Doctor of Philosophy / Department of Statistics / Christopher Vahl / Veterinary drugs are used extensively for disease prevention and treatment in food producing animals. The residues of these drugs and their metabolites can pose risks for human health. Therefore, a withdrawal time is established to ensure consumer safety so that tissue, milk or eggs from treated animals cannot be harvested for human consumption until enough time has elapsed for the residue levels to decrease to safe concentrations. Part of the process to establish a withdrawal time involves a linear regression to model drug residue depletion over time. This regression model is used to calculate a one-sided, upper tolerance limit for the amount of drug residue remaining in target tissue as a function of time. The withdrawal period is then determined by finding the smallest time so that the upper tolerance limit falls below the maximum residue limit. Observations with measured residue levels at or below the limit of quantitation (LOQ) of the analytical method present a special challenge in the estimation of the tolerance limit. Because values observed below the LOQ are thought to be unreliable, they add in an additional source of uncertainty and, if dealt with improperly or ignored, can introduce bias in the estimation of the withdrawal time. The U.S. Food and Drug Administration (FDA) suggests excluding such data while the European Medicine Agency (EMA) recommends replacing observations below the LOQ with a fixed number, specifically half the value of the LOQ. However, observations below LOQ are technically left censored and these methods are do not effectively address this fact. As an alternative, a regression method accounting for left-censoring is proposed and implemented in order to adequately model residue depletion over time. Furthermore, a method based on generalized (or fiducial) inference is developed to compute a tolerance limit with results from the proposed regression method. A simulation study is then conducted to compare the proposed withdrawal time calculation procedure to the current FDA and EMA approaches. Finally, the proposed procedures are applied to real experimental data.