Environmental Health Criteria 214

HUMAN EXPOSURE ASSESSMENT
events increases), the amount of variability should increase as well.
For example, if we expect a count of 1 then it is not too difficult to
imagine observing 0 or 2. Likewise, if we expect a count of 20 000
then it is not difficult to imagine 20 100 or 19 900 as reasonable
observations. However, the variance is definitely larger in the second
case. The formula used to compute the probability of a specific number
of counts being observed over a fixed time interval is listed in Eq.
4.11 of Table 10.
For example, the Poisson distribution can be used in an exposure
model to characterize the frequency with which a person comes in
contact with a contaminant; say, the number of times per day a person
encounters benzo [a]pyrene associated with environmental tobacco
smoke. Assume that based on existing data, the expected number of
encounters is anticipated to be 2 per day. Using Eq. 4.11, with lambda
= 2, there is a 9% chance that an individual will have 4 (i.e.,
n =4) encounters with benzo [a]pyrene on a given day. Subject to
limitations associated with the independence assumption noted above,
the Poisson distribution can be a useful exposure modelling tool.
4.4 Parametric inferential statistics
Inferential statistics is the process of using the observed data
and assumptions about the distribution and variation of the data to
draw conclusions. The two complementary components of inference are
parameter estimation (either point or interval estimation) and
hypothesis testing. Only frequentist, or classical, inference will
be discussed here. However, Bayesian statistical inference, as well as
decision theory, can be valuable for incorporating other aspects such
as prior belief and loss into a statistical analysis, and they are
worth consideration. Further information on Bayesian statistics may be
found in Carlin & Louis (1996).
4.4.1 Estimation
Exposure measurement data can be used to estimate the parameters
of a model (e.g., a probability distribution), especially those that
describe the mean and variance of the variable. The two types of
common reported estimates are point estimates and interval
estimates.
Point estimation for quantities is commonly performed using
maximum likelihood, ordinary least squares or weighted least squares
methods. All estimates are chosen because they optimize (i.e., find
the maximum or minimum of) some objective function such as the
likelihood function or squared error function. One example is the
sample mean for the population mean when the data are normal, using
maximum likelihood, or for any data, using least squares.
Two different forms of interval estimation are used to
characterize variability in point estimates. The first is based on
error propagation and is the result of simulating data to see what
distribution of results might be expected under the model; the second
is the usual statistical notion of confidence intervals. This approach
focuses more on the variability of a modelled outcome due to
variability of the input, and is useful in designing studies and
determining which factors will have the greatest effect on the
variability of the exposures. These procedures are described more
fully in Chapter 6.
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