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A calibration hierarchy for risk models was defined: from utopia to empirical data
Open accessAbstract
Objective Calibrated risk models are vital for valid decision support. We define four levels of calibration and describe implications for model development and external validation of predictions.Study design and setting We present results based on simulated data sets.Results A common definition of calibration is "having an event rate of R% among patients with a predicted risk of R%," which we refer to as "moderate calibration." Weaker forms of calibration only require the average predicted risk (mean calibration) or the average prediction effects (weak calibration) to be correct. "Strong calibration" requires that the event rate equals the predicted risk for every covariate pattern. This implies that the model is fully correct for the validation setting. We argue that this is unrealistic: the model type may be incorrect, the linear predictor is only asymptotically unbiased, and all nonlinear and interaction effects should be correctly modeled. In addition, we prove that moderate calibration guarantees nonharmful decision making. Finally, results indicate that a flexible assessment of calibration in small validation data sets is problematic.Conclusion Strong calibration is desirable for individualized decision support but unrealistic and counter productive by stimulating the development of overly complex models. Model development and external validation should focus on moderate calibration.
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“There was some miscalibration in mean risk for Examples 1-2 where the development sample was small, even though the correct underlying model formulation was estimated.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 3.1. Level 1: mean calibration (Calibration-in-the-large)
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“Enlarging the development dataset alleviated the problems (Example 4 in Table 1, Figure 1D).”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 2.2. Illustration: Examples 1-5 For illustration, we consider five simulated examples, as previously presented [5].We randomly generate four independent predictor variables to .These predictor variables are ordinal with three categories (-1, 0, and 1) that each have 33% prevalence in order to visualize calibration by covariate pattern.Let outcome be generated by an underlying logistic regression model with the true linear predictor , hence the intercept equals 0, and to are assumed to have linear main effects [5].In simulations, the true probability of event is calculated using the linear predictor and the observed outcome is generated as a Bernoulli variable from .Then, we develop a model to predict based on to , which means that we are using the correct model formulation.Finally, we validate on a new dataset that is generated with the exact same procedure.We repeat this process four times to illustrate the influence of random variability: (1) using 100 simulated patients for development and 100 for validation, (2) using 100 for development and 10,000 for validation, (3) using 10,000 for development and 100 for validation, (4) using 10,000 for development and 10,000 for validation, and (5) using 10 million for development and 10 million for validation.Estimation of model coefficients was unstable when the development data contains only 100 patients (Table 1), despite having more than 10 events per variable (4 parameters, 44 non-events, 56 events).Risk estimates were overfitted, as evidenced by the calibration slope and calibration curves at validation (Example 2 in Table 1, and Figure 1B) as well as the calibration curves.Further, the calibration intercept was negative which is suggestive of general overestimation.Enlarging the development dataset alleviated the problems (Example 4 in Table 1, Figure 1D).Similar issues emerge when using a small validation dataset (Figures 1A and 1C.Calibration results for Examples 1 and 3 are unstable, in particular the flexible calibration curve.With 10 million patients the coefficient estimates and calibration results were perfect (Table 1, Figure 1E).
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“The results suggest that the flexible calibration curve is more variable and hence requires more data than the calibration slope for a stable assessment.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 8. Sample size at external validation
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“It is clear that observed calibration curves will easily deviate from the diagonal line even when the model is moderately or strongly calibrated.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 8. Sample size at external validation
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“Nevertheless, the decision curve for the true model is the best one.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 4. Calibration, decision-making, and clinical utility
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“Again, only the flexible calibration curve reveals that calibration is problematic.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 3.4. Missing non-linearity or interaction: examples 6-7
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“The moderately recalibrated version does not have that problem anymore: Net Benefit is now at least as high as that of the default strategies.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 4. Calibration, decision-making, and clinical utility
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“The original model has Net Benefit below that of treat all or treat none for a subset of risk thresholds (Figure 5).”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 4. Calibration, decision-making, and clinical utility
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“For larger development datasets the disagreement between observed and predicted risks disappeared (Examples 3-5).”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 3.1. Level 1: mean calibration (Calibration-in-the-large)
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“results indicate that a flexible assessment of calibration in small validation datasets is problematic.”
- Also stated in: § 2.2. Illustration: Examples 1-5 For illustration, we consider five simulated examples, as previously presented [5].We randomly generate four independent predictor variables to .These predictor variables are ordinal with three categories (-1, 0, and 1) that each have 33% prevalence in order to visualize calibration by covariate pattern.Let outcome be generated by an underlying logistic regression model with the true linear predictor , hence the intercept equals 0, and to are assumed to have linear main effects [5].In simulations, the true probability of event is calculated using the linear predictor and the observed outcome is generated as a Bernoulli variable from .Then, we develop a model to predict based on to , which means that we are using the correct model formulation.Finally, we validate on a new dataset that is generated with the exact same procedure.We repeat this process four times to illustrate the influence of random variability: (1) using 100 simulated patients for development and 100 for validation, (2) using 100 for development and 10,000 for validation, (3) using 10,000 for development and 100 for validation, (4) using 10,000 for development and 10,000 for validation, and (5) using 10 million for development and 10 million for validation.Estimation of model coefficients was unstable when the development data contains only 100 patients (Table 1), despite having more than 10 events per variable (4 parameters, 44 non-events, 56 events).Risk estimates were overfitted, as evidenced by the calibration slope and calibration curves at validation (Example 2 in Table 1, and Figure 1B) as well as the calibration curves.Further, the calibration intercept was negative which is suggestive of general overestimation.Enlarging the development dataset alleviated the problems (Example 4 in Table 1, Figure 1D).Similar issues emerge when using a small validation dataset (Figures 1A and 1C.Calibration results for Examples 1 and 3 are unstable, in particular the flexible calibration curve.With 10 million patients the coefficient estimates and calibration results were perfect (Table 1, Figure 1E). · plain
“Calibration results for Examples 1 and 3 are unstable, in particular the flexible calibration curve.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > Abstract
plainn = 100- Asserted: passes — The authors put it forward as true.
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- Also stated in: § 2.2. Illustration: Examples 1-5 For illustration, we consider five simulated examples, as previously presented [5].We randomly generate four independent predictor variables to .These predictor variables are ordinal with three categories (-1, 0, and 1) that each have 33% prevalence in order to visualize calibration by covariate pattern.Let outcome be generated by an underlying logistic regression model with the true linear predictor , hence the intercept equals 0, and to are assumed to have linear main effects [5].In simulations, the true probability of event is calculated using the linear predictor and the observed outcome is generated as a Bernoulli variable from .Then, we develop a model to predict based on to , which means that we are using the correct model formulation.Finally, we validate on a new dataset that is generated with the exact same procedure.We repeat this process four times to illustrate the influence of random variability: (1) using 100 simulated patients for development and 100 for validation, (2) using 100 for development and 10,000 for validation, (3) using 10,000 for development and 100 for validation, (4) using 10,000 for development and 10,000 for validation, and (5) using 10 million for development and 10 million for validation.Estimation of model coefficients was unstable when the development data contains only 100 patients (Table 1), despite having more than 10 events per variable (4 parameters, 44 non-events, 56 events).Risk estimates were overfitted, as evidenced by the calibration slope and calibration curves at validation (Example 2 in Table 1, and Figure 1B) as well as the calibration curves.Further, the calibration intercept was negative which is suggestive of general overestimation.Enlarging the development dataset alleviated the problems (Example 4 in Table 1, Figure 1D).Similar issues emerge when using a small validation dataset (Figures 1A and 1C.Calibration results for Examples 1 and 3 are unstable, in particular the flexible calibration curve.With 10 million patients the coefficient estimates and calibration results were perfect (Table 1, Figure 1E). · plain
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“This model exhibits perfect strong calibration: both calibration curves and all covariate patterns coincide with the diagonal.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 3.6. Model misspecification: Examples 8-9
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“Thus, the model is weakly but not moderately calibrated on the development data.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 3.4. Missing non-linearity or interaction: examples 6-7
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“This model exhibits perfect moderate calibration but lacks strong calibration because results per covariate pattern are scattered around the diagonal.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 3.6. Model misspecification: Examples 8-9
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“With 10 million patients the coefficient estimates and calibration results were perfect (Table 1, Figure 1E).”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 2.2. Illustration: Examples 1-5 For illustration, we consider five simulated examples, as previously presented [5].We randomly generate four independent predictor variables to .These predictor variables are ordinal with three categories (-1, 0, and 1) that each have 33% prevalence in order to visualize calibration by covariate pattern.Let outcome be generated by an underlying logistic regression model with the true linear predictor , hence the intercept equals 0, and to are assumed to have linear main effects [5].In simulations, the true probability of event is calculated using the linear predictor and the observed outcome is generated as a Bernoulli variable from .Then, we develop a model to predict based on to , which means that we are using the correct model formulation.Finally, we validate on a new dataset that is generated with the exact same procedure.We repeat this process four times to illustrate the influence of random variability: (1) using 100 simulated patients for development and 100 for validation, (2) using 100 for development and 10,000 for validation, (3) using 10,000 for development and 100 for validation, (4) using 10,000 for development and 10,000 for validation, and (5) using 10 million for development and 10 million for validation.Estimation of model coefficients was unstable when the development data contains only 100 patients (Table 1), despite having more than 10 events per variable (4 parameters, 44 non-events, 56 events).Risk estimates were overfitted, as evidenced by the calibration slope and calibration curves at validation (Example 2 in Table 1, and Figure 1B) as well as the calibration curves.Further, the calibration intercept was negative which is suggestive of general overestimation.Enlarging the development dataset alleviated the problems (Example 4 in Table 1, Figure 1D).Similar issues emerge when using a small validation dataset (Figures 1A and 1C.Calibration results for Examples 1 and 3 are unstable, in particular the flexible calibration curve.With 10 million patients the coefficient estimates and calibration results were perfect (Table 1, Figure 1E).
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“First, strong calibration requires that the model form (e.g. a generalized linear model such as logistic regression) is correct.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 5. Strong calibration: realistic or utopic?
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“Refitting should lead to moderate calibration, although this may also require reassessment of nonlinear effects.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 7. A link with model updating
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“in reality, the true model is unknown, may not have a generalized linear form, or may not even exist [22].”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 5. Strong calibration: realistic or utopic?
What the sources say
- Not checked yetcites [22] Chatfield, 1995, Model uncertainty, data mining and statistical inference, J R Stat Soc Ser A source
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“In sum, aiming for strong calibration requires the assumption that the model formulation is fully correct, and that unbiased model coefficients as well as unbiased linear predictors are obtained.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 5. Strong calibration: realistic or utopic?
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In the paper
“This method addresses lack of weak calibration, but does not guarantee moderate calibration unless all coefficients were biased by the same rate.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 7. A link with model updating
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In the paper
“we prove (see Appendix) that clinical utility in terms of Net Benefit will not be lower than the default strategies that either classify everyone as having the event or as not having the event.”
- Also stated in: § 6. Moderate calibration: a pragmatic guarantee for non-harmful decision-making · plain
“moderate calibration guarantees that decision-making based on the model is not clinically harmful.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 4. Calibration, decision-making, and clinical utility
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- Also stated in: § 6. Moderate calibration: a pragmatic guarantee for non-harmful decision-making · plain
In the paper
“For example, strong interactions or nonlinearities may lead to miscalibration in the development sample although weak calibration is perfect [11].”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 3.3. Level 3: moderate calibration
What the sources say
- Not checked yetcites [11] Austin, 2014, Graphical assessment of internal and external calibration of logistic regression models by using loess smoothers, Stat Med source
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In the paper
“Third, strong calibration not only requires correctly estimated regression coefficients for the main effects of model predictors, but also requires fully correct modeling of all nonlinear and non-additive effects.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 5. Strong calibration: realistic or utopic?
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In the paper
“For a moderately calibrated model the following property holds, among patients with an estimated risk of % on average out of 100 of these patients have the event.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > Additional information
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In the paper
“The odds of the risk threshold is the harm-to-benefit ratio, i.e. the ratio of the harm of a false positive and the benefit of a true positive classification[21].”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > cex.d01
What the sources say
- Not checked yetcites [21] Pauker, 1975, Therapeutic decision making: a cost-benefit analysis, N Engl J Med source
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In the paper
“This suggests that moderate calibration prevents the model from becoming harmful [18] but that strong calibration may further enhance the model's clinical utility.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 4. Calibration, decision-making, and clinical utility
What the sources say
- Not checked yetcites [18] Van Calster, 2015, Calibration of risk prediction models: impact on decision-analytic performance, Med Decis Making source
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In the paper
“This will only address calibration-in-the-large, but does not guarantee weak calibration.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 7. A link with model updating
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In the paper
“While still providing sensible risk predictions, simple models that are for example moderately but not strongly calibrated often have many practical advantages such as transparency or ease of use [33].”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 6. Moderate calibration: a pragmatic guarantee for non-harmful decision-making
What the sources say
- Not checked yetcites [33] Marewski, 2012, Heuristic decision making in medicine, Dialogues Clin Neurosci source
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In the paper
“This means that at least part of the measurement error is systematic, and further thwarts the concept of the existence of a true model.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 5. Strong calibration: realistic or utopic?
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In the paper
“Calibration is a pivotal aspect of model performance [2][3][4]”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 1. Introduction
What the sources say
- Not checked yetcites [2] Steyerberg, 2009
- Not checked yetcites [3] Kim, 2011, Probabilistic classifiers with high-dimensional data, Biostatistics source
- Not checked yetcites [4] Pepe, 2013, Methods for evaluating prediction performance of biomarkers and tests
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In the paper
“measurement error of the predictors is in practice often ignored although it is known that this is a common phenomenon that can bias regression coefficients [28].”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 5. Strong calibration: realistic or utopic?
What the sources say
- Not checked yetcites [28] Fuller, 1987
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In the paper
“If the model's Net Benefit is lower than that of a default strategy, then using the model to support decision-making can be considered as clinically harmful.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > cex.d01
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In the paper
“In previous simulations, the Hosmer-Lemeshow test showed such poor performance that it may not be recommended for routine use [7,37].”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 9. Statistical testing for calibration
What the sources say
hedged- Asserted: passes — The authors put it forward as true.
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In the paper
“Net Benefit accounts for the different consequences of true and false positives through the risk threshold, i.e. the risk at which one is indifferent about classifying a patient as having the event (and providing treatment) or not.”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > cex.d01
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Calibrated risk models are vital for valid decision support.
“Calibrated risk models are vital for valid decision support.”
Claims other papers make about this one
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“These approaches can be classified into three increasingly stringent levels, labelled as mean, weak, and moderate calibration. 52”
From Evaluation of performance measures in predictive artificial intelligence models to support medical decisions: overview and guidance (2025) · cites this paper as [52]
In the citing paper
“All discussed calibration measures are semi-proper, 52”
From Evaluation of performance measures in predictive artificial intelligence models to support medical decisions: overview and guidance (2025) · cites this paper as [52]
In the citing paper
“Estimated probabilities with too much spread are on average too close to 0 and 1 (slope < 1), whereas estimated probabilities with too little spread are on average too close to the prevalence (slope > 1).”
From Evaluation of performance measures in predictive artificial intelligence models to support medical decisions: overview and guidance (2025) · cites this paper as [52]
In the citing paper
“Previous recommendations were to include at least 100 to 200 individuals in the smallest outcome category.”
From Evaluation of performance measures in predictive artificial intelligence models to support medical decisions: overview and guidance (2025) · cites this paper as [52]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > 4. Calibration, decision-making, and clinical utility
“Let us consider Example 9 (Figure 3C).The model presented is not calibrated in a weak sense.We derived decision curves to assess the clinical utility of (1) the original miscalibrated model, (2) a recalibrated model to ensure moderate calibration, and (3) the strongly calibrated true model (Figure 5).”
§ A calibration hierarchy for risk models was defined: from utopia to empirical data > cex.d01
“Figure: 5 Figure 5 . Figure 5. Decision curves for Example 8 to assess clinical usefulness.”
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Metadata
- Authors
- Van Calster, B., Nieboer, D., Vergouwe, Y., De Cock, B., Pencina, M. J., & Steyerberg, E. W.
- Affiliations
- Erasmus MC (NL)
- Duke University (US)
- Venue
- Journal of Clinical Epidemiology
- Publisher
- Elsevier BV
- Published
- 2016-06
- Volume
- 74
- Pages
- 167-176
- ISSN
- 0895-4356 (ISSN-L)
- OpenAlex
- W2225109326
- Cited by
- 910
- References
- 42
- Citations by year
- 2026: 192 · 2025: 99 · 2024: 118 · 2023: 93 · 2022: 125 · 2021: 91 · 2020: 58 · 2019: 49 · 2018: 35 · 2017: 30 · 2016: 19
- FWCI
- 44.92
- Type
- journal article
- Language
- en
- Open access
- green
- lirias.kuleuven.be
- Licence
- Locations
- Journal of Clinical Epidemiology (publishedVersion)
- PubMed (publishedVersion)
- Lirias (KU Leuven) (submittedVersion, other-oa)
- EUR Research Repository (Erasmus University Rotterdam) (submittedVersion)
- Funders
- Fonds Wetenschappelijk Onderzoek (G049312N)
- KU Leuven (C24/15/037)
- Vlaamse regering (G049312N, C24/15/037)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods in Epidemiology
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Health Sciences › Health Professions › Health Information Management › Artificial Intelligence in Healthcare
- Keywords
- risk model calibration, external validation, decision support, overfitting, simulation study
- MeSH terms
- Calibration, Computer Simulation, Decision Support Techniques, Humans, Risk, Reproducibility of Results, Models, Statistical, Bias, Risk Assessment
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