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Researchers sometimes argue that their exposure-measurement errors are independent of other errors and are nondifferential with respect to disease, resulting in estimation bias toward the null. Among well-known problems with such arguments are that independence and nondifferentiality are harder to satisfy than ordinarily appreciated (e.g., because of correlation of errors in questionnaire items, and because of uncontrolled covariate effects on error rates); small violations of independence or nondifferentiality may lead to bias away from the null; and, if exposure is polytomous, the bias produced by independent nondifferential error is not always toward the null. The authors add to this list by showing that, in a 2 × 2 table (for which independent nondifferential error produces bias toward the null), accounting for independent nondifferential error does not reduce the p value even though it increases the point estimate. Thus, such accounting should not increase certainty that an association is present.