peated many times, each empirical pseudorandom variable provides information on the standard error or other parameters of interest. In many settings this approach can overcome analytic intractability or uncertain distributional assumptions by massive computer calculation (DiCiccio and Romano, 1988, 1990; Laird and Louis, 1989). Particular success has been achieved for various forms of regression models (Faraway, 1990; Huet et al., 1990), including generalized linear models (Mapleson, 1986; Rothe, 1989; Moulton and Zeger, 1991). Specific bootstrap applications have been developed for survival analyses and mortality estimation (Wahrendorf et al., 1987), longitudinal studies with repeated measurements (Moulton and Zeger, 1989), teratogenicity (Carr and Portier, 1992), and human genetic studies that include investigation of gene-environment interactions (Konigsberg et al., 1991).

Analysis of Data When the Shape of the Dose-Response Relation is Unknown

Traditionally, toxicology has focused on finding the lowest-observed-effects levels (LOELs) and no-observed-effects levels (NOELs). Less emphasis has been placed on identifying the shape of the dose-response curve; typical experiments have compared 3 or 4 doses with a control regimen. In examining data from such studies, unsophisticated analytic techniques may not detect or may not properly define the true dose-response relation. Rothman (1986) cites an example of the relation between water-chlorination levels and brain cancer. Despite a monotonically increasing trend in outcome, an argument was made for a lack of association, against the evidence of simple visual inspection. Analysis was done by using pairwise t-tests against the lowest category, rather than a test for trend. The data were interpreted as suggesting a threshold between the third and fourth exposure categories. Nonparametric smoothing provides an estimate of the expected value of an outcome as a function of the exposure variable of interest, without making a priori judgments on the shape of the dose-response relation. This method can show indications of thresholds and nonlinearities in the dose-response curve. Alternating-conditional-expectations (ACE) regression models and generalized additive models are also valuable for examining dose-response relations. These models make fewer explicit assumptions about functional form than do linear models, for example, and hence their results are less likely to be biased. The price for this reduction in bias is a reduction in sensitivity to detect real effects.

Where data are not necessarily linear and the investigator has some notion about the shape of the underlying relation, parametric nonlinear models can be used. Examples are periodic regression (e.g., fitting sine,

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