1518 Mokhtari and Frey
Fig. 3. Case study scenario for application of sensitivity analysis to the
comingled variability and uncertainty analysis of a model.
based on a “one dimensional” comingling of vari- ability and uncertainty are compared to those from
a “two-dimensional” simulation in which variability and uncertainty are distinguished. The latter is typi-
cally preferred where possible and appropriate to the assessment objective.
Fig. 3 illustrates sensitivity analysis in the context of comingled variability and uncertainty. There is one
data set generated for each of the uncertainty real- izations of the model. Each data set has m rows of
randomly generated samples from variability distri- butions and the corresponding output values. Prior
to application of sensitivity analysis, these data sets are appended together to form a single data set with
n × m
rows. Sensitivity analysis is then applied to the comingled data set and a set of rankings is obtained.
For the comingled analysis in the growth estimation part values of 100 and 650 were selected for n and m,
respectively.
4. RESULTS
This section presents results from application of ANOVA to each of the probabilistic scenarios defined
in Section 3.4. For each probabilistic scenario, results from ANOVA are compared with results from Pear-
son and Spearman correlation analyses. Application of ANOVA to identify a saturation point in the model
response and a methodology for quantifying ambigu- ity in ranks in a one-dimensional analysis is demon-
strated. Ranking of the important factors based on ANOVA also is verified.
4.1. Application of ANOVA
Inputs in the growth estimation part are continu- ous, and hence must be partitioned into levels.
36
Frey et al.
demonstrated three approaches for defining fac- tor levels for continuous inputs based on: 1 evenly
spaced intervals; 2 evenly spaced percentiles; and 3 visual inspection of the cumulative distribution
function CDF for each input.
25
For the first ap- proach, each input domain is classified into equal
ranges. For the second approach, the CDF of the gen- erated values for an input in a probabilistic simula-
tion is used for defining factor levels at evenly spaced percentiles. For the third approach, boundaries for
each factor level are defined corresponding to per- centiles of the CDF that are associated with a sub-
stantial change in the slope.
A key consideration in defining levels is the num- ber of data points within factor levels, which affects
Two-Dimensional Probabilistic Risk Assessment Model 1519
Table III. Levels Defined for Factors
in the Growth Estimation Part of the E. coli
Model Number
Levels and Corresponding Factor
a
of Levels Percentiles
b
Temp1 5
7.5–11, 11–14.5, 14.5–18, 18–21.5, 21.5
c
Temp2 3
7.5–13.5, 13.5–19.5, 19.5
c
Temp3 5
7.5–11, 11–14.5, 14.5–18, 18–21.5, 21.5
c
Time1 12
0–24, 24–48, . . . , 264–288, 288
c
Time2 2
0–3.5, 3.5
c
Time3 12
0–24, 24–48, . . . , 264–288, 288
c
MD 3
{6.5,6.5–8.5, 8.5} {20th, 80th} Percentiles LP1
4 {50, 50–65, 65–95, 95} {20th, 50th, 80th} Percentiles
LP2 4
{35, 35–55, 55–90, 90} {20th, 50th, 80th} Percentiles LP3
4 {45, 45–65, 65–95, 95} {20th, 50th, 80th} Percentiles
GT1 4
{7, 7–9.5, 9.5–12.5, 12.5} {20th, 50th, 80th} Percentiles GT2
4 {4.5, 4.5–8, 8–12, 12} {20th, 50th, 80th} Percentiles
GT3 4
{6.5, 6.5–9.5, 9.5–13, 13} {20th, 50th, 80th} Percentiles
a
The abbreviations used for factors in this table are the same as those defined in Table I.
b
The ranges that define each factor level and the percentiles of the CDF corresponding to the breakpoint between factor levels are given.
c
For this factor equal intervals are used as levels.
the power of statistical tests.
42
There is a tradeoff be- tween selecting a larger number of factor levels, which
can produce more highly resolved insights regarding sensitivity, and getting statistically significant results.
There is also a tradeoff between the desired number of iterations e.g., in a Monte Carlo simulation that
are used to populate factor levels and the computa- tional time.
Table III summarizes the levels defined for factors in the case studies. These levels are used in both proba-
bilistic scenarios. Levels are mostly defined based on visual inspection of the CDF for each factor. Each
CDF is prepared based on the generated values for a factor in the comingled analysis of variability and un-
certainty, since this probabilistic approach gives the widest range of variation for each factor. For storage
times at Stages 1 and 3, levels are defined at equal intervals.
4.2. Variability Analysis for Different Uncertainty Realizations
