Uncertainty analysis, Monte Carlo technique

Figure 22.4 Monte Carlo techniques were used to simulate different hypothetical individuals for different instances of the trial design, using variability and uncertainty distributions from the model analysis. The result is a collection of predicted outcomes, shown as a binned histogram (top figure). Success was defined as a difference in end point measurement of X or smaller between drug and comparator. Likelihood of success (shown in the bottom figure as a cumulative probability) for this example (low/medium drug dose and high comparator dose) is seen to be low, about 33%.

The estimates of an NCE s expected sales trajectory can be analyzed using a Monte Carlo technique, which produces a range of possible outcomes, given uncertainty about the actual magnitudes of various input assumptions. The results of a Monte Carlo analysis... 

Thompson KM, Burmaster DE, and Crouch EAC (1992) Monte Carlo techniques for quantitative uncertainty analysis in public health risk assessments. Risk Analysis 12 53-63. 

Monte Carlo—A statistical technique commonly used to quantitatively characterize the uncertainty and variability in estimates of exposure or risk. The analysis uses statistical sampling techniques to obtain a probabilistic approximation to the solution of a mathematical equation or model. 

First-order error analysis is a method for propagating uncertainty in the random parameters of a model into the model predictions using a fixed-form equation. This method is not a simulation like Monte Carlo but uses statistical theory to develop an equation that can easily be solved on a calculator. The method works well for linear models, but the accuracy of the method decreases as the model becomes more nonlinear. As a general rule, linear models that can be written down on a piece of paper work well with Ist-order error analysis. Complicated models that consist of a large number of pieced equations (like large exposure models) cannot be evaluated using Ist-order analysis. To use the technique, each partial differential equation of each random parameter with respect to the model must be solvable. 

The following section describes the most commonly used technique for propagating variability and parameter uncertainty separately, 2nd-order Monte Carlo analysis. A brief case study illustrating the technique is included in Section 7.3. 

Burmaster and Anderson (1994) have compiled a list of principles of good practice, which were originally aimed at Monte Carlo simulations, but are valuable also for other techniques in uncertainty analysis. These recommendations later appeared in modified and supplemented form in various handbooks and other publications, e.g., the USEPA Guiding Principles for Monte Carlo Analysis (USEPA 1997). 

Monte Carlo simulation An iterative resampling technique frequently used in uncertainty analysis in risk assessments to estimate the distribution of a model s output parameter. 

This section provides an overview of common methods for quantitative uncertainty analysis of inputs to models and the associated impact on model outputs. Furthermore, consideration is given to methods for analysis of both variability and uncertainty. In practice, commonly used methods for quantification of variability, uncertainty or both are typically based on numerical simulation methods, such as Monte Carlo simulation or Latin hypercube sampling. However, there are other techniques that can be applied to the analysis of uncertainty, some of which are non-probabilistic. Examples of these are interval analysis and fuzzy methods. The latter are briefly reviewed. Since probabilistic methods are commonly used in practice, these methods receive more detailed treatment here. The use of quantitative methods for variability and uncertainty is consistent with, or informed by, the key hallmarks of data... 

Monte Carlo simulation modeling represents the next stage or advancement of uncertainty analysis. This computer-aided stochastic (i.e., random, involving chance) probability analysis technique allows one to more transparently and completely present information about the predictions of exposure and the uncertainty associated with these predictions. In this method the predictor variables, in this case G and Q are described as distributions rather than point estimates of best, worst or average. 

Sensitivity analysis is a type of uncertainty analysis that is used to consider the impacts of uncertainty. In such analyses, one input is changed at a time to determine how the results of a model will change over the range of possible values of that single input. Multiple inputs can be varied simultaneously, using a sampling technique called Monte Carlo analysis, to obtain an overall distribution of the result. 

Characterizing the overall uncertainties associated with the PBPK model estimates is also an important component of the PBPK model evaluation and application. This includes characterizing the uncertainties in model outputs resulting from the uncertainty in the PBPK model parameters. Traditionally, Monte Carlo has been employed for performing uncertainty analysis of PBPK models (39, 40). Some of the recent techniques that have been applied for the uncertainty analysis of PBPK models include the stochastic response surface method (SRSM) (38, 41) and the high-dimensional model reduction (HDMR) technique (42). 

Monte Carlo analysis is a specific probabilistic assessment method that can be used to characterize health risks and their likelihood of occurrence based on a wide range of parameters (Shade and Jayjock 1997). The U.S. EPA s Stochastic Human Exposure and Dose Simulation (SHEDS) model allows for the quantification of exposures based on a probabilistic assessment of multiple exposure pathways and multiple routes of exposure (Mokhtari et al. 2006 US EPA 2003b). Additional applications of probabilistic techniques wiU be discussed in the section below on conducting an uncertainty analysis of reconstructed exposure values. 

Regardless of the chosen exposure reconstruction method, it is important to document the exposure estimation process, and, whenever possible, evaluate any uncertainties inherent in the selected methodology and the results obtained using Monte Carlo or a similar technique (Benke et al. 2001 Stewart et al. 1996, 2003). Uncertainty analysis techniques are increasingly common in exposure reconstruction, and can be critical to understanding the variability and uncertainty in the estimates generated. 

Statistical thermal design uncertainty was evaluated by a Monte Carlo sampling technique combined with the subchannel analysis code. The engineering imcer-tainty of the Super LWR is evaluated as 31.88°C based oti the Monte Carlo Statistical Thermal Design Procedure. 

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