1 Introduction↩︎

In recent years, the application of artificial intelligence (AI) in medical screening has grown rapidly. Particularly in areas such as cancer screening, disease detection, and risk assessment, deep learning models have been proven to improve diagnostic accuracy [1], [2]. A high-performance classification model should not only achieve high overall accuracy but also maintain robust classification performance across different cohorts to ensure fairness and reliability [3], [4].

In various research fields (such as decentralized finance, personalized recommendation systems, and medical diagnostics) researchers tend to focus on different target groups [5]. For example, in financial credit assessment, more attention may be paid to the misclassification rate of high-risk users, whereas in medical screening, clinicians are more concerned with the sensitivity for high-risk patients [6]. Therefore, it remains a challenging problem to objectively evaluate the performance of classification models under different application contexts and goals [7], [8].

Current evaluation metrics often show significant limitations when dealing with imbalanced data, noticeable disparities in classification performance across groups, or the specific needs of high-risk populations. Traditional classification metrics, such as accuracy and F1-score, tend to overestimate the actual performance of models in imbalanced datasets, especially in medical screening tasks where the detection precision of minority classes is often overlooked [9], [10]. Consequently, there is a need for a new classification evaluation approach that ensures reliable predictive capability across multi-group data and provides a more interpretable evaluation framework [11], [12].

Widely used classification evaluation metrics, such as Accuracy, Sensitivity, and Specificity, perform well under the assumptions of balanced data and independent samples. These metrics offer a degree of objectivity, intuitiveness, and fairness, providing an initial assessment of a classifier’s performance. For example:

• Sensitivity (Sen) reflects the classifier’s ability to identify true positive cases correctly. In cancer screening tasks, sensitivity measures the model’s ability to detect individuals with the disease. For instance, a sensitivity of 0.95 indicates that out of 100 actual patients, the model successfully detects 95, misclassifying only 5 [13], [14].

• Specificity (Spe) reflects the classifier’s ability to identify true negative cases correctly. In medical screening, specificity is often used to measure the false positive rate. Particularly in early screening tasks, a higher specificity implies fewer misclassifications of healthy individuals. For example, a specificity of 0.95 means that out of 100 healthy individuals, the model correctly classifies 95, while 5 are incorrectly identified as positive [15].

However, in medical research tasks such as cancer screening, these traditional metrics exhibit several limitations. Traditional classification metrics are computed at the individual sample level, treating each sample as an independent instance. This neglects the fact that a patient may undergo multiple tests [16]. In cancer screening, a single patient may receive multiple examinations at different time points or using different devices, and it is the aggregation of these results that ultimately informs a clinician’s diagnosis. Relying solely on sample-level evaluation can lead to several issues [17], [18]:

• Overlooking instability at the subject level: Multiple test samples from the same individual are treated independently. The model may produce inconsistent predictions for a single patient, but traditional evaluations fail to capture this instability.

• Misjudgment affecting overall diagnosis: A patient’s final diagnosis is based on multiple test results. A misclassification of just one sample could lead to an incorrect diagnosis, which traditional metrics do not account for.

• Misclassification risk at the subject level: While the model may perform well on individual samples, it may still present a high risk of misclassification at the patient level, thereby reducing the practical effectiveness of the screening system.

Second, traditional evaluation methods do not account for the importance of different population groups [19]. In medical screening, certain high-risk groups (e.g., elderly patients or individuals with a family history) require more precise detection than the general population [20]. However, traditional metrics are usually computed globally across all samples, without assigning higher weight to these critical subgroups. As a result, a model might perform well overall while underperforming on key populations [21].

Finally, traditional metrics struggle to reflect the impact of data imbalance. In real-world medical screening, positive cases are typically much fewer than negative ones. In such scenarios:

• The model may become biased toward the majority class: Due to the small proportion of positive samples, the model may prioritize optimizing for the negative class during training, at the expense of identifying positive cases—resulting in high miss rates.

• Evaluation metrics may fail to accurately reflect true screening performance: Traditional metrics do not adequately measure the model’s ability to classify the minority class, potentially leading to an overestimation of performance in practical applications [22], [23].

In summary, traditional classification evaluation methods show notable limitations in medical tasks such as cancer screening. These include the reliance on sample-level independent evaluation, the inability to measure the importance of different population groups, and the distortion of evaluation results under imbalanced data conditions [24].

2 Proposed Metrics↩︎

To address the aforementioned limitations, we propose a novel classification evaluation framework called Cohort-Attention Evaluation Metrics (CAT). This framework introduces improvements in the following aspects:

2.0.0.1 Patient-Level Evaluation

The CAT metrics perform classification evaluation at the patient level, ensuring that multiple test results from the same patient are not treated as independent instances. Instead, the final diagnostic result is derived by aggregating multiple prediction outputs. This approach captures the overall classification stability for each patient and reduces the impact of individual sample misclassifications on the final diagnosis.

2.0.0.2 Entropy-Based Distribution Weighting

Within the CAT framework, we calculate the entropy of each cohort’s distribution and adjust the weight of each group of patients accordingly, ensuring that no single group disproportionately influences the overall evaluation.

2.0.0.3 Cohort-Weighted Sensitivity (CATSensitivity, CATSen)

Traditional sensitivity calculations fail to account for the balance of classification performance across different groups. CATSen addresses this by weighting the sensitivity of each cohort, thereby ensuring that the classifier’s predictive performance on minority groups is fairly represented, and mitigating bias caused by imbalanced data.

2.0.0.4 Cohort-Weighted Specificity (CATSpecificity, CATSpe)

Conventional specificity is computed based on all negative samples globally. In contrast, CATSpe applies a cohort-weighted approach to ensure that the specificity for each group of negative samples is appropriately evaluated. This prevents the distortion of specificity due to underrepresented cohorts.

2.0.0.5 Sensitivity-Specificity Balance (CATMean)

In classification tasks, sensitivity and specificity often involve trade-offs. We propose CATMean, which is calculated using the harmonic mean of CATSen and CATSpe. This ensures a balanced evaluation between sensitivity and specificity, providing a more comprehensive measure of model performance.

3 CAT Metric Formulation and Analysis↩︎

In this section, we provide a detailed analysis of the mathematical foundation underlying the CAT evaluation framework, highlighting the significance and innovations of each formula. Building upon traditional classification evaluation methods, CAT introduces a more stable and fair metric system through patient-level evaluation, entropy-based uncertainty weighting, cohort-weighted sensitivity and specificity (CATSensitivity and CATSpecificity), and a harmonic mean-based balance metric (CATMean). This approach is particularly suited for high-risk tasks such as medical screening, ensuring that the model performs not only well in overall evaluation but also maintains stability and reliability across different cohorts and at the patient level.

3.1 Computation of Patient-Level Classification Accuracy↩︎

The CAT metrics adopt a tied ID mechanism to ensure that all test results associated with the same patient are not evaluated independently. Instead, all predictions from the same patient are aggregated to assess the overall classification performance. The specific calculation is as follows: \[\text{Score}_k = \left| \sum_j\widehat{y}_{kj}-(1-y_k) \cdot n_k \right|\]

When the true label of tester \(k\) is positive, i.e., \(y_k = 1\): \[\text{Score}^+_k = \sum\nolimits_j\widehat{y}_{kj}.\] where \(\widehat{y}_{kj}\) denotes the prediction result of the \(j\)-th test for tester \(k\), with \(\widehat{y}_{kj} \in \{0,1\}\). \(\sum_j\widehat{y}_{k,j}\) represents the total number of times the tester was predicted as positive.

In this case, if the model’s predictions are completely correct (i.e., all predictions are \(1\)), then \(\text{Score}_k\) reflects the total number of tests. If any tests are incorrectly predicted as negative, the score will be reduced accordingly.

When the true label of tester \(k\) is negative (\(y_k = 0\)): \[\text{Score}^-_k = \left| \sum_{j} \widehat{y}_{kj} - n_k \right|\]

Where: \(n_k\) is the total number of tests conducted for tester \(k\). \(\sum_j\widehat{y}_{kj}\) still represents the total number of times the tester was predicted as positive.

Since \(y_k = 0\), the ideal case is that all predictions should be \(0\), i.e., \(\sum_j \widehat{y}_{kj} = 0\), and thus \(\text{Score}_k = |0 - n_k| = n_k\). If any tests are incorrectly predicted as positive, then \(\sum_j \widehat{y}_{kj} > 0\), leading to a higher final score, which indicates a more severe error.

Next, the prediction score for each tester is normalized: \[\text{Acc}^*_i=\frac{\text{Score}^*_k}{n^*_i}\; \forall \ast\in\{+,-\}\]

Where \(n_k\) is the total number of tests for tester \(k\), ensuring that the score is normalized within the range [0, 1], so that patients with higher testing frequency do not disproportionately affect the overall evaluation.

3.1.1 Statistical Justification for Aggregation at the Tester Level↩︎

Consider the common scenario of repeated measurements in medical screening. Let the \(n_k\) test results for the \(k\)-th tester be denoted as \({y_{k,j}}_{j=1}^{n_k}\), with statistical properties satisfying:

For any tester \(k\), there exists an intraclass correlation coefficient between the test results: \[\rho = \frac{\text{Cov}(y_{k,j}, y_{k,l})}{\text{Var}(y_{k,j})}, \quad (j\neq l)\] When \(\rho > 0\), the assumption of sample independence does not hold, and aggregation evaluation becomes necessary.

Let the variance for traditional sample-level evaluation be \(\sigma^2_{\text{sample}}\), and the variance for tester-level evaluation be:

\[\sigma^2_{\text{patient}} = \sigma^2_{\text{sample}} \left( \frac{1+\rho(n_k-1)}{n_k} \right)\]

When \(\rho > 0\), \(\sigma^2_{\text{patient}} < \sigma^2_{\text{sample}}\), proving that the aggregation strategy reduces the evaluation variance.

As shown in 1, the x-axis represents the number of tests per tester \(n_k\), and the y-axis represents the ratio of tester-level evaluation variance \(\sigma^2_{\mathrm{patient}}\) to traditional sample-level variance \(\sigma^2_{\mathrm{sample}}\). The results show that:

• When the intraclass correlation coefficient \(\rho = 0\) (independent samples), the variance ratio strictly follows the \(1/n_k\) decay rule (blue solid line), which aligns with the central limit theorem.

• As \(\rho\) increases (red dashed line, green dotted line), the variance reduction effect gradually weakens, but it still satisfies \(\sigma^2_{\mathrm{patient}} < \sigma^2_{\mathrm{sample}}\).

This result validates the theoretical analysis: in scenarios with intraclass correlation, the aggregation evaluation at the tester level can effectively reduce evaluation variance, and an increase in the number of tests \(n_k\) significantly improves evaluation stability.

3.2 Weighting Mechanism Based on Detection Distribution Entropy↩︎

In medical screening scenarios, there may be significant differences in the testing frequency across different testers. Some testers may undergo multiple tests due to high risk (concentrated testing frequency), while others may only undergo a single test. This uneven distribution of test frequencies can affect the fairness of the evaluation. Therefore, we use information entropy theory to weight the distribution of testers’ testing frequencies, aiming to capture the maximum information.

\[A^\ast_k = \frac{\sum_{i\in C^\ast_k} E^\ast_i \cdot \text{Acc}^\ast_i}{\sum_{i\in C^\ast_k} E^\ast_i},\quad \forall \ast\in\{+,-\}\] The key parameters are defined as follows:

• \(E^\ast_i = -p^\ast_i \log p^\ast_i\): The detection distribution entropy of tester \(i\)

• \(p^\ast_i = n^\ast_i / N^\ast\): The proportion of test frequency for tester \(i\)

• \(n^\ast_i\): The number of tests performed by tester \(i\) in the cohort

• \(N^\ast = \sum_{i} n^\ast_i\): The total number of tests in the cohort

3.2.1 Mathematical Properties of Detection Distribution Entropy↩︎

1. Anomaly Detection Suppression: Testers with high testing frequencies (\(p^\ast_i \to 1\)) are given exponentially decaying weights.

2. Weight Normalization: The effect of cohort size is eliminated through the denominator \(\sum E^\ast_i\).

   

3.3 Cohort-Weighted Sensitivity (CATSensitivity, CATSen)↩︎

The traditional method for calculating sensitivity (Sensitivity) is as follows: \[\text{Sensitivity} = \frac{\text{True Positives (TP)}}{\text{True Positives (TP)} + \text{False Negatives (FN)}}\]

To overcome the issue of data imbalance, we propose the cohort-weighted sensitivity (CATSensitivity, CATSen), which is calculated as follows: \[\text{CATSen} = \left(1-\frac{1}{1+e^{0.5-\alpha}}\right)\frac{1}{|K^+_\star|}\sum_{\forall k\in K^+_\star}A^+_k + \left(\frac{1}{1+e^{0.5-\alpha}}\right)\frac{1}{|K^+|-|K^+_\star|}\sum_{\forall k \in K^+/K^+_\star}A^+_k\]

Where:

• \(A^+_k\) represents the positive sample score of cohort \(k\).

• \(K^+_\star\) represents the selected cohort’s positive testers.

• \(K^+/K^+_\star\) represents the positive testers of the non-selected cohort.

• \(\alpha\) is a tuning parameter that controls the weight proportion of the sig group.

• \(|K^+_\star|\) represents the cohort in the sig parameter.

• \(|K| - |K^+_\star|\) represents the non-sig parameter cohort.

3.4 Cohort-Weighted Specificity (CATSpecificity, CATSpe)↩︎

Similar to sensitivity, the traditional method for calculating specificity (Specificity) is as follows: \[\text{Specificity} = \frac{\text{True Negatives (TN)}}{\text{True Negatives (TN)} + \text{False Positives (FP)}}\]

To provide a more reasonable specificity calculation at the cohort level, we propose weighted specificity: \[\text{CATSpe} = \frac{\alpha}{|K^-_\star|}\sum_{\forall k\in K^-_\star}A^-_k + \frac{(1-\alpha)}{|K^-|-|K^-_\star|}\sum_{\forall k \in K^-/K^-_\star}A^-_k\]

3.5 Sensitivity-Specificity Balance (CATMean): A Unified All-Purpose Evaluation Metric↩︎

To provide a unified, stable, and adaptable evaluation metric for all scenarios, we propose CATMean, which simultaneously considers sensitivity, specificity, and the impact of data imbalance, delivering comprehensive information within a single metric: \[\text{CATMean} = \sqrt{\frac{(1+\beta^2) \times \text{CATSen} \times \text{CATSpe}}{\beta^2 \times \text{CATSen} + \text{CATSpe}}}\] where \(\text{CATSen}\) is the cohort-weighted sensitivity, ensuring that data imbalance does not affect the evaluation of positive samples. \(\text{CATSpe}\) is the cohort-weighted specificity, ensuring that the evaluation of negative samples is also fair. \(\beta\) controls the relative importance of sensitivity and specificity. The core advantage of CATMean lies in its versatility, automatic balance between sensitivity and specificity, and adaptability to different application scenarios.

3.5.1 Core Advantages of CATMean↩︎

• Versatility: CATMean unifies the advantages of traditional evaluation metrics. There is no need to separately check sensitivity, specificity, or other metrics. With just CATMean, one can comprehensively evaluate the model’s performance.

• Automatic Balance of Sensitivity and Specificity: The harmonic mean automatically achieves the optimal balance without manual intervention.

• High Adaptability: By adjusting \(\beta\), CATMean can be flexibly applied to different application scenarios, such as cancer screening tasks, health population screening, and more.

• Robustness: The cohort-weighted mechanism prevents evaluation bias in the case of data imbalance, ensuring stable performance of classifiers across different groups and tasks.

3.5.2 CATMean Replacing Traditional Evaluation Systems↩︎

Compared to the traditional approach, which requires reviewing multiple evaluation metrics, CATMean serves as a unified all-purpose metric. With just one value, it can comprehensively measure the model’s classification capability, avoiding the need to repeatedly adjust evaluation standards across different tasks. This makes CATMean a more stable, intuitive, and efficient evaluation method for medical screenings, disease prediction, and other high-risk tasks.

4 Analysis of the Impact of Parameters \(\alpha\) and \(\beta\)↩︎

In the CAT evaluation system, \(\alpha\) and \(\beta\) are two key parameters used to adjust the calculation of Cohort-weighted Sensitivity (CATSen), Cohort-weighted Specificity (CATSpe), and Sensitivity-Specificity Balance (CATMean). This section will analyze, from both a formulaic and graphical perspective, how changes in these two parameters affect the final evaluation scores.

4.1 Impact of Parameter \(\alpha\)↩︎

The parameter \(\alpha\) is mainly used to adjust the calculation of Cohort-weighted Sensitivity (CATSen) and Cohort-weighted Specificity (CATSpe). Its value range is \([0,1]\), and it controls the weight distribution between positive and negative samples in the evaluation.

4.1.1 Impact on CATSen↩︎

The formula for Cohort-weighted Sensitivity (CATSen) is as follows: \[\text{CATSen} = \left(1-w\right)\frac{1}{|K^+_\star|}\sum_{\forall k\in K^+_\star}A^+_k + w\frac{1}{|K|-|K^+_\star|}\sum_{\forall k \in K/K^+_\star}A^+_k.\] Here, \(w=\frac{1}{1+e^{0.5-\alpha}}\) is the weight for the non-sig cohort and \(1-w\) is the weight for the sig cohort. The sigmoid function smoothes the weight transition, and when \(\alpha = 0.5\), the weights for both sig and non-sig cohorts are equal. When \(\alpha\) changes, the weight distribution will change as follows:

• When \(\alpha > 0.5\), the weight of the sig cohort increases, and the weight of the non-sig cohort decreases. In this case, CATSen places more emphasis on the classification ability of the sig cohort, which is suitable for evaluations focused on specific groups.

• When \(\alpha < 0.5\), the weight of the non-sig cohort increases, and the weight of the sig cohort decreases. In this case, CATSen places more emphasis on the classification ability of the general population, which is suitable for health screening tasks.

4.2 Impact of Parameter \(\beta\)↩︎

The parameter \(\beta\) is primarily used to adjust the calculation of Sensitivity-Specificity Balance (CATMean). Its value range is \((0,+\infty)\), and it controls the relative importance of sensitivity and specificity.

4.2.1 Impact on CATMean↩︎

The formula for Sensitivity-Specificity Balance (CATMean) is as follows: \[\text{CATMean} = \sqrt{\frac{(1+\beta^2) \times \text{CATSen} \times \text{CATSpe}}{\beta^2 \times \text{CATSen} + \text{CATSpe}}}\]

Where:

• \(\beta\) controls the relative importance of Sensitivity (CATSen) and Specificity (CATSpe).

When \(\beta\) changes, the calculation of CATMean will change as follows:

• When \(\beta = 1\), CATMean degenerates into the traditional F1-score, where the weights of sensitivity and specificity are equal.

• When \(\beta < 1\), the weight of sensitivity increases, and the weight of specificity decreases. In this case, CATMean places more emphasis on the classification ability for positive samples, making it suitable for evaluating high-risk groups.

• When \(\beta > 1\), the weight of specificity increases, and the weight of sensitivity decreases. In this case, CATMean places more emphasis on the classification ability for negative samples, making it suitable for health screening tasks.

4.3 Joint Impact of \(\alpha\) and \(\beta\)↩︎

In practical applications, the values of \(\alpha\) and \(\beta\) can be jointly adjusted to suit different application scenarios. For example:

• In cancer screening tasks, \(\alpha>0.5\) and \(\beta<1\) can be set to ensure the model’s classification ability for high-risk groups.

• In health population screening tasks, \(\alpha<0.5\) and \(\beta>1\) can be set to ensure the model’s classification ability for healthy populations.

By adjusting \(\alpha\) and \(\beta\), the CAT evaluation system can flexibly adapt to different application needs, providing more fair and reliable evaluation results.

4.4 Chart Analysis↩︎

3 shows the variation of CATMean with \(\beta\), where \(\alpha\) takes different values (\(\alpha=0.0,0.3,0.5,0.7,1.0\)). From the figure, it can be seen that:

• As \(\beta\) increases, the value of CATMean gradually increases, indicating that the model places more emphasis on specificity (the classification ability for negative samples).

• As \(\alpha\) increases, the value of CATMean also increases, indicating that the weight of the sig group increases, further boosting the value of CATMean.

4 shows the variation of CATMean with \(\alpha\), where \(\beta\) takes different values (\(\beta=1.0,1.5,2.0,2.5,5.0,10.0\)). From the figure, it can be seen that:

• As \(\alpha\) increases, the value of CATMean gradually increases, indicating that the weight of the sig group increases, further boosting the value of CATMean.

• As \(\beta\) increases, the value of CATMean also increases, indicating that the model places more emphasis on specificity (the classification ability for negative samples).

5 shows the variation of AUC, sensitivity, specificity, CATSen, CATSpe, and CATMean with \(\alpha\), where \(\beta\) takes different values. From the figure, it can be seen that:

• As \(\alpha\) increases, the values of CATSen and CATMean gradually increase, indicating that the weight of positive samples and the sig group increase, further boosting these metrics.

• As \(\beta\) increases, the value of CATMean also increases, indicating that the model places more emphasis on specificity (the classification ability for negative samples).

5 Case Study: Validation of CAT Metrics Based on Multicenter Cancer Screening Data↩︎

5.1 Experimental Configuration and Data Preparation↩︎

This study uses a clinical multicenter dataset. The data preprocessing and model construction process is as follows:

• Data Features:

  • Genomic structural variations: Fragmentation features (frag.arm), Copy number variations (cnv)

  • Motif break patterns: motif.breakpoint

  • Clinical metadata: Patient ID, detection frequency, cohort origin, etc.

• Cohort Division:

  • Training Set: 1,527 samples from 14 cohorts including K799, R083, etc.

  • Validation Set: 718 samples from 4 cohorts including KAG9, Rep69, etc.

  • Test Set: 221 samples from 4 independent cohorts including KAE3, etc.

  • High-risk Concern Group (sig group): TubeTest, KAG9, Rep69

• Parameter Selection:

  • \(\alpha\) = 0.7

  • \(\beta\) = 0.5

• Model Architecture:

  

5.2 Validation Set and Test Set Group Results↩︎

This table presents the sensitivity and specificity scores across different cohorts for the validation and test sets. Sensitivity values correspond to positive cases, while specificity values represent negative cases.

5.3 Validation and Test Set Comprehensive Indicator Analysis↩︎

In practical medical screening, due to the relatively low sample proportion of high-risk focus groups (sig groups), such as TubeTest, KAG9, and Rep69, traditional classification metrics (e.g., sensitivity and specificity) often result in overestimated scores. For example, in the validation set, the traditional sensitivity is 0.857, and specificity is 0.841, which on the surface suggests good model performance. However, a deeper analysis of the high-risk groups reveals that the sensitivity for KAG9 is only 0.333 (1/3), and for Rep69, the sensitivity is only 0.667 (20/30). This indicates that the model has a high misdiagnosis rate in these high-risk groups. Additionally, all samples in the TubeTest cohort are negative, giving a specificity of 1.000 (78/78). This cohort is selected to demonstrate the effect of the sig group on the score, as typically, the sig group consists of all positive samples, reducing the false negative rate.

To more fairly assess the model’s performance in the sig groups, the CAT metric system applies a tester-level aggregation and population-weighting mechanism, giving higher weight to the classification ability of these key groups. In the validation set, the CAT sensitivity drops to 0.686, while specificity slightly increases, providing a more accurate reflection of the model’s potential misdiagnosis risk in high-risk populations.

In the test set, the overall model performance declines, with traditional sensitivity dropping to 0.518, and specificity remaining high at 0.903. Since there are no sig groups, the score is only weighted by distribution. The accuracy of the xTenHP group is low, with only (9/21), but since its proportion is small, the overall specificity score remains high. After applying information entropy weighting, the CAT specificity drops from 0.903 to 0.827.

The following figure provides a visual comparison of traditional indicators and CAT metrics for both the validation and test sets, further illustrating that traditional evaluation methods may overestimate the overall performance of the model, while the CAT evaluation system provides a more balanced and accurate diagnostic assessment.

In this experiment, there is a significant difference in CATSensitivity between the validation and test sets, while the decrease in CATSpecificity is relatively small. Since \(\beta\) was set to 0.5 in this experiment, it means that the weight of positive samples (sensitivity) is higher, which reduces the missed diagnosis rate. Furthermore, the difference in CATMean scores between the validation and test sets is notable, primarily because, with a smaller \(\beta\) value, CATSen has a greater impact on the final score, making the overall score more sensitive to changes in sensitivity. This indicates that when optimizing the model, selecting an appropriate \(\beta\) value can effectively balance sensitivity and specificity to meet different screening requirements.

\[\text{CATSen} = \left(1-\frac{1}{1+e^{0.5-\alpha}}\right)\frac{1}{|K^+_\star|}\sum_{\forall k\in K^+_\star}A^+_k + \left(\frac{1}{1+e^{0.5-\alpha}}\right)\frac{1}{|K^+|-|K^+_\star|}\sum_{\forall k \in K^+/K^+_\star}A^+_k\]

\[\text{CATSen} = \left(1-\frac{1}{1+e^{0.5-\alpha}}\right)\frac{1}{|K^+_\star|}\sum_{\forall k\in K^+_\star}A^+_k + \left(\frac{1}{1+e^{0.5-\alpha}}\right)\frac{1}{|K|-|K^+_\star|}\sum_{\forall k \in K/K^+_\star}A^-_k\]

\[\text{CATSpe} = \frac{\alpha}{|K^+_\star|}\sum_{\forall k\in K^+_\star}A^+_k + \frac{(1-\alpha)}{|K|-|K^+_\star|}\sum_{\forall k \in K/K^+_\star}A^-_k\]

\[A^\ast_k = \frac{\sum_{\forall i\in C^+_k}(-p^\ast_i\log p^\ast_i)\cdot {Acc}^\ast_i}{\sum_{\forall i\in C^+_k}(-p^\ast_i\log p^\ast_i)},\;\;\forall\ast\in\{+,-\}\]

\[p^\ast_i=\frac{n^\ast_i}{N^\ast_i}\]

\[Acc^\ast_i=\frac{cn^\ast_i}{n^\ast_i}\]

\[\text{CATmean}_\beta = \sqrt{\frac{(1+\beta^2)\cdot\text{CATSen}\cdot\text{CATSpe}}{\beta^2\cdot\text{CATSen}+\text{CATSpe}}}\]

In order to show the influence of each parameter on the values, we set two datasets and print the variation trend under each parameters. The datasets has such attributes, ‘ID’, ‘\(tied ID\)’, ‘cohort’, ‘\(true label\)’, ‘\(pred label\)’, ‘\(pred proba\)’. In the first dataset A, we set 100 items, with 30 different \(tied ID\), 10 different ‘cohort’, a 6 to 4 ratio of positive to negative, a precision about 85% to 90%. In the second dataset B, we set 50 items, with 15 different \(tied ID\), 6 different ‘cohort’, a same ratio of positive to negative and precision to dataset A.

We test AUC, sensitivity, specificity, CATSensitivity, CATSpecificity, CATmean in this experiment, and we made 4 figures to show the result of experiment. The figure 1 shows the experiment result of value variation with the parameter alpha in dataset A. The figure 2 shows the experiment result of value variation with the parameter beta in dataset A. The figure 3 shows the experiment result of value variation with the parameter alpha in dataset B. The figure 4 shows the experiment result of value variation with the parameter beta in dataset B.

From figure 1 and figure 3, we can find out that the AUC, sensitivity, specificity values is not affected by parameter alpha. The CATSensitivity, CATSpecificity values are decreased linear with alpha. The CATmean value is decreasing in non linear, with a larger value of beta, a larger decreasing rate of CATmean.

From figure 2 and figure 4, we can find out that the CATmean value decrease with parameter beta, with a larger value of alpha, a larger decreasing rate of CATmean.

6 Conclusion↩︎

In this work, we have introduced the cohort-attention evaluation metrics for tied data, a novel framework that addresses fundamental limitations of traditional classification metrics in medical screening scenarios. By shifting the evaluation unit from individual samples to individual patients through a tied-ID aggregation mechanism, our approach explicitly accounts for the inherent correlation among repeated measurements from the same subject, thereby reducing evaluation variance and providing a more stable assessment of model performance in real-world clinical settings. The entropy-based distribution weighting further prevents a small number of excessively tested individuals from dominating the overall score, while the cohort-weighted sensitivity and specificity, controlled by interpretable parameters \(\alpha\) and \(\beta\), allow researchers and clinicians to flexibly emphasise either high-risk subgroups or the general population according to specific screening objectives. The harmonic mean-based CATMean serves as a unified, single-value metric that balances sensitivity and specificity without requiring manual inspection of multiple conflicting indicators, offering a practical and efficient tool for model comparison and selection. Our experimental validation on multicenter cancer screening data demonstrates that traditional metrics tend to overestimate performance, particularly when minority high-risk cohorts exhibit poor classification accuracy, whereas CAT metrics reveal these hidden deficiencies and yield more honest and actionable evaluations. The parameter analysis further confirms that CATMean responds consistently to changes in \(\alpha\) and \(\beta\), providing clear guidance for parameter tuning in different application contexts. Despite these advantages, we acknowledge several limitations that merit future investigation. The current entropy weighting relies on a statistical definition of information content that may not perfectly align with clinical notions of risk, and the aggregation strategy assumes equal importance of all tests per patient, whereas in practice, later tests might carry more diagnostic weight. Moreover, the choice of the sigmoid weighting function for CATSen, while theoretically sound, introduces a saturation region that narrows the effective range of \(\alpha\), and we plan to explore alternative monotonic transformations with sharper transition characteristics. Future work will also extend the CAT framework to handle multi-class problems, incorporate temporal dynamics of sequential tests, and develop automated procedures for selecting optimal \(\alpha\) and \(\beta\) based on clinical utility functions rather than heuristic rules. We believe that the proposed metrics represent a significant step toward more rigorous, fair, and clinically meaningful evaluation of AI models in medical diagnostics, and we encourage the community to adopt and further refine these measures to bridge the gap between machine learning benchmarks and patient-centered outcomes.

Declarations↩︎

Data availability The datasets and Python scripts for implementing of our methodology are available publicly on GitHub repository at https://github.com/DeepCox-AI/Stack.

References↩︎