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Analysis of dose-response microarray data using Bayesian Variable Selection (BVS) methods: Modeling and multiplicity adjustments 7th meeting of the Eastern Mediterranean Region of the International Biometric Society (EMR-IBS) Tel – Aviv 22/04 – 25/04,2013 Ziv Shkedy, Martin Otava, Adetayo Kasim and Dan Lin Center for Statistics (CenStat), Hasselt University, Belgium and Durham University, UK

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Research team Dan Lin. Ziv Shkedy. Martin Otava Luc Bijnens. Willem Talloen. Hinrich Gohlmann. Dhammika Amaratunga Hasselt University, BelgiumJohnson & Johnson Pharmaceutical Durham University, UK Adetyo Kasim. Imperial College, UK Bernet Kato.

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Overview Introduction to dose-response modeling in microarray experiments. Primary interest: selection of a subset of genes with significant monotone dose-response relationship. Focus: 1. Estimation and inference under order restrictions. 2. Multiplicity adjustment. Methodology: Bayesian Variable Selection models. 3

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Dose-response microarray experiment 4 Example of four genes. Different dose-response relationships. Primary Interest: detection of genes with monotone dose- response relationship 4 dose levels. 16988 genes.

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Estimation and inference under order restrictions 5 Primary interest: discovery of genes with monotone relationship with respect to dose. Order restricted inference. Simple order (=monotone) alternatives. 16988 null hypotheses to test

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Model formulation (1) Gene specific model One-way ANOVA with order restricted parameters. Simple order (monotone profiles). The order constraints are build into the specification of the prior distributions (Gelfand, Smith and Lee, 1992).

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Model formulation (1) 7 Specification of the prior : unconstrained prior. Likelihood:

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Model formulation (2) dose mean Re formulation of the mean structure: For a dose-response experiment with 4 dose levels (control + 3 doses):

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Example of one gene (13386) We fitted two monotone models: Equality constraints are replaced with a single parameter.

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Inference 10 dose mean Simple order alternative.

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All possible monotone dose-response models 11 We decompose the simple order alternative to all sub alternative. The null model Simple order alternative.

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All possible monotone dose-response models 12 4 dose levels:

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Bayesian variable selection: model formulation for order restricted model 13 The mean structure: included in the model not Included in the model Bayesian Variable Selection: a procedure of deciding which of the model parameters is equal to zero. Define an indicator variable:

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14 Bayesian variable selection: model formulation for order restricted model The mean structure for a candidate model: Order restrictionsVariable selection ESTIMATION INFERENCE and MODEL SELECTION

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The posterior probability of the null model 15 The posterior probability that the triplet equal to zero:

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Example: gene 3413 16 The highest posterior probability is obtained for the null model (0.514). Shrinkage through the overall mean. BVS

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Example: gene 13386 The highest posterior probability is obtained for model g5. Data do not support the null model.

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Multiplicity adjustment gene g is included in the discovery list gene g is not included in the discovery list The number of genes in the discovery list. Primary interest: discovery of subset of genes with monotone relationship with respect to dose.

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Multiplicity adjustment 19 The expected error rate for the list with all genes for which the posterior probability of the null model < 0.102 are included. τ

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Discussion & To Do list BVS methods: estimation and inference. Multiplicity adjustment is based on the posterior probability of the null model. Connection between BVS and MCT. Connection between BVS and Bayesian model averaging. BVS for order restricted but non monotone alternatives (umbrella alternatives/partial order alternatives). Posterior probabilities for the number of levels and the level probabilities for isotonic regressions.

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Thank you! 21

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