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Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood
Open accessAbstract
MOTIVATION: Mathematical description of biological reaction networks by differential equations leads to large models whose parameters are calibrated in order to optimally explain experimental data. Often only parts of the model can be observed directly. Given a model that sufficiently describes the measured data, it is important to infer how well model parameters are determined by the amount and quality of experimental data. This knowledge is essential for further investigation of model predictions. For this reason a major topic in modeling is identifiability analysis. RESULTS: We suggest an approach that exploits the profile likelihood. It enables to detect structural non-identifiabilities, which manifest in functionally related model parameters. Furthermore, practical non-identifiabilities are detected, that might arise due to limited amount and quality of experimental data. Last but not least confidence intervals can be derived. The results are easy to interpret and can be used for experimental planning and for model reduction. AVAILABILITY: An implementation is freely available for MATLAB and the PottersWheel modeling toolbox at http://web.me.com/andreas.raue/profile/software.html. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.
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TL;DR
The paper presents a profile-likelihood method for assessing structural and practical identifiability in partially observed dynamical models and for deriving likelihood-based confidence intervals. Applied to a JAK-STAT signaling model, it identifies four structurally non-identifiable parameters and one practically non-identifiable parameter, and shows how additional measurements can resolve these ambiguities.
Key findings
- Profile likelihood distinguishes structural non-identifiability, seen as a flat profile, from practical non-identifiability, seen as a profile with a minimum that fails to cross the confidence threshold in one or both directions; identifiable parameters cross the threshold on both sides.
- In the JAK-STAT application, parameters p2, x1(0), s1, and s2 were structurally non-identifiable, with functional relations p2·x1(0)=const, s1·x1(0)=const, and s2·x1(0)=const; p3 was practically non-identifiable for increasing values.
- Table 1 reports infinite likelihood-based intervals for the structurally non-identifiable parameters, whereas Hessian-based asymptotic intervals were finite but misleadingly narrow, e.g. p2: −33 to +31, x1(0): −31 to +32, s1: −32 to +31, s2: −32 to +31; for p3, the profile-likelihood interval was −1.14 to +∞ versus −1.14 to +0.15 from the Hessian.
- Adding hypothetical measurements of x1(0)=200±20 nM and x3/(x2+x3)=0.90±0.05 at t=20 min made p2, x1(0), s1, and s2 structurally identifiable and p3 practically identifiable, while p2 remained practically non-identifiable. Profile-likelihood computation took 54±18 s per parameter on a 1 GHz CPU with 2 GB RAM.
Methods
- The approach uses an ODE model with weighted least-squares objective χ², computes the profile likelihood for each parameter by re-optimizing all other parameters at fixed values of the target parameter, and derives likelihood-based confidence intervals from χ² thresholds.
- The application is a JAK-STAT signaling model inspired by Swameye et al., calibrated to experimental data; observables are phosphorylated STAT in cytoplasm (y1) and total STAT in cytoplasm (y2) measured by quantitative western blotting, with cytoplasm and nucleus volumes of 1400 µm³ and 450 µm³.
- Parameters calibrated include p1–p4, s1, s2, and x1(0), with x2(0)–x4(0) set to zero; the model was implemented and fitted with PottersWheel, and experimental planning and model reduction were explored via profile-likelihood trajectories, Monte Carlo simulations, and likelihood-ratio thresholds.
Limitations
- The authors state that structural non-identifiability cannot be resolved by refining existing measurements or reducing noise; it requires a qualitatively new measurement that changes the observation mapping, such as observing additional species. Practical non-identifiability can be resolved by increasing the amount or quality of data and/or selecting better measurement time points.
- The authors note that recovering functional relations unambiguously from parameter changes along the profile likelihood is only possible when the corresponding manifold is one-dimensional; higher-dimensional manifolds require further analysis, such as the approach of Hengl et al.
- Evident from the text: the method is demonstrated on a single JAK-STAT model and dataset, with no broad empirical comparison across many model classes; the approach also assumes a sufficiently adequate model and normally distributed measurement noise.
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In the paper
“The largest discrepancy occurs for practically non-identifiable parameters, where asymptotic confidence intervals are significantly smaller than likelihood-based confidence intervals.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“The variability of the predicted model observables y along the profile likelihood of p 3 remains consistent with the measurement errors, as shown in Figure 6.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“The resulting plots of profile likelihood versus parameter reveal four structurally non-identifiable parameters p 2 , x 1 (0), s 1 , s 2 by their flat profile likelihood, see Figure 3, upper panel.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 5 APPLICATION
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In the paper
“For identifiable parameters the discrepancies are small.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“The functional relations h connecting these structurally non-identifiable parameters can be recovered from the changes of the remaining parameters, while calculating the profile likelihood of the structurally non-identifiable parameters, as shown in Figure 4.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“The recalculated profile likelihood reveals, that parameters p 2 , x 1 (0), s 1 , s 2 become structurally identifiable (Fig. 3, lower panel), by measuring only one of them.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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In the paper
“The profile likelihood reveals that parameter p 3 is practically non-identifiable for increasing parameter values.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: An appropriate model M that sufficiently describes the available experimental data is assumed.
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In the paper
“Nevertheless, a lower confidence bound can be derived.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In this case, asymptotic approximation leads to large but nevertheless finite confidence intervals for the structurally non-identifiable parameters.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“All internal model states can only be identified up to a common factor.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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In the paper
“An approach to detect structural non-identifiability by the corresponding functional relations was introduced by Hengl et al. (2007).It is able to detect flatness of likelihood for arbitrary models, but it is not intended to detect practical non-identifiability.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 3 EXISTING METHODS
What the sources say
- Partly supportedcites Hengl et al., 2007 sourceExperimental — accuracy not yet measured
The Approach and Conclusion describe detecting structural non-identifiability through functionally related parameters in nonlinear models. But the Introduction explicitly lists practical identifiability analysis, accounting for data quality, as an aim of the method, contrary to the claim that it is not intended for that purpose. The passages do not specifically describe the method as detecting likelihood flatness. (LLM, conf=0.90)
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In the paper
“Spots of largest variability suggest where and when a measurement of a species most efficiently determines theses parameters: x 1 at times t = 0 or t > 50; x 2 to x 4 at times 5 < t < 30.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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In the paper
“Calculating the profile likelihood takes 54 ± 18 s per parameter, using an implementation of the approach embedded in the Potters-Wheel fitting toolbox (1 GHz CPU, 2 GB RAM), which is described in the Supplementary Material.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 5 APPLICATION
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In the paper
“Parameter p 3 becomes practically identifiable, only previously structurally non-identifiable parameter p 2 remains practically non-identifiable.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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In the paper
“The trajectories of the internal model states x along the profile likelihood of the practically non-identifiable parameter p 3 shown in Figure 6 comprises large antipodal variability of species x 2 and x 3 , revealing that the experimental setup is inappropriate to estimate this parameter with confidence.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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In the paper
“h 1 (θ sub ) = p 2 •x 1 (0)-const = 0 h 2 (θ sub ) = s 1 •x 1 (0)-const = 0 ( 1 1 ) h 3 (θ sub ) = s 2 •x 1 (0)-const = 0”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The approach was applied to a model of the JAK-STAT signaling pathway.Non-identifiable parameters were detected, revealing limitations in the experimental setup.Additional measurements that efficiently improve parameter identification were suggested and validated.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“They are exact if the observables y depend linearly on θ.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.2.1 Asymptotic confidence intervals
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In the paper
“Unfortunately, these methods become rapidly infeasible with increasing model size (Margaria et al., 2001;White et al., 2001).Practical non-identifiability cannot be detected, since experimental data are disregarded.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 3 EXISTING METHODS
What the sources say
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In the paper
“Even identifiable parameters can only be determined within confidence intervals, which contain the true value of the parameter with a desired probability (Lehmann and Leo, 1983).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 1 INTRODUCTION
What the sources say
- Not matchedcites Lehmann and Leo, 1983
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In the paper
“These methods are appropriate if functional relations h between the parameters emerging from structural non-identifiability are linear.This is often not the case for reaction networks modeled by ODE, because observables depend non-linearly on the parameters.Practical non-identifiability cannot be detected, because a quadratic approximation is not able to explain increasing but limited behavior of”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 3 EXISTING METHODS
What the sources say
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In the paper
“However, even for the simplest reaction network the observables y depend nonlinearly on θ and the amount and quality of experimental data is often insufficient.Therefore, asymptotic confidence intervals might not be appropriate (Joshi et al., 2006).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.2.1 Asymptotic confidence intervals
What the sources say
- Not checked — no full text of the sourcecites Joshi et al., 2006 source
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In the paper
“Likelihood-based confidence intervals are considered superior to asymptotic confidence intervals for finite samples (Neale and Miller, 1997).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.2.2 Finite sample confidence intervals
What the sources say
- Not checked — no full text of the sourcecites Neale and Miller, 1997 source
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In the paper
“Along a practical non-identifiability, the observable y change only negligibly remaining compliant with the given measurement accuracy.Nevertheless, model behavior in terms of internal states x might vary strongly.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > )]:
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In the paper
“This arises frequently if amount and quality of experimental data is insufficient and manifests in a confidence interval that is infinite.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 2.3.2 Practical non-identifiability
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In the paper
“The points of passover represent likelihoodbased confidence intervals as defined in Equation ( 8) (Royston, 2007).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 4 APPROACH
What the sources say
- Not checked — no full text of the sourcecites Royston, 2007 source
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In the paper
“The results are easy to interpret”
- Also stated in: § 6 DISCUSSION · plain
“The approach results in easily interpretable plots of profile likelihood versus parameter.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Abstract
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- Also stated in: § 6 DISCUSSION · plain
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In the paper
“The profile likelihood of a practically non-identifiable parameter has a minimum, but is not excessing a threshold α for increasing and/or decreasing values of θ i (see Definition 1).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 4 APPROACH
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In the paper
“This gives insight into a complex and high-dimensional parameter space.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“It can be automated, but an explicit advantage is that the output might be evaluated visually.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“The approach can be applied to any parameter estimation problem, where a likelihood or a similar objective criterion is available, e.g.partial differential equations (PDE) or stochastic differential equations (SDE).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“The profile likelihood can be calculated for each parameter separately.Thereby it is possible to restrict the analysis to the parameters relevant for the biological question.Moreover, this allows to perfectly parallelize the approach, which is a major benefit for its scalability.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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In the paper
“Working in logarithmic parameter space linearizes the functional relations given in Equation (11).”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Structural non-identifiability
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The impact of new measurements can be evaluated by Monte Carlo simulations.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > Experimental planning
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, identifiability analysis ensures that the model complexity is tailored to the information content given by the experimental data.”
§ Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood > 6 DISCUSSION
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The authors state this as an assumption.
The measurement noise is normally distributed.
“The measurement noise (t) is assumed to be normally distributed.”
The authors state this as an assumption.
An appropriate model M that sufficiently describes the available experimental data is assumed.
“Furthermore, an appropriate model M that sufficiently describes the available experimental data is assumed.”
The paper relies on this without support.
No approach for explicit testing of practical non-identifiability is available.
“Similar to a clear formal definition, an approach for explicit testing of practical non-identifiability is not available to our knowledge.”
The authors state this as an assumption.
The initial values of species x2 to x4 are equal to zero.
“The initial values of species x 2 to x 4 are assumed to be equal to zero.”
The authors state this as an assumption.
For the model-reduction argument, a parameter θi is practically non-identifiable for decreasing parameter value.
“Assume that a parameter θ i is practically non-indentifiable for decreasing parameter value.”
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The source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“Such parameter estimations usually yield statistical information related to parameter precision (importantly, not accuracy) such as standard errors or correlation matrices, and sometimes information derived from more sophisticated approaches such as profile likelihood.39”
From Best Practices to Maximize the Use and Reuse of Quantitative and Systems Pharmacology Models: Recommendations From the United Kingdom Quantitative and Systems Pharmacology Network (2019) · cites this paper as [39]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“Raue et al. (2009) referred to likelihood ratio C.I.s as finite sample C.I.s, to underscore the superiority of likelihood ratio C.I.s over Wald-type C.I.s.”
From rSalvador: An R Package for the Fluctuation Experiment (2017) · cites this paper as Raue et al. 2009
In the citing paper
“When two models are nested (Mð hPL Þ and Mð hopt Þ) their likelihood ratio is approximately distributed according to a v 2 p distribution.Here p refers to the number of degrees of freedom which is defined as the difference in the number of parameters (which in this case is 1).”
From Parameter uncertainty in biochemical models described by ordinary differential equations (2013) · cites this paper as [36]
In the citing paper
“In such cases, more reliable confidence intervals can be obtained using the Profile Likelihood method [36].”
From Parameter uncertainty in biochemical models described by ordinary differential equations (2013) · cites this paper as [36]
In the citing paper
“Techniques for this exist in both a frequentist [13,36,91] as well as a Bayesian setting [87,90].”
From Parameter uncertainty in biochemical models described by ordinary differential equations (2013) · cites this paper as [36]
In the citing paper
“Fortunately, recent literature is correcting this (e.g. [129–140]).”
From Reverse engineering and identification in systems biology: strategies, perspectives and challenges (2014) · cites this paper as [133]
In the citing paper
“Given the data, not all parameters are identifiable (Raue et al., 2009).”
From A bayesian approach to targeted experiment design (2012) · cites this paper as Raue et al., 2009
In the citing paper
“Methods for an approximate sampling of the parameter space, e.g. the Markov Chain Monte Carlo (MCMC) methods [11,12], and bootstrap based approaches [4,13] are numerically demanding and provide only rough approximations for ODE models. Therefore, it is difficult to control the coverage of the prediction confidence intervals for such approaches.”
From Likelihood based observability analysis and confidence intervals for predictions of dynamic models (2012) · cites this paper as [4]
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.
Concentration unit conflicts with figure labels
MinorThe text says concentrations are modeled in nanometers, a unit of length, while the figure labels use nM, a unit of concentration.
“Species concentrations are modeled in nanometer.”
“time / min x 1 / nM”
Citations
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The impact of travelling on the COVID-19 infection cases in Germany
Schäfer, M., Wijaya, K. P., Rockenfeller, R., et al. (2022) · BMC Infectious Diseases · doi:10.1186/s12879-022-07396-1 · Cited by 11
“[37], these authors suggest that the asymptotic confidence interval can be a good approximation of the uncertainty in the optimal parameters u providing that, besides the aforementioned relation, the measurement error is relatively small as compared to the data.”
ResultLie symmetries of nonlinear systems with unknown inputs
Shi, X. F., & Chatzis, M. N. (2022) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2022.110027 · Cited by 11
A Continuation Technique for Maximum Likelihood Estimators in Biological Models
Cassidy, T. (2023) · Bulletin of Mathematical Biology · doi:10.1007/s11538-023-01200-0 · Cited by 11
“(θi,α) = {c|PLEθi(c)−PLEθi(θ ∗ i )< ∆α} where ∆α = χ2(α,d f ) is the χ2 distribution at significance level α and d f degrees of freedom [Raue et al., 2009].”
Yang, H., Meijer, H. G. E., Doll, R. J., Buitenweg, J. R., & van Gils, S. A. (2015) · Biological Cybernetics · doi:10.1007/s00422-015-0656-4 · Cited by 11
“Prior to performing parameter estimation, the identifiability of the hazard model should be explored, i.e., a unique estimate can be determined provided sufficient information is available (Bellu et al. 2007; Raue et al. 2009).”
BackgroundComputational Modeling of Mitochondrial Energy Transduction
Schmitz, J. P. J., Vanlier, J., van Riel, N. A. W., et al. (2011) · Critical Reviews in Biomedical Engineering · doi:10.1615/critrevbiomedeng.v39.i5.20 · Cited by 11
A Parts Database with Consensus Parameter Estimation for Synthetic Circuit Design
Huynh, L. N., & Tagkopoulos, I. (2016) · ACS Synthetic Biology · doi:10.1021/acssynbio.5b00205 · Cited by 11
Brouwer, A. F., Zelner, J., Eisenberg, M. C., Kimmins, L. M., Ladisky, M., Collins, J., & Eisenberg, J. N. S. (2020) · Epidemiology · doi:10.1097/ede.0000000000001218 · Cited by 11
Yi, D. H., Kim, D.-W., & Park, C. S. (2020) · Energy and Buildings · doi:10.1016/j.enbuild.2020.110293 · Cited by 11
Joubert, D., Stigter, J. D., & Molenaar, J. (2021) · Scientific Reports · doi:10.1038/s41598-021-96293-9 · Cited by 13
Aldrin, M., Aanes, F. L., Tvete, I. F., et al. (2021) · Fisheries Research · doi:10.1016/j.fishres.2021.106071 · Cited by 11
Sterile inflammation alters neutrophil kinetics in mice
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“The practical identifiability of the parameter (which depends on the discrete nature and quality of the measured data) appears as the magnitude of spread in the parameter distributions(24, 25).”
BackgroundSpatial Aspects in Biological System Simulations
Resat, H., Costa, M. N., & Shankaran, H. (2010) · Methods in enzymology on CD-ROM/Methods in enzymology · doi:10.1016/b978-0-12-381270-4.00017-2 · Cited by 11
Mathematical model of early Reelin-induced Src family kinase-mediated signaling
Hass, H., Kipkeew, F., Gauhar, A., Bouché, E., May, P., Timmer, J., & Bock, H. H. (2017) · PLoS ONE · doi:10.1371/journal.pone.0186927 · Cited by 10
Self-citationComputational approaches to kinetic model selection
Tsu, J., Díaz, V. H. G., & Willis, M. J. (2018) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2018.12.002 · Cited by 11
Duchesne, R., Guillemin, A., Fabien Crauste, A., & Gandrillon, O. (2019) · In Silico Biology · doi:10.3233/isb-190471 · Cited by 11
A steady-state algebraic model for the time course of covalent enzyme inhibitionPreprint
Kuzmič, P. (2020) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2020.06.10.144220 · Cited by 11
“” Structural identifiability analysis [12] is concerned with idealized data, completely free of the error, whereas practical identifiability analysis takes the inevitable random error into account [13].”
BackgroundGene regulatory networks: from correlative models to causal explanations
Maizels, R. J., & Briscoe, J. (2026) · Nature Reviews Genetics · doi:10.1038/s41576-026-00939-1 · Cited by 21
“Mathematical models can suffer from a phenomenon known as structural non-identifiability 55,56 , wherein different sets of model parameters generate the same output, making it impossible to determine the ‘correct’ parameter solution.”
Befort, B. J., Garciadiego, A., Wang, J., et al. (2023) · Fluid Phase Equilibria · doi:10.1016/j.fluid.2023.113833 · Cited by 10
Inferring delays in partially observed gene regulation processes
Hong, H., Cortez, M. J., Cheng, Y., et al. (2023) · Bioinformatics · doi:10.1093/bioinformatics/btad670 · Cited by 10
Bayesian estimation in veterinary pharmacology: A conceptual and practical introduction
Woodward, A. P. (2024) · Journal of Veterinary Pharmacology and Therapeutics · doi:10.1111/jvp.13433 · Cited by 9
Ribeiro, L. S., de Miranda, G. B., Rocha, B. M., Chapiro, G., & dos Santos, R. W. (2024) · Water Resources Research · doi:10.1029/2023wr036751 · Cited by 10
“The profile likelihood (PL) technique (Raue, Kreutz, Maiwald, et al., 2013) provides a practical approach for conducting identifiability analysis of parameters.”
MethodComparative analysis of practical identifiability methods for an SEIR model
Saucedo, O., Laubmeier, A. N., Tang, T., Levy, B., Asik, L., Pollington, T. M., & Prosper, O. (2024) · AIMS Mathematics · doi:10.3934/math.20241204 · Cited by 10
Haghebaert, M., Varsos, P., Meiburg, R., & Vignon-Clementel, I. E. (2025) · The Journal of Physiology · doi:10.1113/jp287929 · Cited by 9
“‘Identifiability’ refers to the ability to uniquely estimate model parameters based on available data (Little et al., 2010; Raue et al., 2009; Vanlier et al., 2012).”
Nicoulaud-Gouin, V., Garcia-Sanchez, L., Giacalone, M., Attard, J. C., Martin-Garin, A., & Bois, F. Y. (2016) · Journal of Environmental Radioactivity · doi:10.1016/j.jenvrad.2016.06.008 · Cited by 10
“(Raue et al., 2009).”
InfluentialSiripuram, V. K., Wright, D. F. B., Barclay, M. L., & Duffull, S. B. (2017) · Journal of Pharmacokinetics and Pharmacodynamics · doi:10.1007/s10928-017-9530-4 · Cited by 11
Parameter Estimation in the Age of Degeneracy and Unidentifiability
Lederman, D., Patel, R., Itani, O., & Rotstein, H. G. (2022) · Mathematics · doi:10.3390/math10020170 · Cited by 10
Sachak-Patwa, R., Lafferty, E. I., Schmit, C. J., Thompson, R. N., & Byrne, H. M. (2023) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2023.111491 · Cited by 11
Tönsing, C., Steiert, B., Timmer, J., & Kreutz, C. (2023) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1011417 · Cited by 13
Self-citationBasal MET phosphorylation is an indicator of hepatocyte dysregulation in liver disease
de Lara, S. B., Kemmer, S., Biermayer, I., Feiler, S., Vlasov, A., D’Alessandro, L. A., Helm, B., Mölders, C., Dieter, Y., Ghallab, A., et al. (2024) · Molecular Systems Biology · doi:10.1038/s44320-023-00007-4 · Cited by 10
“( A ) The pro fi le likelihood as a measure of parameter identi fi ability (Raue et al, 2009) is depicted for all dysregulated parameters before implementation of the DIA data.”
BackgroundMethodSelf-citationChowell, G., Dahal, S., Bleichrodt, A., et al. (2024) · Infectious Disease Modelling · doi:10.1016/j.idm.2024.02.001 · Cited by 10
Assessing parameter identifiability for dynamic causal modeling of fMRI data
Arand, C., Scheller, E., Seeber, B., Timmer, J., Klöppel, S., & Schelter, B. (2015) · Frontiers in Neuroscience · doi:10.3389/fnins.2015.00043 · Cited by 10
“In total, by analysing the parameter identifiability, the reliability of model predictions can be improved (Raue et al., 2009).”
BackgroundMethodInfluentialSelf-citationChallenging issues in Bayesian calibration of multi-physics models
Ling, Y., & Mahadevan, S. (2013) · AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference (SDM) · doi:10.2514/6.2013-1874 · Cited by 10
Structural vs Practical Identifiability in System Biology.
Saccomani, M. P. (2013) · Padua Research Archive (University of Padova) · Cited by 12
Validation and selection of ODE models for gene regulatory networks
Kim, J. (2016) · Chemometrics and Intelligent Laboratory Systems · doi:10.1016/j.chemolab.2016.06.016 · Cited by 10
Lagergren, J. H., Reeder, A., Hamilton, F., et al. (2018) · Bulletin of Mathematical Biology · doi:10.1007/s11538-018-0421-7 · Cited by 10
“, when every variable in the model is a longitudinal covariate in the available time series data, accurate parameterization may still be a challenge due to identifiability-related issues (Cobelli and DiStefano 1980; DiStefano 2015; Meshkat et al. 2014; Raue et al. 2009).”
BackgroundMeyer, J., Elyorgun, D., Conradi, C., & Drews, A. (2018) · Process Safety and Environmental Protection · doi:10.1016/j.cherd.2018.04.029 · Cited by 10
Modeling nutrient and disease dynamics in a plant-pathogen system
Pell, B., Kendig, A. E., Borer, E. T., & Kuang, Y. (2018) · Mathematical Biosciences & Engineering · doi:10.3934/mbe.2019013 · Cited by 10
How to prevent viremia rebound? Evidence from a PRRSv data-supported model of immune response
Go, N., Touzeau, S., Islam, Z. U., Belloc, C., & Doeschl-Wilson, A. B. (2019) · BMC Systems Biology · doi:10.1186/s12918-018-0666-7 · Cited by 10
Optimisation of anti-cancer peptide vaccines to preferentially elicit high-avidity T cells
Kumbhari, A., Kim, P., & Lee, P. P. (2019) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2019.110067 · Cited by 10
Estimation of Process Model Parameters
Deppe, S., Frahm, B., Hass, V. C., et al. (2019) · Methods in molecular biology · doi:10.1007/978-1-0716-0191-4_12 · Cited by 10
Iwanami, S., Kitagawa, K., Ohashi, H., et al. (2020) · PLoS Biology · doi:10.1371/journal.pbio.3000562 · Cited by 10
“We also tested structural identifiability for all parameter estimates by calculating profile likelihood [23,24] using package dMod [25] in R Statistical Software [26].”
MethodVirtual Patient Generation using Physiological Models through a Compressed Latent Parameterization
Tivay, A., Kramer, G. C., & Hahn, J.-O. (2020) · 2020 American Control Conference (ACC) · doi:10.23919/acc45564.2020.9147298 · Cited by 12
“…mathematical models proposed in biology and physics are known to exhibit the “sloppiness” property [15] (i.e., a lack of practical identifiability [16], [17]) in many directions in their parameter space, which is known to cause parameter estimates to drift out of proportion during model…”
BackgroundOptimization of the Controls against the Spread of Zika Virus in Populations
González‐Parra, G., Díaz‐Rodríguez, M., & Arenas, A. J. (2020) · Computation · doi:10.3390/computation8030076 · Cited by 10
“However, those mathematical models have more parameters, which may cause an identifiability problem when we need to estimate the model’s parameter values using real data of infected cases [2,28,29].”
BackgroundLira-Parada, P. A., Tuveri, A., Seibold, G. M., & Bar, N. (2021) · Biochemical Engineering Journal · doi:10.1016/j.bej.2021.107989 · Cited by 10
“where J(p)) is the Jacobian and the information on the standard deviation of the measurements (σi) is grouped in a diagonal matrix V [33, 37, 36].”
BackgroundBrowning, A. P., Drovandi, C., Turner, I., Jenner, A. L., & Simpson, M. J. (2022) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1010734 · Cited by 11
“Such identifiability analysis is well established for ordinary differential equation models [9, 10, 13], stochastic models [12, 14–16], and, recently, partial differential equation models [8, 17].”
InfluentialMischnik, M., Gambaryan, S. P., Subramanian, H., et al. (2014) · Molecular BioSystems · doi:10.1039/c4mb00170b · Cited by 10
Self-citationDocherty, P. D., Schranz, C., Chase, J. G., et al. (2013) · Computer Methods and Programs in Biomedicine · doi:10.1016/j.cmpb.2013.06.017 · Cited by 10
Optimal Sampling and Estimation in PASL Perfusion Imaging
Santos, N., Sanches, J., Sousa, I., et al. (2011) · IEEE Transactions on Biomedical Engineering · doi:10.1109/tbme.2011.2164916 · Cited by 10
“Practical identifiability, however, may not hold if there are not enough data or if data were not acquired at the appropriated locations [25].”
BackgroundPrediction uncertainty and optimal experimental design for learning dynamical systems
Letham, B., Letham, P. A., Rudin, C. D., & Browne, E. P. (2016) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.4953795 · Cited by 19
Wang, J., Lü, F., Lin, K., Xiao, W., Song, X., & He, Y. (2016) · Stochastic Environmental Research and Risk Assessment · doi:10.1007/s00477-016-1370-z · Cited by 10
“Furthermore, the PLF method has been increasingly applied in recent years (Gilli and Kellezi 2006; Virtanen and Uusipaikka 2008; Raue et al. 2009; Becker et al. 2010; Schaber 2012; Schelker et al. 2012; Kreutz et al. 2013).”
MethodKinetic analysis of the accumulation of a half-sandwich organo-osmium pro-drug in cancer cells
Ballesta, A., Billy, F., Coverdale, J. P. C., Song, J., Sánchez-Cano, C., Romero‐Canelón, I., & Sadler, P. J. (2019) · Metallomics · doi:10.1039/c9mt00173e · Cited by 10
Interrogating theoretical models of neural computation with emergent property inferencePreprint
Bittner, S. R., Palmigiano, A., Piet, A. T., Duan, C. A., Brody, C. D., Miller, K. D., & Cunningham, J. P. (2019) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/837567 · Cited by 14
Quantitative analysis of tumour spheroid structurePreprint
Browning, A. P., Sharp, J. A., Murphy, R. J., Gunasingh, G. P., Lawson, B., Burrage, K., Haass, N. K., & Simpson, M. J. (2021) · bioRxiv (Cold Spring Harbor Laboratory) · doi:10.1101/2021.08.05.455334 · Cited by 10
“Marginal confidence intervals 239 The profile likelihood method [37,59] allows for the construction of univariate confidence interval 240 of each parameter.”
MethodHashemi, M., Vattikonda, A. N., Jha, J., Ší́p, V., Woodman, M., Bartoloméi, F., & Jirsa, V. (2022) · medRxiv · doi:10.1101/2022.06.02.22275860 · Cited by 10
Yang, Y., Noble, D. W. A., Spake, R., Senior, A. M., Lagisz, M., & Nakagawa, S. (2023) · doi:10.32942/x2rg7x · Cited by 12
Minimally sufficient experimental design using identifiability analysis
Gevertz, J. L., & Kareva, I. (2024) · npj Systems Biology and Applications · doi:10.1038/s41540-023-00325-1 · Cited by 15
Shin, S., Chew, N. J., Ghomlaghi, M., Chüeh, A. C., Jeong, Y., Nguyen, L. K., & Daly, R. J. (2024) · Cancer Research · doi:10.1158/0008-5472.can-23-3409 · Cited by 9
Rationalised experiment design for parameter estimation with sensitivity clustering
Chhajer, H., & Roy, R. (2024) · Scientific Reports · doi:10.1038/s41598-024-75539-2 · Cited by 9
“First, it uses the parameter sensitivity of a variable to determine its informativeness for estimating the values of the corresponding parameters 7,33 .”
Relouw, F. J. A., Kox, M., Taal, H. R., Koch, B. C. P., Prins, M. W. J., & van Riel, N. A. W. (2024) · npj Systems Biology and Applications · doi:10.1038/s41540-024-00473-y · Cited by 13
Wang, S., & Hao, W. (2025) · Advanced Science · doi:10.1002/advs.202504346 · Cited by 10
A pluralistic framework for measuring, interpreting and decomposing heterogeneity in meta‐analysis
Yang, Y., Noble, D. W. A., Spake, R., Senior, A. M., Lagisz, M., & Nakagawa, S. (2025) · Methods in Ecology and Evolution · doi:10.1111/2041-210x.70155 · Cited by 11
Dolejsch, P., Hass, H., & Timmer, J. (2019) · BMC Bioinformatics · doi:10.1186/s12859-019-2976-1 · Cited by 9
Self-citationBinns, M., Usai, A., & Theodoropoulos, C. (2024) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2024.108683 · Cited by 9
Ha, S., & Onori, S. (2024) · Journal of The Electrochemical Society · doi:10.1149/1945-7111/ad7292 · Cited by 9
“53 57 Structural identi fi ability refers to the inherent mathematical properties of a model and whether the model allows for unique parameter estimation, regardless of the data or estimation method used.”
A Systems Biology Study on NFκB Signaling in Primary Mouse Hepatocytes
Pinna, F., Sahle, S., Beuke, K., Bissinger, M., Tuncay, S., D’Alessandro, L. A., Gauges, R., Raue, A., Timmer, J., Klingmüller, U., et al. (2012) · Frontiers in Physiology · doi:10.3389/fphys.2012.00466 · Cited by 9
“In order to analyze the uniqueness of the fit and thus parameter identifiability (Raue et al., 2009), the fitting procedure was repeated 30 times with random start values resulting in 30 parameter sets fitting the experimental data well.”
MethodSelf-citationZimmer, C. (2016) · PLoS ONE · doi:10.1371/journal.pone.0159902 · Cited by 9
“A similar phenomenon has been observed previously [42] and the use of the profile likelihood techniques [43] has been suggested to improve the analysis.”
Modeling the temporal periodicity of growth increments based on harmonic functions
Hidalgo-de-la-Toba, J. Á., Morales‐Bojórquez, E., González-Peláez, S. S., Bautista‐Romero, J. J., & Lluch-Cota, D. B. (2018) · PLoS ONE · doi:10.1371/journal.pone.0196189 · Cited by 9
Fitting stochastic predator–prey models using both population density and kill rate data
Barraquand, F., & Giménez, O. (2021) · Theoretical Population Biology · doi:10.1016/j.tpb.2021.01.003 · Cited by 13
“However, as explained in Eisenberg and Hayashi (2014), likelihood profiles can sometimes be ambiguous when combinations of multiple parameters are at play (Raue et al., 2009).”
BackgroundMethodHierarchical Bayesian models of transcriptional and translational regulation processes with delays
Cortez, M. J., Hong, H., Choi, B., Kim, J. K., & Josić́, K. (2021) · Bioinformatics · doi:10.1093/bioinformatics/btab618 · Cited by 9
A method for the inference of cytokine interaction networks
Jansen, J. E., Aschenbrenner, D., Uhlig, H. H., Coles, M. C., & Gaffney, E. A. (2022) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1010112 · Cited by 9
“using the Profile Likelihood Approach [32] (Section “S2 Appendix”).”
InfluentialRosenbaum, B., & Fronhofer, E. A. (2023) · Ecosphere · doi:10.1002/ecs2.4503 · Cited by 9
Scott, F., Li, M., Williams, D. L., Conejeros, R., Hodge, D. B., & Aroca, G. E. (2014) · Bioresource Technology · doi:10.1016/j.biortech.2014.11.062 · Cited by 9
In silico labeling reveals the time-dependent label half-life and transit-time in dynamical systems
Maiwald, T., Blumberg, J., Raue, A., Hengl, S., Schilling, M., Sy, S. K. B., Becker, V., Klingmüller, U., & Timmer, J. (2012) · BMC Systems Biology · doi:10.1186/1752-0509-6-13 · Cited by 9
“We recently suggested a profile likelihood-based approach to determine simultaneous and separate confidence intervals for calibrated unknown model parameters [15].”
MethodSelf-citationAlternative model for precipitation probability distribution: application to Spain
Moncho, R., Caselles, V., & Chust, G. (2011) · Climate Research · doi:10.3354/cr01055 · Cited by 9
Talemi, S. R., Kollárovič, G., Lapytsko, A., & Schaber, J. (2015) · Scientific Reports · doi:10.1038/srep13540 · Cited by 10
Optimal Experiment Design and Leveraging Competition for Shared Resources in Cell-Free Extracts
Halter, W., Allgöwer, F., Murray, R. M., & György, A. (2018) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc.2018.8619039 · Cited by 9
“In what follows, we omit the subscript of θ̂ as under a Gaussian model with i.i.d. noise, as in our case, we have that θ̂LSE = θ̂MLE [21].”
InfluentialBO4IO: A Bayesian optimization approach to inverse optimization with uncertainty quantification
Lu, Y., Hu, W., Paulson, J. A., & Zhang, Q. (2024) · Computers & Chemical Engineering · doi:10.1016/j.compchemeng.2024.108859 · Cited by 10
Wyk, H. V., Brouwer, A. F., Lee, G. O., Márquez, S., Andrade, P., Ionides, E. L., Coloma, J., & Eisenberg, J. N. S. (2025) · Epidemiology · doi:10.1097/ede.0000000000001874 · Cited by 9
Simpson, M. J., & Baker, R. E. (2026) · SIAM Review · doi:10.1137/24m1667968 · Cited by 13
“A common tool for assessing practical identifiability is the profile likelihood [12, 13, 8], which is the main focus of this article.”
Model-based selection of the robust JAK-STAT activation mechanism
Rybiński, M., & Gambin, A. (2012) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2012.04.031 · Cited by 8
“PL-based IA (Raue et al., 2009) is a method of analysis of the relation between parameter and the SSE (χ2) output in terms of possibility of maintaining a low error value with respect to a systematical change of a value of one parameter and the best fit values for remaining parameters.”
BackgroundMethodInfluentialTriemer, S., Schulze, M., Wriedt, B., Schenkendorf, R., Ziegenbalg, D., Krewer, U., & Seidel‐Morgenstern, A. (2021) · Journal of Flow Chemistry · doi:10.1007/s41981-021-00181-2 · Cited by 8
“We used the profile likelihood approach that yields improved confidence intervals for the model parameters besides conclusions about parameter identifiability [60, 61].”
MethodRegulation of STAT3 signaling in IFNγ and IL10 pathways and in their cross-talk
Sarma, U., Maiti, M., Nair, A., et al. (2021) · Cytokine · doi:10.1016/j.cyto.2021.155665 · Cited by 8
Ayalew, M., Mwaura, B. W., Kumbale, C. M., Rono, C., Torres-Patarroyo, N., Vlčko, T., Ohnoutková, L., & Voit, E. O. (2023) · Frontiers in Plant Science · doi:10.3389/fpls.2023.1147598 · Cited by 9
Estimating global identifiability using conditional mutual information in a Bayesian framework
Bhola, S., & Duraisamy, K. (2023) · Scientific Reports · doi:10.1038/s41598-023-44589-3 · Cited by 9
“Another popular identification technique is likelihood profiling 4, 21–23 .”
Exploring tunneling ESEEM beyond methyl groups in nitroxides at low temperatures
Eggeling, A., Ngendahimana, T., Jeschke, G., et al. (2024) · Physical Chemistry Chemical Physics · doi:10.1039/d4cp01212g · Cited by 8
Plank, M. J., & Simpson, M. J. (2024) · Royal Society Open Science · doi:10.1098/rsos.240733 · Cited by 9
Incorporating spatial diffusion into models of bursty stochastic transcription
Miles, C. E. (2025) · Journal of The Royal Society Interface · doi:10.1098/rsif.2024.0739 · Cited by 8
Quantitative Contribution of IL2Rγ to the Dynamic Formation of IL2-IL2R Complexes
Ponce, L. F., García‐Martínez, K., & León, K. (2016) · PLoS ONE · doi:10.1371/journal.pone.0155684 · Cited by 8
“The selected solutions from model fitting show variations of around 2 orders of magnitude in the value of the kinetic coefficients, indicating that they can be considered as practical no identifiable according [33].”
BackgroundAngulo, O., Gandrillon, O., & Crauste, F. (2017) · Journal of Theoretical Biology · doi:10.1016/j.jtbi.2017.10.031 · Cited by 8
Clément, F., Robin, F., & Yvinec, R. (2019) · SIAM Journal on Applied Mathematics · doi:10.1137/17m1161336 · Cited by 8
Kurzhunov, D., Borowiak, R., Reisert, M., et al. (2018) · Magnetic Resonance in Medicine · doi:10.1002/mrm.27224 · Cited by 8
Bianconi, F., Antonini, C., Tomassoni, L., & Valigi, P. (2018) · IEEE Transactions on Control Systems Technology · doi:10.1109/tcst.2018.2844362 · Cited by 8
“Then identifiability analysis is performed through computation of confidence intervals for all parameters [20].”
MethodLill, D., Timmer, J., & Kaschek, D. (2019) · PLoS ONE · doi:10.1371/journal.pone.0217837 · Cited by 8
Self-citationA New Bayesian Methodology for Nonlinear Model Calibration in Computational Systems Biology
Bianconi, F., Tomassoni, L., Antonini, C., & Valigi, P. (2020) · Frontiers in Applied Mathematics and Statistics · doi:10.3389/fams.2020.00025 · Cited by 8
“Common objective functions are the sum of squared residuals or the negative loglikelihood [14].”
BackgroundParameter Estimation in a PDE Model for the Spatial Spread of Cocoa Black Pod Disease
Fomba, C. G. N., Hoopen, G. M. T., Soubeyrand, S., Roques, L., Ambang, Z., & Soh, P. T. (2021) · Bulletin of Mathematical Biology · doi:10.1007/s11538-021-00934-z · Cited by 8
Akimov, V., Fehling-Kaschek, M., Barrio‐Hernandez, I., Puglia, M., Bunkenborg, J., Nielsen, M. M., Timmer, J., Dengjel, J., & Blagoev, B. (2021) · Journal of Molecular Biology · doi:10.1016/j.jmb.2021.167240 · Cited by 8
Self-citationSantos, J. P. G., Pajo, K., Trpevski, D., Stepaniuk, A., Eriksson, O., Nair, A. G., Keller, D. X., Kotaleski, J. H., & Kramer, A. (2021) · Neuroinformatics · doi:10.1007/s12021-021-09546-3 · Cited by 7
An analysis of structural and practical identifiability applied to a transient generator model
Landgraf, T. G., Cari, E. P. T., & Alberto, L. (2022) · Electric Power Systems Research · doi:10.1016/j.epsr.2022.107817 · Cited by 8
Catozzi, S., Hill, R. J. W., Li, X., Dulong, S., Collard, E., & Ballesta, A. (2022) · CPT Pharmacometrics & Systems Pharmacology · doi:10.1002/psp4.12895 · Cited by 9
Metadata
- Authors
- Raue, A., Kreutz, C., Maiwald, T., Bachmann, J., Schilling, M., Klingmüller, U., & Timmer, J.
- Affiliations
- University of Freiburg (DE)
- Corresponding author
- Raue, A.
- Venue
- Bioinformatics
- Publisher
- Oxford University Press (OUP)
- Published
- 2009-06-08
- Volume
- 25
- Issue
- 15
- Pages
- 1923-1929
- ISSN
- 1367-4803 (ISSN-L)
- 1367-4811
- OpenAlex
- W2168648059
- Cited by
- 1,442
- References
- 29
- Citations by year
- 2026: 209 · 2025: 107 · 2024: 103 · 2023: 82 · 2022: 98 · 2021: 118 · 2020: 97 · 2019: 116 · 2018: 94 · 2017: 78 · 2016: 87 · 2015: 64 · 2014: 52 · 2013: 54 · 2012: 39
- FWCI
- 10.99
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Bioinformatics (publishedVersion)
- PubMed (publishedVersion)
- OPUS (Augsburg University) (submittedVersion)
- Funders
- Deutsches Krebsforschungszentrum
- European Commission (223188, EU-FP7, FP7 HEALTH)
- Bundesministerium für Bildung und Forschung (FRISYS 0313921, 0313074D, 0313921)
- FP7 Health
- Topics
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Gene Regulatory Network Analysis
- Physical Sciences › Engineering › Control and Systems Engineering › Control Systems and Identification
- Life Sciences › Biochemistry, Genetics and Molecular Biology › Molecular Biology › Microbial Metabolic Engineering and Bioproduction
- Keywords
- identifiability analysis, profile likelihood, parameter estimation, confidence intervals, experimental design, model reduction
- MeSH terms
- Algorithms, Models, Biological, Probability, Computational Biology
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-11, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-11 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Cited sources are checked with a language model’s help (gpt-6-sol).
Statistics and arithmetic are recomputed, not judged.
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