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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
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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
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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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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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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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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
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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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Immunological heterogeneity informs estimation of the durability of COVID-19 vaccine protectionPreprint
de Cellès, M. D., Wong, A., Guevara, L. B., et al. (2021) · Research Square · doi:10.21203/rs.3.rs-566774/v1 · Cited by 1
“In a first analysis, we assumed that the variability in the duration of vaccine protection was known, and we calculated the profile log-likelihood to estimate the maximum likelihood estimate and the 99% confidence interval (CI) of the average duration of vaccine protection (1/α) [51].”
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“[15] Raue A, Kreutz C, Maiwald T, Bachmann J, Schilling M, Klingmüller U, Timmer J: Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood.”
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“…point of maximum variability in the output (experimental) variable with respect to parameter of interest and adds new (model-estimated) data corresponding to that time point, and then re-runs the parameter identification analysis on the improved data to check for confidence bound improvement 24 .”
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“…the CI given limited available experimental time points; alternative approaches include increasing sequencing times or profiling the likelihoods 72,73 after fitting the model (Supplementary Note 3), at the cost of a complicated experimental design or a linearly increased computational complexity.”
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Dynamic Causal Modeling in Probabilistic Programming LanguagesPreprint
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“…responses, where distinct discrepancies in energy, shape, phase, or feature location can be traded against each other under a single scalar objective, producing “mathematically improved” fits that do not preserve the intended physical interpretation (Raue et al., 2009; Terven et al., 2025).”
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“Practical identifiability was evaluated to determine the recoverability of model parameters under finite and noisy measurements (64; 79).”
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Model Identification by Utilizing Likelihood‐Based Methods
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“To address the problems stated above, Raue et al. (2009) proposed an approach to evaluate both structural and practical identifiability of model parameters based on their profile likelihood.”
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“Practical non-identifiability occurs when experimental data is of insufficient quality or quantity compared to the size of a model (Raue et al., 2009, Docherty et al., 2011).”
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“Typically, for dynamical models representing biochemical processes, PL plots of structurally non-identifiable parameters were flat for the entire interval, which is restricted by lower and upper pre-defined boundaries (Raue et al., 2009).”
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Applications in Cancer Research: Mathematical Models of Apoptosis
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Modeling Heterogeneity in Infectious Disease Systems for Inference and Monitoring
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“…the connection to possibilistic as opposed to probabilistic reasoning may also help explain why profile likelihood has emerged as a particularly promising method of identifiability analysis (Raue et al., 2009), where identifiability is traditionally a prerequisite for probabilistic analysis.”
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Parameter estimation and identifiability in a neural population model for electro-cortical activityPreprint
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“Recent advances from various angles have been made in the construction of nonlinear approximations to regions of practically unidentifiable parameters around a nominal estimate (e.g. Raue et al. (2009); Calderhead and Girolami (2011)).”
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“Since the optimum path does change in one direction, however, we have confidence that we are capturing some of the structure, and often these kind of ‘practical unidentifiabilities’ can be resolved by including more calibration data [130].”
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Structural and Practical Identifiability Analysis
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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.
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