Variance propagation

In order to determine variation in PBLx intake resulting from the variance (due to uncertainty and variability) in the parameters used to describe the source-to-dose model, we first use the 

For the DPD method, as for the factorial design example above, we used three input variables and a High , Medium and Low value to represent each input distribution. The values used to represent these ranges are listed in Table A2.5. In the DPD case, the High and Low values were calculated as the median values of the upper and lower 33rd percentiles of the lognormal distributions used to represent the variance of the input parameters. The Medium value was set at the median value of each input distribution. 

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Sensitivity analysis methods can be used in combination with methods for variance propagation. For example, Cullen Frey (1999) describe how variance in the sum of random numbers can be apportioned among the inputs to the sum. All of the statistical sensitivity methods mentioned above can be applied to the results of Monte Carlo simulation, in which... 

This is an example exposure assessment that illustrates quantitative representations of uncertainty and variability at the higher tiers of an exposure assessment. This case-study is based on human exposures to a persistent, bioaccumulative and lipid-soluble compound through fish consumption. This compound is fictional and referred to here as PBLx, but it has properties that correspond to those of known persistent compounds. Specific goals of this case-study are to illustrate (1) the types of uncertainty and variability that arise in exposure assessments, (2) quantitative uncertainty assessment, (3) how distributions are established to represent variability and uncertainty, (4) differences among alternative variance propagation methods, (5) how to distinguish uncertainty from variability and (6) how to communicate the results of an uncertainty analysis. 

The composition of the case-study includes a conceptual model, the modelling approach, construction of input distributions and variance propagation methods. When evaluating uncertainty, it is important to consider how each of these elements contributes to overall uncertainty. 

For many mathematical operations, including addition, subtraction, multiplication, division, logarithms, exponentials and power relations, there are exact analytical expressions for explicitly propagating input variance and covariance to model predictions of output variance (Bevington, 1969). In analytical variance propagation methods, the mean, variance and covariance matrix of the input distributions are used to determine the mean and variance of the outcome. The following is an example of the exact analytical variance propagation approach. If w is the product of x times y times z, then the equation for the mean or expected value of w, E(w), is ... 

Bevington (1969) lists variance propagation solutions like the one above for several mathematical operations. 

In order to illustrate the use of the variance propagation methods described above, we have selected for the case-study a simple three-input exposure model. The three inputs for this model include water concentration, fish BCF and fish consumption rates. The model output is dose expressed in micrograms per day averaged over a one-year exposure period. This model has the form ... 

A2.3.7 Variance propagation with uncertainty and variability combined... 

Table A2.6 Selected statistics of the PBLx intake distribution obtained from different model variance propagation methods. ...

The differences in estimation of these moments for each scenario are graphically illustrated in Figures A2.4, A2.5 and A2.6, where the CDFs obtained from each numerical variance propagation method are compared with the analytical results. The results of the analytical method are assumed to represent the true moments of the model output and, therefore, the true CDF. Mean and standard deviation of ln(x) are used in plotting the analytical CDF. The equations for the transformation from arithmetic moments to the moments of ln(x) are as follows ... 

So far in the case-study, we have focused on variance propagation methods and have not made an effort to distinguish between the relative contributions to overall variance from uncertainty and variability. In the examples above, the cumulative distributions presented in figures all reflect overall variance that includes the combined contributions from both uncertainty and variability. So our last step is to illustrate a two-dimensional analysis in which we distinguish and display separate contributions from uncertainty and variability. We begin this analysis by going back to our inputs and assessing the relative contributions from uncertainty and variability. 

A parameter such as a rate constant is usually obtained as a consequence of various arithmetic manipulations, and in order to estimate the uncertainly (error) in the parameter we must know how this error is related to the uncertainties in the quantities that contribute to the parameter. For example, Eq. (2-33) for a pseudo-first-order reaction defines k, which can be determined by a semilogarithmic plot according to Eq. (2-6). By a method to be described later in this section the uncertainty in itobs (expressed as its variance associated with cb. Thus, we need to know how the errors in fcobs and cb are propagated into the rate constant k. 

This argument obviously can be generalized to any number of variables. Equation (2-65) describes the propagation of mean square error, or the propagation of variances and covariances. 

If we assume that the residuals in Equation 2.35 (e,) are normally distributed, their covariance matrix ( ,) can be related to the covariance matrix of the measured variables (COV(sy.,)= LyJ through the error propagation law. Hence, if for example we consider the case of independent measurements with a constant variance, i.e. 

The variance characterizes the spread of AA if an infinite number of independent simulations are carried out, each with a finite sample of size N. In practice, usually only one estimate (or a small number of repeats) of free energy differences are taken, and the variance in free energy must be estimated. One way to compute the variance is to use the error propagation formula (for a forward calculation)... 

The variance of the free energy calculation is (with error propagation)... 

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