Epidemiologic studies routinely suffer from bias due to exposure measurement error. In this paper, the authors examine the effect of measurement error when the exposure variable of interest is constrained by a lower limit. This is an important consideration, since often in epidemiologic studies an exposure variable is constrained by a lower limit such as zero or a nonzero detection limit. In this paper, attenuation of exposure-disease associations is defined within the framework of a classical model of uncorrelated additive error. Then, the special case of nonlinearity due to the effect of a lower threshold is examined. A general model is developed to characterize the effect of random measurement error when there is a lower threshold for recorded values. Findings are illustrated under the assumption that the true exposure follows the lognormal and gamma distributions. The authors show that the direction and magnitude of bias in estimated exposure-response associations depends on the population distribution of the exposure, the magnitude of the recording threshold, the value assigned to below-threshold measurement results, and the variance in the measured exposure due to random measurement error.