Ranked set sampling is not well-suited for the design of experiments primarily for two reasons; it requires larger number of experimental units, and the role of randomization is not well defined. To resolve these concerns, recently order restricted randomized (ORR) designs have been developed and its properties have been discussed in the literature.
Often there is information available in certain studies that traditional experimental designs are ill-equipped to use. This information is subjective in nature, but may be extremely useful to conduct the experiment. The order restricted randomized design (ORRD) takes advantage of this subjective information quite naturally in the experimental design setting. Sets of units are taken from the population of interest. Units in each set are judgment ranked from smallest to largest based on some concomitant variable, often derived from the subjective information. The judgment ranking process leads to a positive covariance structure among the within set units. This correlation structure along with a proper randomization is then used to reduce the variance of the estimate of the contrast of interest. This dissertation develops nonparametric statistical inference for linear models in the experimental design setting based on ORRD.
Chapter 1 provides a review and description of existing designs in the literature that are closely related to ORRD. We introduce two designs, Design 1 and Design 2, where Design 1 emphasizes all pairwise comparisons and Design 2 emphasizes all possible contrasts. Chapter 2 introduces the model in detail and develops the asymptotic distribution of the estimator. The ORR design for two treatments that minimizes the asymptotic variance of the contrast estimator is discussed in Section 2.7.
In Chapter 3, we develop testing procedures for general linear hypotheses. Tests include the score, drop and Wald tests. Chapter 4 discusses the simulation results to evaluate the empirical size and power of the tests. It is shown that, in general, Design 2 outperforms Design 1 under a wide range of judgment ranking information. Chapter 5 applies the proposed procedure to a clinical trial data set. Chapter 6 contains a brief summary of our work and provides a list of open problems for future study.
%It is shown that ORR design performs better than its competitor classical designs in linear models. We prove that the optimal allocation for a treatment versus control design using ORR is one where the treatment and control alternate. A simulation study shows that test and estimators performs reasonably well even for moderately large sample sizes.