Abstract
In the development of mathematical models we are plagued with two key concerns: is the model unique and accurate enough? Under the umbrella of the first question we will show whether the extent of the model structure in our data will shed light on all of the model’s parameters. Identifiability in model advancement helps us with the question ‘will a proposed experiment on a system enable us to determine values for the parameters of a model of that system?’ The model is assumed to be known and to reflect the response of the system to the experiment. Identifiability is not concerned with the precision with which the parameters can be estimated. A comprehensive review and demonstrations of the practical considerations regarding identifiability are presented. We further introduced new concepts which when omitted from consideration in the identifiability process can lead to serious misjudgement about the resolution of important aspect of the model under investigation, and its identifiability classification per se. On the second question we discuss current methods and their pitfalls in evaluating model adequacy. The concordance correlation coefficient (CCC) has been commonly used to assess agreement of continuous data, such as agreement of a new assay and a gold-standard assay, observed versus model predicted values, different methods, raters, and reproducibility. The main advantage of CCC is that it incorporates precision and accuracy simultaneously. There are four methods to compute CCC; two of them are extremely dependent on normality whereas the other two are more robust to non-normality. Therefore, datasets that departures from normality or are not independent may pose a significant problem when using CCC with the squared distance payoff function. It has been shown that CCC may indicate an increasing agreement as the marginal distribution becomes different, suggesting the agreement should be compared over a similar range.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Atkinson, G. and Nevill, A., 1997. Comment on the use of concordance correlation to assess the agreement between two variables. Biometrics 53:775–777.
Barnhart, H.X., Haber, M. and Song, J., 2002. Overall Concordance Correlation Coefficient for Evaluating Agreement among Multiple Observers. Biometrics 58:1020–1027.
Barnhart, H.X., Song, J. and Haber, M.J., 2005. Assessing intra, inter and total agreement with replicated readings. Statistics in Medicine 24:1371–1384.
Barnhart, H.X. and Williamson, J.M., 2001. Modeling concordance correlation via GEE to evaluate reproducibility. Biometrics 57:931–940.
Berman, M. and Schonfeld, R.L., 1956. Invariants in experimental data on linear kinetics and the formulation of models. Journal of Applied Physics 27:1361–1370.
Boston, R.C., Wilkins, P.A., and Tedeschi, L.O., 2007. Identifiability and Accuracy: Two critical problems associated with the application of models in nutrition and the health sciences. In: Hanigan, M. (ed.) Mathematical Modeling for Nutrition and Health Sciences. Roanoke, VA, USA. pp.161–193.
Carrasco, J.L. and Jover, L., 2003. Estimating the generalized concordance correlation coefficient through variance components. Biometrics 59:849–858.
Carrasco, J.L. and Jover, L., 2005. Concordance correlation coefficient applied to discrete data. Statistics in Medicine 24:4021–4034.
Carrasco, J.L., Jover, L., King, T.S. and Chinchilli, V.M., 2007. Comparison of concordance correlation coefficient estimating approaches with skewed data. Journal of Biopharmaceutical Statistics, 17:673–684.
Carrasco, J.L., King, T.S. and Chinchilli, V.M., 2009. The concordance correlation coefficient for repeated measures estimated by variance components. Journal of Biopharmaceutical Statistics 19:90–105.
Chinchilli, V.M., Martel, J.K., Kuminyika, S. and Lloyd, T., 1996. A weighted concordance correlation coefficient for repeated measurement designs. Biometrics 52:341–353.
Cobelli, C. and DiStefano, III, J.J., 1980. Parameter and structural identifiability concepts and ambiguities: a critical review and analysis. American Journal of Physiology (Regulatory, Integrative and Comparative Physiology) 239:R7–24.
Deyo, R.A., Diehr, P. and Patrick, D.L., 1991. Reproducibility and responsiveness of health statis measures. Controlled Clinical Trials 12:142S–158S.
Freese, F., 1960. Testing accuracy. Forest Science 6:139–145.
Godfrey, K., 1983. Compartmental Models and their Application. Academic Press, London and New York.
Jacquez, J.A., 1996. Compartmental Analysis in Biology and Medicine (3rd ed.). BioMedware, Ann Arbor, Michigan.
King, T.S. and Chinchilli, V.M., 2001a. A generalized concordance correlation coefficient for continuous and categorical data. Statistics in Medicine 20:2131–2147.
King, T.S. and Chinchilli, V.M., 2001b. Robust estimator of the concordance correlation coefficient. Journal of Biopharmaceutical Statistics 11:83–105.
King, T.S., Chinchilli, V.M. and Carrasco, J.L., 2007. A repeated measures concordance correlation coefficient. Statistics in Medicine 26:3095–3113.
Krippendorff, K., 1970. Bivariate agreement coefficients for reliability of data. Sociological Methodology, 2:139–150.
LeCourtier, Y. and Walter, E., 1981. Comments on ‘On parameter and structural identifiability: non unique observability/reconstructibility for identifiable systems, other ambiguities and new defnitions’. IEEE Transactions and Automatic Control 26:800–801.
Liao, J.J.Z., 2003. An improved concordance correlation coefficient. Pharmaceutical Statistics 2:253–261.
Lin, L., Hedayat, A.S. and Wu, W., 2007. A unified approach for assessing agreement for continuous and categorical data. Journal of Biopharmaceutical Statistics 17:629–652.
Lin, L.I.-K., 1989. A concordance correlation coefficient to evaluate reproducibility. Biometrics, 45:255–268.
Lin, L.I.-K., Hedayat, A.S., Sinha, B. and Yang, M., 2002. Statistical methods in assessing agreement: Models, issues, and tools. Journal of the American Statistical Association 97:257–270.
Müller, R. and Büttner, P., 1994. A critical discussion of intraclass correlation coefficients. Statistics in Medicine 13:2465–2476.
Nickerson, C.A.E., 1997. A note on ‘A concordance correlation coeffcient to evaluate reproducibility’. Biometrics 53:1503–1507.
Stefanovski, D., Moate, P.J. and Boston, R.C., 2003. WinSAAM: A Windows-based compartmental modeling system. Metabolism: Clinical and Experimental 52:1153–1166.
Tedeschi, L.O., 2006. Assessment of the adequacy of mathematical models. Agricultural Systems, 89:225–247.
Williamson, J.M., Lipsitz, S.R. and Manatunga, A.K., 2000. Modeling kappa for measuring dependent categorical agreement data. Biostat 1:191–202.
Author information
Authors and Affiliations
Corresponding author
Editor information
Rights and permissions
Copyright information
© 2011 Wageningen Academic Publishers
About this chapter
Cite this chapter
Tedeschi, L.O., Boston, R. (2011). Identifiability and accuracy: a closer look at contemporary contributions and changes in these vital areas of mathematical modelling. In: Sauvant, D., Van Milgen, J., Faverdin, P., Friggens, N. (eds) Modelling nutrient digestion and utilisation in farm animals. Wageningen Academic Publishers, Wageningen. https://doi.org/10.3920/978-90-8686-712-7_10
Download citation
DOI: https://doi.org/10.3920/978-90-8686-712-7_10
Publisher Name: Wageningen Academic Publishers, Wageningen
Online ISBN: 978-90-8686-712-7
eBook Packages: Biomedical and Life SciencesBiomedical and Life Sciences (R0)