Master of Science / Department of Statistics / Christopher I. Vahl / Chemicals and drugs applied to animals used in meat production often have the potential to cause adverse effects on human consumers. To ensure safety, a withdrawal period, i.e. the minimum time allowed between application of the drug and entry of the animal into the food supply, must be determined for each drug used on food-producing animals. The withdrawal period is based on an upper tolerance limit at a given time point. It is not uncommon that the concentration of the drug in some tissue samples to be measured at a level below the limit of quantitation (LOQ). Because the measurement of the tissue concentration cannot be confidently determined with enough precision, these types of observations are often treated as if they were left censored where the censoring value is equal to the limit of quantitation. Several methods are commonly used in practice to deal with this situation. The simplest methods are either to exclude observations below the limit of quantitation or to replace those values with zero, LOQ or ½ LOQ. Previous studies have shown that these methods result in biased in estimation of the population mean and population variance. Alternatively, one could incorporate censoring into the likelihood and compute the maximum likelihood estimate (MLE) for the population mean and variance assuming a normal or lognormal distribution. These estimates are also biased but it has been shown that they are asymptotically unbiased. However, it is not clear yet how these various methods affect estimation of the upper tolerance limit, especially when the sample size is small, e.g. less than 35. In this report, we will examine the effects of substituting the LOQ or ½ LOQ for censored values as well as using the MLEs of the mean and variance in the construction of an upper tolerance limit for a normal population through simulation. Additionally, we propose a modified substitution method where observations below the LOQ are replaced by functions of the order statistics of non-censored observations under an assumption of symmetry. Its performance relative to the above methods will also be evaluated in the simulation study. In the end, the results from this study will be applied to an environmental study.

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.

對一個母體Ｆ（ｘ）來講，第Ｐ百分位數 就是使得Ｆ（Ｘｐ）＝Ｐ的點，而 的信 賴上、下限就對應到母體的容忍上、下限（Tolerance Limit Lower or Upper）ｒ＝ Ｐ（Ｕ Ｘｐ）or ＝Ｐ（Ｌ Ｘｒ）。古期的無母數作法是根據一個秩統計量（Or der Sta-＆ Koopman及Proschan分別提出不同的無母數方法，二者均是以秩統計量的 線性組合為主。唯Ｈ＆Ｋ的容忍上、下限限於Ψu、ΨL族群，而Ψu＝｛Ｆ：─log（ １－Ｆ）is convex），ΨL＝｛Ｆ：─log F is convex ）；Ｂ＆Ｐ的容忍下限作法 僅適是ＩＦＲ（Increasing Failure Rate） 而且Ｆ（０）＝０的族群。本論文以負 指數分配為母體在參合模式下，εＦ□＋（１－ε）Ｆ□來比較這兩種容忍上、上限 ，同時也覯察了若母體該是參合模式而誤採單一模式時所發生的偏差。 本論文計吊五章，第一章：緒論；第二章：古典的無母數方法；第三章：參合模式下 Hanson & Koopman的方法；第四章：參合模式下Barlow & Proschan 的方法；第五章 ：二種方法的分析比較。

Solving problems of production and calibration spindles water supply river armours by the help of method SPC. Analysis capability of gauge by the help of methodists MSA-Cg,Cgk. Evaluation capability of machanical abnd digital outside micrometer. Verification limiting deviation of micrometrical screw in among - calibration interval. Attestation capability of machine and of the process. Construction of regulation diagram. Project of new control instruments. Technically-economic evaluation:savings 110 560,00Kč, profitability 764,85%, economic return 0,13 year. Successful application of methodists SPC attested by document about adjustability of the process.