  The shape of the relationship between a continuous exposure variable and a
binary disease variable is often central to epidemiologic investigations. This
paper investigates a number of issues surrounding inference and the shape of
the relationship. Presuming that the relationship can be expressed in terms of
regression coefficients and a shape parameter, we investigate how well the
shape can be inferred in settings which might typify epidemiologic
investigations and risk assessment. We also consider a suitable definition of
the median effect of exposure, and investigate how precisely this can be
inferred. This is done both in the case of using a model acknowledging
uncertainty about the shape parameter and in the case of ignoring this
uncertainty and using a two-step method, where in step one we transform the
predictor and in step two we fit a simple linear model with transformed
predictor. All these investigations require a family of exposure-disease
relationships indexed by a shape parameter. For this purpose, we employ a
family based on the Box-Cox transformation.
