18%20Moderated-t%20Test%204_3_08

18%20Moderated-t%20Test%204_3_08 - Consider a CRD with Two...

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1 1 Introduction to Empirical Bayes Models and Moderated t-test Peng Liu 4/3/2008 2 Consider a CRD with Two Treatments 1 2 1 1 2 2 3 Measure Expression with Affy GeneChips 1 2 1 1 2 2 4 A Model for the Log Data from Gene j Treatment 1 observations i.i.d. Treatment 2 observations i.i.d. independent of Mean may be different for each combination of gene and treatment. Parameters: ( μ 1j , μ 2j , and σ j ) 2
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2 5 A Model for the Log Data from Gene j Treatment 1 observations i.i.d. Treatment 2 observations i.i.d. independent of Variance is assumed to be the same for both treatments within each gene, but the variance is allowed to change from gene to gene. 6 Testing for Differential Expression We wish to test for each gene j =1,2,. .., J. 7 Consider a Two-Sample t -Test for Each Gene mean of treatment 1 observations for gene j mean of treatment 2 observations for gene j pooled variance estimate of variance of trt 1 observations for gene j variance of trt 2 observations for gene j 8 Distribution of the t -Statistics under Our Model Assumptions ± Whenever H 0j is true (i.e., whenever gene j is EE), t j will have a t -distribution with d=n 1 +n 2 -2 degrees of freedom. ± Whenever H 0j is false (i.e., whenever gene j is DE), t j will have a non-central t -distribution with d=n 1 +n 2 -2 degrees of freedom and non- centrality parameter
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3 9 Distributions of test statistic under H 0 and H a H 0j is true μ 1j μ 2j = 0 σ j 2 =1 n 1 = n 2 =5 H 0j is false μ 1j μ 2j = 1 σ j 2 =1 n 1 = n 2 =5 t -statistic density 10 The above test uses “frequentist” idea ± The previous 2-sample t-test is based on the reasoning of “frequentist” approach. ± With this approach, the parameters ( μ 1j , μ 2j , and σ j ) are parameters that are unknown but fixed at some values. ± Inferences are done for those parameters based on our observed data. E.g., we either reject H 0 or accept H 0 based on the p-value that is calculated from our data under the assumed model. 2 11 Bayesian statistics ± Another branch of statistical inference uses the “Bayesian” idea. ± For the parameters such as σ j , instead of treating them as unknown fixed values, they are assumed to follow some kind of distribution. ± Then inferences are done based on both the data and the assumed distribution of the parameters. 2 12 Motivating vignette ± Suppose you are going to submit a manuscript to a journal whose acceptance rate is 10% for all submission. ± What would you expect of the result for your first
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18%20Moderated-t%20Test%204_3_08 - Consider a CRD with Two...

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