Interval estimation of the attributable risk in case-control studies with matched pairs

Objective—The attributable risk (AR), which represents the proportion of cases who can be preventable when we completely eliminate a risk factor in a population, is the most commonly used epidemiological index to assess the impact of controlling a selected risk factor on community health. The goal of this paper is to develop and search for good interval estimators of the AR for case-control studies with matched pairs. Methods—This paper considers five asymptotic interval estimators of the AR, including the interval estimator using Wald’s statistic suggested elsewhere, the two interval estimators using the logarithmic transformations: log(x) and log(1–x), the interval estimator using the logit transformation log(x/(1–x)), and the interval estimator derived from a simple quadratic equation developed in this paper. This paper compares the finite sample performance of these five interval estimators by calculation of their coverage probability and average length in a variety of situations. Results—This paper demonstrates that the interval estimator derived from the quadratic equation proposed here can not only consistently perform well with respect to the coverage probability, but also be more eYcient than the interval estimator using Wald’s statistic in almost all the situations considered here. This paper notes that although the interval estimator using the logarithmic transformation log(1–x) may also perform well with respect to the coverage probability, using this estimator is likely to be less eYcient than the interval estimator using Wald’s statistic. Finally, this paper notes that when both the underlying odds ratio (OR) and the prevalence of exposure (PE) in the case group are not large (OR <2 and PE <0.10), the application of the two interval estimators using the transformations log(x) and log(x/(1–x)) can be misleading. However, when both the underlying OR and PE in the case group are large (OR >4 and PE >0.50), the interval estimator using the logit transformation can actually outperform all the other estimators considered here in terms of eYciency. Conclusions—When there is no prior knowledge of the possible range for the underlying OR and PE, the interval estimator derived from the quadratic equation developed here for general use is recommended. When it is known that both the OR and PE in the case group are large (OR >4 and PE >0.50), it is recommended that the interval estimator using the logit transformation is used. (J Epidemiol Community Health 2001;55:885–890) To assess the public health importance of controlling a selected risk factor, the attributable risk (AR), which represents the proportion of cases who could be preventable if we completely eliminated this risk factor in a population, is probably one of the most commonly used epidemiological indices. When studying a rare disease in the presence of nuisance confounders, we may often use matched pair casecontrol study design to increase the eYciency. In fact, the estimation of the AR using the retrospective data has recently received intensive discussions. There are, however, only a few papers that discuss estimation of the AR in matched case-control studies. Whittemore included a brief discussion on estimation of the AR for frequency matching, but noted that her approach would not be appropriate for the matched pair study, in which each stratum consisted of only one case and one control. Using Wald’s statistic, Kuritz and Landis derived an asymptotic interval estimator of the AR. Kuritz and Landis further extended their result to the case of more than one matched control per case, but found that the coverage probability of their interval estimator might be less than the desired confidence level by >2% even when the number of matched pairs was as large as 100. The purpose of this paper is to search for other better alternative interval estimators of the AR to the one using Wald’s statistic for the matched pair case-control study. This paper considers five interval estimators of the AR, including the estimator using Wald’s statistic, the two interval estimators using the logarithmic transformation : log(x) and log(1–x), the interval estimator using the logit transformation: log(x/(1–x)), and the interval estimator derived from a simple quadratic equation developed here. To compare the finite sample performance of these estimators, this paper calculates the exact coverage probability and the average length in a variety of situations. Finally, this paper includes an example taken from a study of oral conjugated oestrogens and endometrial cancer 21 to illustrate the use of these interval estimators. Methods Consider a case-control study, in which we take a random sample of n subjects from the case J Epidemiol Community Health 2001;55:885–890 885 Department of Mathematical and Computer Sciences, College of Sciences, San Diego State University, San Diego, CA 92182-7720, USA