We investigated bias produced by incorrectly identifying the group with zero exposure prevalence. The observed difference in bias sensitivity between the two AF estimators was small and may not be of practical importance. Nevertheless, the bias in FIGURET (Eq 14) can be viewed as the upper limit of the (negative) bias in FIGUREL in the absence of confounding and effect modification. This is true unless errors in the estimated exposure prevalences reverse the direction of the estimated effect. Even small proportions of exposed subjects within the group regarded as truly unexposed produced severe bias of both AF estimates and decreased coverage of corresponding CIs. As highlighted by the empirical example, it is crucial when establishing exposure databases that the exposure prevalence is assessed also for groups in which the exposure is expected to be rare, and not regard these simply as zero.
When estimating the AF based on the exposure proportion of the cases, lacking representativeness of the case series may produce attenuations as well as exaggerations of the AF estimate even if the OR estimate is unbiased. 27 Confounding across groups can be adjusted for in partially ecologic settings if individual data on confounders are available. It should be noted, however, that the additive-relative OR model (Eq 13) provides a valid confounder adjustment only if each exposure probability x is constant across various levels of the confounders, ie, when there is no confounding within groups. Any residual confounding across or within groups may hide, alter, or even spuriously create a nonzero AF. However, the partially ecologic case-control design facilitates a more detailed grouping of the population, which may reduce such ecologic bias 43 and makes the design far more attractive than the traditional pure ecologic design.
Maria Albin and Timo Kauppinen gave access to the empirical dataset presented in the text.
Appendix A: Derivation of vâr(AFL)
Let Sca and Sco denote the sum of the observed exposure probabilities among the cases and controls, respectively, and let MATH
The mean exposure probability among the controls is MATHwhere n is the number of controls. The logit transformation of AFL (Eq 6) is MATH
Thus, by assuming that the control selection is such that is a valid estimate of the exposure prevalence JOURNAL/epide/04.02/00001648-200207000-00015/ENTITY_OV0335/v/2017-07-26T080015Z/r/image-png in the population, it follows that MATH
If lnˆ;β is an unbiased estimator of the true value lnβ and U( lnβ) is the efficient score evaluated at lnβ, then it can be shown that asymptotically, subject to certain regularity conditions, 31,44MATH
For binary regression models, 45 such as the linear OR model MATH
Thus, MATHand hence MATH
Using the delta method, 32 it follows that MATHand, furthermore, MATHand MATH
Accordingly, var(FIGUREL) can be estimated as MATH
Appendix B: The Distribution of the Exposure Probabilities Estimated from the Case Series
The exposure probability for the cases in group j, x ′j, satisfies 27MATHand MATHsuch that MATH
Thus, x ′j >xj if β > 0 and 0 <xj < 1, ie, when there is an harmful effect of exposure, cases are more likely to have been exposed than controls within the same group. As a result, the overall exposure prevalence among the cases is MATHwhere η′j is the proportion of cases that belongs to group j.
Under the linear OR model (Eq 4), and as a reasonable approximation under the additive-relative OR model (Eq 13), the group-specific proportions satisfy 12MATHwhich is equivalent with MATH
Given that ∑j = 0J − 1 ηj = 1, some algebraic manipulations yield MATH
Thus, ηj (j = 0, 1, ..., J − 1) and hence var(x) can be estimated on the basis of the observed distribution of the cases in the various groups.
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