Here are some major results obtained in this research project during the past year: 1. Motivated by a problem encountered in the analysis of cell cycle gene expression data, we developed a methodology for estimating parameters subject to order constraints on a unit circle. A normal eukaryotic cell cycle has four major phases during cell-division and a cell cycle gene has its peak expression (phase angle) during the phase that may correspond to its biological function. Since the phases are ordered along a circle, the phase angles of cell cycle genes are ordered unknown parameters on a unit circle. The problem of interest is to estimate the phase angles using the information regarding the order among them. We address this problem by developing a circular version of the well known isotonic regression for Euclidean data. Due to the underlying geometry, the standard pool adjacent violator algorithm (PAVA) can not be used for deriving the circular isotonic regression estimator (CIRE). However, PAVA can be modified to obtain a computationally efficient algorithm for deriving CIRE. We illustrate CIRE by estimating the phase angles of some of well known cell cycle genes using the unrestricted estimators obtained in Liu et al. (2004). 2. Microarray gene expression studies over ordered categories are routinely conducted to gain insights into biological functions of genes and the underlying biological processes. Some common experiments are time-course/dose-response experiments where a tissue or cell-line is exposed for different doses and/or durations of time to a chemical. A goal of such studies is to identify gene expression patterns/profiles over the ordered categories. This problem can be formulated as a multiple testing problem where for each gene the null hypothesis of no difference between the successive mean gene expressions are tested and further directional decisions are made if it is rejected. Much of the existing multiple testing procedures are devised for controlling the usual false discovery rate (FDR) rather than the mixed directional FDR, the expected proportion of Type I and directional errors among all rejections. Benjamini and Yekutieli (2005) proved that an augmentation of the usual Benjamini-Hochberg (BH) procedure can control the mixed directional FDR while testing simple null hypotheses against two-sided alternatives in terms of one dimensional parameters. In this article, we consider the problem of controlling the mixed directional FDR involving multidimensional parameters. To deal with this problem, we develop a procedure extending that of Benjamini and Yekutieli based on the Bonferroni test for each gene. A proof is given for its mixed directional FDR control when the underlying test statistics are independent across the genes. The results of a simulation study evaluating its performance under independence as well as under dependence of the underlying test statistics across the genes relative to other relevant procedures are reported. Finally, the proposed methodology is applied to a time-course microarray data obtained by Lobenhofer et al. (2002). We identified several important cell-cycle genes that were not identified by the previous analyses of the same data by Lobenhofer et al. (2002) and Peddada et al. (2003). Although some of our findings overlap with previous findings, we identify several other genes that compliment the results of Lobenhofer et al. (2002). 3. Researchers routinely use historical control data (HCD) when analyzing rodent carcinogenicity data obtained in a particular study. Although the concurrent control group is considered to be the most relevant group to compare with the dose groups, the HCD provides a broader perspective to assist in understanding the significance of the current study. The HCD is used to provide information about the incidences of spontaneous tumors and malignant systemic disorders such as lymphoma and leukemia. This article presents some possible ways of incorporating the HCD when analyzing data from a rodent cancer bioassay. Specifically, exploratory (informal) and formal statistical procedures for analyzing such data are reviewed. The boxplot is presented as an exploratory tool that describes the current data in the context of the distribution of the HCD. It will also identify potential outliers that would not be otherwise be flagged using standard methods such as the mean, standard deviation, and range.
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