Likelihood based observability analysis and confidence intervals for predictions of dynamic models

BMC Systems Biology

Jens Timmer²  Andreas Raue³  Clemens Kreutz¹ 

[1] Freiburg Centre for Biosystems Analysis (ZBSA), University of Freiburg, Habsburgerstraße 49, 79104 Freiburg, Germany;Department of Clinical and Experimental Medicine, Universitetssjukhuset, 58183 Linköping, Sweden;Institute of Bioinformatics and Systems Biology, Helmholtz Zentrum München, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany

关键词: Uncertainty;    Statistical inference;    Signal transduction;    Ordinary differential equations;    Optimal experimental design;    Profile likelihood;    Prediction;    Parameter estimation;    Likelihood;    Identifiability;    Confidence intervals;   

Others  :  1143604
DOI  :  10.1186/1752-0509-6-120

 received in 2012-03-15, accepted in 2012-08-24,  发布年份 2012

【 摘 要 】

Background

Predicting a system’s behavior based on a mathematical model is a primary task in Systems Biology. If the model parameters are estimated from experimental data, the parameter uncertainty has to be translated into confidence intervals for model predictions. For dynamic models of biochemical networks, the nonlinearity in combination with the large number of parameters hampers the calculation of prediction confidence intervals and renders classical approaches as hardly feasible.

Results

In this article reliable confidence intervals are calculated based on the prediction profile likelihood. Such prediction confidence intervals of the dynamic states can be utilized for a data-based observability analysis. The method is also applicable if there are non-identifiable parameters yielding to some insufficiently specified model predictions that can be interpreted as non-observability. Moreover, a validation profile likelihood is introduced that should be applied when noisy validation experiments are to be interpreted.

Conclusions

The presented methodology allows the propagation of uncertainty from experimental to model predictions. Although presented in the context of ordinary differential equations, the concept is general and also applicable to other types of models. Matlab code which can be used as a template to implement the method is provided athttp://www.fdmold.uni-freiburg.de/∼ckreutz/PPL webcite.

【 授权许可】

   
2012 Kreutz et al.; licensee BioMed Central Ltd.

【 图 表 】

Figure 1.

【 参考文献 】

• [1]Sachs L: Applied Statistics. Springer, New York; 1984.
• [2]Davison A, Hinkley D: Bootstrap Methods and Their Application. Cambridge University Press, Cambridge; 1997.
• [3]DiCiccio T, Tibshirani R: Bootstrap confidence intervals and bootstrap approximations. J Am Stat Assoc 1987, 82(397):163-170.
• [4]Joshi M, Seidel-Morgenstern A, Kremling A: Exploiting the bootstrap method for quantifying parameter confidence intervals in dynamical systems. Metab Eng 2006, 8(5):447-455. [ http://dx.doi.org/10.1016/j.ymben.2006.04.003 webcite]
• [5]Venzon DJ, Moolgavkar SH: A Method for Computing Profile-Likelihood-Based Confidence Intervals. Appl Statist 1988, 37:87-94.
• [6]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. Bioinformatics 2009, 25:1923-1929. [ doi:10.1093/bioinformatics/btp358 webcite]
• [7]Hlavacek WS: How to deal with large models? Mol Syst Biol 2009, 5:240-242.
• [8]Swameye I, Müller T, Timmer J, Sandra O, Klingmüller U: Identification of nucleocytoplasmic cycling as a remote sensor in cellular signaling by data-based modeling. Proc Natl Acad Sci 2003, 100(3):1028-1033.
• [9]Marimont RB, Shapiro MB: Nearest neighbour searches and the curse of dimensionality. IMA J Appl Mathematics 1979, 24:59-70. [ http://imamat.oxfordjournals.org/content/24/1/59.abstract webcite]
• [10]Scott DW, Wand MP: Feasibility of multivariate density estimates. Biometrika 1991, 78:197-205. [ http://www.jstor.org/stable/2336910 webcite]
• [11]Gelman A, Carlin JB, Stern HS, Rubin DB: Bayesian Data Analysis, Second Edition (Chapman & Hall/CRC Texts in Statistical Science),. Chapman and Hall/CRC, Boca Raton; 2003.
• [12]Kass R, Carlin B, Gelman A, Neal R: Markov Chain Monte Carlo in practice: a roundtable diskussion. Am Stat 1998, 52:93-100.
• [13]Molinaro AM, Simon R, Pfeiffer RM: Prediction error estimation: a comparison of resampling methods. Bioinformatics 2005, 21(15):3301-3307. [ http://dx.doi.org/10.1093/bioinformatics/bti499 webcite]
• [14]Bayarri M, Berger J: The interplay of Bayesian and frequentist analysis. Stat Sci 2004, 19:58-80.
• [15]Hinkley D: Predictive likelihood. The Ann Stat 1979, 7(4):718-728.
• [16]Booth JG, Hobert JP: Standard Errors of Prediction in Generalized Linear Mixed Models. J Am Stat Assoc 1998, 93:262-267.
• [17]Butler RW: Predictive likelihood inference with applications. J Roy Stat Soc B 1986, 48:1-38.
• [18]Cooley TF, Chib S, Parke WR: Predictive efficiency for simple nonlinear models. J Econometrics 1989, 40:33-44.
• [19]Bjornstad JF: Predictive likelihood: a review. Stat Sci 1990, 5(2):242-254. [ http://www.jstor.org/stable/2245686 webcite]
• [20]Feder PI: On the distribution of the Log Likelihood Ratio test statistic when the true parameter is “near” the boundaries of the hypothesis regions. Ann Math Stat 1968, 39(6):2044-2055.
• [21]Seber G, Wild C: Nonlinear regression. Wiley, New York; 1989.
• [22]Cox D, Hinkley D: Theoretical Statistics. Chapman & Hall, London; 1994.
• [23]Meeker W, Escobar L: Teaching about approximate confidence regions based on maximum likelihood estimation. Am Stat 1995, 49:48-53.
• [24]Kreutz C, Bartolome-Rodriguez MM, Maiwald T, Seidl M, Blum HE, Mohr L, Timmer J: An error model for protein quantification. Bioinformatics 2007, 23(20):2747-2753. [ http://dx.doi.org/10.1093/bioinformatics/btm397 webcite]
• [25]Kholodenko BN: Negative feedback and ultrasensitivity can bring about oscillations in the mitogen-activated protein kinase cascades. Eur J Biochem 2000, 267(6):1583-1588.
• [26]Marjoram P, Molitor J, Plagnol V, Tavare S: Markov chain Monte Carlo without likelihoods. PNAS 2003, 100(26):15324-15328.
• [27]Kreutz C, Timmer J: Systems biology: experimental design. FEBS J 2009, 276(4):923-942. [ http://dx.doi.org/10.1111/j.1742-4658.2008.06843.x webcite]
• [28]Smith R, Tebaldi C, Nychka D, Mearns L: Bayesian modeling of uncertainty in ensembles of climate models. J Am Stat Assoc 2009, 104(485):97-116.
• [29]Allison J, Banks J, Barlow RJ, Batley JR, Biebel O, Brun R, Buijs A, Bullock FW, Chang CY, Conboy JE, Cranfield R, Dallavalle GM, Dittmar M, Dumont JJ, Fukunaga C, Gary JW, Gascon J, Geddes NI, Gensler SW, Gibson V, Gillies JD, Hagemann J, Hansroul M, Harrison PF, Hart J, Hattersley PM, Hauschild M, Hemingway RJ, Heymann FF, Hobson PR, et al.: The detector simulation program for the OPAL experiment at LEP. Nuc Instr Meth A 1992, 317:47-74.
• [30]Ideker T, Thorsson V, Ranish JA, Christmas R, Buhler J, Eng JK, Bumgarner R, Goodlett DR, Aebersold R, Hood L: Integrated genomic and proteomic analyses of a systematically perturbed metabolic network. Science 2001, 292:929-934. [ http://dx.doi.org/10.1126/science.292.5518.929 webcite]