Each simulation trial begins with the simulation of variance components to be taken as the true values. These true values are then used to compute expected means squares. The variance and variance proportion simulations are computed using simulated mean square terms. The mean square terms are simulated from chi-square distributions scaled by the associated expected mean square. The simulated variances are computed using method of moments estimators and are then used in computing the confidence bounds for the variance proportions.
For example, consider the variance for Part. The simulation computations follow these steps:
1. Simulate the mean square for Part, the mean squares for each two-way interaction involving Part, as well as any mean squares for higher order interactions involving Part.
2. Compute the variance using inclusion-exclusion:
variance of Part = mean square of Part – mean square of two-way interactions + mean square of three-way interactions -…+/-mean square of highest order interaction
3. The simulated variance proportion for Part is then computed as Variance of Part/Total Variance.
Note: Only the distributions of the mean square terms are known directly. Specifically, Mean Square = Chi-Squared Distribution(df)*Expected Mean Square/df. The distributions of the variance and variance proportions are approximated by the simulated values. Because of this, the estimated bias and standard errors can differ each time a variance term is updated or each time the platform is run.