Improving the Estimation of Prediction Increment Measures in Logistic and Survival Analysis
Simple Summary
Abstract
1. Introduction
1.1. ΔAUC for Evaluating Discrimination Improvement
1.2. NRI, NRI > 0, and IDI: Definitions and Controversaries
1.3. Challenges in Estimation of Discrimination Improvement Measures
2. Methods
2.1. General Framework: Logistic Regression
2.2. General Framework: Time-to-Event Regression
2.3. Measures of Discrimination Improvement
2.3.1. AUC and Overall C
2.3.2. NRI, NRI > 0, and NRI(t)
2.3.3. IDI and IDI(t)
2.4. Confidence Interval Estimation
3. Simulation Study Results
4. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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n = 2000 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Strong New Marker (μY=1 = 0.8) | Moderate New Marker (μY=1 = 0.5) | |||||||||
∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Asymptotic | 93.7 | 84.4 | 93.4 | 94.4 | 76.1 | 94.3 | 81.2 | 95.1 | 94.5 | 75.2 |
Bias Corrected | 94.6 | 94.5 | 91.0 | 94.6 | 95.5 | 94.8 | 92.8 | 92.4 | 94.7 | 94.2 |
Percentile | 94.7 | 97.0 | 97.5 | 95.5 | 95.4 | 94.7 | 97.4 | 99.1 | 96.2 | 94.0 |
Bootstrap-t | 95.1 | 94.4 | 89.3 | 93.7 | 95.5 | 95.3 | 90.7 | 89.8 | 94.1 | 96.1 |
Weak New Marker (μY=1 = 0.2) | Null New Marker (μY=1 = 0.0) | |||||||||
∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Asymptotic | 90.6 | 86.4 | 93.5 | 95.4 | 71.0 | 100.0 | 94.6 | 94.1 | 95.8 | 94.0 |
Bias Corrected | 93.3 | 88.0 | 87.6 | 93.8 | 92.7 | 98.4 | 90.5 | 89.7 | 96.0 | 95.5 |
Percentile | 95.5 | 99.5 | 99.4 | 97.7 | 94.8 | 98.9 | 100.0 | 100.0 | 98.4 | 95.9 |
Bootstrap-t | 87.4 | 84.8 | 83.6 | 88.7 | 87.9 | 98.6 | 88.1 | 86.0 | 90.0 | 98.4 |
n = 300 | ||||||||||
Strong New Marker (μY=1 = 0.8) | Moderate New Marker (μY=1 = 0.5) | |||||||||
∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Asymptotic | 92.6 | 80.9 | 93.3 | 94.2 | 76.4 | 88.7 | 78.6 | 92.2 | 93.5 | 71.8 |
Bias Corrected | 94.3 | 91.2 | 91.8 | 95.3 | 95.4 | 92.3 | 87.1 | 87.0 | 92.3 | 92.9 |
Percentile | 94.4 | 97.6 | 99.2 | 97.2 | 94.9 | 95.9 | 99.1 | 99.7 | 98.4 | 94.8 |
Bootstrap-t | 96.0 | 88.8 | 87.5 | 93.6 | 97.3 | 86.4 | 82.8 | 83.0 | 87.3 | 90.5 |
Weak New Marker (μY=1 = 0.2) | Null New Marker (μY=1 = 0.0) | |||||||||
∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Asymptotic | 88.4 | 79.1 | 91.8 | 96.4 | 75.3 | 99.5 | 89.9 | 93.2 | 93.9 | 96.8 |
Bias Corrected | 91.0 | 83.1 | 85.5 | 89.7 | 85.6 | 98.4 | 88.0 | 87.4 | 94.6 | 94.5 |
Percentile | 97.1 | 99.9 | 100.0 | 97.7 | 97.3 | 98.8 | 100.0 | 100.0 | 98.4 | 95.2 |
Bootstrap-t | 86.1 | 85.0 | 82.8 | 82.2 | 73.3 | 96.9 | 89.7 | 86.7 | 89.0 | 97.8 |
3catNRI categories: [0.00, 0.05), [0.05, 0.20), and [0.20+). | ||||||||||
2catNRI categories: [0.00, 0.10) and [0.10+). |
n = 2000 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Strong New Marker (μY=1 = 0.8) | Moderate New Marker (μY=1 = 0.5) | |||||||||
∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Asymptotic | 95.5 | 89.7 | 93.8 | 95.7 | 70.4 | 94.5 | 90.4 | 93.9 | 93.6 | 69.9 |
Bias Corrected | 95.5 | 94.2 | 94.2 | 96.3 | 96.0 | 94.2 | 93.6 | 93.0 | 94.4 | 94.4 |
Percentile | 95.5 | 96.3 | 96.9 | 96.4 | 96.1 | 93.9 | 97.1 | 96.8 | 95.7 | 94.3 |
Bootstrap-t | 95.5 | 93.4 | 93.4 | 94.5 | 96.6 | 94.6 | 91.7 | 91.2 | 94.5 | 95.2 |
Weak New Marker (μY=1 = 0.2) | Null New Marker (μY=1 = 0.0) | |||||||||
∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Asymptotic | 95.2 | 92.0 | 95.3 | 94.5 | 70.1 | 99.9 | 95.5 | 92.9 | 93.8 | 91.8 |
Bias Corrected | 96.0 | 91.8 | 91.4 | 94.1 | 96.1 | 97.6 | 90.5 | 91.2 | 95.8 | 79.4 |
Percentile | 95.4 | 99.6 | 99.3 | 96.1 | 95.8 | 97.8 | 100.0 | 100.0 | 98.0 | 78.4 |
Bootstrap-t | 95.4 | 88.8 | 88.4 | 94.0 | 96.2 | 97.8 | 88.3 | 89.1 | 90.6 | 96.9 |
n = 300 | ||||||||||
Strong New Marker (μY=1 = 0.8) | Moderate New Marker (μY=1 = 0.5) | |||||||||
∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Asymptotic | 94.4 | 89.8 | 96.1 | 93.8 | 71.7 | 93.1 | 89.8 | 95.0 | 94.3 | 71.4 |
Bias Corrected | 94.9 | 93.5 | 94.0 | 94.0 | 94.3 | 94.9 | 90.6 | 91.5 | 94.9 | 94.5 |
Percentile | 95.0 | 98.4 | 98.7 | 95.9 | 93.8 | 95.2 | 98.2 | 99.1 | 96.5 | 94.8 |
Bootstrap-t | 95.9 | 91.0 | 90.7 | 93.7 | 96.1 | 95.6 | 88.6 | 88.0 | 94.4 | 96.4 |
Weak New Marker (μY=1 = 0.2) | Null New Marker (μY=1 = 0.0) | |||||||||
∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆AUC | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Asymptotic | 86.6 | 90.0 | 93.3 | 96.3 | 68.4 | 99.9 | 94.8 | 92.2 | 94.1 | 93.6 |
Bias Corrected | 89.9 | 86.8 | 89.8 | 90.8 | 87.0 | 98.2 | 90.2 | 92.1 | 94.3 | 76.9 |
Percentile | 98.8 | 100.0 | 100.0 | 99.1 | 98.1 | 98.4 | 100.0 | 100.0 | 97.3 | 72.1 |
Bootstrap-t | 78.7 | 82.1 | 85.0 | 84.4 | 77.3 | 98.6 | 86.9 | 89.0 | 89.0 | 97.9 |
3catNRI categories: [0.00, 0.40), [0.40, 0.60), and [0.60+). | ||||||||||
2catNRI categories: [0.00, 0.50) and [0.50+). |
n = 2000 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Strong New Marker (Adjusted HR = 2.0) | Moderate New Marker (Adjusted HR = 1.5) | |||||||||
∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Bias Corrected | 95.1 | 94.3 | 93.1 | 93.9 | 95.7 | 95.6 | 92.1 | 90.5 | 95.1 | 94.8 |
Percentile | 95.2 | 96.9 | 98.4 | 95.5 | 95.8 | 95.3 | 97.5 | 98.4 | 96.5 | 94.8 |
Bootstrap-t | 95.5 | 93.2 | 92.4 | 93.5 | 96.2 | 96.2 | 90.8 | 90.2 | 94.1 | 96.2 |
Hybrid | 93.6 | 91.8 | 92.5 | 93.3 | 92.8 | 89.9 | 89.6 | 90.5 | 93.7 | 89.1 |
Weak New Marker (Adjusted HR = 1.2) | Null New Marker (Adjusted HR = 1.0) | |||||||||
∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Bias Corrected | 93.2 | 86.0 | 89.2 | 92.5 | 93.2 | 98.3 | 89.1 | 90.1 | 95.0 | 91.2 |
Percentile | 96.9 | 99.6 | 99.9 | 97.6 | 95.2 | 98.8 | 100.0 | 100.0 | 98.2 | 92.0 |
Bootstrap-t | 91.7 | 85.2 | 86.9 | 91.7 | 95.0 | 97.9 | 84.9 | 85.1 | 91.9 | 98.7 |
Hybrid | 75.7 | 86.8 | 90.0 | 91.1 | 76.6 | 100.0 | 94.2 | 90.9 | 90.7 | 100.0 |
n = 300 | ||||||||||
Strong New Marker (Adjusted HR = 2.0) | Moderate New Marker (Adjusted HR = 1.5) | |||||||||
∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Bias Corrected | 94.3 | 90.4 | 91.0 | 92.6 | 94.3 | 87.9 | 85.4 | 90.0 | 91.7 | 91.6 |
Percentile | 94.2 | 97.8 | 99.2 | 97.1 | 94.5 | 95.6 | 99.7 | 99.8 | 98.9 | 96.7 |
Bootstrap-t | 95.3 | 90.9 | 89.1 | 93.1 | 95.2 | 87.3 | 80.5 | 85.5 | 90.0 | 92.1 |
Hybrid | 83.6 | 84.7 | 88.2 | 90.4 | 81.3 | 77.8 | 81.5 | 88.9 | 88.3 | 73.1 |
Weak New Marker (Adjusted HR = 1.2) | Null New Marker (Adjusted HR = 1.0) | |||||||||
∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Bias Corrected | 92.6 | 82.5 | 89.1 | 89.2 | 84.5 | 98.5 | 85.7 | 84.3 | 93.4 | 84.8 |
Percentile | 98.3 | 100.0 | 100.0 | 98.1 | 97.5 | 99.0 | 99.8 | 100.0 | 97.7 | 83.2 |
Bootstrap-t | 82.1 | 73.3 | 79.0 | 87.3 | 82.2 | 96.9 | 78.1 | 77.9 | 90.8 | 95.2 |
Hybrid | 92.4 | 91.0 | 87.9 | 86.5 | 66.3 | 99.7 | 93.3 | 84.1 | 89.6 | 99.7 |
3catNRI categories: [0.00, 0.05), [0.05, 0.20), and [0.20+). | ||||||||||
2catNRI categories: [0.00, 0.10) and [0.10+). |
n = 2000 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|
Strong New Marker (Adjusted HR = 2.0) | Moderate New Marker (Adjusted HR = 1.5) | |||||||||
∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Bias Corrected | 95.6 | 94.1 | 95.2 | 94.5 | 94.5 | 95.0 | 93.4 | 92.9 | 94.5 | 95.7 |
Percentile | 95.9 | 96.0 | 97.0 | 94.9 | 94.6 | 94.6 | 96.9 | 97.3 | 96.5 | 95.3 |
Bootstrap-t | 95.7 | 93.4 | 95.3 | 94.2 | 94.7 | 94.9 | 93.0 | 92.6 | 94.8 | 95.4 |
Hybrid | 95.9 | 93.8 | 95.0 | 94.3 | 94.2 | 94.0 | 93.2 | 92.7 | 94.9 | 95.0 |
Weak New Marker (Adjusted HR = 1.2) | Null New Marker (Adjusted HR = 1.0) | |||||||||
∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Bias Corrected | 94.8 | 91.8 | 92.7 | 93.1 | 94.3 | 99.3 | 92.8 | 86.1 | 96.2 | 97.1 |
Percentile | 94.8 | 98.2 | 99.1 | 94.9 | 94.4 | 99.4 | 100.0 | 100.0 | 98.5 | 97.9 |
Bootstrap-t | 95.7 | 91.7 | 90.5 | 93.4 | 95.3 | 97.9 | 85.8 | 79.4 | 91.9 | 98.6 |
Hybrid | 90.9 | 90.5 | 92.1 | 93.2 | 90.1 | 100.0 | 95.4 | 87.0 | 91.9 | 100.0 |
n = 300 | ||||||||||
Strong New Marker (Adjusted HR = 2.0) | Moderate New Marker (Adjusted HR = 1.5) | |||||||||
∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Bias Corrected | 95.2 | 92.7 | 92.8 | 94.9 | 95.0 | 95.4 | 92.5 | 93.4 | 93.9 | 94.8 |
Percentile | 94.8 | 97.6 | 99.1 | 96.0 | 95.4 | 95.3 | 98.8 | 99.4 | 95.7 | 94.9 |
Bootstrap-t | 95.1 | 92.9 | 91.6 | 94.5 | 95.2 | 95.8 | 92.5 | 91.9 | 93.9 | 96.3 |
Hybrid | 93.3 | 92.0 | 91.8 | 94.1 | 92.6 | 88.3 | 91.3 | 92.3 | 93.6 | 88.5 |
Weak New Marker (Adjusted HR = 1.2) | Null New Marker (Adjusted HR = 1.0) | |||||||||
∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | ∆C | 3catNRI | 2catNRI | NRI > 0 | IDI | |
Bias Corrected | 90.4 | 87.0 | 91.4 | 92.8 | 90.4 | 99.4 | 88.7 | 76.2 | 95.0 | 96.9 |
Percentile | 98.0 | 99.8 | 99.9 | 98.9 | 96.8 | 99.5 | 100.0 | 100.0 | 98.6 | 97.6 |
Bootstrap-t | 86.8 | 85.3 | 86.2 | 91.7 | 90.3 | 97.9 | 77.0 | 67.7 | 91.5 | 98.2 |
Hybrid | 80.2 | 89.6 | 89.3 | 91.4 | 73.1 | 99.9 | 93.6 | 74.2 | 91.2 | 100.0 |
3catNRI categories: [0.00, 0.40), [0.40, 0.60), and [0.60+). | ||||||||||
2catNRI categories: [0.00, 0.50) and [0.50+). |
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Enserro, D.M.; Miller, A. Improving the Estimation of Prediction Increment Measures in Logistic and Survival Analysis. Cancers 2025, 17, 1259. https://doi.org/10.3390/cancers17081259
Enserro DM, Miller A. Improving the Estimation of Prediction Increment Measures in Logistic and Survival Analysis. Cancers. 2025; 17(8):1259. https://doi.org/10.3390/cancers17081259
Chicago/Turabian StyleEnserro, Danielle M., and Austin Miller. 2025. "Improving the Estimation of Prediction Increment Measures in Logistic and Survival Analysis" Cancers 17, no. 8: 1259. https://doi.org/10.3390/cancers17081259
APA StyleEnserro, D. M., & Miller, A. (2025). Improving the Estimation of Prediction Increment Measures in Logistic and Survival Analysis. Cancers, 17(8), 1259. https://doi.org/10.3390/cancers17081259