Statistical methods for estimating attributable risk from retrospective data

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Abstract
This paper extends Levin's measure of attributable risk1 to adjust for confounding by aetiologic factors other than the exposure of interest. One can estimate this extended measure from case‐control data provided either (i) from the control data one can estimate exposure prevalence within each stratum of the confounding factor; or (ii) one has additional information available concerning the confounder distribution and the stratum‐specific disease rates. In both cases we give maximum likelihood estimates and their estimated asymptotic variances, and show them to be independent of the sampling design (matched vs. random). Computer simulations investigate the behaviour of these estimates and of three types of confidence intervals when sample size is small relative to the number of confounder strata. The simulations indicate that attributable risk estimates tend to be too low. The bias is not serious except when exposure prevalence is high among controls. In this case the estimates and their standard error estimates are also highly unstable. In general, the asymptotic standard error estimates performed quite well, even in small samples, and even when the true asymptotic standard error was too small. By contrast, the bootstrap estimate2 tended to be too large. None of the three confidence intervals proved superior in accuracy to the other two. Thus there appears no advantage in using the log‐based interval suggested by Walter3′4 which is always longer than the simpler symmetric interval.