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...statistics. The idea of using a t-statistic with a Bayesian adjusted denominator was also proposed by Baldi and Long (2001) who developed the useful cyberT program. Their work was limited though to two-sample control versus treatment designs and their model did not distinguish between differentially and non-differentially expressed genes. They also did not develop consistent estimators for the hyperparameters. The degrees of freedom associated with the prior distribution of the variances was set to a default value while the prior variance was simply equated to locally pooled sample variances. Tusher et al (2001), Efron et al (2001) and Broberg (2003) have used t statistics with offset standard deviations. This is similar in principle to the moderated t-statistics used here but the offset t-statistics are not motivated by a model and do not have an associated distributional theory. Tusher et al (2001) estimated the offset by minimizing a coefficient of variation while Efron et al (2001) used a percentile of the distribution of sample standard deviations. Broberg (2003) considered the two sample problem and proposed a computationally intensive method of determining the offset by minimizing a combinatio...

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...a set of n microarrays yielding a response vector yTg = (yg1, . . . , ygn) for the gth gene. The responses will usually be log-ratios for twocolor data or log-intensities for single channel data, although other transformations are possible. The responses are assumed to be suitably normalized to remove dye-bias and other technological artifacts; see for example Huber et al (2002) or Smyth and Speed (2003). In the case of high density oligonucleotide array, the probes are assumed to have been normalized to produce an expression summary, represented here as ygi, for each gene on each array as in Li and Wong (2001) or Irizarry et al (2003). We assume that E(yg) = Xαg where X is a design matrix of full column rank and αg is a coefficient vector. We assume var(yg) = Wgσ 2 g where Wg is a known non-negative definite weight matrix. The vector yg may contain missing values and the matrix Wg may contain diagonal weights which are zero. Certain contrasts of the coefficients are assumed to be of biological interest and these are defined by βg = C T αg. We assume that it is of interest to test whether individual contrast values βgj are equal to zero. For example, with design (d) above the experimenter might want...

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...proved test statistics. In many gene discovery experiments for which microarrays are used the primary aim is to rank the genes in order of evidence against H0 rather than to assign absolute p-values (=-=Smyth et al, 2003-=-). This is because only a limited number of genes may be followed up for further study regardless of the number which are significant. Even when the above distributional assumptions fail for a given d...

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...alysis of microarray experiments for the amount of multiple testing, perhaps by controlling the familywise error rate or the false discovery rate, even though this reduces the power available to detect changes in expression for individual genes (Ge et al, 2002). On the other hand, the parallel nature of the inference in microarrays allows some compensating possibilities for borrowing information from the ensemble of genes which can assist in inference about each gene individually. One way that this can be done is through the application of Bayes or empirical Bayes methods (Efron, 2001, 2003). Efron et al (2001) used a non-parametric empirical Bayes approach for the analysis of factorial data with high density oligonucleotide microarray data. This approach has much potential but can be difficult to apply in practical situations especially by less experienced practitioners. Lonnstedt and Speed (2002), considering replicated two-color microarray experiments, took instead a parametric empirical Bayes approach using a simple mixture of normal models and a conjugate prior and derived a pleasingly simple expression for the posterior odds of differential expression for each gene. The posterior odds express...

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...ferential Expression Published by The Berkeley Electronic Press, 2004 the posterior odds of reducing the number of hyperparameters which need to estimated under the hierarchical model; in particular, knowledge of the non-null prior for the fold changes are not required. The moderated t-statistic is shown to follow a t-distribution with augmented degrees of freedom. The moderated t inferential approach extends to accommodate tests involving two or more contrasts through the use of moderated F -statistics. The idea of using a t-statistic with a Bayesian adjusted denominator was also proposed by Baldi and Long (2001) who developed the useful cyberT program. Their work was limited though to two-sample control versus treatment designs and their model did not distinguish between differentially and non-differentially expressed genes. They also did not develop consistent estimators for the hyperparameters. The degrees of freedom associated with the prior distribution of the variances was set to a default value while the prior variance was simply equated to locally pooled sample variances. Tusher et al (2001), Efron et al (2001) and Broberg (2003) have used t statistics with offset standard deviations. This is ...

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... yielding a response vector yTg = (yg1, . . . , ygn) for the gth gene. The responses will usually be log-ratios for twocolor data or log-intensities for single channel data, although other transformations are possible. The responses are assumed to be suitably normalized to remove dye-bias and other technological artifacts; see for example Huber et al (2002) or Smyth and Speed (2003). In the case of high density oligonucleotide array, the probes are assumed to have been normalized to produce an expression summary, represented here as ygi, for each gene on each array as in Li and Wong (2001) or Irizarry et al (2003). We assume that E(yg) = Xαg where X is a design matrix of full column rank and αg is a coefficient vector. We assume var(yg) = Wgσ 2 g where Wg is a known non-negative definite weight matrix. The vector yg may contain missing values and the matrix Wg may contain diagonal weights which are zero. Certain contrasts of the coefficients are assumed to be of biological interest and these are defined by βg = C T αg. We assume that it is of interest to test whether individual contrast values βgj are equal to zero. For example, with design (d) above the experimenter might want to make all the pairwise...

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...) give a review of test statistics for 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...

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...odel and do not have an associated distributional theory. Tusher et al (2001) estimated the offset by minimizing a coefficient of variation while Efron et al (2001) used a percentile of the distribution of sample standard deviations. Broberg (2003) considered the two sample problem and proposed a computationally intensive method of determining the offset by minimizing a combination of estimated false positive and false negative rates over a grid of significance levels and offsets. Cui and Churchill (2003) give a review of test statistics for 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 Lonn...

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...ations are absent. For such microarrays, design matrices can be formed exactly as in classical linear model practice from the biological factors underlying the experimental layout. In general we assume that we have a set of n microarrays yielding a response vector yTg = (yg1, . . . , ygn) for the gth gene. The responses will usually be log-ratios for twocolor data or log-intensities for single channel data, although other transformations are possible. The responses are assumed to be suitably normalized to remove dye-bias and other technological artifacts; see for example Huber et al (2002) or Smyth and Speed (2003). In the case of high density oligonucleotide array, the probes are assumed to have been normalized to produce an expression summary, represented here as ygi, for each gene on each array as in Li and Wong (2001) or Irizarry et al (2003). We assume that E(yg) = Xαg where X is a design matrix of full column rank and αg is a coefficient vector. We assume var(yg) = Wgσ 2 g where Wg is a known non-negative definite weight matrix. The vector yg may contain missing values and the matrix Wg may contain diagonal weights which are zero. Certain contrasts of the coefficients are assumed to be of biologic...

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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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... similar in principle to the moderated t-statistics used here but the offset t-statistics are not motivated by a model and do not have an associated distributional theory. Tusher et al (2001) estimated the offset by minimizing a coefficient of variation while Efron et al (2001) used a percentile of the distribution of sample standard deviations. Broberg (2003) considered the two sample problem and proposed a computationally intensive method of determining the offset by minimizing a combination of estimated false positive and false negative rates over a grid of significance levels and offsets. Cui and Churchill (2003) give a review of test statistics for 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 K...

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...test statistics for 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 c...

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...y. Tusher et al (2001) estimated the offset by minimizing a coefficient of variation while Efron et al (2001) used a percentile of the distribution of sample standard deviations. Broberg (2003) considered the two sample problem and proposed a computationally intensive method of determining the offset by minimizing a combination of estimated false positive and false negative rates over a grid of significance levels and offsets. Cui and Churchill (2003) give a review of test statistics for 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 linea...

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...n 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 observations on the sam...

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...proved test statistics. In many gene discovery experiments for which microarrays are used the primary aim is to rank the genes in order of evidence against H0 rather than to assign absolute p-values (=-=Smyth et al, 2003-=-). This is because only a limited number of genes may be followed up for further study regardless of the number which are significant. Even when the above distributional assumptions fail for a given d...

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...ownregulated as expected. Several of the other genes are closely related to ApoAI. The top eight genes here have been confirmed to be differentially expressed in the knockout versus the control line (=-=Callow et al, 2000-=-). For these data the top eight genes stand out clearly from the other genes and all methods clearly separate these genes from the 21sTable 4: Top 15 genes from the ApoAI data Annotation M-value Ord t...

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... Bgj. Even when not on the boundary, the estimator for pj is likely to be sensitive to the particular form of the prior distribution assumed for βgj and possibly also to dependence between the genes (=-=Ferkingstad et al, 2003-=-). A practical strategy to bypass these problems is to set the pj to values chosen by the user, perhaps pj = 0.01 or some other small value. Since v 1/2 0j σg is the standard deviation of the log-fold...

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...with a Bayesian adjusted denominator was also proposed by Baldi and Long (2001) who developed the useful cyberT program. Their work was limited though to two-sample control versus treatment designs and their model did not distinguish between differentially and non-differentially expressed genes. They also did not develop consistent estimators for the hyperparameters. The degrees of freedom associated with the prior distribution of the variances was set to a default value while the prior variance was simply equated to locally pooled sample variances. Tusher et al (2001), Efron et al (2001) and Broberg (2003) have used t statistics with offset standard deviations. This is similar in principle to the moderated t-statistics used here but the offset t-statistics are not motivated by a model and do not have an associated distributional theory. Tusher et al (2001) estimated the offset by minimizing a coefficient of variation while Efron et al (2001) used a percentile of the distribution of sample standard deviations. Broberg (2003) considered the two sample problem and proposed a computationally intensive method of determining the offset by minimizing a combination of estimated false positive and false...

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...e microarrays used in this experiment were printed with 8448 probes (spots) including 768 control spots. The hybridized microarrays were scanned with an Axon scanner and SPOT image analysis software (=-=Buckley, 2000-=-) was used to capture red and green intensities for each spot. The data was normalized using print-tip loess normalization and between arrays scale normalization using the LIMMA package (Smyth, 2003)....

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...sample problem and proposed a computationally intensive method of determining the offset by minimizing a combination of estimated false positive and false negative rates over a grid of significance levels and offsets. Cui and Churchill (2003) give a review of test statistics for 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 ...

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...xperiments. 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 observations on the same spot. Chu et al (2002...

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...in Table 2 are not affected by the prior limits on v0s 2 0 discussed in Section 6. 9 Data Examples 9.1 Swirl Consider the Swirl data set which is distributed as part of the marrayInput package for R (=-=Dudoit and Yang, 2003-=-). The experiment was carried out using zebrafish as a model organism to study the early development in vertebrates. Swirl is a point mutant in the BMP2 gene that affects the dorsal/ventral body axis....

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...proved test statistics. In many gene discovery experiments for which microarrays are used the primary aim is to rank the genes in order of evidence against H0 rather than to assign absolute p-values (=-=Smyth et al, 2003-=-). This is because only a limited number of genes may be followed up for further study regardless of the number which are significant. Even when the above distributional assumptions fail for a given d...

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...an associated distributional theory. Tusher et al (2001) estimated the offset by minimizing a coefficient of variation while Efron et al (2001) used a percentile of the distribution of sample standard deviations. Broberg (2003) considered the two sample problem and proposed a computationally intensive method of determining the offset by minimizing a combination of estimated false positive and false negative rates over a grid of significance levels and offsets. Cui and Churchill (2003) give a review of test statistics for 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 ...

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...istics and posterior odds, are implemented in the software package Limma for the R computing environment (Smyth et al, 2003). Limma is part of the Bioconductor project at http://www.bioconductor.org (=-=Gentleman et al, 2003-=-). The Limma software has been tested on a wide range of microarray data sets from many different facilities and has been used routinely at the author’s institution since the middle of 2002. Colophon ...