## Mathematics Ph.D. Dissertations

#### Title

Dose-Response Analysis for Time-Dependent Efficacy

2016

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Statistics

Hanfeng Chen (Committee Member)

Junfeng Shang (Committee Member)

Vipaporn Phuntumart (Other)

#### Abstract

In dose-response studies, a critical research issue is to estimate the minimum effective dose (MED) and the Maximum Tolerated dose (MTD) of a drug. The problems of identifying the minimum effective dose and the maximum tolerated dose of a drug have been studied by many researchers when the endpoints are continuous and binary, and are measured at a particular time point. However, in recent dose-response related research, the responses are measured over a sequence of time points. In this situation, the previously developed procedures for the continuous and binary outcomes at a single time point are not applicable for the estimations of MED and the MTD of a drug when the longitudinal effect of the drug is taken into consideration.

In this dissertation, we developed statistical procedures to find the MED and the MTD of a drug when the responses are observed over a period of time at different dose levels. Since finding the time-dependent MED and MTD of a drug is a multiple comparison problem, we need to control the family-wise error rate, the probability of incorrectly declaring any ineffective doses of a drug as effective for MED (or any unsafe doses as safe for MTD) at a pre-specified level of significance (alpha) for the adjustment of multiplicity.

Two types of statistical procedures are developed to address the problem of time dependent MED (and MTD) in this dissertation. One type is with multiplicity adjustment such as the Bonferroni Correction method for MED (and for MTD, respectively). And another is without multiplicity adjustment such as the partitioning method for MTD (and for MTD, respectively).

In our study, we assumed that both the efficacy and the toxicity of a drug increase with the dose level over time. The consequence of this assumption is that if a dose is not declared as efficacious, then we stop checking the lower doses when evaluating efficacy (or if a dose is not declared as safe, we do not need to test the higher doses for toxicity investigation). In this dissertation, we used the partitioning principle to propose confidence-set based procedures for estimating the minimum effective dose (MED) and the maximum tolerated dose(MTD) of a drug when the responses are measured over time at different dose levels. The proposed procedures are compared by simulation studies, which cast new lights on the power performance of different innovative procedures proposed in this dissertation. We proved that the simultaneous confidence regions have the correct coverage probability 1 - alpha, and applied these procedures to analyze two real data sets. One is for the beetle killing effect on a plant based insecticide (Pyrethrurm); and another is for the hind-limb grip strength of rats under different levels of toxicity over time. The new confidence procedures proposed in this dissertation reveal new insights on the efficacy for insecticide over time, and neurotoxic effects on nervous system of rats over time. They also enhance the literature on statistical methodologies for time dependent dose-response research.

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