The results are given in Table 2, which lists the main and total effects for each epistemic factor. Recall that the main effects indicate, on average, how much the variance could be reduced with perfect information for that factor, and that differences between main and total effects represent the degree to which a factor is involved with interaction effects. A histogram of the reliability index is also given in Figure 2, which can be used to gauge the overall impact of epistemic uncertainty on the reliability assessment. Table 2 Epistemic factor sensitivities for beam example L w b E ε Main effect 0.40 0.18 0.15 0.03 0.01 Total effect 0.63 0.40 0.36 0.25 0.23 Fig. 2 Histogram of the reliability index for the beam example 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Reliability Index 0 1000 2000 3000 4000 5000 6000 Frequency The results indicate that on average, 40% of the variance in the reliability index could be reduced by gaining perfect information about the distribution of L. Thus, the best way to reduce uncertainty would be to collect additional samples to characterize the distribution of L. The analysis indicates that uncertainty in the distributions for wand b also contribute significantly to uncertainty in the reliability index. The uncertainty in E, however, is negligible, even though from a deterministic standpoint, the parameter E plays a very important role in the behavior of the beam. The reason that E shows a small main effect is because its distribution is already wellcharacterized based on 200 samples, so little epistemic uncertainty remains. The results also indicate that the deterministic model error, ε, has a negligible impact on the overall uncertainty: in this case, improving the confidence in the deterministic model would not help reduce the uncertainty in the reliability index. The total effects, by comparison with the main effects, may be used to gauge the role that interactions play for each factor. In this case, each factor shows a significant difference between main and total effects, which indicates that interactions John McFarland and David Riha 426
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