Approximations
Mathematical models, even very good ones, provide estimates
that are approximately true. The world
isn’t perfectly round, but a mathematical model that assumes that it is will
still tell you about how far you will need to travel to get to the other
side. A flat screen representation of a
little part of the world isn’t correct either, but it can still get you around
town.
Characterizing a risk is theoretically not difficult
either. Plugging an estimated dose into
a simple dose-response model will yield an estimate of the magnitude of the undesirable
response. But, for most food safety
issues that are usually concerned with effects that are too small to measure
reliably there is a problem: The simple
dose-response model probably isn’t a very good approximation. In fact, there usually are range of estimates
that could be correct, and sometimes that range can be very wide. Therefore, an uncertainty analysis may be
necessary to show just how wide the estimates can be. Public health risk models often must have
another dimension to them as well. Because
of differences in food consumption patterns and individual differences in what
a chemical does after it is ingested, a dietary risk model will usually
incorporate statistical components that describe population variability. As a result of all that added complexity, getting
an approximately correct estimate for a problem that is worthy of quantitation
may require a complex model or simulation.
Monte Carlo Simulation
When two or more mathematical probability “input” distributions
are present in a model, the most common method for estimating the probability
of an estimated value is called Monte Carlo
simulation. This “iterative” method uses
a computer to repeatedly sample random draws from each input distribution to
calculate the model output value; the resulting distribution is used to characterize
the distribution of the final estimate.
A one-dimensional Monte Carlo simulation can be used to calculate
the uncertainty associated with a population estimate. For example, the impact of the uncertainty of
a per capita exposure assessment and the uncertainty associated with a
dose-response estimate can be combined to estimate the uncertainty associated with
a disease frequency. Uncertainty
simulations may include both statistical uncertainties (e.g. sampling error) as
well as probability trees that represent theoretical uncertainty.
A one-dimensional Monte Carlo simulation can also be used to
characterize the uncertainty of the magnitude of an effect in an individual person. In this case, statistical distributions that describe
population characteristics must either be replaced with individual values (i.e.
a distribution describing body weights can be replaced with the actual body
weight for the individual) or be treated as statistical uncertainties when
individual values are unknown.
But, to characterize the uncertainty associated with an
effect of varying magnitude in a population, then one dimension is not enough. Two dimensional public health computer simulations
that employ one iterative loop embedded inside another one. The inner loop typically randomly samples
from distributions that describe the frequency of occurrence of a value in a
population (e.g. a distribution of body weights), while the outer loop samples probability
distributions and probability trees that are intended to represent
uncertainty. The end result is a two
dimensional array of values that characterize the uncertainty associated with a
frequency distribution. In order to simplify presentation, the
uncertainty dimension is often represented with a central value (i.e. the median
or arithmetic average) and confidence intervals (e.g. the 5th and 95th
percentiles) for each population estimate.
Other Simulation Methods
An alternative method for generating estimates from
uncertain and variable input values is to use stratified sampling. If the distributions aren’t comprised of
discrete values already (e.g. an empirical distribution), this involves breaking
each continuous distribution down into a set of fixed intervals with an average
value for each, and then calculating all possible permutations. This technique is not as easy to implement,
but since it doesn’t use random numbers it gives a simulation result that is
the same every time.
Latin Hypercube sampling
is a synthesis of Monte Carlo and stratified sampling. As in stratified sample each distribution in
broken down into fixed intervals, but then each interval is sampled with an
equal number of random draws. This
method has the advantage of sampling the entire distribution, but also produces
more reproducible results with fewer iterations.
Probability Trees vs. Sensitivity Analysis
A probability tree is a tool for including the uncertainty
arising from alternative theoretical assumptions in a quantitative uncertainty
analysis. Sensitivity analysis is a widely used alternative
method for representing theoretical uncertainty that depends on providing
entirely different estimates for each alternative assumption.
What almost always happens with sensitivity analysis is that
one assumption is used to for the primary estimate that is actually used to
inform subsequent decision making, while the others are recognized as “also
rans”. As a result, for all practical
purposes, this means that one theory is assigned a probability of one, while
the others have a probability of zero.
As a form of political rhetoric, a sensitivity analysis is often used as
a technique for acknowledging scientific criticism without really giving it any
credence. If the other theories really are
crackpot ideas that have no scientific credibility whatsoever anyway, that is
not a bad thing. Im amy case, including the alternative
estimates does give a reader the option of reversing the probability assignment
of the author. Furthermore, a
sensitivity analysis may be used to show that an estimate is not heavily influenced
by a particular assumption. But, if the alternative
assumptions really are both plausible and influential, then not including them
in the uncertainty analysis will result in confidence intervals that are
unjustifiably narrow.
General References
Cullen AC and Frye HC (1999). Probabilistic
Techniques in Exposure Assessment: A Handbook for Dealing with Variability and
Uncertainty in Models and Inputs. Plenum,
New York.
Morgan MG, Henrion M, and Small M (1992). Uncertainty:
A Guide to Dealing with Uncertainty in Quantitative Risk and Policy Analysis. Cambridge University Press, Cambridge.
Kraftwerk (1981). Computer World. In: Computer World, Track 1
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