Some assumptions contain elements of both uncertainty and variability. For instance, an assumption might describe the distribution of body weights in a population, but the parameters of the distribution might be uncertain. These types of assumptions are called second-order assumptions (also, second-order random variables; see Burmaster and Wilson, 1996, in the Bibliography). You can model these types of assumptions in Crystal Ball by placing the uncertain parameters of the distribution in separate cells and defining these cells as assumptions. You then link the parameters of the variability assumption to the uncertainty assumptions using cell references.
To illustrate this for the Toxic Waste Site.xls spreadsheet:
Enter the values 70 and 10 into cells G4 and G5, respectively.
These are the mean and standard deviation of the Body Weight assumption in cell C4, which is defined as a normal distribution.
Define an assumption for cell G4 using a normal distribution with a mean of 70 and a standard deviation of 2.
Define an assumption for cell G5 using a normal distribution with a mean of 10 and a standard deviation of 1.
Enter references to these cells in the Body Weight assumption.
For an example, see Figure 86, Assumption Using Cell References for the Mean and Standard Deviation, following.
When you run the tool for second-order assumptions, the uncertainty of the assumptions’ parameters is modeled in the outer simulation, and the distribution of the assumption itself is modeled (for different sets of parameters) in the inner simulation.
Often, the parameters of assumptions are correlated. For example, you would correlate a higher mean with a higher standard deviation or a lower mean with a lower standard deviation. Defining correlation coefficients between parameter distributions can increase the accuracy of the two-dimensional simulation. With data available, as in sample body weights of a population, you can use the Bootstrap tool to estimate the sampling distributions of the parameters and the correlations between them.
