Interval estimation of the attributable risk for multiple exposure levels in case-control studies with confounders

The attributable risk (AR) is one of the most important and commonly-used epidemiological indices to assess the public health importance of an association between a risk factor and a disease. When the underlying risk factor has multiple exposure levels in the presence of confounders, we consider the case-control studies using random sampling to collect the cases and controls here. We develop four asymptotic interval estimators for AR, including the interval estimator using Wald's statistic, the interval estimator using the logarithmic transformation, the interval estimator using the logit transformation, and the interval estimator derived from a quadratic equation. We apply Monte Carlo simulation to evaluate the finite-sample performance of these interval estimators in a variety of situations. We demonstrate that given an adequately large sample size, all the estimators developed here can actually perform reasonably well. We note that the interval estimator using the logit transformation may be of limited use when the number of studied subjects is not large. We also note that the interval estimator using the logarithmic transformation can lose efficiency compared to the interval estimator using Wald's statistic or the interval estimator derived from a quadratic equation developed in this paper. Finally, we use the data taken from a case control study of the oral contraceptive use in myocardial infarction patients with various smoking levels to illustrate he practical usefulness of these estimators.