Indiana University-Purdue University Indianapolis (IUPUI) / Cluster randomization is frequently used in clinical trials for convenience of inter ventional implementation and for reducing the risk of contamination. The opera tional convenience of cluster randomized trials, however, is gained at the expense of reduced analytical power. Compared to individually randomized studies, cluster randomized trials often have a much-reduced power. In this dissertation, I consider ways of enhancing analytical power with historical trial data. Specifically, I introduce a hierarchical Bayesian model that is designed to incorporate available information from previous trials of the same or similar interventions. Operationally, the amount of information gained from the previous trials is determined by a Kullback-Leibler divergence measure that quantifies the similarity, or lack thereof, between the histor ical and current trial data. More weight is given to the historical data if they more closely resemble the current trial data. Along this line, I examine the Type I error rates and analytical power associated with the proposed method, in comparison with the existing methods without utilizing the ancillary historical information. Similarly, to design a cluster randomized trial, one could estimate the power by simulating trial data and comparing them with the historical data from the published studies. Data analytical and power simulation methods are developed for more general situations of cluster randomized trials, with multiple arms and multiple types of data following the exponential family of distributions. An R package is developed for practical use of the methods in data analysis and trial design.

This research is motivated by an issue frequently encountered in environmental water quality evaluation. Many times, the sample size of water monitoring data is too small to have adequate power. Here, we present a Bayesian power prior approach by incorporating the current data and historical data and/or the data collected at neighboring stations to make stronger statistical inferences on the parameters of interest. The elicitation of power prior distributions is based on the availability of historical data, and is realized by raising the likelihood function of the historical data to a fractional power. The power prior Bayesian analysis has been proven to be a useful class of informative priors in Bayesian inference. In this dissertation, we propose a modified approach to constructing the joint power prior distribution for the parameter of interest and the power parameter. The power parameter, in this modified approach, quantifies the heterogeneity between current and historical data automatically, and hence controls the influence of historical data on the current study in a sensible way. In addition, the modified power prior needs little to ensure its propriety. The properties of the modified power prior and its posterior distribution are examined for the Bernoulli and normal populations. The modified and the original power prior approaches are compared empirically in terms of the mean squared error (MSE) of parameter estimates as well as the behavior of the power parameter. Furthermore, the extension of the modified power prior to multiple historical data sets is discussed, followed by its comparison with the random effects model. Several sets of water quality data are studied in this dissertation to illustrate the implementation of the modified power prior approach with normal and Bernoulli models. Since the power prior method uses information from sources other than current data, it has advantages in terms of power and estimation precision for decisions with small sample sizes, relative to methods that ignore prior information. / Ph. D.

The most common type of confirmatory trial is a randomised trial comparing the experimental treatment of interest to a control treatment. Confirmatory trials are expensive and take a lot of time in the planning, set up and recruitment of patients. Efficient methodology in clinical trial design is critical to save both time and money and allow treatments to become available to patients quickly. Often there are data available on the control treatment from a previous trial. These historical data are often used to design new trials, forming the basis of sample size calculations, but are not used in the analysis of the new trial. Incorporating historical control data into the design and analysis could potentially lead to more efficient trials. When the historical and current control data agree, incorporating historical control data could reduce the number of control patients required in the current trial and therefore the duration of the trial, or increase the precision of parameter estimates. However, when the historical and current data are inconsistent, there is a potential for biased treatment effect estimates, inflated type I error and reduced power. We propose two novel weights to assess agreement between the current and historical control data: a probability weight based on tail area probabilities; and a weight based on the equivalence of the historical and current control data parameters. For binary outcome data, agreement is assessed using the posterior distributions of the response probability in the historical and current control data. For normally distributed outcome data, agreement is assessed using the marginal posterior distributions of the difference in means and the ratio of the variances of the current and historical control data. We consider an adaptive design with an interim analysis. At the interim, the agreement between the historical and current control data is assessed using the probability or equivalence probability weight approach. The allocation ratio is adapted to randomise fewer patients to control when there is agreement and revert back to a standard trial design when there is disagreement. The final analysis is Bayesian utilising the analysis approach of the power prior with a fixed weight. The operating characteristics of the proposed design are explored and we show how the equivalence bounds can be chosen at the design stage of the current study to control the maximum inflation in type I error. We then consider a design where a treatment arm is added to an ongoing clinical trial. For many disease areas, there are often treatments in different stages of the development process. We consider the design of a two-arm parallel group trial where it is planned to add a new treatment arm during the trial. This could potentially save money, patients, time and resources. The addition of a treatment arm creates a multiple comparison problem. Dunnett (1955) proposed a design that controls the family-wise error rate when comparing multiple experimental treatments to control and determined the optimal allocation ratio. We have calculated the correlation between test statistics for the method proposed by Dunnett when a treatment arm is added during the trial and only concurrent controls are used for each treatment comparison. We propose an adaptive design where the sample size of all treatment arms are increased to control the family-wise error rate. We explore adapting the allocation ratio once the new treatment arm is added to maximise the overall power of the trial.