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...r differential expression for microarray experiments. Newton et al (2001), Newton and Kendziorski (2003) and Kendziorski et al (2003) have considered empirical Bayes models for expression based on gamma and log-normal distributions. Other authors have used Bayesian methods for other purposes in microarray data analysis. Ibrahim et al (2002) for example propose Bayesian models with correlated priors to model gene expression and to classify between normal and tumor tissues. Other approaches to linear models for microarray data analysis have been described by Kerr et al (2000), Jin et al (2001), Wolfinger et al (2001), Chu et al (2002), Yang and Speed (2003) and Lonnstedt et al (2003). Kerr et al (2000) propose a single linear model for an entire microarray experiment whereas in this paper a separate linear model is fitted for each gene. The single linear model approach assumes all equal variances across genes whereas the current paper is designed to accommodate different variances. Jin et al (2001) and Wolfinger et al (2001) fit separate models for each gene but model the individual channels of two color microarray data requiring the use of mixed linear models to accommodate the correlation between obser...

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...reen channels intensities separately, i.e., to analyze log-intensities instead of log-ratios. It is possible to do this using a mixed model representation which treats each spot as a randomized block =-=[48, 37]-=-. Limma implements mixed model methods for separate channel analysis which make use of shrinkage methods to ensure stable and reliable inference with small numbers of arrays [37]. Limma also provides ...

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...e from the multiplicity of genes and the possibilities for parallel inference. A standard analysis method is to fit the same statistical model separately to the expression measurements for each gene (=-=Wolfinger et al., 2001-=-; Yang and Speed, 2003). A number of authors have noted that inference for each individual gene can be made more reliable by making use of information generated from the whole ensemble of genes (Newto...

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...ormation before analyzing microarray data, investigators have, implicitly or explicitly, assumed a multiplicative measurement error model (Dudoit et al., 2002; Newton et al., 2001; Kerr et al., 2000; =-=Wolfinger et al., 2001-=-). The fact, seen in Figure 1, that observed 5intensity increases linearly with concentration in the original scale but not in the log-scale suggests that background noise is additive with non-zero m...

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... such as sampling from an effectively infinite population. We have used fixed effect models as a starting point, but agree with those who argue that random effects are more appropriate in some cases (=-=Wolfinger et al., 2000-=-). Our hesitation has been that the empirical distributions of parameter estimates and residuals we have seen have been decidedly non-normal, but standard methods for random effects assume underlying ...

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...ut many other estimation methods have been proposed. Chen et al [6] proposed iterative estimation of as part of one of the first proposed normalization procedures. Kerr et al [21] and Wolfinger et al =-=[22]-=- have proposed the use of ANOVA models for normalization. These methods are equivalent to subtracting a global constant as above. Global normalization is still the most widely used in spite of the evi...

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...ly distributed. By using a log-scale transformation before analyzing microarray data, a great number of investigators have, implicitly or explicitly, proposed a multiplicative measurement error model =-=[7, 19, 15, 20]-=- for microarray data. A slightly more complicated additive background multiplicative measurement error model has been proposed by, for example, [11, 6, 8]. In Figure 1a we see observed P M log (base 2...

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...d and scanned. There were 5776 probe spots on each array. Only the spots that passed a quality control of image analysis were used. Expression levels were normalized and generated using ANOVA models (=-=Wolfinger et al., 2001-=-). Probe spots with less than two valid expression numbers at any time point were further filtered out, leaving 2936 spots for differential expression analysis. The ANOVA F -statistic was used to test...

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...one will clearly be forced to compare samples tagged with different color dyes. METHODS AND RESULTS Our approach to design comparisons utilizes analysis of variance (ANOVA) models (Kerr et al., 2001; =-=Wolfinger et al., 2001-=-; Lee et al., 2000). For each gene, a separate ANOVA model is fit to the arrays; most effects in the model are not of interest, but are included because this automatically adjusts the estimates of int...