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Modifications of BME Analysis 187

thus offering a measure of the strength and consistency of the association. An
exposure may not be considered as a possible causal factor if the vector health-
effect predictions provide no improvement in the rate prediction compared to
the scalar health-effect predictions.
The quantitative assessment of the prediction accuracy of Postulate 9.1
can be made in terms of the prediction errors at a set of "control points,"
i.e., points where the actual health effects (e.g., death rates) are known and
can be compared with the health-effect predictions obtained from scalar vs.
vector BME techniques. Analysis of an exposure-mortality association, e.g., is
based on the successful prediction of mortality at a set of control points using
the combined exposure (X) and death-rate (D) data. Providing that there
are no strong effects due to hidden confounding factors, an improved mortality
prediction obtained from the combination of death rate and exposure data as
compared to that obtained merely from death-rate data should support the
existence of an association between X and D. Hence, a suitable PEP measure
of the strength of the exposure-mortality association is defined as

(in %), where EDX is the mortality prediction error (in the stochastic sense)
based on death-rate D and exposure data X, and ED, the mortality prediction
error based on death-rate data only [the extension of Equation 9.48 in the
case of confounding factors is discussed later]. A consistently negative PDX-
value supports the existence of an exposure-mortality association. In terms
of mathematical logic, the analysis above may be expressed by the material
conditional of the form "X implies PDX < 0," for short, X —» (flux < 0).
The material conditional is a logical structure that is based on truth-functional
concepts (see Chapter 4, "Material and strict map conditionals," p. 98) and is
equivalent to the statement "it is not the case that X and not Pox < 0," in
short, -<[XI\~>(PDX < 0)]. It is worth mentioning that the scientific reasoning
of the PEP criterion violates neither Hume's nor Mill's rules of cause and
effect (as discussed, e.g., in Harris, 1996). Also, since the essential structure
underlying PEP is predictability, it is in agreement with Popper's concept of
scientific reasoning (Popper, 1962).

C O M M E N T 9.3: In addition t o suggesting possible exposure-effect associa-

tions, the PEP criterion canbe helpful in testing or confirming association

hypotheses developed from other investigations (medical, biological, toxico-

logical, etc.J. In some cases sufficient information may be obtained on the
relation between exposure and effect to set realistic exposure standards.

The following example presents an application of the PEP criterion by
Christakos and Serre (2000a), in the case of cold temperature data (which is
considered as the exposure variable in this case) vs. mortality data (which is the
health-effect variable) in the State of North Carolina. This example illustrates
the use of the PEP criterion to obtain a quantitative assessment of the asso-
ciation between cold temperature and mortality distributions in space/time, if