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Bayesian Analysis of Fmri Data and RNA-Seq Time Course Experiment Data

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Bayesian Analysis of Fmri Data and RNA-Seq Time Course Experiment Data Bayesian Analysis of fMRI Data and RNA-Seq Time Course Experiment Data A Dissertation presented to the Faculty of the Graduate School at the University of Missouri In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by YUAN CHENG Advisor: Marco A. R. Ferreira December 2015 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled: Bayesian Analysis of fMRI Data and RNA-Seq Time Course Experiment Data presented by Yuan Cheng, a candidate for the degree of Doctor of Philosophy and hereby certify that, in their opinion, it is worthy of acceptance. Dr. Marco A. R. Ferreira Dr. Paul Speckman Dr. TieMing Ji Dr. Subharup Guha Dr. Jeff Rouder ACKNOWLEDGMENTS I would never have been able to finish my dissertation without the guidance of my advisor and my committee members, help from friends, and support from my family and husband. I would like to express my sincere gratitude and thanks to my advisor, Marco A. R. Ferreira, for introducing me to the research of Bayesian statistics and all the challenging yet interesting topics. Without his continuous encouragement and inspi- ration, all the work will never be possible. I also thank him for his kind help and wise guidance through out my PhD study, which make my research life here much easier and more enjoyable. I am grateful to Dr. Rouder for his help in conquering hard background materials of functional MRI and his suggestions in the development of this work. I extend my thanks to my committee members: Dr. Paul Speckman, Dr. Subharup Guha and Dr. Tieming Ji for their insightful comments and suggestions on this work. I would like to thank Dr. Shiqi Cui for his tremendous help in my research related to RNA-Seq Time Course experiments. I would also like to thank my parents. They were always supporting me and encouraging me with their best wishes. Finally, I would like to thank my husband, Yuelei Sui. He was always there cheering me up and stood by me through the good times and bad. ii TABLE OF CONTENTS ACKNOWLEDGMENTS . ii LIST OF TABLES . iv LIST OF FIGURES . .v ABSTRACT . vi CHAPTER 1 Introduction . .1 1.1 Introduction of Functional MRI . .2 1.1.1 Background . .4 1.2 Introduction of Time Course RNA-Seq Experiments . .9 1.2.1 Background . .9 Part I: Inference 2 A New Hemodynamic Response Function for Modeling Functional MRI Data . 13 2.1 Modeling The Hemodanymic Response Function . 14 2.1.1 fMRI Experiment Designs and Hemodanymic Response Function 14 2.1.2 Reviews on Hemodanymic Response Function Models . 15 2.2 New HRF Model . 18 2.2.1 General Linear Model and Assumptions . 18 2.2.2 The Triple Gamma Hemodynamic Response Function . 21 2.3 Inference . 24 iii 2.3.1 Mixture Priors for β ....................... 24 2.3.2 Model Selection on β by Zellner's-g prior . 25 2.3.3 The MCMC Algorithm for Estimating HRF Parameters . 27 3 Bayesian Model Selection . 32 3.1 Bayesian Model Selection and Nonlocal Priors . 32 3.2 Model Selection on β ........................... 33 3.2.1 Nonlocal Positive Truncated pMOM . 33 3.2.2 Bayesian Model Selection for β ................. 36 3.2.3 The MCMC Algorithm for Posterior Simulations . 41 3.3 Multiple Tests and FDR Control . 46 3.4 Simulation Study . 47 3.5 Real Data Applications . 50 3.5.1 A Preliminary Experiment . 51 3.5.2 Main Experiment . 53 4 Timing Brain Activation . 73 4.1 Model Selection Strategy for Timing Brain Activation . 74 4.1.1 Model Selection on β and c2 ................... 74 4.1.2 Model Selection Criteria . 80 4.1.3 Comparing DIC and Marginal Likelihood . 83 4.2 Testing the Model Selection Results . 92 4.2.1 Linear Model Fitting Exploring . 93 4.3 One Model Fitting Procedure of Selected Voxels . 95 iv 5 Bayesian Model Selection by Using Nonlocal Prior in Time Course RNA-seq Experiments . 107 5.1 Introduction . 107 5.2 Principal Component Regression Model . 109 5.3 Prior Specification . 111 5.4 Posterior Inference . 111 5.4.1 Full Conditionals and MCMC Algorithm . 111 5.5 Application to Vaccination Data . 115 5.5.1 Preprocessing and Empirical Priors . 115 5.5.2 Differentially expressed genes identification . 117 6 Summary . 139 BIBLIOGRAPHY . 141 VITA ...................................... 150 v LIST OF TABLES Table Page 3.1 Comparison MSE of γ .......................... 49 3.2 Comparison MSE of β .......................... 49 4.1 Different β Combined Models . 79 4.2 Model Selection Result 1 . 85 4.3 Model Selection Result 2 . 86 4.4 Model Selection Result 3 . 87 4.5 Linear Fixed Model Fitting Result . 94 4.6 Linear Mixed Model Fitting Result . 94 4.7 Estimated d3 of Visual and Motor of Subject 7 . 97 4.8 Estimated d3 of Visual and Motor of Subject 8 . 99 4.9 Estimated d3 of Visual and Motor of Subject 9 . 101 4.10 Estimated d3 of Visual and Motor of subject 10 . 103 4.11 Estimated d3 of Visual and Motor of subject 11 . 105 5.1 DE Genes for Vaccinated Subjects 1 . 133 5.2 DE Genes for Vaccinated Subjects 2 . 134 5.3 DE Genes for Vaccinated Subjects 3 . 135 vi 5.4 DE Genes for Non-vaccinated Subjects . 138 vii LIST OF FIGURES Figure Page 1.1 fMRI terminology . .6 1.2 fMRI experiment set-up and BOLD response data . .7 2.1 canonical HRF . 18 2.2 visual and motor HRF . 19 2.3 Linear Invariant System(convolution model) . 20 2.4 Triple Gamma HRF . 23 3.1 nonlocal pMOM . 35 3.11 BIC-scanner . 51 3.2 Noisy Synthetic Data . 56 3.3 Map of Posterior Probability of Alternative Model . 56 3.4 Simulated Left HRF . 57 3.5 Simulated MiddleBack HRF . 57 3.6 Simulated Right HRF . 57 3.7 standardized γ for canonical HRF GLM . 58 3.8 standardized β for canonical HRF GLM . 59 3.9 standardized γ for new method . 60 viii 3.10 standardized β for new method . 61 3.12 PCA Result on HRF parameters' estimations from angle 1 . 62 3.13 PCA Result on HRF parameters' estimations from angle 2 . 62 3.14 PCA Result on HRF parameters' estimations from angle 3 . 63 3.15 PCA Result on HRF parameters' estimations from angle 4 . 63 3.16 HRF Shapes Correspond to PCA Results . 64 3.17 First Session Histograms of d3 and c2 of Subject 7 . 65 3.18 3D Plot of First Session c2 of Subject 7 . 65 3.19 3D Plot of First Session d3 of Subject 7 . 65 3.20 Second Session Histograms of d3 and c2 of Subject 7 . 66 3.21 3D Plot of Second Session c2 of Subject 7 . 66 3.22 3D Plot of Second Session d3 of Subject 7 . 66 3.23 First Session Histograms of d3 and c2 of Subject 8 . 67 3.24 3D Plot of First Session c2 of Subject 8 . 67 3.25 3D Plot of First Session d3 of Subject 8 . 67 3.26 Second Session Histograms of d3 and c2 of Subject 8 . 68 3.27 3D Plot of Second Session c2 of Subject 8 . 68 3.28 3D Plot of Second Session d3 of Subject 8 . 68 3.29 First Session Histograms of d3 and c2 of Subject 9 . 69 3.30 3D Plot of First Session c2 of Subject 9 . 69 3.31 3D Plot of First Session d3 of Subject 9 . 69 3.32 Second Session Histograms of d3 and c2 of Subject 9 . 70 3.33 3D Plot of Second Session c2 of Subject 9 . 70 3.34 3D Plot of Second Session d3 of Subject 9 . 70 ix 3.35 First Session Histograms of d3 and c2 of Subject 10 . 71 3.36 3D Plot of First Session c2 of Subject 10 . 71 3.37 3D Plot of First Session d3 of Subject 10 . 71 3.38 Second Session Histograms of d3 and c2 of Subject 10 . 72 3.39 3D Plot of Second Session c2 of Subject 10 . 72 3.40 3D Plot of Second Session d3 of Subject 10 . 72 4.1 DIC Selection of Sub7 Session A . 88 4.2 BF Selection of Sub7 Session A . 88 4.3 DIC Selection of Sub7 Session B . 88 4.4 BF Selection of Sub7 Session B . 88 4.5 DIC Selection of Sub8 Session A . 89 4.6 BF Selection of Sub8 Session A . 89 4.7 DIC Selection of Sub8 Session B . 89 4.8 BF Selection of Sub8 Session B . 89 4.9 DIC Selection of Sub9 Session A . 90 4.10 BF Selection of Sub9 Session A . 90 4.11 DIC Selection of Sub9 Session B . 90 4.12 BF Selection of Sub9 Session B . 90 4.13 DIC Selection of Sub10 Session A . 91 4.14 BF Selection of Sub10 Session A . ..
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