When will nondifferential misclassification of an exposure preserve the direction of a trend?

Dosemeci et al. (Am J Epidemiol 1990; 132:746-8) gave examples in which nondifferential misclassification of exposure reversed the direction of a trend. Gilbert (Am J Epidemiol 1991; 134:440-1) proposed that these examples occurred because the errors in exposure were systematic, and she pointed out that the relation between the measured and the true exposure was not monotonic. Assuming that the mean response either monotonically increases or decreases with the true exposure and that the exposure misclassification is nondifferential, the authors show that if the mean value of the measured exposure increases with the true exposure, then the direction of the trend cannot be reversed. Consequently, Gilbert's intimation that reversal of trend can only occur when errors are systematic is correct. However, the present authors' result is stronger in that even when errors in assessing exposure do include a systematic component, if monotonicity can be assumed, reversal of trend cannot occur. The weaker condition of positive correlation between the measured and true exposure is not sufficient to guarantee nonreversal of trend, as they show by example.