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"It has been claimed and demonstrated that many of the conclusions drawn from biomedical research are probably false" (Button et al, 2013). ``Toth et al. (2010) make a common mistake of seeing structure where none exists" Chowdhury et al. (2015). "...the performance of their particular set of 150 probe-sets does not stand out compared to that of randomly sampled sets of 150 probe-sets from the same array” Jacob et al. (2016). These are statements that have emerged over the last few years in the biomedical and bioinformatics literature. Many of the studies in bioinformatics fall into the “small n, large p” category, where numerous statistical difficulties emerge. Rather than identifying studies that have false results, this thesis provides a unifying, theoretical and computational solution to two high dimensional problems that arise in these studies: the problem of variable selection-dimension reduction; and, that of sample size determination. We first consider these problems separately and from a theoretical viewpoint. Next we show how our methods can be applied to different applications in bioinformatics. For the first problem, we revisit and explain the concepts of central subspaces and conditional independence to develop a novel matrix decomposition - the ST matrix decomposition. This decomposition forms the basis of our two new computational methods that involve covariance matrices and covariance operators, the stMVN and the stKDR methods, respectively. For the second problem, we review the Kullback-Leibler divergence and use independent mixture models to develop a new formula for sample size determination for the high dimensional setting where p>>n. We then combine the solutions to these two problems to demonstrate through simple simulations, that when sample sizes are determined according to our formula, it is possible to detect the true predictors that are associated with a response variable. Finally, we show how our unifying solution can be applied in practi

When computing a confidence interval for a binomial proportion p one must choose between using an exact interval, which has a coverage probability of at least 1 a for all values of p, and a shorter approximate interval, which may have lower coverage for some p but that on average has coverage equal to 1 a. We investigate the cost of using the exact one and two-sided Clopper-Pearson confidence intervals rat her than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymptotic expansions, we also give a closed-form formula for determining the sample size for the exact Clopper-Pearson methods. For two-sided intervals, our investigation reveals an interesting connection between the frequentist Clopper-Pearson interval and Bayesian intervals based on noninformative priors.

A phase III clinical trial is a comparative study in which one assesses the relative efficacy of a treatment or treatments. It generally has one of the following objectives: 1. To determine the effectiveness of a new treatment relative to the natural history of the disease, for example when comparing a new treatment to not treatment or to placebo in an adjuvant setting. 2. To determine if a new treatment is more effective than the best current standard therapy (at the risk of increasing the toxicity). 3. To determine if a neu treatment is as effective as the best current standard therapy but is associated with less severe toxicity or a better quality of life (equivalence trial). Clinical trials must be properly designed in order to asnwer such questions with a high degree of certainty. The purpose of this paper is to present several concepts which must be taken into account during the process of designing a clinical trial.

This report documents progress in validating two empirical models of the distribution of total sky cover over lines and areas. The Burger Area Algorithm (BAA), as applied to sky cover, is a mathematical model of the probability that fraction C or less of area A is covered by clouds, given scale length r and mean clear pO (one minus the mean sky cover). The Burger Line Algorithm (BLA) is similar except it gives the probability of linear coverage rather than areal coverage. The validation procedure adopted relies on sky cover distributions over lines and areas of various sizes at three U.S. sites representative of several homogeneous climatic regions. The distributions will be obtained empirically from satellite imagery. The six tasks identified are: 1) Select test areas; 2) Determine sample sizes; 3) Acquire satellite imagery; 4) Develop cloud detection and analysis algorithms; 5) Create a sky cover database; and 6) Determine error bounds for the BAA and BLA models. This report discusses the output of Task 1 through 4 which are currently complete (April 1985). The methodology employed during Task 2 (Monte Carlo simulation of the validation process) is described in detail.