Parametric Sensitivity analysis of nonlinear multiparmeter
Regression model systems

H. Sulieman1*, P.J. McLellan2
1American University of Sharjah, UAE; 2Queen’s University, Canada
hsulieman@aus.edu

Parametric sensitivity analysis is applied here in the context of nonlinear regression parameter estimation in multi-response models. The predictions from these models are subject to uncertainties propagated from the estimated parameters using available experimental data. When the uncertainties associated with these estimates are unacceptably large, an alternative purpose of the experimental effort could be to reduce some or all of the parameter uncertainties that propagate into the model predictions. This situation motivates the parametric sensitivity analysis in regression parameter estimation.

The conventional approach for summarizing parametric sensitivity of the predicted responses is through the use of the first-order derivatives of the predicted response(s) with respect to each parameter. These sensitivity coefficients can be termed as “marginal” sensitivities because they focus on individual parameters only. While they provide helpful insight into the estimation characteristics, they can be misleading when correlations among parameters exist. 
 
Sulieman et al. (Technometrics 43 (4), 2001) developed a new parametric sensitivity measure based on profiling algorithm used to assess parameter nonlinearities in regression models and to construct likelihood intervals for the individual parameters. They applied the measure to single-response nonlinear models and later extended the approach to assess parameter sensitivities in multi-response regression models. The profile-based sensitivity coefficient can be defined as the total derivative of the model response predicted function with respect to a parameter of interest, with the remaining parameters held at their conditional estimates. Consequently and in contrast with marginal sensitivity coefficient, the profile-based sensitivity measure accounts for both correlation structure among the parameters and model nonlinearity. It also provides sensitivity information over wide range of parameter uncertainties.

In this paper, an overview of profile-based sensitivity measure for multi-response regression models and some related computational aspects are given. The new sensitivity approach is applied to regression models described by systems of Ordinary Differential Equations using Box-Draper determinant criterion for parameter estimation. Application of the approach is illustrated using a model formulated by the Dow Chemical Company to describe the kinetics of an isothermal batch reactor system. The following formulation of the model was studied by Biegler and Tjoa (1991) and is used here.

Where 

The model includes eight parameters to be estimated using three sets of experimental data taken at different settings and three response variables. A wide range of sensitivity behaviour is observed, including instances in which the traditional and profile-based sensitivity coefficients lead to different conclusions concerning the impact of a parameter of interest on the predicted responses from the model.

References  

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[2]   Biegler, L.T. and Tjoa, I.B., 1991. "Simultaneous solution and optimization strategies for parameter 
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[3]   Saltelli, A., Tarantola, S., Campolongo, F. and Ratto, M., 2004.Sensitivity Analysis in Practice. A Guide to 
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[4]   Saltelli, A., Chan, K. and Scott, E.M., 2000. Sensitivity Analysis, Wiley, New York, USA.

[5]   Sulieman, H., McLellan, P.J. and Bacon, D.W., 2004. "A Profile-based approach to parametric sensitivity 
        in multi-response regression models", Computational Statistics & Data Analysis, 45, 721-740.

[6]   Sulieman, H., McLellan, P.J. and Bacon, D.W., 2001. "A Profile-Based Approach to Parametric Sensitivity 
       Analysis of Nonlinear Regression Models", Technometrics, 43(4), 425-33.

[7]  Turanyi, T., 1990. "Sensitivity Analysis of Complex Kinetic Systems. Tools and Applications", Journal of 
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