includes relatively low levels of exposure to complexes of poorly defined materials. Thus, an environmental pollutant is likely to be associated with relatively small risks, though it could affect large numbers of people.

At any given level of statistical significance, there is a relation among study power, sample size, prevalence of exposure, and expected rate of a given outcome. In general, studies of larger numbers of persons over longer periods are more likely to yield positive results than those involving smaller populations for shorter periods. However, even large studies with long followup will result in uncertain findings if exposure is poorly measured or misclassified (see chapter 3). The sample size needed to achieve a given study power is also related to whether exposure is measured as a dichotomous or continuous variable, to the variability in distribution of the exposure, and to the effects of confounders and errors in the measure of exposure. In general, larger samples are needed when exposure measures are not continuous, when the effects of confounders and errors of measurement cannot be taken into account, and when the adverse outcome is a rare event (Greenland, 1983; McKeown-Eyssen and Thomas, 1985; Lubin et al., 1988; Lubin and Gail, 1990). Finally, all statistical-power calculations depend on the critical assumption that bias in both exposure and outcome can be ignored; this assumption may be rarely true in practice.

Statistical-significance testing is used to assess the likelihood that positive results of any given study represent a "real" association. However, no matter which statistical tests are employed, common research designs all produce studies with fixed, known chances of making a type I error, that is, of finding a positive result when one does not really exist. This probability is called alpha and is generally determined by a statistician at the time the protocol is drafted. It is commonly set at 5%.

Of equal importance for environmental epidemiology is a consideration of the probability that a failure to find an effect is a false negative, or type II error. This often occurs when small numbers of persons are studied and when relatively low risks are involved. Statistical tests cannot determine whether or not an error has been made but can indicate the probability that an error could occur, called beta, if the effect is of some hypothetical size specified by the investigator. The power to detect an effect of that size, defined as 1-beta, depends on the alpha level of significance testing and the unknown relative risk. Tables have been devised to help determine the number of observations required to have specified power to detect an effect of specified size if an association exists (Fleiss, 1981). For any specific size of effect, the power of a study increases as the study size increases.

Many episodes of environmental contamination involve low relative risks and small numbers of people, so environmental-epidemiology stud-



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