Abstract: Linear regression models are often useful tools for exploring the relationship between a response and a set of explanatory (predictor) variables. When both the observed response and the predictor variables are contaminated/distorted by unknown functions of an observable confounder, inferring the underlying relationship between the latent (unobserved) variables is more challenging. Recently, S¸ent¨urk and M¨uller (2005) proposed the method of covariate-adjusted regression (CAR) analysis for this distorted data setting. In this paper, we describe graphical techniques for assessing departures from or violations of specific assumptions regarding the type and form of the data distortion. The type of data distortion consists of multiplicative, additive or no-distortion. The form of the distortion encompasses a class of general smooth distorting functions. However, common confounding adjustment methods in regression analysis implicitly make distortion assumptions, such as assuming additive or multiplicative linear distortions. We illustrate graphical detection of departures from such assumptions on the distortion. The graphical diagnostic techniques are illustrated with numeri cal and real data examples. The proposed graphical assessment of distortion assumptions is feasible due to the CAR estimation method, which utilizes a local regression technique to estimate a set of transformed distorting functions (S¸ent¨urk and Nguyen, 2006).
Abstract: The problem of detecting differential gene expression with mi croarray data has led to further innovative approaches to controlling false positives in multiple testing. False discovery rate (FDR) has been widely used as a measure of error in this multiple testing context. Direct estima tion of FDR was recently proposed by Storey (2002, Journal of the Royal Statistical Society, Series B 64, 479-498) as a substantially more powerful al ternative to the traditional sequential FDR controlling procedure, pioneered by Benjamini and Hochberg (1995, Journal of the Royal Statistical Society, Series B 57, 289-300). Direct estimation to FDR requires fixing a rejection region of interest and then conservatively estimating the associated FDR. On the other hand, sequential FDR procedure requires fixing a FDR control level and then estimating the rejection region. Thus, sequential and direct approaches to FDR control appear very different. In this paper, we intro duce a unified computational framework for sequential FDR methods and propose a class of more powerful sequential FDR algorithms, that link the direct and sequential approaches. Under the proposed unified compuational framework, both approaches simply approximate the least conservative (op timal) sequential FDR procedure. We illustrate the FDR algorithms and concepts with some numerical studies (simulations) and with two real ex ploratory DNA microarray studies, one on the detection of molecular signa tures in BRCA-mutation breast cancer patients and another on the detection of genetic signatures during colon cancer initiation and progression in the rat.