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This chapter discusses probabilistic methods for conducting uncertainty analysis, methods that can be use to evaluate both local and global sensitivity of models to parameters, and issues related to the validation of models that express uncertainty in their results. Analytical and Monte Carlo methods for propagating uncertainty through models are described, along with potential limitations of these methods and the problems that can be encountered. The chapter introduces methods for assigning distributions to model parameters. Statistical methods that can be used to help interpret and express the results of probabilistic uncertainty analyses, such as confidence and tolerance intervals, are introduced and their pertinent assumptions are described. Various statistical analyses that can be used for sensitivity analysis and their associated sampling designs are reviewed.

The effective management of natural resources use requires a mechanistic understanding of the system, not just correlations between variables of the kind discussed in Chapter 4. Understanding may simply be in the form of a conceptual model, but is much more powerful when formalized as a mathematical model. This chapter introduces methods for building a model of the system that can be used to predict future sustainability with or without management interventions. The emphasis is on the simulation of biological and bioeconomic dynamics, for which step-by-step worked examples are given. These examples start with conceptual models, then show how to formalize these as mathematical equations, build these into computer code; test model sensitivity, validity, and alternative structures; and finally, explore future scenarios. Methods for modelling stochasticity and human behaviour are also introduced, as well as the use of Bayesian methods for understanding dynamic systems and exploring management interventions.

This chapter discusses incomplete sensitivity. Incomplete sensitivity means that during sensitivity evaluation, only the deformation of the geometry is accounted for and the change of the state variable due to the change of geometry is ignored. The chapter gives the class of functionals for which this assumption can be made. Incomplete sensitivity calculations are illustrated on several model problems, giving the opportunity of introducing low-complexity models for sensitivity analysis. The chapter shows by experience that the accuracy is sufficient for quasi-Newton algorithms, and also that the complexity of the method is drastically reduced making possible real time sensitivity analysis latter used for unsteady applications.

This chapter continues the discussion of the preceding chapter on how to deal with hidden bias caused by unobserved differences between the treatment (T) and control (C) groups. The preceding chapter presented practical and basic approaches; this chapter shows other approaches for hidden bias. Sensitivity analysis examines how a finding obtained under no hidden bias assumption changes as hidden bias is allowed; sensitivity analysis is informative but stops short of giving a definite answer to hidden bias. The more conventional approach would be parametric ‘selection-correction’ or ‘control-function’ methods; these are relatively straightforward to implement, but may be too restrictive. Nonparametric ‘bounding’ approaches would perhaps be too unrestrictive, providing bounds on the treatment effect of interest which are typically too wide to be useful. An approach to avoid hidden bias by controlling for post-treatment covariates is available in some special circumstances.

There are several ways to detect publication bias and assess its influence in a meta-analysis. This chapter identifies out-dated methods and describes current approaches. It discusses cumulative meta-analysis and subgroup analysis. Moderator analysis and meta-regression are used to explore variations in effect sizes due to methodological features and other variables in the primary studies. Sensitivity analysis is used to discover the extent to which results of meta-analysis are robust for outliers, and for decisions and assumptions made during the analysis. Finally, statistical power analysis is an important tool for planning and evaluating meta-analyses.

This chapter describes sensitivity analysis and automatic differentiation (AD). These include the theory, then an object oriented library for AD by operator overloading, and finally the authors' experience with AD systems using code generation operating in both direct and reverse modes. The chapter describes the different possibilities and through simple programs, gives a comprehensive survey of direct AD by operator overloading and for the reverse mode, the adjoint code method. Several elementary and more advanced examples help the understanding of this central concept.

Next to preventing injuries, it is of increasing interest to have some idea on the cost aspects of injuries and the prevention thereof. What does an injury cost in terms of direct medical costs and indirect (e.g. work absenteeism) costs? Additionally, how much can be saved when injuries are prevented, taking into account that preventive measures have certain costs attached as well? This chapter focuses on this relatively new concept for sports injury research and provides background for gathering and analyzing cost data related to sports injury (prevention).

In some decision problems, even after reflection, relevant probabilities and utilities are indeterminate. Then a rational decision maximizes utility with respect to a quantization of beliefs and desires. It is robust with respect to a sensitivity analysis. Rationality does not require meeting the maximin rule, which in its standard form expresses excessive aversion to risk. Multiple decisions must form a coherent set, however.

This paper explains the results of Zangemeister et al. The experimental and modeling results of Hannaford et al. were also reassessed with respect to normally fast and very fast time-optimal movements. More importantly, this study employs mathematical and manipulation analysis of the model, specifically the threefold approach of sensitivity analysis to gain valuable insights about the pathologic features of clinical neurologic deficits. The modification of an existing model is also presented and briefly discussed in this chapter. This chapter concludes that applying powerful mathematical tools such as threefold sensitivity analysis to analytic models is helpful in explaining and treating disorders of motor control.

This chapter provides a framework for the prevention and control of bias due to unequal attrition in cohort studies. Like the effects of third-variable confounding, the effects of unequal attrition can be limited by the design, as well as the analysis, of a cohort study. The most problematic kind of attrition is differential. This kind of attrition causes a bias that may not be remediable. The chapter uses sensitivity analysis to gauge the potential bias.

This chapter describes the implications of network stoichiometry for sensitivity analysis. It reinterprets the main results of metabolic control analysis (MCA) from the point of view of engineering control theory. The equivalent results are derived from a control-engineering viewpoint, culminating in a control-theoretic interpretation of the main results of metabolic control analysis: the summation and connectivity theorems. It illustrates that MCA offers an extension of standard sensitivity analysis. This chapter bridges the gap between the fields of metabolic control analysis and engineering control theory, further advancing cross-fertilization between these complementary fields.

Tanzania has traditionally been an egalitarian society. However, the debt crisis of the 1980s and the ensuing adjustment policies period proved detrimental to equity. The economic recovery recently enjoyed by Tanzania has not resulted in a significant reduction of income poverty. While there has been progress in some social indicators it has been unconscionably slow for many dimensions of well-being. These observations call for a deeper analysis of the characteristics of poverty and disparities, moving beyond income poverty and income distribution. Thus, this chapter approaches the issue from the perspective of child poverty and its characteristics. Using a multidimensional approach, the analysis is centered on the depth and severity of child poverty and on the changes between 1999 and 2004/5. The sensitivity analysis indicates the estimation of child poverty is robust to changes in thresholds of various dimensions of deprivation. The analysis of the evidence gives indications on how to pursue effective policies that could improve the situation of children, especially girls, not only in education and nutrition but in the other dimensions as well.

The chapter “Sensitivity Analysis” reviews why sensitivity analysis is a critical component of mathematical modeling, and the different ways of approaching it. A sensitivity analysis is an attempt to identify the parts of the model (i.e. structure, parameter values) that are most important for governing the output. It is an important part of modeling because it is used to quantify the degree of uncertainty in the model prediction and, in many cases, is the main goal of the model (i.e. the model was developed to identify the most important ecological processes). The chapter covers the idea of “local” versus “global” sensitivity analysis via individual parameter perturbation, and how interactive effects of parameters can be revealed via Monte Carlo analysis. Structural versus parameter uncertainty is also explained and explored.

The key objective of uncertainty analysis is to support better decision making. Uncertainty analysis can help inform the standard decision options of “accept” or “reject,” but extend these options to include, for example, adoption alongside research or adoption only in the context of research. Since the original Panel’s report, a range of analytical methods has emerged to guide these decisions. We note that deterministic sensitivity analysis can provide useful insights into model behavior and validation, but emphasize that probabilistic sensitivity analysis provides stronger analytical support for decision making. The Second Panel therefore urges that structural uncertainties be tested in analyses, that decision uncertainty be presented using probabilities for specified cost-effectiveness thresholds or cost-effectiveness acceptability curves (CEACs), and that expected value-of-information analysis be used to guide decision making fully by quantifying the value of generating additional evidence.

This chapter describes a method for constructing confidence intervals for general equilibrium model results using the covariance matrices routinely computed in the course of estimating the model's parameters. The method is then applied to IGEM to produce confidence intervals for several types of results: (i) the levels of variables in IGEM's base case, (ii) deviations in variables resulting from a policy experiment consisting of a carbon tax used to reduce the rate of tax on capital income, and (iii) the effects of the carbon tax swap on the components of social welfare. For each case, a decomposition analysis is used to identify the key uncertainties underlying the confidence interval. The carbon tax swap is shown to produce statistically significant gains in key variables, including social welfare, along with significant reductions in pollution. The method is also shown to have important advantages over alternative approaches. The confidence intervals it produces are very similar to those generated by Monte Carlo simulation but are much faster to compute for large models. In addition, the decomposition analysis shows that off-diagonal terms in the parameter covariance matrices often contribute significantly to the confidence intervals for key results. By capturing these effects, the method provides better characterization of uncertainty than alternative methodologies that rely on independent perturbations of parameters.

This chapter introduces the useful model-analytic tools of both sensitivity analysis and structured singular value analysis and their application to cellular networks. It reviews the Nyquist stability criterion and extends it to conditions guaranteeing robust stability (RS). It then examines the structured singular value analysis for robust performance. This chapter shows that control-theoretic tools, such as sensitivity analysis and phase sensitivity, offer powerful means for network elucidation and manipulation in biological systems. It suggests that using tools from control theory to guide both mathematical modeling and experimental design can facilitate the iterative paradigm of systems biology and shed light on the complex network behavior underlying biological organisms.

This chapter examines the properties of game-theoretic solutions in RICE through sensitivity analysis from an incentive perspective. The issues include intertemporal stability of the grand coalition under the Lindahl social welfare weights, the range of solutions with the core properties or having the Lindahl equilibrium properties, and incentive reactions to false perception of climate change by individual regions.

This chapter discusses some of the reasons for seeking to model complex cellular biological systems. It begins by presenting a philosophical overview and historical context. It then considers the purposes and implications of modeling; the different kinds of models; and sensitivity analysis.

In his treatise on electricity and magnetism, Maxwell (1873) published an equation that described the conductivity of an electrolyte that contained nonconducting spheres as: . . . Ψ = co/cw = 2Φ/(3-Φ) . . . where the “meaning” of Ψ (psi) has been most commonly interpreted as some expression of tortuosity, Co and Cw are the conductivity of the medium and the electrolyte, respectively, and Φ is the proportion of the medium that is occupied by the electrolyte. Since that time, considerable efforts have been devoted to elucidation of the electrical properties of porous materials, particularly with the advent of the first resistivity log in 1927, which founded an entire industry focused on estimating fluid saturations in hydrocarbon reservoirs from downhole measurements. To some degree, spirited discussions in the literature reflect two schools of thought, one that considers the role of the resistive framework from a primarily empirical point of view, and the other that models the conductive fluid phase in terms of electrical efficiency. Clearly, the two concepts are intertwined because resistivity is the reciprocal of conductivity and the pore network is the complement of the rock framework. If the solid part of the rock is nonconductive, then the ability of a rock to conduct electricity is controlled by the conductive phase in the pore space, which should make the case for equations to be formulated from classical physical theory. This approach is typically developed using electrical flow through capillary tubes as a starting point. Unfortunately, the topological transformation of a capillary tube model to a satisfactory representation of a real pore network is a formidable challenge, so that mathematical solutions may not be acceptable, even though they are grounded in basic physics. The most successful model along these lines has been proposed by Herrick and Kennedy (1994), who maintain that while the Archie equation is a useful parametric function, it has no physical basis. Some of their conclusions are reviewed at the end of this chapter.