# UNCERTAINTY ANALYSIS AND RESULTS

## Introduction

[Chapter 11](../Chapter_11_ready%20for%20CDC%20031405.doc),
*Point-Estimate Results*, provided the results of the base-case dose
assessment. In that particular analysis, a single value of dose was
determined for each member of 7 hypothetical 4-member families (a total
of 28 receptors). Each family was subjected to a different exposure
scenario. These doses were obtained based on point estimates (single
representative values) assigned to the variables involved in the
analyzed models. This point-estimate analysis is frequently referred to
as a deterministic analysis.

While the single-point estimates used in a deterministic estimate of
dose may be considered "representative" or "realistic," they do not
capture the inherent uncertainty in the variables of interest. By
nature, many of the natural processes and phenomena that investigators
attempt to represent with equations and numbers will vary with time,
space, and environmental conditions. Likewise, many of the variables
used to represent or describe a natural process will change with time,
space, and the environmental conditions.

In addition to the uncertainty in the natural phenomena, there is
uncertainty associated with our understanding of how the phenomena used
to represent natural processes are related (the conceptual models),
their mathematical descriptions, and their computational implementation.
When these uncertainties are considered, the reconstructed dose is
uncertain; instead of a single-point estimate of dose for each receptor
(as described in [Chapter
11](../Chapter_11_ready%20for%20CDC%20031405.doc)), multiple values (a
probability distribution of dose) result. Simply, the uncertainty
approach involves treating specified input variables as sets of values
rather than as single values in the computation. The computation is
repeated multiple times using various combinations of the input variable
values. This results in a set of dose estimates rather than a single
value. This set can then be used to describe the uncertainty associated
with the result from the deterministic approach.

In the dose reconstruction for the Savannah River Site, the analysts
focused on the uncertainty associated with the variables that are used
to describe a particular process (e.g., the bio-uptake factors,
consumption of foodstuffs, and particle size). This approach is the most
common way to address uncertainty because it does not require
development of alternative conceptual models or computational codes. In
addition, this approach enables an analysis of the sensitivity (i.e.,
what changes in the results \[dose\] are produced by specific changes in
the input variables). In this report, the term "uncertainty analysis" is
used to describe three types of evaluation:

1\. Description of the uncertainty in dose (i.e., how much variability
is estimated for dose given uncertainty in the input variables).

2\. Description of how the uncertainty in dose depends on the
uncertainty for each uncertain input variable.

3\. Description of the sensitivity of dose to variations in input
variables.

Sometimes the terms "probabilistic" or "stochastic" analysis are used to
describe these evaluations.

### Overall Description of Uncertainty Analysis

Uncertainty analysis is sometimes referred to as probabilistic analysis
or Monte Carlo analysis. It involves multiple computations that use the
same code but change the values of the input variables with each
computation. The input values are selected randomly from either sets of
observations or distributions, using observations and judgment based
upon experience. Section 12.1.1.2 describes this selection process. The
selected values for the input variables of interest are then used in the
computation, and the value for each of the sampled variables changes
with each computation. For example, if 10 values are selected for each
variable, the computation is repeated 10 times. It should be noted that
each individual computation is a deterministic analysis similar to the
point-estimate case but with different values for the variables.

In this analysis, the effect of 14 variables on dose was investigated.
This required a series of computations that changed the values of these
variables with each computation. As explained below, each computation is
called a realization and yields a dose estimate based on a particular
choice of values for all the input variables.

In an uncertainty analysis, the aggregate results are generally of
interest rather than the result from any one computation. This set of
results can be described in terms of simple statistics to give the
representative values (e.g., mean), range, and nature of the
distribution. In exercises such as dose reconstruction or risk analysis,
uncertainty analysis provides insight into the uncertainty associated
with the results.

Uncertainty analysis can be approached in a variety of ways. The most
common approach is the Monte Carlo approach, which is based on random
selection of the variable values from specified distributions. Section
12.1.1.2 describes the details of the Monte Carlo approach.

#### Rationale -- Why We Do It; What Are We Trying to Determine?

The primary task of this project was to estimate the dose to 28
hypothetical individuals resulting from exposure to releases from the
Savannah River Site (SRS). This estimation was performed using two
methods: a point-estimate analysis and an uncertainty analysis. The
point-estimate analysis used a representative value for each of the
model input variables and resulted in an estimate of dose, cancer
incidence risk, and cancer fatality risk for each of the 28 hypothetical
receptors specified in the 7 scenarios. The point-estimate analysis
included all credible exposure pathways given the behaviors specified in
the scenarios and the land-use, water-use, and agricultural practices in
the region surrounding the SRS during the period studied. The
point-estimate analysis included only those radionuclides that survived
the Level 1 screening of the Phase II report. [Chapter
11](../Chapter_11_ready%20for%20CDC%20031405.doc) describes the
point-estimate results. The point-estimate results provided a very
detailed picture of what caused the dose and risk to each receptor
including:

-   How doses and risks changed in time.

-   What radionuclides contributed most to dose and risks for a
    particular receptor.

-   What pathways contributed most to dose and risks for a particular
    receptor.

-   How the behaviors specified in the various scenarios influenced
    doses and risks.

An uncertainty analysis was undertaken with the objective of quantifying
the uncertainty associated with the estimated doses. To reduce the
computational and data-handling burden, risks were not carried along as
part of the calculation; however, risks can be estimated from the doses
using adult risk factors. Because the uncertainty analysis involves
repeating the dose assessments of the point-estimate analysis many
times, a daunting amount of computation would be required if all input
variables were considered in the uncertainty analysis. To reduce the
effort associated with defining a distribution for each variable, the
dose-assessment model used for the uncertainty analysis was simplified.
Section 12.2 describes this simplification.

Because the results of the point-estimate analysis were used to guide
these simplifications, the point-estimate analysis played the role of a
screening analysis. As such, the point-estimate analysis was used to
determine which pathways and radionuclides were very minor contributors
to dose and could be excluded from the uncertainty analysis. In
addition, the point-estimate results helped to determine which variables
could be considered to be fixed, rather than uncertain, in the
uncertainty analysis as described in Section 12.3.

#### Approach -- Monte Carlo Sampling of Inputs to Provide Distributions of Dose

In uncertainty analysis, a randomly generated value selected from the
probability distribution of each uncertain variable is assigned to that
variable; other variables considered to be certain are assigned their
nonrandom values. The values assigned to all variables comprise the set
of values for one realization (i.e., one set of input variables for one
computer run) that results in a random value as the outcome for that
particular realization. Depending on the number of uncertain variables
involved in the model and the sampling method adopted, the number of
realizations required for good statistical estimates may range from a
few to a few thousand. A distribution of dose would be obtained after
estimating the dose for each of the realizations considered.

The process of random sampling falls into two categories: simple random
sampling (Monte Carlo sampling \[MCS\]) and stratified sampling (e.g.,
Latin Hypercube Sampling \[LHS\]) (1). In MCS, each uncertain variable
is assigned a range and probability distribution (mathematically
described as a probability density function) that may be based on
observations, judgment, or a combination of the two. The probability
density functions for each of the uncertain variables of interest are
then randomly sampled, resulting in a set of values for each of the
input variables of interest. Generally, many thousands of samples are
required to adequately represent the probability density function when
using simple random sampling. A large number of samples is needed to
obtain values over the entire range of the variable.

The LHS technique divides the probability density function associated
with an uncertain variable into several strata ("bins") to ensure the
probability of choosing a random value from each stratum is the same.
Compared to the Monte Carlo sampling technique, LHS is more efficient
and enables more complete sampling from the probability density function
with a limited number of samples. This technique requires more computer
memory because all samples are generated at once to ensure appropriate
distribution and independence or correlation among the variables. In
this analysis, the computation time for each realization was
sufficiently long to make LHS the preferred sampling technique because
fewer realizations are required.

## Development of a Simplified Model

Uncertainty analyses can be very computationally intensive, requiring a
number of individual computations (realizations) that is many times the
number of input variables. Each realization requires another computation
using the preprocessor, GENII v.2 Code, and postprocessor as described
in [Chapter 4](../Chapter_04_ready%20for%20CDC_031405.doc). To make the
problem more manageable, it was decided to focus the uncertainty
analysis on those radionuclides and pathways that were most important in
the determination of the point estimate. This simplification reduced the
computation time and the amount of data handling.

All the various contributors to dose, as described in [Chapter
11](../Chapter_11_ready%20for%20CDC%20031405.doc), were evaluated. These
include specific pathways such as dose from eating vegetables or dose
from a specific radionuclide such as Cs-137. Each pathway and/or
radionuclide contribution was compared with the total dose for a
particular receptor. The contributors to dose by pathway and
radionuclide were arranged in order from least to greatest. The smallest
contributor was eliminated, then the next smallest was eliminated, and
the process was repeated until the cumulative contribution of eliminated
pathways and/or radionuclides to the total dose was just under 5
percent. This process was performed on all 28 receptors.

The set of radionuclides and pathways satisfying the 5-percent criterion
for all receptors was chosen as the set for simplifying the analysis.
Because the pathways and radionuclides were not mutually exclusive
(e.g., removing both soil ingestion and uranium takes out uranium dose
from soil ingestion only once, but it was counted twice for the
cumulative percent), the actual reduction in total dose was always less
than 3 percent. With the removal of a particular radionuclide or
pathway, a number of input variables were also eliminated from the
computation. This helped to reduce the number of variables that needed
to be evaluated for inclusion in the uncertainty analysis.

It should be noted that eliminating radionuclides and pathways from the
analysis of a particular scenario in effect changed the conceptual model
that was evaluated for the scenario. The effect of this change in the
conceptual model was evaluated by comparing the point estimate of total
dose (all pathways and radionuclides) to the estimate of total dose from
the simplified model (reduced set of pathways and radionuclides). The
total dose (using the simplified model but using the exact same input
variable values as the point estimate) is designated "Run 0."

Table 12-1 presents the comparison of dose for each receptor from both
the original point estimate and from the simplified computation ("Run
0"). Because the simplified computation provides doses within 3 percent
of the more complete model, this simplification was considered to be
acceptable (Table 12-1 shows that the largest deviation is 2.6 percent
for the Child Born in 1964 for the Migrant Family scenario).

Table 12‑1 Comparison of Dose Estimates from Complete and Simplified
Models

+--------------+------------+-------------+----------+---------------+
| Scenario     | Family     | Point       | Run 0    | Ratio, Run 0  |
|              | Member     | Estimate    |          | over Point    |
|              |            |             | (milliS  | Estimate      |
|              |            | (mil        | ieverts) |               |
|              |            | liSieverts) |          |               |
+==============+============+=============+==========+===============+
| Delivery     | Adult      | 6.106       | 6.091    | 0.997         |
| Family       | Female     |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Delivery     | Adult Male | 6.283       | 6.266    | 0.997         |
| Family       |            |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Delivery     | Child Born | 5.200       | 5.180    | 0.996         |
| Family       | 1955       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Delivery     | Child Born | 2.090       | 2.081    | 0.996         |
| Family       | 1964       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Migrant      | Adult      | 0.447       | 0.438    | 0.979         |
| Family       | Female     |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Migrant      | Adult Male | 0.624       | 0.614    | 0.983         |
| Family       |            |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Migrant      | Child Born | 2.178       | 2.160    | 0.992         |
| Family       | 1955       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Migrant      | Child Born | 0.083       | 0.081    | 0.974         |
| Family       | 1964       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Near Water   | Adult      | 2.091       | 2.057    | 0.984         |
| Family       | Female     |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Near Water   | Adult Male | 2.205       | 2.170    | 0.984         |
| Family       |            |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Near Water   | Child Born | 3.137       | 3.099    | 0.988         |
| Family       | 1955       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Near Water   | Child Born | 1.759       | 1.734    | 0.986         |
| Family       | 1964       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Outdoor      | Adult      | 3.030       | 3.001    | 0.990         |
| Family       | Female     |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Outdoor      | Adult Male | 4.216       | 4.169    | 0.989         |
| Family       |            |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Outdoor      | Child Born | 9.435       | 9.383    | 0.994         |
| Family       | 1955       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Outdoor      | Child Born | 1.826       | 1.810    | 0.991         |
| Family       | 1964       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Rural Family | Adult      | 0.303       | 0.299    | 0.985         |
| One          | Female     |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Rural Family | Adult Male | 0.423       | 0.418    | 0.987         |
| One          |            |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Rural Family | Child Born | 1.589       | 1.580    | 0.994         |
| One          | 1955       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Rural Family | Child Born | 0.072       | 0.071    | 0.981         |
| One          | 1964       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Rural Family | Adult      | 0.696       | 0.685    | 0.985         |
| Two          | Female     |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Rural Family | Adult Male | 0.974       | 0.961    | 0.987         |
| Two          |            |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Rural Family | Child Born | 3.751       | 3.729    | 0.994         |
| Two          | 1955       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Rural Family | Child Born | 0.140       | 0.137    | 0.979         |
| Two          | 1964       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Urban Family | Adult      | 0.330       | 0.325    | 0.986         |
|              | Female     |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Urban Family | Adult Male | 0.731       | 0.723    | 0.989         |
+--------------+------------+-------------+----------+---------------+
| Urban Family | Child Born | 2.686       | 2.675    | 0.996         |
|              | 1955       |             |          |               |
+--------------+------------+-------------+----------+---------------+
| Urban Family | Child Born | 0.107       | 0.106    | 0.983         |
|              | 1964       |             |          |               |
+--------------+------------+-------------+----------+---------------+

## Input Variables and Realizations

As discussed in the previous sections, the input variables involved in
the computational model for the uncertainty analysis fall into two
categories: 1) input variables that are considered as uncertain and 2)
input variables that are considered as certain or fixed. While all input
variables have an inherent uncertainty, the analysis described in
Section 12.4 identified those variables that have the largest effect on
the uncertainty of the resultant dose. This analysis reduced the number
of input variables that will be considered as uncertain in the
uncertainty analysis to 14, leaving the remaining input variables to be
considered as certain (fixed) and thus treated as point estimates or
fixed values.

Certain and uncertain input variables are introduced to the
computational model differently. Certain input variables are represented
by a single value, whereas an uncertain input variable is represented by
a probability distribution and its associated statistics. Because an
uncertain input variable cannot be represented by a single value, a set
of values sampled from the variable's probability distribution is used
to represent the variable of interest. The degree of representativeness
depends on the sampling technique and the number of samples taken from
the probability distribution.

As noted in Section 12.1.1.2, random sampling techniques fall into two
categories: 1) simple random sampling (SRS), also known as MCS, and 2)
the stratified sampling, also known as the LHS. The MCS technique does
the sampling completely at random with each sample taken sequentially,
whereas the LHS technique performs the sampling by a constraining value
selection based on previously constructed realizations. By not allowing
samples to be drawn from previously sampled intervals for a variable,
LHS provides a more representative sampling of the distributions with a
smaller sample size.

MCS is a computationally time-intensive sampling technique. The samples
are chosen completely at random within the range of the probability
distribution. This necessitates large numbers of random samples for
highly skewed or long-tailed probability distributions to reasonably
represent the uncertain nature of the variable under consideration.

LHS is more efficient than MCS because it is designed to accurately
recreate the input distribution with fewer samples than the MCS method.
However, LHS is a memory-intensive technique because the entire set of
samples for all variables is collected at once.

All input values (fixed and sampled) are collected together in a matrix
format. One set of input values from this collection, with a single
value for each variable, is referred to as a realization. Each
realization, when used in the computer code, results in a single value
for the output variable (e.g., dose). When an uncertainty analysis is
performed, the computer code is run iteratively, each time using a
different combination of input variable values as determined by the
sampling. For an uncertainty analysis involving n uncertain variables,
3n realizations (computer runs) are generally considered to be adequate
when using LHS sampling.

### Description

LHS was adopted as the sampling technique for the uncertainty analysis.
After examining two general-purpose commercial software packages for
risk analysis―Crystal Ball 2000 (2) and \@RISK (3), it was decided to
use the LHS computer code developed by Sandia National Laboratories for
the U.S. Nuclear Regulatory Commission (4). This decision was based on
the need to be able to define correlations among the uncertain
variables. The LHS computer code requires each uncertain variable to be
identified by its probability density function (probability
distribution) and two quantiles for that distribution: the 0.001
quantile (0.1 percentile) and the 0.999 quantile (99.9 percentile).

Table 12-2 provides the 14 variables that were considered to be
uncertain for the uncertainty analysis. For each variable, Table 12-2
contains the type of probability distribution and some of the statistics
for that distribution, including the 0.001 and 0.999 quantiles. These
quantiles were obtained from built-in functions of Microsoft Excel
although these quantiles may lay outside the range of reality for a
particular variable they were the required input for the LHS code

The 14 uncertain variables involved in the uncertainty analysis were
considered to be independent of one another. Therefore, during the
preparation of the input data file for the LHS code (discussed in the
previous section), a particular "flag" was used to indicate that no
correlation existed among the variables involved. This flag resulted in
small correlation coefficients (less than 0.2) when the correlation
coefficient matrix associated with the 40 X 14 output matrix (the output
matrix with 40 realizations) was examined.

## Overall Description of the Simplification Approach 

This section describes the process by which 14 variables were selected
for uncertainty/sensitivity analysis from an initial list of 331
possible variables.

### Reasons for Limiting the Scope 

A standard reference on radiological assessment (Till and Meyer, 1983)
states:

"The first step in an uncertainty analysis is to limit the scope. This
requires an explicit statement of the objectives of the assessment and a
determination of relevant radionuclides, exposure pathways, and model
parameters. Limiting the scope of an uncertainty analysis avoids
exhausting financial, physical, and human resources on aspects of
assessment models that are not significant."

For this analysis, the scope was limited to focus attention on variables
with the greatest potential for affecting variation in the dose to
receptors. This avoided unnecessary use of resources for insignificant
aspects of the models.

Table 12‑2 The Fourteen Uncertain Variables Considered for Uncertainty
Analysis

<table>
<colgroup>
<col style="width: 14%" />
<col style="width: 18%" />
<col style="width: 8%" />
<col style="width: 9%" />
<col style="width: 9%" />
<col style="width: 6%" />
<col style="width: 7%" />
<col style="width: 7%" />
<col style="width: 8%" />
<col style="width: 10%" />
</colgroup>
<tbody>
<tr class="odd">
<td>Variable</td>
<td>Description</td>
<td>Units</td>
<td>Dist. Type</td>
<td>Pt. Estimate</td>
<td>Log Mean</td>
<td>Log Std. Dev.</td>
<td>Median</td>
<td><p>0.001</p>
<p>Quantile</p></td>
<td><p>0.999</p>
<p>Quantile</p></td>
</tr>
<tr class="even">
<td><em><strong>Water Pathway</strong></em></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td>CLBFF, Cs-137</td>
<td>Bioconcentration in fish</td>
<td>L/kg</td>
<td>Lognormal</td>
<td>4,700</td>
<td>8.46</td>
<td>1.20</td>
<td>4.70E+03</td>
<td>116</td>
<td>1.90E+05</td>
</tr>
<tr class="even">
<td>CLBFF, Sr-90</td>
<td>Bioconcentration in fish</td>
<td>L/kg</td>
<td>Lognormal</td>
<td>450</td>
<td>6.11</td>
<td>2.03</td>
<td>4.50E+02</td>
<td>0.849</td>
<td>2.39E+05</td>
</tr>
<tr class="odd">
<td>CLBFF, P-32</td>
<td>Bioconcentration in fish</td>
<td>L/kg</td>
<td>Lognormal</td>
<td>50,000</td>
<td>10.82</td>
<td>0.89</td>
<td>5.00E+04</td>
<td>3.15E+03</td>
<td>7.93E+05</td>
</tr>
<tr class="even">
<td><em><strong>Air Pathway</strong></em></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td>LEAFRS, RESFAC;1</td>
<td>Soil resuspension factor―farm</td>
<td>1/m</td>
<td>Lognormal</td>
<td>0.00001</td>
<td>-11.51</td>
<td>2.62</td>
<td>1.00E-05</td>
<td>3.03E-09</td>
<td>3.30E-02</td>
</tr>
<tr class="even">
<td>RADIUS</td>
<td>Particle radius</td>
<td>µm</td>
<td>Lognormal</td>
<td>0.5</td>
<td>-0.69</td>
<td>0.41</td>
<td>5.00E-01</td>
<td>0.143</td>
<td>1.75</td>
</tr>
<tr class="odd">
<td>WTIM</td>
<td>Weathering rate constant from plant</td>
<td>d</td>
<td>Lognormal</td>
<td>14</td>
<td>2.64</td>
<td>1.40</td>
<td>1.40E+01</td>
<td>0.183</td>
<td>1.07E+03</td>
</tr>
<tr class="even">
<td>DPVRES</td>
<td>Deposition velocity from air to plant surfaces</td>
<td>m/s</td>
<td>Lognormal</td>
<td>0.001</td>
<td>-6.91</td>
<td>1.26</td>
<td>1.00E-03</td>
<td>2.07E-05</td>
<td>4.84E-02</td>
</tr>
<tr class="odd">
<td>DEPFR1, 2</td>
<td>Deposition retention fraction for plants</td>
<td>Fraction</td>
<td>Lognormal</td>
<td>0.25</td>
<td>-1.39</td>
<td>0.59</td>
<td>2.50E-01</td>
<td>4.07E-02</td>
<td>1.54</td>
</tr>
<tr class="even">
<td>GRWPA, 5&amp;6</td>
<td>Growing period for animal forage</td>
<td>d</td>
<td>Lognormal</td>
<td>30</td>
<td>3.40</td>
<td>0.44</td>
<td>3.00E+01</td>
<td>7.65</td>
<td>118</td>
</tr>
<tr class="odd">
<td>CLFMT, I-131</td>
<td>Cattle intake-to-beef transfer factor</td>
<td>d/kg</td>
<td>Lognormal</td>
<td>0.04</td>
<td>-3.22</td>
<td>0.50</td>
<td>4.00E-02</td>
<td>8.49E-03</td>
<td>0.188</td>
</tr>
<tr class="even">
<td>F</td>
<td>Release factor</td>
<td></td>
<td>Lognormal</td>
<td>1</td>
<td>0.00</td>
<td>0.20</td>
<td>1.00E+00</td>
<td>0.539</td>
<td>1.86</td>
</tr>
<tr class="odd">
<td>CONSUM, 5</td>
<td>Animal feed consumption rate― beef animal forage</td>
<td>kg/d</td>
<td>Normal</td>
<td>36</td>
<td>36.00*</td>
<td>8.63*</td>
<td></td>
<td>9.33</td>
<td>62.7</td>
</tr>
<tr class="even">
<td>CLFMK, I-131</td>
<td>Cow intake-to-milk transfer factor</td>
<td>d/L</td>
<td>Lognormal</td>
<td>0.01</td>
<td>-4.61</td>
<td>0.91</td>
<td>1.00E-02</td>
<td>6.06E-04</td>
<td>0.165</td>
</tr>
<tr class="odd">
<td>BIOMA2, 5&amp;6</td>
<td>Animal forage standing biomass (wet)</td>
<td>kg/m<sup>2</sup></td>
<td>Lognormal</td>
<td>0.3</td>
<td>-1.20</td>
<td>0.08</td>
<td>3.00E-01</td>
<td>0.233</td>
<td>0.386</td>
</tr>
</tbody>
</table>

\* Mean and standard deviation for normal probability distribution.

By limiting the analysis scope to those radionuclides, pathways, and
input variables most influencing dose and dose variation, the
computational burden was focused on the most important model aspects
(some of this simplification is described in Section 12.2). By screening
the input variables to select those influencing variations in dose most,
the development of probability distributions was focused on the most
important variables.

### Variable Selection Process―Overview

The point estimates provided in [Chapter
11](../Chapter_11_ready%20for%20CDC%20031405.doc) are based on a single
value chosen for each input variable. All input variables were
considered to be fixed, or "certain," even though a range of values may
have been considered in determining the single representative value for
each variable. These representative values were chosen to be consistent
with the specified scenarios, and site characteristics. When a range of
values was available for a variable, the representative value was chosen
in such a way that it did not intentionally underestimate or
overestimate the dose.

In the uncertainty analysis, all of the "certain" input variables used
to generate the point estimates could be considered as uncertain
variables with their own probability distributions because, in fact, all
input variables are uncertain. However, in most examples of
environmental analysis, only a few input variables dominate the results.
This variable selection process is intended to reduce the number of
variables treated as uncertain to focus attention on those that appear
to have the most effect on variations in dose.

Two categories of input variables were categorically excluded from
consideration as uncertain:

1\. The variable values representing the behavior of the receptors
(e.g., the amount of milk consumed at a particular age) were derived
from the scenario specifications. Because the scenarios were
hypothetical and specified by the CDC and the SRSHES, and because the
specified behaviors represented a range of plausible variation, these
variable values were considered to be fixed for the purposes of this
study.

2\. This study has used a set of dose and risk factors established by
national and international radiation protection organizations over many
decades. These factors were treated as fixed because the uncertainty in
these factors was considered small compared to that in other input
variables and the uncertainty in these variables would be similar for
any dose estimate.

Figure 12-1 shows the sequence of steps used by the variable selection
approach to reduce the number of input variables considered in the
uncertainty analysis.

At the beginning of the selection process, all input variables except
those categorically excluded were considered to be candidates for
treatment as uncertain variables. This stepwise process was intended to
eliminate variables as candidates for the "uncertain" category based on
a defined and defensible criterion at each step in the process. The
following criteria were used at the six steps of the selection process:

1\. Eliminate variables concerned with exposure pathways that are not
used for modeling the SRS.

2\. Eliminate variables that are only used for radionuclides and
pathways that are minor contributors to dose.

3\. Eliminate variables that are only used for radionuclides and
pathways whose fractional contribution to dose squared is small.

4\. Eliminate any remaining variables in categories already determined
as fixed, such as variables associated with scenario specifications or
dose and risk coefficients.

5\. Combine input variables that may be specified separately but are
more appropriately considered the same for this study.

6\. Eliminate variables with small contributions to variance.

![](media/image1.wmf){width="6.495833333333334in"
height="3.9229166666666666in"}

**Figure 12‑1 Steps Involved in Variable Reduction and Uncertainty Model
Simplification**

Each of these steps is described in more detail below. In accounting for
the number of variables, [Appendix
F](../Appendix_F%20tables%20-%20Ready%20for%20CDC%20031405.doc) provides
tables that list 331 separate variables. These lists are taken as the
starting point of this process to reduce the number of variables
considered as uncertain. However, there is a subjective element in
counting the variables. For example, the variable "ARMETFILE" is the
name of the file containing the 20-year average meteorological data that
is read into GENII. These meteorological data were not counted among the
input variables required to be specified because 1) they were computed
separately and not included in the variable tables and 2) they were kept
constant for all computations.

The main reason for performing the uncertainty analysis is to estimate
the confidence interval for the estimated dose to each receptor. The
confidence interval is a commonly used measure to describe the
uncertainty in a variable. A key factor in determining the confidence
interval is the variance of the dose (or equivalently the standard
deviation of the dose, which is the square root of variance). Some of
the screening steps are based on the fact that even though a variable
may be important in determining the value of dose, it may have little
influence on the variance of the dose.

#### Start

The 331 variables were specified to obtain the point estimate of dose
for each receptor. Although each variable was carefully evaluated and
specified, counting them is somewhat subjective. As stated in Section
12.4.2, some variables are just the name of a file containing extensive
but fixed values―these variables were not included in the count. Other
variables were indexed by exposure location or radionuclide and were
specified for each location and radionuclide with potentially different
values. In this description, they were counted as a single variable if a
single value was used for every instance. Additional inputs to the dose
modeling that were not counted as variables include 1) air releases by
year, isotope, and source; 2) computer grid coordinates; and 3) water
concentrations by year, isotope, and location. In addition to the
variable lists in [Appendix
F](../Appendix_F%20tables%20-%20Ready%20for%20CDC%20031405.doc), a
variable "F" was added to the lists which accounts for uncertainty in
radionuclide releases.

#### Step 1―Eliminate Unused Variables

Because of the scenario specifications and the assumptions made for
modeling, the point-estimate analysis did not use all the input
variables employed by GENII to model the transport and exposure
processes. For example, [Appendix
F](../Appendix_F%20tables%20-%20Ready%20for%20CDC%20031405.doc) shows
that food chain transport variables associated with irrigation are not
used due to the lack of evidence that the Savannah River had ever been
used for irrigation in the region. Variables such as irrigation time
(IRTIMA, IRTIMR, IRTIMT) and irrigation rate (RIRR, RIRRA, RIRRR) are
not specified for the point-estimate analysis and need not be considered
in the uncertainty analysis. As part of the scenario implementation,
beef cattle were assumed to consume no contaminated feed (beef cattle
consumed contaminated forage-grass); therefore, about 30 food chain
variables were eliminated. Also note that cerium, technetium, niobium,
zinc, cobalt, sulphur, phosphorus, yttrium (a decay product of niobium),
and zirconium are only released to water, so their terrestrial uptake
factors may be eliminated; this eliminates eight variables for each
element. Approximately 102 variables were eliminated in this step,
leaving approximately 229 potential uncertain variables.

#### Step 2―Screen Out Minor Contributors to Dose

As described in detail in Section 12.2, a simplified model was obtained
by screening out radionuclides and pathways whose combined contribution
to total dose was less than 5 percent. The actual reduction in dose was
always less than 3 percent for any receptor. The following radionuclides
and pathways were eliminated from the model on this basis:

1\. Air Release:

> a\. Radionuclides: americium-241, cesium-137, iodine-129,
> strontium-89, strontium-90, and all isotopes of uranium.
>
> b\. Pathways: ground contamination, grain, and soil ingestion.

2\. Water Release:

> a\. Radionuclides: iodine-129, niobium-95, strontium-89, and all
> isotopes of uranium.
>
> b\. Pathways: Boating, swimming immersion, and swimming inadvertent
> ingestion.

By eliminating these pathways and isotopes, it was unnecessary to
consider about 60 associated variables as uncertain. For example,
terrestrial uptake factors for various forms of vegetation and animals,
and for the isotopes americium-241, cesium-137, iodine-129,
strontium-89, strontium-90, and all uranium isotopes, could be
eliminated from consideration as uncertain variables. Because uptake
factors are generally element-specific and not isotope-specific, the
uptake factors for iodine remained because iodine-131 was not screened
out. Therefore, 8 terrestrial uptake factors for each of the elements
americium, cesium, strontium, thorium (a decay product of uranium), and
uranium were eliminated, thereby reducing the number of potential
uncertain variables by 40. In addition, six variables related to grain
and eight variables associated with direct exposure to contaminated soil
were reclassified. Another six variables related to grain and aquatic
uptake were eliminated. Approximately 60 variables were eliminated in
this step, leaving approximately 169 potential uncertain variables.

#### Step 3―Screen Out Instances Where Fraction Squared Is Small

[Appendix L](../Appendix_L_Ready%20for%20CDC%20031405.doc) discusses a
mathematical approach in which the dose for any receptor is considered
to be approximated by a linear function of each of the input variables,
expanded about their nominal values (i.e., the point-estimate values).
Quite a bit is known about the point-estimate dose, including the
fractional contribution to the total dose from each radionuclide and
through each pathway. Mathematically, the fraction of dose F(ij) is
known for each pathway (i) and each radionuclide (j). This information
has been used in the variable selection process described in Section
12.2 to eliminate pathways and radionuclides from the model.

The approach discussed in [Appendix
L](../Appendix_L_Ready%20for%20CDC%20031405.doc) shows that the variance
of the dose is proportional to the square of the sum of the fractional
contributions to dose, where the sum is taken over all pathways and all
radionuclides for which the variable has an impact on dose. Note that a
particular variable may not have an impact on the dose resulting from a
particular radionuclide or pathway. For example, the variable
characterizing the uptake of cesium-137 by fish does not have any impact
on the dose from iodine from drinking milk. As an approximation, it was
assumed that if the sum of the squares of the fractions for all isotopes
of a given pathway were small, then the pathway and its associated
variables would not contribute significantly to the variance of dose.
Similarly, it was assumed that if the sum of the squares of the
fractions for all pathways of a given isotope were small, then the
radionuclide and its associated variables would not contribute
significantly to the variance of dose.

For air releases, four receptors were used as indicators to test for
these conditions. The receptors were the Adult Female and Child Born in
1964 for Rural Family #2 and for the Migrant Worker Family. As discussed
in [Chapter 11](../Chapter_11_ready%20for%20CDC%20031405.doc), for air
releases, the two adults and Child Born in 1955 had similar contributors
to total dose; however, the Child Born in 1964 was different because
that receptor missed the large iodine releases early in the SRS history.
Consequently, fractional dose contributions to both the Adult Female and
Child Born in 1964 were examined for this step of the variable
reduction. Rural Family #2 was selected because it had fractional
contributions to dose typical for most of the scenarios receiving doses
from air releases only; however, the Migrant Family was somewhat of an
outlier and was included for that reason. From these instances, certain
radionuclides and pathways were found to be insignificant contributors
to the variance of dose. The incremental contribution to total dose by
every radionuclide-pathway pair was computed and then renormalized by
the total dose from the pathways and radionuclides remaining after the
model simplification described in Section 12.2. (As shown by the ratios
in Table 12-1, this adjustment in fractional contribution was small when
compared to the fractional contributions stated in [Chapter
11](../Chapter_11_ready%20for%20CDC%20031405.doc).) These renormalized
fractional values were then squared, and the squared fractional values
were renormalized by the largest value. The renormalized squared
fractions were then summed over pathways and radionuclides. In general,
if any of the resulting sums were smaller than 0.01, then the variables
associated with those radionuclides and pathways were eliminated. On
this basis the variables associated with the following pathways or
radionuclides were thus eliminated:

-   Carbon-14.

-   Plutonium-238,239.

-   Ruthenium-106.

-   Resuspended soil inhalation.

-   Poultry ingestion.

-   Eggs ingestion.

-   Fruit ingestion.

-   Root vegetable ingestion.

For the Child Born in 1964 for Rural Family #2, the sum for root
vegetable ingestion slightly exceeded the 0.01 criterion.

Two receptors included in water exposure scenarios were used as
indicators to test for these conditions. These were the Adult Female and
Child Born in 1964 for the Delivery Family. The Delivery Family was
chosen because it had the largest doses from water releases; the Adult
Female and Child Born in 1964 were chosen for the reasons stated
previously. From these instances in which water release doses dominated,
the following pathways and radionuclides were found to be insignificant
contributors (as stated above) to the variance of dose:

-   Shoreline direct exposure.

-   Cesium-144.

-   Cobalt-60.

-   Cesium-134.

-   Hydrogen-3.

-   Iodine-131.

-   Plutonium-238,239.

-   Ruthenium-106.

-   Zinc-65.

-   Zirconium-95.

By eliminating variables associated with these pathways and
radionuclides that are insignificant contributors to the variance of
dose, approximately 58 variables were reclassified as fixed. For
example, 7 uptake factors each for the elements plutonium and ruthenium
were eliminated, thereby reducing the number of potential uncertain
variables by 14. Elimination of the egg and poultry pathways reduced the
number of variables by about 22. Elimination of water-release
radionuclides reduced the number of uncertain variables by nine.
Elimination of variables associated with ingestion of fruit and root
vegetables reduced the variable count by about 12. This left about 111
potential uncertain variables.

#### Step 4―Eliminate Health Effects Variables and Scenario-Based Variables

A number of the remaining potential uncertain variables were
reclassified as variables to be considered certain or fixed for a
variety of reasons. The following represent some of the important
reasons for these categorical reassignments:

-   Holdup times are unimportant for long-lived radionuclides; the dose
    from phosphorus-32 could be affected because it has a relatively
    short half-life (14.5 days), but it is a relatively small
    contributor.

-   Holdup times for food delivery and processing were assigned as part
    of completion of the scenario specifications; because the scenarios
    are mandated, these variables are considered to be certain.

-   It was independently determined that, for these scenarios,
    radionuclide uptake by vegetation from soil is small compared to
    radionuclide uptake by vegetation from direct deposition; therefore,
    all variables associated with modeling soil uptake are considered
    fixed. Approximately 48 variables were reclassified on this basis.

-   Variables associated with household use of river water were
    considered to be fixed; although volatilization of radionuclides was
    considered for the Near Water Family, it accounted for a very small
    fraction of the total dose.

The variables that relate to these points were considered fixed and
therefore were eliminated at this step. Approximately 82 variables were
thus eliminated, thereby leaving approximately 29 potentially uncertain
variables.

#### Step 5―Combine Variables that May be Considered Together 

Further reduction in the number of variables considered to be uncertain
was accomplished by noting that for three pairs of variables only one
value, albeit an uncertain value, need be assigned to the pair:

1\. BIOMA2, 5 and BIOMA2, 6.

2\. GRWPA, 5 and GRWPA, 6.

3\. DEPFR1 and DEPFR2.

For the first two pairs, the variables referred to values for milk and
beef animal forage. Because the same grass was used for both milk and
beef animal forage, these variables could be assumed to be equal. For
the third pair of variables, dry- and wet-deposition fractions were
assumed to be equal.

In the following two cases, the value of one variable in a pair was
derived from the sampled value of the other variable in the pair by
multiplying the first variable by a constant; essentially, the pair of
variables is assumed to be different by a multiplicative constant and
perfectly correlated as explained in the following:

1\. CONSUM,5―the consumption rate of forage by beef cattle was assumed
to be the fraction 36/29 the value of CONSUM,6―the consumption rate of
forage by milk cattle. This is the ratio of the point estimate values.
The assumption means that whatever causes the beef cattle to eat more,
it will have the same effect on dairy cows.

2\. RESFAC,1―the resuspension factor for rural locations is assumed to
be 100 times the value of RESFAC,2―the resuspension factor for urban
locations. It is essentially assumed that whatever forces produce more
resuspension in the city and suburbs (e.g., higher wind) will do the
same on the farm. For the point-estimate case, it was assumed that the
resuspension factor in the rural areas was 100 times that in the city
due to plowing and other agricultural activities. LEAFRS is essentially
the same variable as RESFAC, so those variables were set equal to the
corresponding values for RESFAC.

Both particle radius and density are factors in determining deposition
of radionuclides, but the equation uses radius squared. Because radius
should have a more significant effect if varied, it was chosen as the
uncertain variable, and density was considered fixed.

The net effect of these various consolidations was to eliminate 8
variables and leave 21 independent candidates for variables to be
treated as uncertain. These variables are listed in Table 12-3.

Table 12‑3 Characteristics of Input Variable Distributions

  ------------- ------------- ----------------------------- ---------- ----------- ------------ ------------
                Variable      Description                   Units      Type        p1\*         p2\*

  ***Water                                                                                      
  Pathway***                                                                                    

  1             CLBFF, Cs-137 Bioconcentration in fish      L/kg       Lognormal   8.4553       1.1975

  2             CLBFF, Sr-90  Bioconcentration in fish      L/kg       Lognormal   6.1092       2.0300

  3             CLBFF, P-32   Bioconcentration in fish      L/kg       Lognormal   10.8198      0.8945

  ***Air                                                                                        
  Pathway** *                                                                                   

  4             CLFMK, I-131  Cow intake-to-milk transfer   d/L        Lognormal   -4.6051702   0.9069765
                              factor                                                            

  5             CLFMT, I-131  Cattle intake-to-beef         d/kg       Lognormal   -3.2188758   0.5015594
                              transfer factor                                                   

  6             ABSHUM        Absolute Humidity             kg/m^3^    Normal      0.01125      0.00053

  7             BIOMA2, 5&6   Animal feed standing biomass  kg/m^2^    Lognormal   -1.20397     0.0813929
                              (wet)-beef & milk animal                                          
                              forage                                                            

  8             BIOMA2, 3     Animal feed standing biomass  kg/m^2^    Lognormal   1.43156^†^   0.02909^†^
                              (wet)-Milk animal feed                                            

  9             BIOMAS, 1     Standing biomass (wet) -      kg/m^2^    Lognormal   0.03486^‡^    
                              Leafy vegetables                                                  

  10            WTIM          Weathering rate constant from d          Lognormal   2.63906      1.40311
                              plant                                                             

  11            CONSUM,       Animal feed consumption       kg/d       Normal      36.0000      8.6300
                5&6^§^        rate-Beef & milk animal                                           
                              forage                                                            

  12            CONSUM, 3     Animal feed consumption       kg/d       Lognormal   1.71^†^      0.262^†^
                              rate-Milk animal feed                                             

  13            DRYFAC, 1     Dry/wet ratio - Leafy         fraction   Lognormal   0.10875^†^   0.00218^†^
                              vegetables                                                        

  14            GRWP, 1       Growing period - Leafy        d          Lognormal   0.16861^‡^    
                              vegetables                                                        

  15            GRWPA, 5&6    Growing period for animal     d          Lognormal   3.40119738   0.4423365
                              feed - Beef & milk animal                                         
                              forage                                                            

  16            GRWPA, 3      Growing period for animal     d          Lognormal   0.05103^‡^    
                              feed - Milk animal feed                                           

  17            DEPFR1,2      Dry & wet deposition          fraction   Lognormal   -1.3862944   0.5873942
                              retention fraction to plants                                      

  18            LEAFRS,       Resuspension factor from soil 1/m        Lognormal   -11.512925   2.6214129
                RESFAC;1^§^   to plant surfaces - farm                                          

  19            DPVRES        Deposition velocity from soil m/s        Lognormal   -6.9077553   1.2555349
                              to plant surfaces                                                 

  20            RADIUS        Particle Radius               µm         Lognormal   -0.6931      0.4055

  21            F             Release Factor                           Lognormal   0            0.2
  ------------- ------------- ----------------------------- ---------- ----------- ------------ ------------

^§^ Related by a constant factor.

\* p1 and p2 are the log mean and log standard deviation of their
corresponding lognormal distributions, respectively, except for
variables No. 6 and 11 that they are the arithmetic mean and the
arithmetic standard deviation of the normal distribution, respectively.

†Arithmatic mean and arithmetic standard deviation.

‡Ratio of arithmetic standard deviation over arithmetic mean.

#### Step 6 -- Retain Only Variables Contributing Substantially to Variance

The final step in reducing the number of input variables to be used in
the uncertainty analysis was based on a factor (Column D in Table 12-4)
that is the squared product of 1) the coefficient of variation for the
variable (Column A in Table 12-4 and 2) the fractional contribution to
dose from all pathways and radionuclides (Column B in Table 12-4) for
which the variable is involved.

The coefficient of variation is equal to the standard deviation of the
distribution describing the variable divided by the arithmetic mean of
the distribution. The coefficient of variation is a relative measure of
the variability of a random variable with respect to its mean value.
Because the coefficient of variation (CV) measures relative variability,
it allows comparison of variability across variables with different
units and different absolute magnitudes. A variable with a small CV is
very repeatable and reliable, experiencing little relative change from
one measurement to another. A variable with a large CV is much more
uncertain.

The data used to determine probability distributions for the 21
variables remaining at this step were gathered from a variety of
published sources (with the exception of one case for which a personal
communication was used). [Appendix
M](../Appendix_M_Ready%20for%20CDC%20031405.doc) provides details of the
development of these probability distributions.

Table 12-4 shows the 21 variables, the coefficient of variation, the
fractional contribution, the product of these squared, and the rank of
the variable based on the combined factor. The following observations
are based on this table:

-   For the water-release variables, the range in the determining factor
    (Column D) is less than two orders of magnitude between the largest
    and smallest value; therefore, all three variables were retained.

-   For the air-release variables, the highest ranked variable was the
    resuspension factor (the determining factor is 664) because the
    fractional contribution was relatively large and the coefficient of
    variation was over 31. This is an extremely large value that ranges
    over several orders of magnitude.

-   Consider eliminating any variable with a determining factor less
    than 0.01: For the air release, the determining factor value falls
    off rapidly, and if the criterion of less than 0.01 was used, all
    variables of rank less than 10 would be excluded.

-   The cutoff for rank was chosen to be 10. However, the variable of
    rank 11 was also included because it participated in dose from milk
    and meat as well as the companion variables―growing period (GRWPA,5)
    and forage consumption (CONSUM,5).

Based on these considerations, a final number of 14 variables were
considered to be uncertain. Table 12-2 provides the statistical
characteristics of these 14 variables, and Section 12.3 discusses how
they were used.

[Appendix N](../Appendix_N_ready%20for%20CDC%20031405.doc) provides the
input data used for the computations (LHS and GENII computer codes).

Table 12‑4 Final Screening Step

  ------------- ---- ------------- ------------- -------------- -------------- -------------- --------
                     Variable      Coefficient   Fractional     Approximate    Square of      Rank for
                                   of Variation  Contribution   Contribution   Contribution   Medium
                                   (Standard     to Total for   to Variance    to Variance    
                                   Deviation     Medium                                       
                                   /Mean)                                                     

  ***Water                                                                                    
  Pathway A B C                                                                               
  D***                                                                                        

               1    CLBFF, Cs-137 1.788E+00     5.25E-01       9.38E-01       8.79E-01       2

               2    CLBFF, Sr-90  7.786E+00     1.59E-01       1.24E+00       1.53E+00       1

               3    CLBFF, P-32   1.107E+00     2.68E-01       2.96E-01       8.79E-02       3

  ***Air                                                                                      
  Pathway** *                                                                                 

               4    CLFMK, I-131  1.130E+00     1.38E-01       1.56E-01       2.43E-02       10

               5    CLFMT, I-131  5.348E-01     5.09E-01       2.72E-01       7.41E-02       7

                6    ABSHUM        4.711E-02     6.05E-01       2.85E-02       8.12E-04       13

               7    BIOMA2, 5&6   8.153E-02     6.93E-01       5.65E-02       3.19E-03       11

                8    BIOMA2, 3     2.032E-02     3.45E-01       7.01E-03       4.91E-05       16

                9    BIOMAS, 1     3.486E-02     6.09E-02       2.12E-03       4.50E-06       17

               10   WTIM          2.482E+00     8.16E-01       2.03E+00       4.10E+00       3

               11   CONSUM, 5     2.397E-01     6.93E-01       1.66E-01       2.76E-02       9

                12   CONSUM, 3     1.532E-01     3.45E-01       5.28E-02       2.79E-03       12

                13   DRYFAC, 1     2.000E-02     6.09E-02       1.22E-03       1.48E-06       18

                14   GRWP, 1       1.686E-01     6.09E-02       1.03E-02       1.05E-04       15

                15   GRWPA, 5&6    4.649E-01     6.93E-01       3.22E-01       1.04E-01       6

                16   GRWPA, 3      5.103E-02     3.45E-01       1.76E-02       3.10E-04       14

               17   DEPFR1,2      6.419E-01     8.16E-01       5.24E-01       2.74E-01       5

               18   LEAFRS,       3.104E+01     8.30E-01       2.58E+01       6.64E+02       1
                     RESFAC;1                                                                 

               19   DPVRES        1.959E+00     8.16E-01       1.60E+00       2.55E+00       4

               20   RADIUS        2.913E+00     1.00E+00       2.91E+00       8.49E+00       2

               21   F             2.000E-01     1.00E+00       2.00E-01       4.00E-02       8
  ------------- ---- ------------- ------------- -------------- -------------- -------------- --------

 = retain for final analysis

## Results of Uncertainty Analysis 

The computer runs provided 40 dose values (corresponding to 40
realizations) for each receptor, as provided in [Appendix
O](../Appendix_O_ready%20for%20CDC%20031405.doc). In other words, the
dose for each receptor was computed 40 times as a function of 40
(random) realizations of the variables considered to be uncertain plus
all the other input variables considered to be fixed. Each set of 40
output values was examined statistically to determine an empirical
probability distribution and values representative of the central
tendency (e.g., mean and median) for the dose to each receptor.

### Description of Dose Distributions

The lognormal probability distribution was assumed to be a potentially
good fit to the 40 random dose values associated with each receptor.
This was assumed because all except 1 of the 14 probability
distributions for the 14 uncertain input variables considered were
lognormal (see Table 12-2). Also, the point-estimate value for each of
these 14 variables from the deterministic analysis had been set equal to
the median of the distribution for the corresponding variable in the
uncertainty analysis. For this reason, the medians of the 28
distributions (1 for each receptor) were expected to be "close" to those
point estimates found in the deterministic analysis contained in
[Chapter 11](../Chapter_11_ready%20for%20CDC%20031405.doc). In addition,
the following is a well-known property of lognormal distributions: If a
random variable is defined as the product of two or more independent
random variables and each of these is described by a lognormal
distribution, then the product will also be described by a lognormal
distribution.

Using the computer software "Crystal Ball" (1), the best fit to the
probability distributions for the total dose to 14 out of the 28
receptors was determined to be lognormal. The lognormal distribution was
the second or third best fit to the dose distributions for the remaining
receptors. However, even when the lognormal distribution was not the
first choice, the best-fit distributions were similar to the lognormal.
The fact that the dose distributions for all the receptors are not
lognormal is attributed to the fact that the dose is computed by a
complex mathematical model of the uncertain input variables involved in
the uncertainty analysis. Except for very simple functions of random
variables (e.g., multiplication by a constant), the function of several
random variables is seldom characterized by the same probability
distribution as the underlying input variables, even if they are all the
same. For example, for the Adult Female member of the Delivery Family,
the best fit to the distribution of dose from only air releases and the
best fit to the distribution of dose from only water releases are gamma
distributions; however, the best fit to the total dose from all
releases, which is the sum of these air and water doses, is a lognormal
distribution.

Table 12-5 displays descriptive statistics for the sample of 40 total
dose realizations for each of the 28 receptors. The table lists the
mean, median, minimum, maximum, and the standard deviation of total dose
for each receptor. Note that the mean dose is larger than the median
dose for every receptor. Because we have a sample of 40 values of dose,
there are 20 values smaller than the median and 20 values larger than
the median. If we had a symmetric distribution of dose, the smaller
values and larger values would extend about equally on either side of
the median value. In our case, however, the lower values are limited by
zero because dose cannot be negative and the higher values extend out to
relatively high values. For example, for the Child Born in 1955 for the
Urban Family, the difference between the maximum value of dose (30.8
milliSieverts) and the median dose (2.55 milliSieverts) is 28.3
milliSieverts; however, the difference between the median dose (2.55
milliSieverts) and the minimum dose (0.345 milliSieverts) is only 2.20
milliSieverts. Figure 12-2 and Figure 12-3 show this skewing toward
higher values, depicting as examples the dose distributions for the
Adult Female in the Rural Family 2 and the Delivery Family,
respectively. The distributions are displayed both as histograms and as
a fitted lognormal distribution. Because the distribution is not
symmetrical and is skewed toward higher values, the mean dose is larger
than the median dose.

The range of the distribution of dose depends on the choice of both the
scenario and family member. For example, for the Child Born in 1955 for
the Urban Family, the ratio of maximum dose to minimum dose is over 89;
for the Child Born in 1964 for the same family, the ratio of maximum
dose to minimum dose is only about 4. Similarly, the ratio by which the
mean dose exceeds the median dose depends on both the scenario and
family member. The minimum value of this ratio is 1.07 for the Child
Born in 1964 for the Urban Family, and the maximum value of this ratio
is 1.69 for the Child Born in 1955 for the same family.

Table 12‑5 Statistics on Total Effective Dose Equivalent for Different
Receptors\*

  ------------- ------------- -------- --------- ---------- ---------- -----------
  Family        Family Member Mean     Median    Minimum†   Maximum†   Standard
                                                                       Deviation

                              (mSv)‡   (mSv)‡    (mSv)‡     (mSv)‡     (mSv)

  Delivery      Adult Female  12.812   9.072     1.813      60.911     12.23
  Family                                                               

  Delivery      Adult Male    13.001   9.281     1.737      61.183     12.26
  Family                                                               

  Delivery      Child Born    10.164   7.993     1.767      35.010     7.76
  Family        1955                                                   

  Delivery      Child Born    4.498    3.159     0.712      15.146     3.73
  Family        1964                                                   

  Migrant       Adult Female  0.793    0.562     0.138      4.615      0.90
  Family                                                               

  Migrant       Adult Male    1.117    0.756     0.184      6.770      1.29
  Family                                                               

  Migrant       Child Born    3.676    2.489     0.417      24.269     4.45
  Family        1955                                                   

  Migrant       Child Born    0.127    0.093     0.043      0.732      0.13
  Family        1964                                                   

  Near Water    Adult Female  3.431    2.738     1.138      9.896      2.04
  Family                                                               

  Near Water    Adult Male    3.574    2.929     1.183      10.084     2.06
  Family                                                               

  Near Water    Child Born    4.815    4.311     1.293      18.333     3.03
  Family        1955                                                   

  Near Water    Child Born    2.850    2.290     0.867      14.090     2.31
  Family        1964                                                   

  Outdoor       Adult Female  4.687    4.263     1.272      11.751     2.46
  Family                                                               

  Outdoor       Adult Male    6.055    5.546     2.026      14.246     2.95
  Family                                                               

  Outdoor       Child Born    13.331   10.988    2.529      60.270     10.21
  Family        1955                                                   

  Outdoor       Child Born    2.951    2.309     0.893      14.158     2.31
  Family        1964                                                   

  Rural Family  Adult Female  0.502    0.387     0.090      3.005      0.53
  One                                                                  

  Rural Family  Adult Male    0.712    0.538     0.120      4.406      0.76
  One                                                                  

  Rural Family  Child Born    2.681    1.697     0.281      17.410     3.34
  One           1955                                                   

  Rural Family  Child Born    0.093    0.077     0.037      0.340      0.06
  One           1964                                                   

  Rural Family  Adult Female  1.174    0.890     0.199      7.162      1.28
  Two                                                                  

  Rural Family  Adult Male    1.655    1.198     0.267      10.502     1.85
  Two                                                                  

  Rural Family  Child Born    6.362    4.006     0.642      41.579     0.798
  Two           1955                                                   

  Rural Family  Child Born    0.190    0.153     0.072      0.796      0.14
  Two           1964                                                   

  Urban Family  Adult Female  0.447    0.284     0.083      2.204      0.39

  Urban Family  Adult Male    0.895    0.698     0.263      3.276      0.55

  Urban Family  Child Born    4.314    2.551     0.345      30.820     5.57
                1955                                                   

  Urban Family  Child Born    0.115    0.107     0.054      0.215      0.03
                1964                                                   
  ------------- ------------- -------- --------- ---------- ---------- -----------

\* The number of decimal places for values in the table are to allow
easy display; the values should be considered to have a precision no
greater than two significant digits.

† These minimum and maximum values are for this sample; another set of
realizations will likely have different values. However, any sample
minimum value is greater than or equal to the population minimum value,
while any sample maximum value is less than or equal to the population
maximum value.

‡ mSv = milliSieverts.

![](media/image2.wmf){width="3.8680555555555554in"
height="2.9347222222222222in"}

> Figure 12‑2 Histogram and Fitted Distribution for Dose to Adult
> Female, Rural Family 2

![](media/image3.wmf){width="3.9159722222222224in"
height="2.9916666666666667in"}

> Figure 12‑3 Histogram and Fitted Distribution for Dose to the Adult
> Female, Delivery Family

### Quantified Confidence Bands for Doses

A primary motivation for the uncertainty analysis is to quantify the
uncertainty associated with the dose to each receptor. One way to
characterize this uncertainty is to state the confidence limits around
the mean dose. Confidence limits define an interval around a parameter
(e.g., the mean) so that the parameter is expected to be within the
interval to a specified probability. Another way of looking at this is
that the mean is estimated based on a finite set of values randomly
sampled from a continuous distribution (i.e., in our case, we have a
sample of 40 doses for each receptor out of an infinite number of
possible values of dose). The estimated parameter has two sources of
uncertainty: 1) the underlying variability of the distributed values and
2) the finite sample size.

A standard statistical text (2) states the following:

$\overline{X} - t_{1 - \frac{\alpha}{2}}(n - 1)\frac{S}{\sqrt{n}} \leq \mu \leq \overline{X} + t_{1 - \frac{\alpha}{2}}(n - 1)\frac{S}{\sqrt{n}}$
(12-1)

where, µ is the population mean

$\overline{X}$ is the sample mean

> $t_{1 - \frac{\alpha}{2}}(n - 1)$ is the Student's t-distribution with
> (n-1) degrees of freedom for a 100(1-α)% total confidence level
> \[100(α/2)% and 100(1-α/2)% confidence limits on the left and right
> sides of the distribution, respectively\]
>
> S is the unbiased estimator of the standard deviation
>
> n is the number of observations (sample size).

If the sample variance is computed by the formula:

${\hat{\sigma}}^{2} = \sum_{i = 1}^{n}\frac{(X_{i} - \overline{X})^{2}}{n}$
(12-2)

then the unbiased estimator for the population variance is given by:

$S^{2} = \frac{n}{n - 1}{\hat{\sigma}}^{2}$ (12-3)

and the standard deviation, of either the sample or population, may be
found by taking the square root of the corresponding variance.

For our computations, n=40. Then the fraction in equation (12-3) is
40/39 = 1.0256; if we take the square root to find S, then the fraction
becomes 1.0256^1/2^= 1.0127. This relates the unbiased estimator to the
sample standard deviation. This is a small correction, but it is made
due to its ease of accomplishment.

Table 12-6 shows the values for t distribution for some typical
confidence levels (1-α) and the 40 realizations.

Table 12‑6 Values for t Distribution for Some Typical Confidence
Intervals

  ------ -------------- ---------------- -------------------- --------------
  α      Lower Bound    Upper Bound      Confidence Level     t(40-1)

  0.1    5%             95%              90%                  1.684875315

  0.05   2.50%          97.50%           95%                  2.022688932

  0.02   1%             99%              98%                  2.425840648
  ------ -------------- ---------------- -------------------- --------------

Since n=40, 40^1/2^ = 6.32455532. Then for α = 0.05,

$\overline{X} - 0\text{.}\text{319815} \cdot S \leq \mu \leq \overline{X} + 0\text{.}\text{319815} \cdot S$
(12-4)

Or the actual mean of dose is between the sample mean plus and minus
about 1/3 of the sample standard deviation with a 95-percent confidence
level. For example, for the Adult Female of the Delivery Family, the
sample mean dose is 0.012812 Sievert; the actual mean dose is expected
to be between 0.008899 Sievert and 0.016724 Sievert with a confidence
level of 95 percent.

The median dose is closer to the point-estimate dose than to the mean
dose because the medians of the lognormal distributions of input
variables were set equal to the point-estimate values for those input
variables. One way to make the same type of confidence statements about
the median dose that have been made about the mean dose is to assume
that the total dose distribution for each receptor is described by a
lognormal distribution. A special property of the lognormal distribution
is that the mean of the natural logarithm of the sampled doses is equal
to the natural logarithm of the median dose. Therefore, equation 12-4
may be used to establish confidence limits around the ln (ln is used to
represent the natural logarithm) median; by taking the inverse
logarithm, one obtains the median and its confidence limits (i.e.,
raising e (the base of the natural logarithms) to the ln median and to
the ln confidence limits provides the median and its associated
confidence limits).

Table 12-7 displays the mean and median for all 28 receptors and also
gives the upper and lower confidence bounds for each statistic at the
95-percent level. Figure 12-4 shows the confidence intervals around the
median for one receptor (the Adult Female for Rural Family 2) overlaid
on the histogram of dose.

![](media/image4.wmf){width="3.9402777777777778in"
height="3.8152777777777778in"}

U= Upper Bound

L= Lower Bound

Figure 12‑4 Confidence Intervals Overlaid on Histogram of Dose

For the mean values, the confidence bounds are almost symmetrical; the
confidence intervals on either side of the mean range from 9- to
41-percent of the mean value depending upon the scenario and family
member. For the median values, assuming a lognormal distribution, the
confidence bounds are skewed toward higher values (i.e., the difference
between the upper bound and the median is generally a higher fraction of
the median than the difference between the median and the lower bound).
The lower bound of the confidence interval ranges from 9- to 26-percent
of the median value; the upper bound of the confidence interval ranges
from 9- to 36-percent of the median value.

Table 12‑7 95% Confidence Intervals for Mean and Median of Total
Effective Dose Equivalent for Each Receptor

+---------+--------+---+--------+--------+----+---------+---------+
| Family  | Family | M | Lower  | Upper  | Me | Lower   | Upper   |
|         | Member | e | Limit  | Limit  | di | Limit   | Limit   |
|         |        | a | of 95% | of 95% | an | of 95%  | of 95%  |
|         |        | n | CI\*   | CI for | †  | CI for  | CI for  |
|         |        |   | for    | Mean   |    | Median  | Median  |
|         |        | ( | Mean   |        | (m |         |         |
|         |        | m | (mSv)‡ | (mSv)‡ | Sv | (mSv)‡  | (mSv)‡  |
|         |        | S |        |        | )‡ |         |         |
|         |        | v |        |        |    |         |         |
|         |        | ) |        |        |    |         |         |
|         |        | ‡ |        |        |    |         |         |
+=========+========+===+========+========+====+=========+=========+
| D       | Adult  | 1 | 8.899  | 16.724 | 8  | 6.464   | 11.608  |
| elivery | Female | 2 |        |        | .6 |         |         |
| Family  |        | . |        |        | 62 |         |         |
|         |        | 8 |        |        |    |         |         |
|         |        | 1 |        |        |    |         |         |
|         |        | 2 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| D       | Adult  | 1 | 9.081  | 16.921 | 8  | 6.643   | 11.849  |
| elivery | Male   | 3 |        |        | .8 |         |         |
| Family  |        | . |        |        | 72 |         |         |
|         |        | 0 |        |        |    |         |         |
|         |        | 0 |        |        |    |         |         |
|         |        | 1 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| D       | Child  | 1 | 7.681  | 12.646 | 7  | 6.173   | 9.923   |
| elivery | Born   | 0 |        |        | .8 |         |         |
| Family  | 1955   | . |        |        | 26 |         |         |
|         |        | 1 |        |        |    |         |         |
|         |        | 6 |        |        |    |         |         |
|         |        | 4 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| D       | Child  | 4 | 3.305  | 5.691  | 3  | 2.541   | 4.259   |
| elivery | Born   | . |        |        | .2 |         |         |
| Family  | 1964   | 4 |        |        | 90 |         |         |
|         |        | 9 |        |        |    |         |         |
|         |        | 8 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Migrant | Adult  | 0 | 0.504  | 1.082  | 0  | 0.431   | 0.713   |
| Family  | Female | . |        |        | .5 |         |         |
|         |        | 7 |        |        | 54 |         |         |
|         |        | 9 |        |        |    |         |         |
|         |        | 3 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Migrant | Adult  | 1 | 0.705  | 1.529  | 0  | 0.595   | 0.997   |
| Family  | Male   | . |        |        | .7 |         |         |
|         |        | 1 |        |        | 70 |         |         |
|         |        | 1 |        |        |    |         |         |
|         |        | 7 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Migrant | Child  | 3 | 2.252  | 5.100  | 2  | 1.871   | 3.246   |
| Family  | Born   | . |        |        | .4 |         |         |
|         | 1955   | 6 |        |        | 65 |         |         |
|         |        | 7 |        |        |    |         |         |
|         |        | 6 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Migrant | Child  | 0 | 0.086  | 0.167  | 0  | 0.085   | 0.122   |
| Family  | Born   | . |        |        | .1 |         |         |
|         | 1964   | 1 |        |        | 02 |         |         |
|         |        | 2 |        |        |    |         |         |
|         |        | 7 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Near    | Adult  | 3 | 2.778  | 4.083  | 2  | 2.480   | 3.517   |
| Water   | Female | . |        |        | .9 |         |         |
| Family  |        | 4 |        |        | 53 |         |         |
|         |        | 3 |        |        |    |         |         |
|         |        | 1 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Near    | Adult  | 3 | 2.916  | 4.232  | 3  | 2.618   | 3.677   |
| Water   | Male   | . |        |        | .1 |         |         |
| Family  |        | 5 |        |        | 03 |         |         |
|         |        | 7 |        |        |    |         |         |
|         |        | 4 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Near    | Child  | 4 | 3.846  | 5.784  | 4  | 3.493   | 4.932   |
| Water   | Born   | . |        |        | .1 |         |         |
| Family  | 1955   | 8 |        |        | 50 |         |         |
|         |        | 1 |        |        |    |         |         |
|         |        | 5 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Near    | Child  | 2 | 2.113  | 3.588  | 2  | 1.913   | 2.826   |
| Water   | Born   | . |        |        | .3 |         |         |
| Family  | 1964   | 8 |        |        | 26 |         |         |
|         |        | 5 |        |        |    |         |         |
|         |        | 0 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Outdoor | Adult  | 4 | 3.900  | 5.475  | 4  | 3.505   | 4.865   |
| Family  | Female | . |        |        | .1 |         |         |
|         |        | 6 |        |        | 29 |         |         |
|         |        | 8 |        |        |    |         |         |
|         |        | 7 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Outdoor | Adult  | 6 | 5.113  | 6.997  | 5  | 4.676   | 6.316   |
| Family  | Male   | . |        |        | .4 |         |         |
|         |        | 0 |        |        | 34 |         |         |
|         |        | 5 |        |        |    |         |         |
|         |        | 5 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Outdoor | Child  | 1 | 10.065 | 16.596 | 10 | 8.640   | 13.222  |
| Family  | Born   | 3 |        |        | .6 |         |         |
|         | 1955   | . |        |        | 88 |         |         |
|         |        | 3 |        |        |    |         |         |
|         |        | 3 |        |        |    |         |         |
|         |        | 1 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Outdoor | Child  | 2 | 2.214  | 3.688  | 2  | 2.017   | 2.945   |
| Family  | Born   | . |        |        | .4 |         |         |
|         | 1964   | 9 |        |        | 37 |         |         |
|         |        | 5 |        |        |    |         |         |
|         |        | 1 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Rural   | Adult  | 0 | 0.332  | 0.672  | 0  | 0.289   | 0.465   |
| Family  | Female | . |        |        | .3 |         |         |
| One     |        | 5 |        |        | 67 |         |         |
|         |        | 0 |        |        |    |         |         |
|         |        | 2 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Rural   | Adult  | 0 | 0.470  | 0.954  | 0  | 0.410   | 0.662   |
| Family  | Male   | . |        |        | .5 |         |         |
| One     |        | 7 |        |        | 21 |         |         |
|         |        | 1 |        |        |    |         |         |
|         |        | 2 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Rural   | Child  | 2 | 1.612  | 3.750  | 1  | 1.333   | 2.337   |
| Family  | Born   | . |        |        | .7 |         |         |
| One     | 1955   | 6 |        |        | 65 |         |         |
|         |        | 8 |        |        |    |         |         |
|         |        | 1 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Rural   | Child  | 0 | 0.074  | 0.113  | 0  | 0.072   | 0.096   |
| Family  | Born   | . |        |        | .0 |         |         |
| One     | 1964   | 0 |        |        | 83 |         |         |
|         |        | 9 |        |        |    |         |         |
|         |        | 3 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Rural   | Adult  | 1 | 0.767  | 1.582  | 0  | 0.658   | 1.076   |
| Family  | Female | . |        |        | .8 |         |         |
| Two     |        | 1 |        |        | 41 |         |         |
|         |        | 7 |        |        |    |         |         |
|         |        | 4 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Rural   | Adult  | 1 | 1.064  | 2.246  | 1  | 0.905   | 1.501   |
| Family  | Male   | . |        |        | .1 |         |         |
| Two     |        | 6 |        |        | 66 |         |         |
|         |        | 5 |        |        |    |         |         |
|         |        | 5 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Rural   | Child  | 6 | 3.808  | 8.915  | 4  | 3.121   | 5.519   |
| Family  | Born   | . |        |        | .1 |         |         |
| Two     | 1955   | 3 |        |        | 50 |         |         |
|         |        | 6 |        |        |    |         |         |
|         |        | 2 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Rural   | Child  | 0 | 0.144  | 0.236  | 0  | 0.141   | 0.192   |
| Family  | Born   | . |        |        | .1 |         |         |
| Two     | 1964   | 1 |        |        | 64 |         |         |
|         |        | 9 |        |        |    |         |         |
|         |        | 0 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Urban   | Adult  | 0 | 0.322  | 0.572  | 0  | 0.281   | 0.433   |
| Family  | Female | . |        |        | .3 |         |         |
|         |        | 4 |        |        | 49 |         |         |
|         |        | 4 |        |        |    |         |         |
|         |        | 7 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Urban   | Adult  | 0 | 0.718  | 1.073  | 0  | 0.670   | 0.918   |
| Family  | Male   | . |        |        | .7 |         |         |
|         |        | 8 |        |        | 84 |         |         |
|         |        | 9 |        |        |    |         |         |
|         |        | 5 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Urban   | Child  | 4 | 2.532  | 6.096  | 2  | 1.955   | 3.616   |
| Family  | Born   | . |        |        | .6 |         |         |
|         | 1955   | 3 |        |        | 59 |         |         |
|         |        | 1 |        |        |    |         |         |
|         |        | 4 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+
| Urban   | Child  | 0 | 0.104  | 0.125  | 0  | 0.101   | 0.121   |
| Family  | Born   | . |        |        | .1 |         |         |
|         | 1964   | 1 |        |        | 11 |         |         |
|         |        | 1 |        |        |    |         |         |
|         |        | 5 |        |        |    |         |         |
+---------+--------+---+--------+--------+----+---------+---------+

\* CI = confidence interval.

† Calculated based on the natural log values of total effective dose
equivalent.

‡ mSv = milliSievert.

Although the confidence intervals may be a substantial fraction of the
central tendency value (the mean or the median), the upper confidence
limits are not larger than the central tendency value (i.e., the
confidence interval is always smaller than the interval 0 to twice the
mean). This may seem counterintuitive to many who consider the dose
estimates of this kind to be accurate only to an order of magnitude, if
that. However, these confidence intervals reflect only the uncertainty
in the input variables to the models and do not reflect the uncertainty
in the models themselves or the approximations made in applying the
models. In a sense, it is more correct to characterize these confidence
bounds as a measure of precision rather than as a measure of accuracy.

For the dose distributions, the largest value of the coefficient of
variation is approximately 1.3. This value is much smaller than the
largest value of the coefficient of variation for the input variables,
which was approximately 31 for the resuspension factor. The reason the
uncertainty in resuspension factor is attenuated when processed by the
dose models is that resuspension does not cause much dose for these
scenarios. Inhalation of resuspended contamination has already been
shown to be a minor pathway (see Section 12.2). In addition, resuspended
contamination can cause dose by depositing on growing vegetation and
becoming incorporated into the food chain. For the scenarios analyzed,
however, direct deposition of contamination is always much more
effective in causing dose than resuspended deposition.

### Comparison of Results from Point Estimate and Uncertainty Analyses 

Table 12-8 and Table 12-9 show for each receptor from the deterministic
analysis the point-estimate dose compared to the results of the
uncertainty analysis―the run 0 dose, the mean dose, and the median dose.
As discussed previously in Section 12.2, the point-estimate dose is very
close to the run 0 dose for each receptor \-- as would be expected
because the same values for the input variables are used. However, the
mean dose from the uncertainty analysis is generally larger (and in some
cases, substantially so) than the point-estimate dose for each receptor.
The ratio of the mean dose to the point-estimate dose ranges from a high
value of 2.15 (115 percent higher) for the Delivery Family Child Born in
1964 to a low value of 1.07 (7 percent higher) for the Urban Family
Child Born in 1964. The median dose from the uncertainty analysis is
generally higher than the point-estimate dose, but usually by a smaller
amount than the corresponding mean dose. However, for the Urban Family
receptors, the median doses are smaller than the point-estimate dose.
The ratio of the median dose to the point-estimate dose ranges from a
high value of 1.54 (54 percent higher) for the Delivery Family Child
Born in 1955 to a low value of 0.86 (14 percent smaller) for the Urban
Family Adult Female.

As discussed in Section 12.4.1, for the dose distributions obtained from
the uncertainty analysis, it is not surprising that the median doses are
generally smaller than the mean doses. However, it was somewhat
surprising that the median doses were mostly larger (sometimes
substantially so) than the point-estimate doses. After all, the point
estimates of dose were developed using "representative" values for all
the input variables, and a selected set of input variables were
represented by distributions. All but one of the input distributions
were chosen to be lognormal distributions (the other was a normal
distribution). The median of each lognormal distribution was set equal
to the point-estimate value of the corresponding input variable. Then
the distribution of doses was generated based on 1) the distributions of
input variables for those chosen to be uncertain and 2) the
point-estimate values for those input variables chosen to be fixed.
Because the medians of the input distributions were the point-estimate
values, it could be expected (in a very approximate fashion) that the
medians of the uncertain doses would be closer to the point-estimate
values than the means of the uncertain doses. In some cases, however,
they are greater than the point estimates by over 50 percent.

To examine this result, some additional analysis was performed. The dose
to the Adult Female for the Delivery Person Family has been a focus of
consideration because the water release dose (water dose) for the point
estimates is known to be 94 percent of the total dose. This receptor was
chosen for closer scrutiny because it has dose dominated by this single
release mode and because the median of the uncertain dose―9.072
milliSieverts―is 1.486 times the point-estimate dose of 6.106
milliSieverts.

An initial conjecture is that the air doses add to the water doses in a
random fashion and provide a "floor" for the total dose. This
more-or-less average addition could cause the low values of the total
doses to be elevated and therefore increase the median. This conjecture
does not seem to be confirmed by the data. The median for just the water
doses is 8.539 milliSieverts. This implies that only 5.9 percent of the
median dose is contributed by the air releases. Therefore, the air
releases hardly seem capable of elevating the median of the dose. To
look at it another way, the mean value of the air release dose is 0.469
milliSieverts. If this mean value of air dose is added to the water
doses, it will not substantially affect the median total dose. Although
the fraction of the air dose increases substantially for lower total
doses, the highest it ever gets is a little over 0.4. For most values,
the fraction is less than 0.1 and for many it is less than 0.05. This is
not the answer.

Table 12‑8 Statistics on Total Effective Dose Equivalent for Different
Receptors

+---------------+-------------+--------------+-------+-------+-------+
| Family        | Family      | Point        | Run 0 | Mean  | M     |
|               | Member      | Estimate     |       |       | edian |
|               |             |              | (m    | (m    |       |
|               |             | (mSv)\*      | Sv)\* | Sv)\* | (m    |
|               |             |              |       |       | Sv)\* |
+---------------+-------------+--------------+-------+-------+-------+
| Delivery      | Adult       | 6.106        | 6.091 | 1     | 9.072 |
| Family        | Female      |              |       | 2.812 |       |
+---------------+-------------+--------------+-------+-------+-------+
| Delivery      | Adult Male  | 6.283        | 6.266 | 1     | 9.281 |
| Family        |             |              |       | 3.001 |       |
+---------------+-------------+--------------+-------+-------+-------+
| Delivery      | Child Born  | 5.200        | 5.180 | 1     | 7.993 |
| Family        | 1955        |              |       | 0.164 |       |
+---------------+-------------+--------------+-------+-------+-------+
| Delivery      | Child Born  | 2.090        | 2.081 | 4.498 | 3.159 |
| Family        | 1964        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Migrant       | Adult       | 0.447        | 0.438 | 0.793 | 0.562 |
| Family        | Female      |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Migrant       | Adult Male  | 0.624        | 0.614 | 1.117 | 0.756 |
| Family        |             |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Migrant       | Child Born  | 2.178        | 2.160 | 3.676 | 2.489 |
| Family        | 1955        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Migrant       | Child Born  | 0.083        | 0.081 | 0.127 | 0.093 |
| Family        | 1964        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Near Water    | Adult       | 2.091        | 2.057 | 3.431 | 2.738 |
| Family        | Female      |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Near Water    | Adult Male  | 2.205        | 2.170 | 3.574 | 2.929 |
| Family        |             |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Near Water    | Child Born  | 3.137        | 3.099 | 4.815 | 4.311 |
| Family        | 1955        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Near Water    | Child Born  | 1.759        | 1.734 | 2.850 | 2.290 |
| Family        | 1964        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Outdoor       | Adult       | 3.030        | 3.001 | 4.687 | 4.263 |
| Family        | Female      |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Outdoor       | Adult Male  | 4.216        | 4.169 | 6.055 | 5.546 |
| Family        |             |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Outdoor       | Child Born  | 9.435        | 9.383 | 1     | 1     |
| Family        | 1955        |              |       | 3.331 | 0.988 |
+---------------+-------------+--------------+-------+-------+-------+
| Outdoor       | Child Born  | 1.826        | 1.810 | 2.951 | 2.309 |
| Family        | 1964        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Rural Family  | Adult       | 0.303        | 0.299 | 0.502 | 0.387 |
| One           | Female      |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Rural Family  | Adult Male  | 0.423        | 0.418 | 0.712 | 0.538 |
| One           |             |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Rural Family  | Child Born  | 1.589        | 1.580 | 2.681 | 1.697 |
| One           | 1955        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Rural Family  | Child Born  | 0.072        | 0.071 | 0.093 | 0.077 |
| One           | 1964        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Rural Family  | Adult       | 0.696        | 0.685 | 1.174 | 0.890 |
| Two           | Female      |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Rural Family  | Adult Male  | 0.974        | 0.961 | 1.655 | 1.198 |
| Two           |             |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Rural Family  | Child Born  | 3.751        | 3.729 | 6.362 | 4.006 |
| Two           | 1955        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Rural Family  | Child Born  | 0.140        | 0.137 | 0.190 | 0.153 |
| Two           | 1964        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Urban Family  | Adult       | 0.330        | 0.325 | 0.447 | 0.284 |
|               | Female      |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Urban Family  | Adult Male  | 0.731        | 0.723 | 0.895 | 0.698 |
+---------------+-------------+--------------+-------+-------+-------+
| Urban Family  | Child Born  | 2.686        | 2.675 | 4.314 | 2.551 |
|               | 1955        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+
| Urban Family  | Child Born  | 0.107        | 0.106 | 0.115 | 0.107 |
|               | 1964        |              |       |       |       |
+---------------+-------------+--------------+-------+-------+-------+

Table 12‑9 Comparison of Mean Dose and Median Dose with Point-Estimate
Dose for each Receptor

+-------------+-----------+--------+-----+---------+-----+-----------+
| Family      | Family    | Pt.    | M   | Ratio,  | Med | Ratio,    |
|             | Member    | Es     | ean | Mean    | ian | Median    |
|             |           | timate |     | over    |     | over      |
|             |           |        | (   | Point   | (   | Point     |
|             |           | (      | mSv | E       | mSv | Estimate  |
|             |           | mSv)\* | )\* | stimate | )\* |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Delivery    | Adult     | 6.106  | 12. | 2.098   | 9.  | 1.486     |
| Family      | Female    |        | 812 |         | 072 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Delivery    | Adult     | 6.283  | 13. | 2.069   | 9.  | 1.477     |
| Family      | Male      |        | 001 |         | 281 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Delivery    | Child     | 5.200  | 10. | 1.955   | 7.  | 1.537     |
| Family      | Born 1955 |        | 164 |         | 993 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Delivery    | Child     | 2.090  | 4.  | 2.153   | 3.  | 1.512     |
| Family      | Born 1964 |        | 498 |         | 159 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Migrant     | Adult     | 0.447  | 0.  | 1.773   | 0.  | 1.255     |
| Family      | Female    |        | 793 |         | 562 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Migrant     | Adult     | 0.624  | 1.  | 1.790   | 0.  | 1.211     |
| Family      | Male      |        | 117 |         | 756 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Migrant     | Child     | 2.178  | 3.  | 1.688   | 2.  | 1.143     |
| Family      | Born 1955 |        | 676 |         | 489 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Migrant     | Child     | 0.083  | 0.  | 1.524   | 0.  | 1.119     |
| Family      | Born 1964 |        | 127 |         | 093 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Near Water  | Adult     | 2.091  | 3.  | 1.641   | 2.  | 1.309     |
| Family      | Female    |        | 431 |         | 738 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Near Water  | Adult     | 2.205  | 3.  | 1.621   | 2.  | 1.328     |
| Family      | Male      |        | 574 |         | 929 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Near Water  | Child     | 3.137  | 4.  | 1.535   | 4.  | 1.374     |
| Family      | Born 1955 |        | 815 |         | 311 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Near Water  | Child     | 1.759  | 2.  | 1.621   | 2.  | 1.302     |
| Family      | Born 1964 |        | 850 |         | 290 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Outdoor     | Adult     | 3.030  | 4.  | 1.547   | 4.  | 1.407     |
| Family      | Female    |        | 687 |         | 263 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Outdoor     | Adult     | 4.216  | 6.  | 1.436   | 5.  | 1.316     |
| Family      | Male      |        | 055 |         | 546 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Outdoor     | Child     | 9.435  | 13. | 1.413   | 10. | 1.165     |
| Family      | Born 1955 |        | 331 |         | 988 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Outdoor     | Child     | 1.826  | 2.  | 1.616   | 2.  | 1.264     |
| Family      | Born 1964 |        | 951 |         | 309 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Rural       | Adult     | 0.303  | 0.  | 1.656   | 0.  | 1.277     |
| Family One  | Female    |        | 502 |         | 387 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Rural       | Adult     | 0.423  | 0.  | 1.683   | 0.  | 1.270     |
| Family One  | Male      |        | 712 |         | 538 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Rural       | Child     | 1.589  | 2.  | 1.687   | 1.  | 1.067     |
| Family One  | Born 1955 |        | 681 |         | 697 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Rural       | Child     | 0.072  | 0.  | 1.295   | 0.  | 1.067     |
| Family One  | Born 1964 |        | 093 |         | 077 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Rural       | Adult     | 0.696  | 1.  | 1.688   | 0.  | 1.279     |
| Family Two  | Female    |        | 174 |         | 890 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Rural       | Adult     | 0.974  | 1.  | 1.700   | 1.  | 1.231     |
| Family Two  | Male      |        | 655 |         | 198 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Rural       | Child     | 3.751  | 6.  | 1.696   | 4.  | 1.068     |
| Family Two  | Born 1955 |        | 362 |         | 006 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Rural       | Child     | 0.140  | 0.  | 1.362   | 0.  | 1.094     |
| Family Two  | Born 1964 |        | 190 |         | 153 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Urban       | Adult     | 0.330  | 0.  | 1.353   | 0.  | 0.859     |
| Family      | Female    |        | 447 |         | 284 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Urban       | Adult     | 0.731  | 0.  | 1.224   | 0.  | 0.954     |
| Family      | Male      |        | 895 |         | 698 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Urban       | Child     | 2.686  | 4.  | 1.606   | 2.  | 0.950     |
| Family      | Born 1955 |        | 314 |         | 551 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+
| Urban       | Child     | 0.107  | 0.  | 1.068   | 0.  | 0.996     |
| Family      | Born 1964 |        | 115 |         | 107 |           |
+-------------+-----------+--------+-----+---------+-----+-----------+

\*mSv = milliSieverts.

The components of the water dose seem to provide a more plausible
explanation of the behavior of uncertain doses. For each realization,
the doses resulting from cesium-137, strontium-90, and phosphorus-32
were listed as well as their sum. The median of the "sum" is 8.419
milliSieverts. This sum accounts for most of the water-dose median
(8.419/8.539 = 0.986) and most of the total-dose median (8.419/9.072 =
0.928). The median of doses for each isotope were found separately, and
the sum of these medians is 5.857 milliSieverts. This is much closer to
the point-estimate dose of 6.106 milliSieverts. This appears to suggest
that if the three isotopic components of dose were summed separately,
they would correspond closely to the base case. However, because these
doses are summed for each realization, the sum is, on average, larger
than the point estimate. Because the uptake factors for these three
isotopes vary independently and randomly, the total dose can be
considered approximately to be the sum of three lognormal distributions.
The smaller, but substantial, doses from strontium-90 and phosphorus-32
essentially make a "floor" on the dose from cesium-137 and boost the
median (as shown) by a factor of about 50 percent over what would occur
if the uptake factors were correlated.

In summary, it appears that the confidence intervals do not include the
point estimates. This appears to be a result of both the assumptions
regarding the input variable distributions and the mathematical form of
the dose model. This also shows the utility of performing an uncertainty
analysis, which may give a different perspective on the problem than a
point estimate.

## Secondary Results

The primary result of the overall uncertainty analysis was to establish
distributions of dose estimates that incorporated uncertainty in the
input variables and to establish the statistical confidence intervals
about the mean values of those distributions. However, the large amount
of data generated in performing this uncertainty analysis can be used to
provide additional insights into how doses and the uncertainties in
doses depend on the uncertain variables analyzed. This additional
information can be useful in determining how to refine the modeling
approaches or how to prioritize the need for additional site data.

In [Chapter 11](../Chapter_11_ready%20for%20CDC%20031405.doc) dose to
various receptors was explored by identifying important factors such as
radionuclides, year of exposure, exposure pathways, and exposure routes.
These results provided information on the [structure]{.underline} of the
model. These point estimate dose results and the conclusions drawn from
them also depended on the choices made for the value of each input
variables, i.e., one value for each input variable. In this chapter the
dependence of dose on the distributions chosen to describe each
uncertain input variable is explored; i.e., the attention here is on the
dependence of dose on the uncertain input variables. These analyses are
based upon the sampled input variables and results of the uncertainty
analysis described in previous sections. By aligning the input variable
values with the respective resultant dose values, several statistical
techniques can be used to evaluate how a change in an input variable
changes the output (dose).

The dependence of dose on the uncertain input variables can be
considered to have two components: (1) how effective a change in an
input variable is in producing a change in dose and (2) how widely a
particular input variable changes. Big variations in dose may be
produced in three ways: (1) dose may be very sensitive to a particular
variable, so even modest variations in that input variable produce
substantial variations in dose; (2) dose may be moderately sensitive to
a particular variable, but the variable is very uncertain, so large
variations in dose are produced; (3) dose may be very sensitive to a
highly variable input, so extremely large variations in dose are
produced. The two components describing the dependence of dose on
uncertain input variable are sometimes characterized by two
coefficients: (1) a sensitivity coefficient and (2) a variance or
uncertainty coefficient. Broadly speaking, the sensitivity coefficient
is the ratio of the fractional change in dose to the fractional change
of an input variable. The variance or uncertainty coefficient is the
fraction of uncertainty in dose attributable to the uncertainty in an
input variable. Both of these aspects are explored in the Sections that
follow.

Sections 12.6.1 and 12.6.2 discuss how some general statistical methods
can be used to describe the relationships between dose and input
variables. Scatter plots are used to give a graphical assessment of the
input-dose relationship. The correlation coefficient (assuming a linear
relationship) was used to quantify the relationship between dose and
input variables. Section 12.6.3 describes how the variability in dose
depends on input variables (i.e., variance or uncertainty
considerations); Section 12.6.4 describes how the dose depends on input
variables (i.e., sensitivity considerations). Only a few examples were
analyzed and described here. An extensive analysis of this type is
beyond the scope of the current study. More extensive analyses of this
type may be appropriate in the future. The limited analyses described
here generally reinforce the understanding of the generation of doses by
SRS releases obtained from the modeling and results described in
previous Chapters.

The names and characteristics of the uncertain variables analyzed are
described in Table 12-2. However, to facilitate this discussion, the
variable names and their physical meaning are repeated in Table 12-10
here.

  ------------------ ----------------------------------------------------
  Table 12-10 Names  
  of Uncertain       
  Variables and      
  Their Physical     
  Meaning            

  Variable Name      Physical Meaning

  ***Water           
  Pathway***         

  CLBFF, Cs-137      Bioconcentration factor for Cs-137 in fish

  CLBFF, Sr-90       Bioconcentration factor for Sr-90 in fish

  CLBFF, P-32        Bioconcentration factor for P-32 in fish

  ***Air Pathway***  

  LEAFRS             Soil resuspension factor; determines the amount of
                     radioactivity deposited on the ground that is
                     resuspended; affects urban and rural scenarios.

  RADIUS             Radius of particles that can deposit on the soil and
                     plant surfaces.

  WTIM               Weathering rate constant determines the rate at
                     which deposited radioactivity is removed by
                     weathering processes from crop surfaces.

  DPVRES             Deposition velocity from air to plant surfaces for
                     resuspended activity.

  DEPFR1, 2          Dry and wet deposition fraction; how much of the
                     radioactivity deposited on plant surfaces is
                     retained and absorbed by the plant.

  GRWPA, 5&6         Growing period for animal forage; the longer the
                     growing period, the more exposure the plant has to
                     air deposition of radionuclides.

  CLFMT I-131        Uptake of I-131 by beef muscle; transfer to meat
                     from cattle food.

  F                  Release factor characterizes the uncertainty in air
                     and water releases of radionuclides from the SRS.

  CONSUM, 5          Animal forage consumption rate; scales forage
                     consumption by milk and beef animals.

  CLFMK I-131        Uptake of I-131 by milk; transfer to milk from cow
                     food.

  BIOMA2             Animal forage standing biomass (wet) describes how
                     much grass is in the field per unit area.
  ------------------ ----------------------------------------------------

### Scatter Plots 

Figure 12-5 shows examples of scatter plots for the Delivery Family
Adult Female. Simple scatter plots of the dose versus a sampled input
variable (e.g., CLBFF-Cs, the bioconcentration of cesium in fish) can
give a quick visual display of the importance of a particular variable
in affecting the dose. For example, the left and right plots are,
respectively, the plot of dose versus the cesium-bioconcentration in
fish and the plot of dose versus the suspended particle radius. The
pattern in the left plot resembles that of a line, indicating a strong
linear relationship between the uptake of cesium in fish and the
resultant dose to the Delivery Family Adult Female. The much more random
"snow" pattern in the right plot indicates that the dose to the Delivery
Family Adult Female is not significantly affected by the resuspended
soil-particle radius.

![](media/image5.wmf){width="6.25in" height="4.21875in"}

Figure 12‑5 Examples of Scatter Plots for the Delivery Family Adult
Female: Dose vs. Bioconcentration of Cesium in Fish (Left Plot) and Dose
vs. Particle Radius (Right Plot)

### Correlation Coefficients

Correlation coefficients are another way of expressing the relationship
between input variables and the dose. Again, this analysis focused on
linear relationships and did not consider nonlinear relationships such
as dose as a function of the square of the particle radius. The
correlation coefficient is a quantitative, statistical measure that
represents the strength of the linear relation between two random
variables (5). In this case, the two random variables are the dose and
the input variable selected for examination. Correlation coefficients
range between -1.0 and +1.0. A value of +1.0 indicates a perfect direct
relationship; a value of -1.0 indicates a perfect negative relationship.
For example, the size of one's bank account may have a correlation of +1
with the size of one's paycheck, but a correlation of -1 with the size
of one's expenditures.

Correlation coefficients for the CLBFF-Cs and radius scatter plots given
above are 0.864 and -0.032, respectively. This quantitatively
demonstrates that CLBFF-Cs is over 10 times more effective in changing
the dose than is the radius.

### Stepwise Regression Analysis

The dose to the 28 hypothetical individuals considered in this analysis
is the summation of incremental doses through various exposure pathways
and from various radionuclides. These doses depend on many environmental
and behavioral variables. Thus, in determining the most important
variables, it is important to consider them together for the purpose of
establishing an order of most to least significant within the group.
This can be accomplished with the stepwise regression analysis
technique. In this type of analysis, the relationship of the dose to all
of the sampled input variables is represented as a polynomial; that is,
the statistical model for the dose is a sum of linear terms where each
term is a product of a coefficient and an input variable.

A main focus of the stepwise regression analysis is to determine the
best value of the coefficient for each input variable. This statistical
model is built in a stepwise fashion by sequentially adding (or
subtracting) one independent variable at a time. The result of this
analysis is a statistical model that describes the dose as a function of
a small set of independent variables (e.g., Dose \~ f(CLBFF-Cs
+CLBFF-Sr)) which are the most important in affecting dose; input
variables with coefficients that are relatively small are generally
deleted from the model. Because this kind of analysis can be very
computationally intensive, statistical software (S-Plus; Insightful,
2002) was used. As the software performed the stepwise regression, it
automatically checked the coefficient of determination (R^2^) and the
statistical level of significance. Variables that do not have
significant correlation with the dose or result in an increase in the
R^2^ are generally left out.

The statistic R^2^, the coefficient of determination, is a "measure of
the proportion of the total variability in the dependent variable that
is accounted for by the regression equation" (6). A value of 1 for R^2^
suggests that all the variability in the dependent variable (dose) is
accounted for by the variability in the selected independent variables.
A low value for R^2^ is often interpreted to mean that the variability
in the dependent variable is not linked to the variation in the
independent variables. This is true to some extent. A low R^2^ is many
times a reflection of a small range of variation of the independent
variable. Or, it might indicate that some of the variables should be
considered in a nonlinear fashion (e.g., X^2^ or 1/x).

The stepwise regression analysis was conducted for the hypothetical
families that consume fish (Delivery, Near Water, and Outdoor Family) by
considering all 14 of the sampled input variables. For those
hypothetical families that did not consume fish (Migrant, Rural Families
1 and 2, Urban/Suburban), the stepwise regression analysis was conducted
without any of the bioconcentration factors in fish (i.e., CLBFF for
cesium, strontium, and phosphorus).

Table 12-10 and Table 12-11 present the results of the stepwise
regression analysis, respectively, for the Delivery Family (which ate
fish) and Rural Family 2 (which did not). Table 12-12 presents the
stepwise regression analysis results for each of the Adult Females
representing all scenarios. The five most important variables are listed
in order of importance. The rank ordering is based upon the correlation
coefficient and the contribution to the R^2^. In general, for those
variables that contribute more than 0.1 to the multiple R^2^, there is
little uncertainty concerning their rank. In this analysis, this
generally applies to the top three variables. Variables ranked fourth
and fifth made a very small contribution to the multiple R^2^ and thus
might change with a different sampling using the Latin Hypercube
Sampling program.

Table 12‑10 Delivery Family Stepwise Regression Results Listing Input
Variables in Order of Influence and the Sum of R^2^ for the Variables
Listed

  ----------------- ---------------- ----------------- -------------------
  Adult Female      Adult Male       Child Born 1955   Child Born 1964

  CLBFF-Cs          CLBFF-Cs         CLBFF-Sr          CLBFF-Sr

  CLBFF-Sr          CLBFF-Sr         CLBFF-Cs          CLBFF-Cs

  F                 F                CLFMK I-131       CLBFF-P

  CLBFF-P           BIOMA2           CLFMT I-131       BIOMA2

  BIOMA2            CLBFF-P          LEAFRS            F

  R^2^ = 0.98       R^2^ = 0.98      R^2^ = 0.96       R^2^ = 0.97
  ----------------- ---------------- ----------------- -------------------

Note: Refer back to Table 12-2 for descriptions of the variables.

Table 12‑11 Rural Family 2 Regression Results Listing Input Variables in
Order of Influence

and the Sum of R^2^ for the Variables Listed

  ------------------ --------------- ----------------- -------------------
  Adult Female       Adult Male      Child Born 1955   Child Born 1964

  DEPRF1,2           LEAFRS          LEAFRS            LEAFRS

  LEAFRS             DEPRF1,2        F                 F

  F                  F               DEPFR 1,2         DEPFR 1,2

  DPVRES             DPVRES          DPVRES            DPVRES

  CONSUM             CONSUM          CONSUM            CONSUM

  R^2^ = 0.597       R^2^ = 0.576    R^2^ = 0.793      R^2^ = 0.5793
  ------------------ --------------- ----------------- -------------------

Note: Refer back to Table 12-2 for full descriptions of the variables.

Table 12‑12 Stepwise Regression Results for the Adult Female of Each
Scenario

  ----------- --------- ----------- ----------- ----------- ----------- ---------
  Delivery    Migrant   Near Water  Outdoor     Rural       Rural       Urban
  Female^1^   Female    Female^1^   Female^1^   Family 1    Family 2    Female
                                                Female      Female      

  CLBFF-Cs    LEAFRS    LEAFRS      CLBFF-Cs    DEPRF1,2    DEPRF1,2    DEPRF 1,2

  CLBFF-Sr    DEPFR 1,2 DEPRF 1,2   CLBFF-Sr    LEAFRS      LEAFRS      WTIM

  F           F         F           CLBFF-P     F           F           CLFMK
                                                                        I-131

  BIOMA2      DPVRES    DPVRES      DEPRF1,2    DPVRES      DPVRES      CLMFT
                                                                        I-131

  CLBFF-P     CONSUM    CONSUM      CLFMT I-131 CONSUM      CONSUM      CONSUM

  R^2^ = 0.98 R^2^      R^2^ =0.692 R^2^ =0.939 R^2^ =0.593 R^2^ =0.597 R^2^
              =0.692                                                    =0.731
  ----------- --------- ----------- ----------- ----------- ----------- ---------

^1^ Ate Fish

Note: Refer back to Table 12-2 for full descriptions of the variables.

In the case of the Delivery Family (Table 12-11), the most important
variables to the dose are the bioconcentration factors for cesium,
strontium, and phosphorus in fish, suggesting that much of the
variability in dose to the Delivery Family is from the variability of
the uptake factors for cesium, strontium, and phosphorus by fish. As
described in [Chapter 3](../Chapter_03_ready%20for%20CDC%20031405.doc),
the Delivery Family gets 50 percent of its fish from Lower Three Runs
Creek, which drains from the area with P and R reactors. This list of
variables may be shortened even more by dropping CLBFF-P, because
CLBFF-Cs and CLBFF-Sr yield an R^2^ of 0.91, which is most of the
variation.

After strontium and cesium in fish, the important variables for the
Delivery Family Child Born in 1955 are those variables associated with
uptake of iodine-131by beef cattle and milk cows (CLFMT I-131 and CLFMK
I-131), reflecting the relatively large releases of iodine-131 in the
early years of site operation.

The most important variables for the Rural Family 2 (Table 12-12) are
related, as expected, to ingestion. However, the variable LEAFRS
(resuspension factor) could also impact inhalation dose. The variable
CONSUM (forage consumption rate) is associated with the uptake of
radionuclides (especially I-131) by beef cattle and milk cows. The R^2^
values for this set of stepwise regression analyses (approximately 0.6
for each receptor) indicate that the relationship between the
independent variables and the dose may not be linear or that there is a
small range for one of the variables (CONSUM has a range of 9 to 62;
less than 1 order of magnitude).

Table 12-13 compares the stepwise regression results for the Adult
Female in each of the seven scenarios; this is intended to show how
results are scenario dependent, but comparisons based on other family
members may identify different variables. In general, the results of the
stepwise regression analysis are consistent with the scenarios as they
are defined. Specifically, of the hypothetical families that eat fish,
the Delivery Family got its fish from Lower Three Runs Creek and the
Savannah River, which exposed them to cesium, strontium, and phosphorus
releases from the SRS at relatively higher concentrations than those
families obtaining fish from only the Savannah River. The members of the
Outdoor Family fished along the Savannah River shoreline adjacent to the
SRS and were thus exposed to radionuclides transported through the
wetlands area. Thus, it is reasonable that the most significant
variables in determining the variation in dose to these families are the
variations in bioconcentration factors for cesium and strontium.

In contrast, although the Near Water Family got its fish from the
Savannah River, the consumption of fish was smaller than for the Outdoor
Family. The bioconcentration factors cesium, strontium, and phosphorus
in fish were not among the most important variables for this scenario.
Instead, the Near Water and Migrant families exhibit a similar ranking
of important variables. The most important variables for these families
are related to exposure by air and consumption of locally grown fruits,
vegetables, milk and beef. Although these families lived on opposite
sides of the SRS, the similarity in the sensitivity rankings suggest
that exposure through pathways related to air transport did not vary
substantially around the circumference of the SRS.

The two Rural families showed the same sensitivities to input variables.
Like the Migrant and Near Water families, the most important variables
affecting variation in dose are those related to airborne transport and
uptake through farm products.

The first-ranked variable for the Migrant and Near Water families is
LEAFRS. This variable contributed 0.41 to the total R^2^. The
second-ranked variable DEPRF1,2 contributed 0.15. By contrast, the
contribution to the total R^2^ for the two Rural families was 0.23 for
DEPFR1,2 and 0.21 for LEAFRS. The reasons for these differences in
variable ranking and their relative contribution are not clear.

The ranking of variables for the Urban/Suburban Family is different from
those families living in more rural settings closer to the SRS. This
family's primary exposure to radionuclides from the site was through the
milk produced in the New Ellenton area. DEPRF1,2 is the most significant
variable with a contribution of 0.4 to the total R^2^. Other variables
are for removal of deposited radionuclides from plant surfaces (WTIM)
and the uptake of I-131 into milk and beef (CLFMT I-131 and CLFMK
I-131).

Several scenarios show sensitivity to the release factor, an expected
result: the more released, the higher the dose.

### Sensitivity Coefficients

In Section 12.6.3, the most important variables were ranked in order of
their contribution to the coefficient of determination, R^2^. This
ranking actually is for the contribution to the variability in the dose,
not the sensitivity, per se. To rank the variables in terms of their
sensitivity coefficient (how a fractional change in the dose reflects a
fractional change in the input variable), the uncertain variables were
transformed to dimensionless quantities by dividing by the mean of each
respective distribution. The transformed variables were used in the
stepwise regression analysis, and while the variables found to be
dominant remained the same, the ranking changed. In this analysis, the
sensitivity coefficients are based on the slope of a least-squares-fit
between dose and the input variable. presents the variable rankings and
their sensitivity coefficients for the Delivery Family and Rural Family
2 Adult Females. These rankings are related to how important the
variables are in determining the dose; these rankings reflect only
slightly the degree of uncertainty in the input variables themselves.

**Table 12‑13 Variable Rankings and Their Sensitivity Coefficients**

+-------------------+------------------+---------------+--------------+
| **Delivery Family |                  | **Rural       |              |
| Adult Female**    |                  | Family 2      |              |
|                   |                  | Adult         |              |
|                   |                  | Female**      |              |
+-------------------+------------------+---------------+--------------+
| **Variable**      | **Sensitivity    | **Variable**  | *            |
|                   | Coefficient**    |               | *Sensitivity |
|                   |                  |               | C            |
|                   |                  |               | oefficient** |
+-------------------+------------------+---------------+--------------+
| > F               | 0.66             | > F           | 0.92         |
+-------------------+------------------+---------------+--------------+
| > BIOMA2          | -0.65            | > DEPFR 1,2   | 0.84         |
+-------------------+------------------+---------------+--------------+
| > CLBFF-Cs        | 0.64             | > CONSUM      | 0.38         |
+-------------------+------------------+---------------+--------------+
| > CLBFF-Sr        | 0.13             | > LEAFRS      | 0.15         |
+-------------------+------------------+---------------+--------------+
| > CLBFF-P         | 0.01             | > DPRVES      | 0.09         |
+-------------------+------------------+---------------+--------------+

Note: Change in sign for BIOMA2 that is more consistent with the
analytical model.

## Summary

The following are some of the key aspects of the uncertainty analysis:

-   Forty realizations were used to investigate the uncertainty behavior
    of dose for each of 28 hypothetical receptors.

-   In general, the distribution of dose for each receptor behaved
    similar to a lognormal distribution.

-   Consistent with the shape of a lognormal distribution, the mean of
    each dose distribution was higher than the median of the
    distribution.

-   The confidence intervals were estimated for the mean and median. The
    sizes of these intervals were not large compared to the
    corresponding mean and median. This appears to reflect the sample
    size (40) and the underlying uncertainty quantified for the dose
    distributions. Also, modeling uncertainty was not estimated; only
    uncertainty related to the input variables was quantified.

-   In general, the mean and median for the uncertainty analysis are
    larger than the corresponding point estimate of dose. This appears
    to reflect the interaction among the uncertain variablesand the
    complexity of the dose model.

-   The calculated confidence intervals do not contain the point
    estimates of dose for the reasons cited in the previous bullet.

-   The sensitivity analysis generally shows the same dominance of the
    milk and beef ingestion pathways for those scenarios dominated by
    dose from air release (as did the point-estimate analysis).
    Similarly, for doses from water releases, the fish ingestion
    dominates―specifically, the fish bioconcentration factors for
    cesium-137, strontium-90, and phosphorus-32.

## References

1\. Decision Engineering, Inc. Crystal ball 2000, user manual. 2000.

2\. Bowen WM, Bennett CA. Statistical methods for nuclear material
management. NUREG/CR-4604. U.S. Nuclear Regulatory Commission,
Washington D.C. December 1988.

3\. Morgan, MG, Henrion M. Uncertainty. Cambridge University Press,
1990: 198-209.

4\. Palisade Corporation. \@RISK, advanced risk analysis for
spreadsheets. Version 4.5. February 2002.

5\. Lurie D, Moore RH. Applying statistics. NUREG-1475. U.S. Nuclear
Regulatory Commission, Washington D.C. December 1994.

6\. Mac Berthouex P, Brown LC. Statistics for environmental engineers.
Lewis Publishers: Boca Raton, Florida. 1994.

7\. Iman RL, Shortencarier, MJ. A FORTRAN 77 program and user's guide
for the generation of latin hypercube and random samples for use with
computer models. NUREG/CR-3624 (SAND83-2365). U.S. Nuclear Regulatory
Commission, Washington, DC. March 1984. Not Referenced.

8\. Till & Meyer. Till, J.E. and H.R. Meyer, eds. Radiological
Assessment, A Textbook on Environmental Dose Analysis. NUREG/CR-3332.
ORNL-5968. September 1983.

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