions are provided as the model output, which is a characteristic of EAs. Therefore, a post-processing process consisting of five sequential steps is designed. In the first step, the diagnostic rules whose PVA on the test set is γ percent higher than PVA on the training set are discarded. Then, a denormalization process is applied to map the values of variables in each rule from the [0,1] interval to the interval of the original values of the variables in the dataset.

In the second step, to increase the readability of the extracted rules, a process of reducing the digits of the variables in the antecedent of the rule is carried out by removing the decimal points. This process is iterative, and in each iteration, the last decimal digit is removed as long as the rule’s performance on the training and test datasets is unchanged. This step is called the decimal truncation step and is applied to all extracted rules.

In the third step, the rules of all diagnostic classes are merged to examine the generalizability of the model. As a result, the output of the third step is a set of rule bases, each containing one rule for each diagnostic class. In binary classification problems, each rule base includes two rules, one for the positive class and one for the negative class. In the fourth step, appropriate thresholds are set for the accuracy, support, and rule length of the rules participating in the construction of the rule bases. Only those rules that meet the threshold requirements will proceed to the next step.

In the fifth step, the rule bases are sorted using the non-dominated sorting method. As a result, the rule bases at the Pareto front are identified. These rule bases are compared with some commonly used black-box methods to evaluate the computational power of the model. Additionally, individual rules of each diagnostic class participating in the rule bases lying on the Pareto front are extracted. These rules, along with the other individual rules that had the highest PVA and highest support, are entered into external evaluation and compared with the white-box methods in the literature. The criteria for selecting the research works in the literature are to report the rules or provide performance criteria for each individual rule.

Humans confirm, reject or label output: clinical utility (interaction space)

Since attention to meaningful human control and inclusion of human-centered aspects in requirements engineering (RE) when building AI-based software can lead to more responsible, unbiased, and inclusive intelligent systems [44, 45], in this study, for the first time to the best of our knowledge, the team of domain experts (healthcare professionals) actively participates in decision-making and evaluating the results of the proposed model.

At first, the maximum number of conditions of each rule (Nc) and the maximum number of rules in the rule bases (Nr) coded in the proposed chromosome structure are determined. Then, in the post-processing stage, the threshold value for the length of the diagnostic rules is determined by experts because of its effect on the interpretability of the rules. Finally, the selected individual rule sets are evaluated by clinical experts. Only those rules approved by the clinical team will be reported as the output of the proposed model.

For the clinical evaluation of the rules, the Waltz and Basili method [56] is used. In this method, domain experts assess the relevance of each rule based on a Likert scale with 4 options: “Not relevant”, “Somewhat relevant”, “Relevant”, and “Highly relevant”. Then, the content validity index (CVI) is calculated based on the following formula:

10

CVI measures the proportion of experts who rated the item as relevant out of the total number of experts. It provides a quantitative measure of the agreement among experts regarding the relevance of the rules. A higher CVI indicates stronger agreement among experts on the relevance of the rules. If the value obtained is less than 0.7, the rule is rejected. If it falls between 0.7 and 0.79, a review is required, and if it is greater than or equal to 0.79, it is considered acceptable.

The pseudo-code shown in Fig. 7 illustrates the execution process of model design through evaluation.

Fig. 7.

The pseudocode of the proposed HCAI model

Statistical significance analysis

In addition to computational and clinical evaluation, Phi correlation coefficient (φ) [57] (also known as Mathew’s correlation coefficient (MCC) [58]) has been used to validate the significance of the results and also to measure the prediction quality of the proposed model from a statistical point of view. The φ (or MCC) can be calculated directly from the confusion matrix using the formula:

11

where TP is the number of true positives, TN is the number of true negatives, FP is the number of false positives, and FN is the number of false negatives. Phi correlation coefficient ranges from −1 to 1: 0 is no relationship. 1 is a perfect positive relationship and −1 is a perfect negative relationship. The interpretation of this score is as follows: If Phi is 0.7 or higher (−0.7 or lower), it indicates a very strong positive (negative) relationship. If it is 0.4 to 0.69 (−0.4 to −0.69), it indicates a strong positive (negative) relationship. If it is 0.3 to 0.39 (−0.3 to −0.39), it indicates a moderate positive (negative) relationship. If it is 0.2 to 0.29 (−0.2 to −0.29), it indicates a weak (negative) positive relationship. If it is 0.1 to 0.19 (−0.1 to −0.19), it means that there is no relationship or it is negligible. And if it is zero, it indicates a lack of relationship between the target and the predicted values [59].

Results

Case study: breast cancer and diabetes

To implement and evaluate the proposed model, the domain of clinical rule extraction was chosen for two diseases: breast cancer and diabetes. The reason for selecting these two diseases is their high prevalence and the crucial role of early diagnosis in their timely treatment (Also, from the perspective of implementation, in the field of extracting diagnostic rules, there are more related works for these two diseases than others in the literature1). In 2020, breast cancer was reported as the most common cancer worldwide, surpassing lung cancer, with 2.3 million affected women [60]. It is one of the leading causes of death among women globally [61]. However, early detection can contribute to reducing the mortality rate associated with this disease [62].

Additionally, diabetes, as a metabolic disease, poses a serious threat to human health, often leading to complications such as kidney disease and retinopathy. According to WHO, approximately 422 million people worldwide have diabetes, and it directly causes 1.5 million deaths annually [63]. The International Diabetes Federation (IDF) Diabetes Atlas reports that one of the warning signs of diabetes is that approximately half of the diabetes cases, mainly type 2, remain undiagnosed [64]. Although diabetes is not currently completely curable, timely interventions can prevent serious consequences. By predicting the likelihood of complications and providing effective preventive treatments, the risk of death can be significantly reduced [65].

Implementation space

The performance of the proposed model was evaluated using three benchmark datasets: Breast Cancer Wisconsin dataset2 (WDBC), Breast Cancer Wisconsin original dataset3 (WBC), and Pima Diabetes dataset. These datasets are among the most commonly used datasets in the research domain. All three datasets are binary classification problems. For WDBC and WBC, malignant samples were considered as the positive class (label 1), and benign samples were considered as the negative class (label 0). In the Pima dataset, samples with diabetes were labeled as the positive class, and those without diabetes were labeled as the negative class. In all three datasets, 70% of the samples were used for training, and the remaining 30% were used for testing.

During the model’s training phase, in collaboration with the design and evaluation team, the following parameters were set: the maximum number of rules in each chromosome (Nr) and the maximum number of conditions in the antecedent of each rule (Nc) were set to 1 and 5, respectively. The selection, crossover, and mutation operators were set to tournament selection, single-point crossover, and inversion, respectively. The population size was set to 100 individuals, with a crossover rate of 0.8 and a mutation rate of 0.3. A rate of the number of genes affected by mutation in a chromosome was set to 0.4. Based on the fitness value, those chromosomes with PVA lower than 70% were considered invalid and removed from the population. Additionally, the minimum support threshold for the rules encoded in each chromosome was set to 70% for the WDBC and WBC datasets and 50% for the Pima dataset. The inclusion thresholds (PVA > 70% and support ≥ 50%) were established via systematic consensus by the multidisciplinary team and 13 external physician evaluators, who determined these values as the minimum acceptable levels for clinical trustworthiness. Rules with low PVA can be misleading, and rules with low support may be rare or spurious. Rules falling below these thresholds can be retained for combination or secondary evaluation. However, in this implementation the chromosomes that did not meet the support thresholds were considered invalid and removed.

The termination criterion for the algorithm was either reaching 300 generations or no improvement in the fitness of the best chromosome for 90 consecutive generations. In step 1 of the post-processing, the γ value was set to 0.15%. As a result, those diagnostic rules that their test PVA outperforms training PVA by 0.15 were discarded. The remaining rules, based on the accuracy and interpretability thresholds determined by the development team, were used for clinical and computational evaluations.

To assess the interpretability and performance of the extracted rules, similar rule extraction methods in the literature were applied to the datasets. The results were then evaluated based on support and PVA metrics. Additionally, to evaluate the computational power of the model compared to the black-box methods, a set of rule bases was created by combining two rules: one for the positive class and another for the negative one. If a sample was covered by both rules, it was assigned to the diagnostic class with the higher PVA.

Nine common black-box algorithms, including k-nearest neighbors (k-NN), Support Vector Machines (SVM) with linear, radial basis function (RBF), and polynomial kernels, Bayesian, Linear Discriminant Analysis (LDA), random forest, multi-layer perceptron (MLP) with 10 hidden units, and deep neural network (DNN) with 100 hidden units were run on the datasets. Since the strength of each proposed diagnostic rule is assessed based on the covered data, the covered data by the proposed rule base was used for both training and testing the black-box methods. Each method was executed 25 times using 10-fold cross-validation, and the average results were used for comparison with the proposed model. Two rule bases were compared with the black-box methods: the rule base with the highest coverage but the lowest accuracy, and the rule base with the highest accuracy and the lowest coverage. It is important to restate that all reported performance metrics in the following subsections—for both the proposed model and the black-box methods—are based exclusively on the subset of samples covered by the extracted rule bases, not on the entire dataset.

The results of running the model on each of the mentioned datasets are presented in detail below. The proposed model was implemented in MATLAB, and the experiments were conducted on a platform with an Intel® Core™ i7-8550 U CPU @ 1.80 GHz 1.99 GHz CPU and 12.0 GB of RAM running a 64-bit MS Windows 10 operating system. It is worth noting that we neither performed fine-tuning of NSGA-II parameters nor investigated adaptive threshold settings to optimize rule extraction and support dynamic tuning based on ensemble features.

Performance on the WBC dataset

The original WBC dataset [66] contains 699 records and 11 features, including a patient identification code, which is omitted in this study. Out of these records, 458 samples belong to the benign class (66%), and 241 samples belong to the malignant tumor class (34%). In this dataset, 16 records in the column related to the “Bare Nuclei” feature have missing values, which have been removed in this study. After removing the records with missing values, the dataset contains 683 samples, including 239 malignant and 444 benign samples. Table 1 presents information about the dataset features.

Table 1.

The feature description of the WBC dataset [67]

No	Attribute name	Values	Description
1)	Clump Thickness	1–10	Measures the relative thickness of the cell clump
2)	Uniformity of Cell Size	1–10	Consistency of cell size across the sample
3)	University of Cell Shape	1–10	Consistency of cell shape across the sample
4)	Marginal Adhesion	1–10	Degree of cell-to-cell adhesion at the margins
5)	Single Epithelial Cell Size	1–10	Size of individual epithelial cells
6)	Bare Nucleoli	1–10	Presence of nuclei not surrounded by cytoplasm
7)	Bland Chromatin	1–10	Texture and appearance of the chromatin in the nucleus
8)	Normal Nucleoli	1–10	Presence and appearance of nucleoli
9)	Mitoses	1–10	Degree of mitotic activity (cell division)
10)	Class	2, 4	Diagnostic outcome: Malignant: 4[4] or Benign: 2[2]

The best diagnostic rules for the malignant and benign classes obtained from the model’s execution on the training dataset were 34 and 52 rules, respectively. After removing the diagnostic rules that had higher PVA on the test set compared to the training set and de-normalizing the remaining rules (step 1 of the post-processing), applying the decimal truncation step (step 2 of the post-processing), and combining the remaining rules (step 3 of the post-processing), 1380 rule bases were obtained, each of which includes one positive and one negative rule.

The rule bases were applied to all 683 samples in the dataset. Then, the accuracy threshold and the lengths of each rule were set to 98 and 3, respectively (step 4 of the post-processing). Out of the 1380 rule bases, 44 rule bases satisfied the mentioned criteria. Subsequently, based on the accuracy and support of rule bases, the Pareto front was plotted (step 5 of the post-processing). Four rule bases were lied on the Pareto front. Figure 8 illustrates the proposed rule bases on the WBC as well as the Pareto front. Based on the results presented in Fig. 8, on average, the proposed model achieved a sensitivity of 98.08%, specificity of 98.79%, accuracy of 98.61%, PPV of 97.49%, negative predictive value (NPV) of 99.18%, and F-measure of 97.77% on the WBC dataset.

Fig. 8.

Providing insights into the trade-off between support and accuracy of the rule bases on the covered samples of the WBC dataset. a) non-dominated sorting of rule bases. b) The Non-dominated rule bases, referred as the pareto front

After finding the Pareto front, the individual diagnostic rules that form the rule bases of the Pareto front, the individual diagnostic rules with the highest support, and those with the highest PVA were extracted for clinical evaluation. After removing duplicate rules, 2 diagnostic rules for the positive class (i.e., for identifying malignant tumors) and 4 diagnostic rules for the negative class (i.e., for identifying benign tumors) were obtained. The information about these unique diagnostic rules and the clinical evaluation results based on the CVI criterion is provided in Tables 2 and 3, respectively.

Table 2.

The proposed diagnostic rules to classify benign and malignant tumors on the WBC dataset

Rule Code	Diagnostic rule	Training Set (Malignant: 168, Benign: 312)			Test Set (Malignant: 71, Benign: 132)
		Covered	Support	PVA	Covered	Support	PVA
O_PPV_1	If (2 ≤ Bare Nuclei And 3 ≤ Uniformity of Cell Shape) THEN Sick	167	92.26	92.81	63	84.51	95.24
O_PPV_2	if (5 ≤ Uniformity of Cell Size And 0.9 ≤ Bare Nuclei) THEN Malignant	127	73.81	97.64	48	67.61	100
O_NPV_1	if (Normal Nucleoli ≤ 3 And Uniformity of Cell Size ≤ 3 And Bare Nuclei ≤ 2) THEN Benign	286	91.67	100.00	59	79.17	96.61
O_NPV_2	if (Single Epithelial Cell Size ≤ 3 And Bare Nuclei ≤ 2) THEN Benign	287	91.99	100.00	120	87.59	100
O_NPV_3	if (Normal Nucleoli ≤ 3 And Uniformity of Cell Size ≤ 3 And Bare Nuclei ≤ 4) THEN Benign	297	94.55	99.33	116	87.88	100
O_NPV_4	If (Uniformity of Cell Size ≤ 3 And Normal Nucleoli ≤ 3 And Bare Nuclei ≤ 8) THEN not-Sick	301	95.51	99.00	115	85.61	98.26

Table 3.

The result of clinical evaluation of the proposed diagnostic rules on the WBC by 13 domain experts

Rule Code	WBC Dataset (Malignant: 239, Benign: 444)			# Opt. 1 or 2 (Out of 13)	# Opt. 3 or 4 (Out of 13)	CVI	Confirmed or reject
	Covered	Support	PVA
O_PPV_1	230	89.96	93.48	0	13	1	confirmed
O_PPV_2	175	71.97	98.29	2	11	0.85	confirmed
O_NPV_1	402	90.54	100.00	2	11	0.85	confirmed
O_NPV_2	402	90.09	99.50	1	12	0.92	confirmed
O_NPV_3	423	94.14	98.82	1	12	0.92	confirmed
O_NPV_4	432	95.72	98.38	0	13	1.00	confirmed

After obtaining validation from domain experts for the individual diagnostic rules, for the purpose of evaluating the computational performance of the proposed model, these rules were compared with the rules presented in the literature. Despite similar criteria being used to report results in the proposed model compared to [67], due to the utilization of the entire dataset with 699 samples, the single diagnostic rules presented in [8] were re-executed on the WBC dataset after removing missing data (683 samples). Furthermore, considering the differences in result reporting methods in [33, 38–40, 42, 68–70], the presented rules in these eight studies were re-evaluated on the WBC database, and their results were reported. For these mentioned studies, the results of combining all their diagnostic rules for one class were compared with the single diagnostic rule of the proposed model.

It is also worth noting that in the works [33] and [39], two and three diagnostic rules for detecting the positive class (malignant tumors) were reported, respectively, and no diagnostic rules were provided for the benign class. Therefore, these two methods are only included in the comparison in Fig. 9a. In [70], after training a neural network on the WBC dataset, three sets of diagnostic rules were presented by pruning the network; We used the rule base consisting of 3 if-else-if rules, which had the highest accuracy, for comparison (rule set 3 from section 4.3 in [70]). In both diagnostic classes, the proposed diagnostic rules outperformed the white-box methods mentioned in the previous paragraph, as shown in Fig. 9.

Fig. 9.

Comparison of the performance of the proposed diagnostic rules with the rules reported in the literature on the WBC dataset

Ultimately, the goal was to approach the performance of the proposed white-box model to the performance of the black-box models. After verifying all individual rules by healthcare professionals, for evaluating the computational power of the proposed model, two rule bases from the five proposed ones in Fig. 8 (rule bases 1 and 4) were compared with nine black-box methods. Tables 4 and 5 show the performance comparison results of rule base 1, covering 577 samples (84.48% of the data) with an accuracy of 99.48 (the highest accuracy), and rule base 4, covering 627 samples (91.80% of the data, the highest coverage) with an accuracy of 97.93 with the black-box methods. In each column of the table, the darkest red indicates that the performance of the black-box method, based on that metric, is more than 3 units lower than the proposed method’s performance. White color indicates that the performance of the black-box method and the proposed method is almost the same, with less than 1 unit difference. The darkest blue indicates that the performance of the black-box method is more than 3 units higher than the proposed method.

Table 4.

Comparison of the performance of the proposed rule base 1 with the most common black-box methods on the covered samples of the WBC dataset

Table 5.

Comparison of the performance of the proposed rule base 4 with the most common black-box methods on the covered samples of the WBC dataset

Performance on the WDBC dataset

The WDBC dataset contains information on 569 breast cancer tumors, comprising 357 (63%) benign and 212 (37%) malignant tumors. Each tumor is characterized by 32 features, with the first feature denoting patient information and the last feature indicating the tumor type (benign or malignant). The patient characteristic feature has been omitted in this study. The remaining features are derived from digital images obtained through fine needle aspiration (FNA) from breast masses. These features articulate the attributes of the cell nuclei present in the images. For each sample, 10 different features describing the size, shape, and texture of the tumor nuclei have been extracted. Subsequently, for each feature, the standard error, mean, and maximum (or worst) value were computed across individual cell nuclei. All these features are numerical, and larger values suggest a higher likelihood of malignancy in the tumor. Table 6 presents information about the dataset features.

Table 6.

The characteristics of the attributes of the WDBC dataset

Feature name	Explanation
Radius	Average distances from center to points on the perimeter
Texture	Standard deviation of gray-scale values
Perimeter	Total distance between consecutive snake points
Area	The number of pixels inside the snake and half of the pixels in its perimeter
Compactness	Perimeter^2/area-1.0
Smoothness	Local changes along radii
Concavity	The intensity of the concave sections of the contour
Concave points	The number of concave points on the contour
Symmetry	Cell nucleus symmetry
Fractal_dimension	“Coastline approximation” − 1

The best diagnostic rules for classifying malignant and benign tumors, obtained by running the model on the training dataset, resulted in 91 and 42 rules, respectively. After removing the diagnostic rules that had higher PVA on the test set compared to the training set and de-normalizing the remaining rules, applying the decimal truncation step, and combining the remaining rules, 3256 rule bases were obtained each of which includes one positive and one negative rule.

The rule bases were applied to all 569 samples in the dataset. Subsequently, the accuracy threshold and the length of each rule were set to 98 and 3, respectively. Out of the 3256 rule bases, 41 rule bases satisfied the specified criteria. Following this, the Pareto front was plotted based on the accuracy and support of rule bases, with eight rule bases lying on the Pareto front. Figure 10 illustrates the proposed rule bases on the WDBC as well as the Pareto front. According to the results presented in Fig. 10, on average, the proposed model achieved a sensitivity of 97.39%, specificity of 99.15%, accuracy of 98.54%, Positive Predictive Value (PPV) of 98.55%, Negative Predictive Value (NPV) of 98.59%, and an F-measure of 97.96% on the WDBC dataset.

Fig. 10.

Providing insights into the trade-off between support and accuracy of the rule bases on the covered samples of the WDBC dataset. a) Non-dominated sorting of rule bases. b) The non-dominated rule bases, referred as the pareto front

After identifying the Pareto front, individual diagnostic rules constituting the rule bases on the Pareto front, as well as unique rules with the highest support and the highest PVA, were extracted for clinical evaluation. Following the removal of duplicate rules, three diagnostic rules for the positive class (identifying malignant tumors) and four diagnostic rules for the negative class (identifying benign tumors) were obtained. Information about these unique diagnostic rules and the clinical evaluation results based on the CVI criterion is provided in Tables 7 and 8, respectively.

Table 7.

The proposed diagnostic rules to classify benign and malignant tumors on the WDBC dataset

Rule Code	Diagnostic rule	Training Set (Malignant: 148, Benign: 250)			Test Set (Malignant: 64, Benign: 107)
		Covered	Support	PVA	Covered	Support	PVA
D_PPV_1	If (21 ≤ texture_worst And 0.05 ≤ concavepoints_mean And 791 ≤ area_worst) THEN Malignant	118	79.73	100.00	48	75	100
D_PPV_2	If (0.05 ≤ concavepoints_mean And 111 ≤ perimeter_worst) THEN Malignant	123	81.08	97.56	51	78.13	98.04
D_PPV_3	If (0.22 ≤ concavity_worst And 101.6 ≤ perimeter_worst And 14.2 ≤ texture_mean) THEN Malignant	152	93.24	90.79	62	90.63	93.55
D_NPV_1	If (area_se ≤ 93 And radius_worst ≤ 16.2 And concavepoints_worst ≤ 0.135) THEN Benign	214	84.80	99.07	99	91.59	98.99
D_NPV_2	If (area_se ≤ 93 And radius_worst ≤ 16.6 And concavepoints_worst ≤ 0.135) THEN Benign	227	89.60	98.68	100	92.52	99.00
D_NPV_3	If (area_se ≤ 93 And perimeter_worst ≤ 113 And concavepoints_worst ≤ 0.135) THEN Benign	230	90.80	98.70	103	92.52	96.12
D_NPV_4	If (area_se ≤ 93 And perimeter_mean ≤ 98.3 And concavepoints_worst ≤ 0.135) THEN Benign	233	91.60	98.28	105	93.46	95.24

Table 8.

The result of clinical evaluation of proposed diagnostic rules on the WDBC by 13 domain experts

Rule Code	WDBC Dataset (Malignant: 212, Benign: 357)			# Opt. 1 or 2 (Out of 13)	# Opt. 3 or 4 (Out of 13)	CVI	Confirmed or reject
	Covered	Support	PVA
D_PPV_1	166	78.30	100.00	1	12	0.92	confirmed
D_PPV_2	174	80.19	97.70	2	11	0.85	confirmed
D_PPV_3	214	92.45	91.59	2	11	0.85	confirmed
D_NPV_1	313	86.84	99.04	2	11	0.85	confirmed
D_NPV_2	327	90.48	98.78	2	11	0.85	confirmed
D_NPV_3	333	91.32	97.90	2	11	0.85	confirmed
D_NPV_4	338	92.16	97.34	2	11	0.85	confirmed

After the individual diagnostic rules were validated by domain experts, a comparison was made with rules presented in studies [8] and [71] to assess the computational performance of the proposed model. Due to similar evaluation criteria used in the proposed method and [8], the results reported in the mentioned research were directly incorporated into the analysis. However, considering the difference in reporting the results of diagnostic rules in [71], which was specifically written for the malignant class, the proposed rule in [71] was re-run on the WDBC database, and the results were entered into the comparison. In both diagnostic classes, the proposed rule bases outperformed other white-box methods, as illustrated in Fig. 11.

Fig. 11.

Comparison of the performance of the proposed diagnostic rules with the rules reported in the literature on the WDBC dataset

Considering the validation of all individual rules by healthcare professionals, to evaluate the computational power of the proposed model in comparison to the black-box methods, among the eight proposed rule bases, rule base 1 with a coverage of 479 samples (84.18% of the data) and an accuracy of 99.37% (the highest accuracy), and rule base 8 with coverage of 523 samples (91.92% of the data, the highest coverage) and an accuracy of 97.90% were selected from Fig. 10. Tables 9 and 10, respectively, display the performance comparison results of rule bases 1 and 8 with the black-box methods.

Table 9.

Comparison of the performance of the proposed rule base 1 with the most common black-box methods on the covered samples of the WDBC dataset

Table 10.

Comparison of the performance of the proposed rule base 8 with the most common black box methods on the covered samples of the WDBC dataset

Performance on the Pima dataset

The Pima dataset is one of the challenging and widely used datasets for evaluating diabetes diagnostic systems. It comprises 768 records, with 268 cases indicating diabetes and 500 individuals classified as not having diabetes. Each record includes 9 features, along with the corresponding class label. A notable challenge associated with this dataset is the presence of missing values and outliers. In this study, following methodologies employed in our prior works [72, 73], the KNN imputation method (KNNi) was utilized for handling missing values, and the K-means clustering algorithm was employed to eliminate data outliers during the preprocessing phase.

To clarify the rationale behind these preprocessing choices, the primary goal of preprocessing was to create a uniform platform for fair comparison across all algorithms. The KNNi and K-means algorithms were selected because they are well-established and reproducible. However, any other standard method for imputation or outlier detection could have been used, as long as all preprocessing steps were applied identically to all models on the same raw data. This ensures that the comparison of results among algorithms is fair and not affected by differences in preprocessing. Subsequently, after preprocessing, the dataset was reduced to 590 records, encompassing 212 individuals with diabetes and 378 individuals without diabetes (The preprocessed Pima dataset is provided as a supplementary file).

Table 11 provides a description of the variables based on descriptive statistics, including mean, standard deviation, and intervals, for the two groups of non-diabetic and diabetic individuals.

Table 11.

Pima diabetes dataset information after preprocessing [72]

No	Attribute name	Unit	Mean		[Min,Max]
			Not-Diabetic N = 378	Diabetes N = 212	Not-Diabetic N = 378		Diabetes N = 212
	Number of Pregnancies (Prg)	-	3.27 ± 2.99	4.94 ± 3.60	[0,13]	[0,15]
	PG Concentration (PG)	-	84.80 ± 28.10	125.76 ± 37.08	[23,193]	[44,199]
	Diastolic BP (BP)	mm Hg	57.88 ± 12.52	63.28 ± 12.62	[25,102]	[30,102]
	Tri Fold Thick (TrFT)	mm	21.79 ± 8.87	26.87 ± 7.42	[1,51]]	[8,53]
	Serum Ins (SrIns)	Mu U/ml	112.19 ± 63.55	181.01 ± 78.30	[1,479]	[22,575]
	BMI	kg/m2	17.24 ± 8.28	23.00 ± 8.07	[0,41]	[7,48]
	DP Function (DP)	-	0.4 ± 0.25	0.51 ± 0.31	[0.08,1.73]	[0.09,1.89]
	Age	-	30.67 ± 11.45	36.83 ± 10.91	[21,72]	[21,70]

The best pre-awareness diagnostic rules for the Diabetes and Not-Diabetic classes, obtained from running the model on the training set, were 2 and 36 rules, respectively. After removing diagnostic rules that had higher PVA on the test set compared to the training set and de-normalizing the remaining rules, applying the decimal truncation step, and combining the rules, 60 rule bases were obtained, each of which contains a positive diagnostic rule and a negative one.

The rule bases were applied to all 590 samples in the dataset. Then, the accuracy threshold and the length of rules were set to 80 and 5, respectively. Out of the 60 rule bases, 40 rule bases met the threshold requirements. Then, based on accuracy and support, the Pareto front was plotted for these eligible rule bases. Seven rule bases were included in the Pareto front. Figure 12 illustrates the proposed rule bases and the Pareto front on the Pima dataset. Considering the results presented in Fig. 12, on average, the proposed model achieved a sensitivity of 76.34%, specificity of 86.78%, accuracy of 83.35%, PPV of 74.33%, NPV of 88.15%, and an F-measure of 75.25% on the Pima dataset.

Fig. 12.

Providing insights into the trade-off between support and accuracy of the rule bases on the covered samples of the pima dataset. a) Non-dominated sorting of rule bases. b) The non-dominated rule bases, referred as the pareto front

After determining the Pareto front, the individual diagnostic rules that form the rule bases of the Pareto front, the individual diagnostic rules with the highest support, and also unique rules with the highest PVA were extracted for clinical evaluation. After removing duplicate rules, two diagnostic rules for the positive class (i.e., for identifying individuals with a high probability of having diabetes) and six diagnostic rules for the negative class (i.e., for identifying individuals with a high probability of being not-Diabetic) were obtained. The unique diagnostic rules and clinical evaluation results based on the CVI criterion are presented in Tables 12 and 13, respectively.

Table 12.

The proposed diagnostic rules to classify diabetes and not-diabetic on the Pima dataset

Rule Code	Diagnostic rule	Training Set (Diabetes: 148, not-diabetic: 265)			Test Set (Diabetes: 64, not-diabetic: 113)
		Covered	Support	PVA	Covered	Support	PVA
P_PPV_1	If (16 ≤ BMI And Age ≤ 53 And 0.31 ≤ DP Function And 129 ≤ Serum Ins)) THEN Diabetes	102	50.00	72.55	32	37.50	75.00
P_PPV_2	if (Pregnancies ≤ 12 And 120 ≤ PG Concentration) THEN Diabetes	110	52.70	70.91	46	53.13	73.91
P_NPV_1	If (Serum Ins ≤ 231 And Age ≤ 37 And Pregnancies ≤ 6 And PG Concentration ≤ 110 And Tri Fold Thick ≤ 28) THEN not-Diabetic	156	53.59	91.03	71	55.75	88.73
P_NPV_2	if (Serum Ins ≤ 110 And Age ≤ 66 And PG Concentration ≤ 110) THEN not-Diabetic	166	56.98	90.96	67	54.87	92.54
P_NPV_3	if (Serum Ins ≤ 149 And Age ≤ 37 And Pregnancies ≤ 6 And PG Concentration ≤ 110 And Diastolic BP ≤ 100) THEN not-Diabetic	177	60.00	89.83	77	61.95	90.91
P_NPV_4	if (Serum Ins ≤ 177 And Age ≤ 40 And Pregnancies ≤ 6 And PG Concentration ≤ 110 And 26 ≤ Diastolic BP)) THEN not-Diabetic	194	63.77	87.11	86	69.03	90.70
P_NPV_5	if (Serum Ins ≤ 126 And Pregnancies ≤ 14 And BMI ≤ 35 And PG Concentration ≤ 123)) THEN not-Diabetic	201	65.28	86.07	81	63.72	88.89
P_NPV_6	if (Serum Ins ≤ 126 And Pregnancies ≤ 14 And Age ≤ 55 And PG Concentration ≤ 129)) THEN not-Diabetic	205	66.04	85.37	81	63.72	88.89

Table 13.

The result of clinical evaluation of proposed diagnostic rules on the Pima by 13 domain experts

Rule Code	Pima Dataset (Diabetes: 212, not-Diabetic: 378)			# Opt. 1 or 2 (Out of 13)	# Opt. 3 or 4 (Out of 13)	CVI	Confirmed or reject
	Covered	Support	PVA
P_PPV_1	134	46.23	73.13	1	12	0.92	Confirmed
P_PPV_2	156	52.83	71.79	2	11	0.85	Confirmed
P_NPV_1	227	54.23	90.31	2	11	0.85	Confirmed
P_NPV_2	233	56.35	91.42	2	11	0.85	Confirmed
P_NPV_3	254	60.58	90.16	1	12	0.92	Confirmed
P_NPV_4	280	65.34	88.21	1	12	0.92	Confirmed
P_NPV_5	282	64.82	86.88	1	12	0.92	Confirmed
P_NPV_6	286	65.34	86.36	1	12	0.92	Confirmed

After verifying individual diagnostic rules by domain experts, to evaluate the computational performance of the proposed model, the validated, proposed rules were compared with the rules presented in [38] and [68]. Considering the preprocessing applied to this dataset, as well as the differences in how the results of diagnostic rules were reported in our study, the rules presented in these two works were re-executed on the preprocessed Pima dataset. The result of the union of diagnostic rules for the classes of diabetes and not-Diabetic provided in [38] and [68], compared with the single proposed rules, is reported in Fig. 13. Since the union of the two diagnostic rules for diabetes in [68] was unable to cover any of the 590 preprocessed Pima data, the result is not shown in Fig. 13. In both diagnostic classes, the proposed rule patterns are significantly superior to the rules presented in [38] and [68].

Fig. 13.

Comparison of the the performance of the proposed diagnostic rules with the rules reported in the literature on the PIMA dataset

Finally, the goal of approaching the performance of the proposed white-box model to the black-box models was investigated. Considering the validation of all individual rules by healthcare professionals, to assess the computational power of the proposed model, from among the 7 proposed rule bases, rule base 1 with coverage of 344 samples (58.31% of the data) with an accuracy of 86.34% (the highest accuracy), and rule base 7 with coverage of 439 samples (74.41% of the data, the highest coverage) with an accuracy of 81.55% were selected from Fig. 12. Tables 14 and 15 respectively depict the results of comparing the performance of rule bases 1 and 7 with the black-box methods.

Table 14.

Comparison of the performance of the proposed rule base 1 with the most common black-box methods on the covered samples of the pima dataset

Table 15.

Comparison of the performance of the proposed rule base 7 with the most common black box methods on the covered samples of the PIMA dataset

Summarizing the results of the experiments

Table 16 provides a summary comparison of the white-box methods on all three datasets. For the proposed method and the method presented by Wang et al. [8], the statistics of the rules with the highest predictive accuracy that covered more than 50% of the samples in the target class were used. As a result, for the WBC dataset, the results of executing the rule for the benign class: “Normal Nucleoli ≤ 2.5 & Uniformity of Cell Size ≤ 2.5” (row 2 of Table 12 in [8]) and the rule for the malignant class: “Bare Nuclei > 2.5 & Uniformity of Cell Shape > 2.5” (row 6 of Table 12 in [8]) were entered into the analyses. For the WDBC dataset, the results of executing the rule for the benign class: “Worst Perimeter ≤ 101.65 & Mean Perimeter ≤ 96.405” (row 5 of Table 11 in [8]) and the rule for the malignant class: “Worst Perimeter > 116.05” (row 2 of Table 11 in [8]) were used. For other methods, the combination of rules reported in the corresponding works was used.

Table 16.

Comparison of the proposed diagnostic rules with the rules proposed by other white-box methods in the literature based on predictive value accuracy (PVA)

Dataset		Measures	De Falco [38]	Chen and Hsu [39]	Du et.al [33]	Mena and Gonzalez [71]	Hsieh et.al [40]	Hayashi and Nakano [42]	Koklu et.al. [68]	Ye et.al. [69]	Setiono [70]	Wang et.al. [8]	Proposed Model
Pima	class 1	#Covered	13	-	-	-	-	-	446	-	-	-	233
		#Rules	4	-	-	-	-	-	2	-	-	-	1
		#Cond.	22	-	-	-	-	-	8	-	-	-	3
		PVA	61.54	-	-	-	-	-	75.54	-	-	-	91.42
	class 2	#Covered	91	-	-	-	-	-	0	-	-	-	134
		#Rules	2	-	-	-	-	-	2	-	-	-	1
		#Cond	7	-	-	-	-	-	7	-	-	-	4
		PVA	63.74	-	-	-	-	-	-	-	-	-	73.13
WBC	class 1	#Covered	431	-	-	-	419	447	443	313	437	401	402
		#Rules	1	-	-	-	2	2	2	1	3	1	1
		#Cond	4	-	-	-	18	4	9	9	12	2	3
		PVA	96.98	-	-	-	98.81	95.30	97.52	100	98.63	99	100
	class 2	#Covered	140	261	291	-	100	236	313	91	-	227	175
		#Rules	2	3	2	-	2	2	2	2	-	1	1
		#Cond	11	6	4	-	18	4	5	4	-	2	2
		PVA	95	91.19	81.44	-	94	92.37	75.72	97.80	-	92.5	98.29
WDBC	class 1	#Covered	-	-	-	-	-	-	-	-	-	316	313
		#Rules	-	-	-	-	-	-	-	-	-	1	1
		#Cond	-	-	-	-	-	-	-	-	-	2	3
		PVA	-	-	-	-	-	-	-	-	-	97.5	99.04
	class 2	#Covered	-	-	-	82	-	-	-	-	-	170	166
		#Rules	-	-	-	1	-	-	-	-	-	1	1
		#Cond	-	-	-	4	-	-	-	-	-	1	3
		PVA	-	-	-	100	-	-	-	-	-	98.2	100

On the Pima dataset, the proposed method achieved the best diagnostic accuracy among all diagnostic rules presented in the literature for both the not-Diabetic and Diabetes classes, with PVA of 91.42% and 73.13%, respectively. On the WBC dataset, the proposed method achieved the highest PVA of 100% for the benign tumor class and 98.29% for the malignant tumor class among all diagnostic rules presented in the literature. On the WDBC dataset, the proposed method obtained the highest PVA of 99.04% for the benign tumor class and 100% for the malignant tumor class among all diagnostic rules presented in the literature.

Additionally, to assess the performance of the proposed method compared to the black-box ones, Table 17 summarizes the results presented in Tables 4, 9, and Table 14 on the test sets for all three datasets. On the Pima dataset, the proposed method outperformed all the black-box methods in terms of sensitivity, accuracy, and NPV. The KNN method performed better than other methods in terms of specificity and PPV. Furthermore, the SVM Linear and Bayesian methods achieved the highest f-measure on this dataset. On the WBC and WDBC datasets, the proposed method demonstrated equal or superior performance compared to the black-box methods in all evaluation metrics. It is crucial to note that the results reported in Table 17—and consequently the comparisons drawn with black-box methods—are based exclusively on the subset of data samples covered by the selected proposed rules, and not on the entire data set. Therefore, the observed superior or comparable performance of the proposed model should be interpreted within this limited context and does not imply generalizable superiority across all data samples.

Table 17.

Comparing the performance of the proposed model with some of the most common black-box methods in the field of medical data mining on the samples covered by the extracted rule bases for the Pima, WBC, and WDBC datasets

Dataset	Measures	KNN	Polynomial	SVM RBF	SVM Linear	Bayesian	LDA	Random Forest	MLP	DNN	Proposed Model
PIMA	Sensitivity	54.05	59.05	67.05	67.57	67.57	64.86	69.16	61.43	71.45	73.33
	Specificity	96.97	88.73	95.53	95.45	95.45	95.45	93.5	96.08	93.58	91.3
	Accuracy	81.55	78.07	85.3	85.44	85.44	84.47	84.76	83.63	81.54	85.86
	PPV	90.91	75.28	89.38	89.29	89.29	88.89	85.8	90.01	83.01	78.57
	NPV	79.01	79.54	83.8	84	84	82.89	84.43	81.69	86.58	88.73
	F-Measure	67.8	65.8	76.61	76.92	76.92	75	76.49	72.79	74.34	75.86
WBC	Sensitivity	93.75	97.67	99.27	100	100	93.75	92.44	94.21	100	100
	Specificity	100	100	99.88	100	100	100	100	100	100	100
	Accuracy	98.17	99.32	99.7	100	100	98.17	97.79	98.3	100	100
	PPV	100	100	99.71	100	100	100	100	100	100	100
	NPV	97.48	99.04	99.7	100	100	97.48	96.99	97.66	100	100
	F-Measure	96.77	98.82	99.48	100	100	96.77	96.03	97.02	100	100
WDBC	Sensitivity	89.8	91.84	89.8	93.88	93.88	89.8	90.45	89.84	90.82	97.96
	Specificity	100	100	99.93	100	100	100	99.84	100	100	100
	Accuracy	96.6	97.28	96.55	97.96	97.96	96.6	96.71	96.61	99.23	99.32
	PPV	100	100	99.84	100	100	100	99.65	100	100	100
	NPV	95.15	96.08	95.14	97.03	97.03	95.15	95.44	95.17	97.28	98.99
	F-Measure	94.62	95.74	94.55	96.84	96.84	94.62	94.82	94.64	98	98.97

Besides, to evaluate the significance of the results, the Phi correlation coefficient (MCC score) was calculated. For each of the data sets, the selected rule bases presented in Figs. 8, 10 and 12 were entered into the statistical analysis. Table 18 shows the correlation between the actual and predicted output of the examples covered by the rule base.

Table 18.

Statistical significance analysis of the results on WBC, WDBC and Pima datasets using phi correlation coefficient

Dataset	#Rule Set	Phi correlation coefficient (MCC score)	p-value	Interpretation
WBC	1	0.99	<0.001	Very strong positive relationship
	2	0.98	<0.001	Very strong positive relationship
	3	0.97	<0.001	Very strong positive relationship
	4	0.97	<0.001	Very strong positive relationship
WDBC	1	0.99	<0.001	Very strong positive relationship
	2	0.98	<0.001	Very strong positive relationship
	3	0.97	<0.001	Very strong positive relationship
	4	0.97	<0.001	Very strong positive relationship
	5	0.97	<0.001	Very strong positive relationship
	6	0.79	<0.001	Very strong positive relationship
	7	0.97	<0.001	Very strong positive relationship
	8	0.96	<0.001	Very strong positive relationship
Pima	1	0.75	<0.001	Very strong positive relationship
	2	0.71	<0.001	Very strong positive relationship
	3	0.74	<0.001	Very strong positive relationship
	4	0.69	<0.001	Strong positive relationship
	5	0.71	<0.001	Very strong positive relationship
	6	0.69	<0.001	Strong positive relationship
	7	0.69	<0.001	Strong positive relationship

On the WBC and WDBC datasets, all the proposed rule bases showed very strong positive relationships with the actual values (MCC ≥ 0.96). On the Pima dataset, four rule bases showed very strong relationships (MCC ≥ 0.71) and three showed strong positive correlations with the actual values (MCC ≥ 0.69). Furthermore, for all rule bases, the statistical significance of the Phi correlation coefficient (MCC) was assessed by computing the corresponding p-value. A p-value < 0.001 indicates that the observed relationship is statistically significant.

Discussion

The power of ML-based clinical predictive models holds promise for integrating data-derived real-world evidence into clinical decision-making [43]. Systematic collection and analysis of the evidence enable continuous evolution in understanding diseases and treatment responses, particularly in the context of emerging diseases [43, 74]. Furthermore, due to the increasing importance and integration of AI in people’s lives, there has been a movement towards HCAI [3]. Therefore, this study focused on designing a diagnostic system with three main objectives: 1) A model designed in a human-centered manner, involving domain experts to prioritize clinical aspects in model design and parameter tuning. 2) A model capable of knowledge enhancement through the extraction of precise rules from data with high interpretability, surpassing other white-box interpretable models in performance. 3) A white-box model with high accuracy, approaching the performance of black-box models. These three criteria of interpretability, accuracy, and clinical evaluation, within the framework of a human-centered design, increase the chances of adoption of these systems in clinical practice guidelines and, consequently, gaining acceptance in real-world settings. The reason for our emphasis on presenting a powerful white box method is that for the adoption of sophisticated and state-of-the art artificial intelligence methods including ensemble and boosting ones in medicine, the output of these models must be based on evidence. In other words, AI methods should follow the same path that statistics took to reach its current important position in medicine. We believe that strengthening this pathway requires providing models such as the proposed one that are transparent while achieving appropriate efficiency and can be easily evaluated clinically.

The results obtained from running the proposed model on the WDBC, WBC, and Pima datasets, clinical evaluation by 13 healthcare professionals, and comparison with interpretable rules provided in 10 existing research studies demonstrate that the study achieved the three mentioned objectives. Among the 73 studies examined in the field of clinical rule extraction, only the mentioned studies have presented diagnostic rules. Other studies mainly fell into three categories: 1) Studies that reported only the model execution results without providing extracted rule sets [34, 36, 41, 75–80]. 2) Studies that presented fuzzy rules [21, 35, 81–86]. 3) Studies that presented rules in the form of mathematical equations using coefficients [87].

While fuzzy single rules with short enough length can be interpreted by humans, the entire fuzzy process, including rule aggregation and subsequent defuzzification, cannot be evaluated by humans without computer assistance. Moreover, interpreting and executing rules involving mathematical relationships with multiplication or exponentiation operators is challenging. Determining the threshold boundaries for each variable in the AI/ML-based extracted rules remains a topic of debate, and assigning coefficients to each variable adds another layer of complexity. Therefore, this study considered algorithms as white-box approaches that have the capability for physicians to evaluate and operationally execute the extracted rules without the need for a computer. We refer to these methods as “clinical white box algorithms.”

It is worth noting that in the literature, to achieve model transparency, explainability is a concept close to interpretability and is sometimes used interchangeably with it. Both techniques—by supporting results and ensuring fairness of decisions, enabling control and correction of behavioral errors, improving model performance through understanding outputs, and gaining insights and discovering new knowledge—introduce model transparency as a powerful tool for achieving trustworthy AI [88]. However, the reason we have not considered explainable AI (XAI) methods in this work is that most XAI methods are approximations of black-box models and do not calculate the exact inner workings of those models; thus, they can, in some cases, introduce unnecessary complexity or mislead and misrepresent the problem’s characteristics [12, 89]. Consequently, in high-stakes scenarios such as medicine, interpretable methods should be used wherever possible instead of explainable methods [89].

Considering the results presented in Figs. 9, 10, 11, 13 and Table 16, the accuracy of individual diagnostic rules generated by the proposed model in both disease classes (malignant tumor class for the WBC and WDBC and diabetes class for the Pima) and non-disease classes (benign tumor class for the WBC and WDBC and not-Diabetic class for the Pima dataset) is higher than all diagnostic rules provided in the literature. This capability demonstrates the strength and comprehensiveness of the proposed model. The high accuracy of a white-box model in all diagnostic classes is crucial in the medical field because the inability to classify a data point into one class (positive or negative) should not lead to the opposite conclusion. Therefore, models that provide only diagnostic rules for one class or make decisions for the other class using an “else” statement (as used in [33, 39, 70, 71]) practically miss the purpose of using white-box methods since the result will be similar to black-box methods but with lower accuracy. The absence of disease by a diagnostic rule does not necessarily mean that the individual is healthy. Instead, uncovered cases need to be examined separately, and more parameters should be investigated. In such cases, domain experts should make decisions based on scientific evidence.

Among the three datasets under investigation, the proposed method performed close to optimal on the WBC and WDBC datasets. According to Table 2, each of the individual diagnostic rules for benign and malignant tumors on the WBC test set had an accuracy of over 95%. In this table, the diagnostic rule “O_PPV_2,” which examines only 2 variables, achieved 100% accuracy in correctly classifying 67.61% of malignant samples in the test set. Additionally, the “O_NPV_2,” examining 2 variables, and the “O_NPV_3,” examining 3 variables, identified 87.59% and 87.88%, respectively, of benign samples on the test set with 100% accuracy. Moreover, according to Table 7, each of the individual diagnostic rules for benign and malignant tumors on the WDBC dataset had an accuracy of over 93%. In this table, the diagnostic rule “D_PPV_1,” which examines 3 variables, successfully classified 75% of malignant samples in the test set with 100% accuracy. Furthermore, the “D_NPV_2” rule, examining 3 variables, identified 92.52% of benign samples on the test set with 99% accuracy. These results provide reliability for the proposed individual diagnostic rules for covered breast cancer data samples. The existence of several different high-accuracy rules for identifying one diagnostic class increases confidence in the diagnostic outcome, as clinical repetition of tests and consistent results lead to clinical decision confirmation. While diagnosing malignant tumors promptly is essential for treatment, diagnosing benign tumors accurately is also vital to prevent unnecessary medical expenses and anxiety arising from misdiagnosis.

According to Table 17, on the covered samples from the WBC and WDBC datasets, the proposed model showed higher observed accuracy all six standard evaluation metrics compared to the nine black-box methods. Additionally, on the challenging Pima dataset, it also showed higher observed sensitivity, accuracy, and NPV on the covered samples. These results indicate competitive performance of the proposed model within the subset of data covered by the extracted rules, which—alongside its unique clinical evaluation feature by 13 domain experts—suggests promising practical applications. The existence of a white-box model with computational capabilities close to black-box methods is crucial because full confidence in the outputs of complex black-box methods can only be achieved when it is proven that the training data includes all features relevant to the disease diagnosis. Given that the medical field is empirical and it cannot be claimed that all factors affecting a disease have been identified so far. Hence, as research based on intelligent methods tries to introduce mathematical and algorithmic models in the clinical domain, some margin of error must be accepted, as uncertainty is also accepted in medical decision-making [90]. Achieving this point requires gaining the trust of medical professionals in intelligent methods. It seems that the way to gain the trust of clinicians in the output of black-box methods goes through the path paved by interpretable rule extraction methods, which are acceptable by providing insight into the decision-making process from a medical perspective.

Furthermore, unlike the methods in [8, 32, 34–37, 40], in this study, the proposed model, based on the NSGA-II algorithm, adjusts the threshold values of the variables, generates rules, and performs classification independently without combining with other methods. Solving a problem independently highlights the importance of meta-heuristic algorithms in data mining, which had not received much attention in the literature before.

Eventually, the main contributions of the study can be summarized as follows: realizing the concept of human-in-the-loop in the development of intelligent diagnostic models in clinical applications, providing a new chromosomal structure to generate optimal rule bases in classification problems, presenting a solution to automatically adjust the threshold values of variables in antecedents of rules, and overall offering a knowledge enhancement system through extracting accurate, hidden rules from data for categorization. The study’s achievements include developing a white-box model with higher performance than similar rule extraction methods and one that is close to common black-box ones, extracting precise and interpretable single rules for each diagnostic class, and providing multiple rules with high coverage and performance for each diagnostic class to increase the performance confirmation of decision-making outcomes.

Limitations:

Although the results of this study are juxtaposed with all related articles in this field retrieved by the search strategy, the lack of similar articles in the field of extracting diagnostic rules for further evaluation of the model is one of the limitations of this study. In addition, according to the authors’ definition of clinical white box methods, models based on meta-heuristic methods were investigated. Therefore, considering other definitions of clinical white box methods with diverse and different algorithmic bases that require exhaustive analysis concerning their comparative strengths and weaknesses is another limitation of this study. Furthermore, the proposed model was applied on clean and complete data. Hence, implementing techniques to handle incomplete data as well as class imbalance in the dataset to improve model performance in minority classes will be considered in future research.

Another important limitation concerns the coverage of the extracted rule bases. By design, the proposed model prioritizes simplicity, interpretability, and clinical usability over full dataset coverage. The goal was to provide individual diagnostic rules that are simple enough and sufficiently small to be evaluated and used by physicians without the need for computers or complex calculations. Uncovered samples are not necessarily errors—they may be complex cases requiring further clinical investigation. Therefore, all performance metrics are based exclusively on covered samples, not the entire dataset. This does not imply generalizable superiority, but reflects a deliberate design choice aligned with human‑centered AI. Future work should develop auxiliary strategies for uncovered cases, such as hierarchical decision frameworks or flagging for clinical review.

It seems necessary to mention that ensemble methods have not been considered in this research because basically the reason for the emergence of these methods is to combine the collective power of multiple models in an effort to achieve a better performance than any single model. This power comes at the cost of losing the simplicity of results interpretation and clinical evaluation without the need for computer implementation. Similarly, this study did not consider XAI methods due to their current limitations—including lack of verifiability and usability by non-technical clinical users [91, 92]. Nevertheless, given the growing attention to XAI as an opportunity to harness black-box predictive power without drawbacks, future work could consider their usability as these methods mature and become more clinically applicable.

Conclusions

Clinical guideline flowcharts are built on if-then logic, translating expert knowledge into structured decision pathways, and use rule-based systems to enhance decision-making in healthcare [93–95]. Additionally, as noted in [96], the interpretability of rule-based systems plays a crucial role in facilitating learning, fostering public trust in algorithmic predictions, and ensuring the seamless integration of intelligent models into clinical practice; this interpretability also enhances patient safety by simplifying testing, auditing, debugging, and identifying model errors; furthermore, rule-based systems enable knowledge extraction from data and the continuous acquisition of new insights. Therefore, this research aimed to develop a more accurate rule extraction system than existing ones, a goal successfully achieved through the conducted experiments.

Moreover, according to the ethical principles of protecting human autonomy and ensuring transparency of AI development in health endorsed by WHO [27], this study dealt with HCAI development for clinical applications. A transparent white-box model with computational power close to black-box algorithms was proposed based on the NSGA-II meta-heuristic algorithm for data classification. On the subset of data covered by the extracted rule bases, the model achieved performance approaching that of traditional black-box algorithms while maintaining interpretability. This finding, however, is conditional on data coverage and does not extend to the entire dataset. Simultaneous attention to computational power and clinical utility in the HCAI approach formed the basis for the design and implementation of the proposed model. The performance of the proposed model was evaluated on three common datasets: WBC, WDBC, and Pima. The results of running the proposed model on all three datasets were evaluated by 13 domain experts. A scrutiny of 383 systematic reviews, 71 meta-analyses, and 73 articles published on AI/ML-based diagnostic models showed that in none of them were the extracted diagnostic rules clinically evaluated, and there was a lack of involvement of healthcare professionals and clinicians in this field of research. Therefore, leveraging the insights of clinical experts and implementing the HCAI framework, based on our knowledge, have been used for the first time in the field of extracting clinical diagnostic rules in the clinical evaluation phase of the results.

The results of implementing the proposed method on three datasets and comparing it with existing rule extraction methods and common black-box methods demonstrated the high computational power of the proposed approach. By implementing the rule extractor separately, the proposed model succeeded in providing diagnostic rules for each diagnostic class. Furthermore, the statistical analysis bolsters the empirical integrity of the presented results. Interpretability, diagnostic accuracy, reliability through the presentation of multiple rules for each diagnostic class, and clinical confirmation of the results introduce the proposed model as a powerful clinical decision support system and a knowledge enhancement source in the clinical domain. However, in this study, the proposed model was evaluated on the public datasets to enable comparison with existing works (It is worth noting that the results of running the algorithm on 6 other datasets are presented in Appendices D and E. The reason for choosing the three datasets mentioned in the paper is their extensive use in research, which facilitates demonstrating the power of the model). Given the remarkable results of the proposed model on public datasets, in future work, the proposed model will be applied to local, real-world datasets; expanding the scope to include additional diversified datasets would provide a more robust demonstration of the model’s generalizability and effectiveness across various medical contexts. On the other hand, because of the separate implementation of the proposed model for each diagnostic class in the data, the applicability of the model on multi-class data will be evaluated in future works. In addition, the use of feature extraction or space transformation techniques and their effect on improving model efficiency will be considered in future works. Finally, in future research, we will develop additional techniques to visualize the extracted rules and create user-friendly interfaces for clinicians to interact with the model to create a systematic feedback loop with them to continuously refine the rules based on real-world application and insights. Future studies may investigate adaptive or automated parameter tuning strategies for NSGA-II to further optimize rule set quality for specific clinical datasets.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Abbreviations

AI

Artificial intelligence

ML

Machine Learning

PPV

Positive Predictive Value

NPV

Negative Predictive Value

EA

Evolutionary Algorithm

ANN

Artificial Neural Networks

NSGA-II

Non-dominated Sorting Genetic Algorithm

MOEA

Multi Objective Evolutionary Algorithm

GA

Genetic Algorithm

TFAMT

Temporal Fuzzy Ant Miner Decision Tree

HCAI

Human-Centered Artificial Intelligence

WHO

World Health Organization

IOM

Institute of Medicine

PVA

Predictive Value Accuracy

PE

Predictive Error

RE

Requirements Engineering

CVI

Content Validity Index

NDR

Non-dominated Diagnostic Rule

IDF

International Diabetes Federation

WDBC

Breast Cancer Wisconsin dataset

WBC

Breast Cancer Wisconsin original dataset

K-NN

k-nearest neighbors

SVM

Support Vector Machines

LDA

Linear Discriminant Analysis

KNNi

K-nearest neighbors Imputation

Authors’ contributions

AG conceived the original idea, and all authors contributed to developed it. AG and FA did the search, selected the studies for inclusion, analyzed the literature, and designed the HCAI model. MBS directed the focus group sessions and managed the external validation process. AG develop the computational model. AG and FA implement the model and did the programming. FA wrote the original draft of the manuscript. AG and FA reviewed and edited the manuscript. AG supervised the study. All authors provided critical feedback and contributed to the final manuscript, and were responsible for the decision to submit for publication.

Funding

This study has received no funding.

Data availability

All data generated or analyzed during this study are included (or cited) in the article and the supplementary materials (Pima Diabetes dataset after Preprocessing.xlsx and Appendix A to E).

Declarations

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Competing interests

The authors declare no competing interests.