Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2008 Spatiotemporal Bayesian Hierarchical Models, with Application to Birth Outcomes Jonathan D. (Jonathan David) Norton Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES SPATIOTEMPORAL BAYESIAN HIERARCHICAL MODELS, WITH APPLICATION TO BIRTH OUTCOMES By JONATHAN D. NORTON A Dissertation submitted to the Department of Statistics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Spring Semester, 2008 The members of the Committee approve the Dissertation of Jonathan D. Norton defended on November 16, 2007. Xufeng Niu Professor Directing Dissertation Isaac Eberstein Outside Committee Member Fred Huffer Committee Member Daniel McGee Committee Member The Office of Graduate Studies has verified and approved the above named committee members. ii This dissertation is dedicated to my wife Lin and my parents Steven and Nancy. iii ACKNOWLEDGEMENTS I would like to thank my advisor Xufeng Niu for his steady hand on the till as I developed my thesis topic. My other committee members also have my gratitude. A special mention goes to Fred Huffer for his theoretical insights. Despite his duties as the department chair, Dan McGee’s door was (literally) almost always open and he was full of advice on my career quandaries. Best of all, I didn’t feel obligated to take it. Isaac Eberstein shared his valuable experience with geographical data. I would also like to thank my fellow students for their companionship and support during my time at FSU. The most appreciation goes to my wife Lin, for providing emotional support and “putting me through”. I also thank her father, Prof. Dong Bao Gong of Xi’an Jiaotong University, for imparting a bit of his vast mathematical wisdom when I visited this summer. In the words of the XJTU motto, “When you drink water, think of the source”. iv TABLE OF CONTENTS List of Tables ...................................... vi List of Figures ..................................... viii Abstract ........................................ x 1. INTRODUCTION ................................. 1 1.1 Disease mapping ............................... 2 1.2 Ecological, Individual, and Contextual Effects ............... 4 2. SPATIAL AND SPATIOTEMPORAL MODELS ................ 8 2.1 Conditional Autoregressive (CAR) Models ................. 8 2.2 Intrinsically Autoregressive (IAR) Models ................. 15 2.3 Convolution Model .............................. 17 2.4 A Correlation Matrix View of CAR, IAR, and Convolution Models ... 19 2.5 Spatiotemporal Extensions .......................... 26 2.6 Models for Birth Outcomes ......................... 29 3. CRITERIA FOR MODEL SELECTION ..................... 32 3.1 Simulations .................................. 34 4. APPLICATION TO BIRTH OUTCOMES IN ARKANSAS .......... 49 4.1 Data ...................................... 49 4.2 Methods .................................... 49 4.3 Results for Preterm Birth .......................... 51 4.4 Results for Low Birth Weight ........................ 63 4.5 Summary of Arkansas results ........................ 67 5. CONCLUSIONS AND FUTURE WORK .................... 69 A. WINBUGS CODE ................................. 71 B. INSTITUTIONAL REVIEW BOARD APPROVAL LETTER ......... 74 REFERENCES ..................................... 76 BIOGRAPHICAL SKETCH ............................. 79 v LIST OF TABLES 3.1 Spatial correlation for simulated draws from hierarchical spatial models, 8 by 8 regular lattice .................................. 36 3.2 Contingency table for model with lowest DIC, τh = 1, τc = 1, n generated from exponential distribution with mean 1000, 8 by 8 regular lattice ..... 40 3.3 Contingency table for model with lowest DIC, IAR data, n exponential with mean 1000, 8 by 8 regular lattice. *One case was scored as a tie because the DIC values were equal 001. .......................... 42 ± 3.4 Contingency table for model with lowest DIC, Theta data, n exponential with mean 1000, 8 by 8 regular lattice ........................ 43 3.5 Contingency table for model with lowest DIC, Convolution data, n exponential with mean 1000, 8 by 8 regular lattice ..................... 44 3.6 Moran’s I for standardized residuals from fitting models to IAR (τc = 1) data, n exponential with mean 1000, 8 by 8 regular lattice ............. 46 3.7 Contingency table for model with lowest DIC, τc = 1, n exponential with mean 1000, 8 by 8 regular lattice. ........................ 46 3.8 Contingency table for model with lowest DIC, IAR data, n exponential with mean 1000, 16 by 16 regular lattice. ....................... 47 3.9 Contingency table for model with lowest DIC, Theta data, n exponential with mean 1000, 16 by 16 regular lattice. *One case was scored as a tie because the DIC values were equal 001. ........................ 47 ± 3.10 Contingency table for model with lowest DIC, Convolution data, n exponential with mean 1000, 16 by 16 regular lattice .................... 48 4.1 DIC for Preterm Birth models. Unless otherwise specified, φ is an IAR process. 52 4.2 Posterior means for random coefficients and τc ................ 53 4.3 DIC for spatiotemporal Preterm Birth models, no covariates ......... 57 4.4 DIC for spatiotemporal Preterm Birth models, with covariates ........ 58 vi 4.5 Coefficients from selected spatiotemporal models for preterm birth ...... 59 4.6 Odds ratio for preterm birth with 10% absolute increase in covariate, with credible interval .................................. 59 4.7 DIC for spatiotemporal LBW models, no covariates .............. 63 4.8 DIC for spatiotemporal LBW models, with covariates ............. 64 4.9 Coefficients from selected spatiotemporal model for LBW ........... 64 vii LIST OF FIGURES 2.1 Correlation matrix from uncentered CAR model on regular 8 by 8 lattice, ρ = .9 20 2.2 Correlation matrix from uncentered CAR model on regular 8 by 8 lattice, ρ = .99 ....................................... 21 2.3 Correlation matrix from uncentered CAR model on regular 8 by 8 lattice, ρ = .999 ...................................... 21 2.4 Correlation matrix from centered CAR model on regular 8 by 8 lattice, ρ = .9 22 2.5 Correlation matrix from centered CAR model on regular 8 by 8 lattice, ρ = .99 23 2.6 Correlation matrix from centered CAR model on regular 8 by 8 lattice, ρ = .999 23 2.7 Correlation matrix from IAR model on regular 8 by 8 lattice. ......... 24 2.8 Correlation matrix from IAR convolution model on regular 8 by 8 lattice. .. 25 3.1 Binomial random field, median p = .1, n = 1000, sampled from IAR with τ =1 37 3.2 Binomial random field, median p = .1, n = 1000, sampled from IAR with τ = 10 ....................................... 38 3.3 Probability of correct classification of Phi random field as function of τc, with 95% confidence interval ............................. 42 4.1 Log-odds of preterm birth (Phi + intercept), Late period, adjusted for rates of multiple birth, black motherhood, poverty, and smoking .......... 54 4.2 Raw log-odds of preterm birth, Late period ................... 55 4.3 Posterior distribution (Median, 2.5th percentile, 97.5th percentile) of baseline probability of preterm birth. ........................... 60 4.4 Phi plus intercept from spatiotemporal model for preterm birth, Periods 1-6 61 4.5 Model-based estimate of probability of preterm birth, Period 6 ........ 62 4.6 Phi from selected model for LBW, all periods ................. 65 viii 4.7 Model-based estimate of probability of LBW, Period 6 ............ 66 4.8 Map of Arkansas counties (Source: US Census Bureau) ............ 68 ix ABSTRACT A class of hierarchical Bayesian models is introduced for adverse birth outcomes such as preterm birth, which are assumed to follow a conditional binomial distribution. The log-odds of an adverse outcome in a particular county, logit(pi), follows a linear model which includes observed covariates and normally-distributed random effects. Spatial dependence between neighboring regions is allowed for by including an intrinsic autoregressive (IAR) prior or an IAR convolution prior in the linear predictor. Temporal dependence is incorporated by including a temporal IAR term also. It is shown that the variance parameters underlying these random effects (IAR, convolution, convolution plus temporal IAR) are identifiable. The same results are also shown to hold when the IAR is replaced by a conditional autoregressive (CAR) model. Furthermore, properties of the CAR parameter ρ are explored. The Deviance Information Criterion (DIC) is considered as a way to compare spatial hierarchical models. Simulations are performed to test whether the DIC can identify whether binomial outcomes come from an IAR, an IAR convolution, or independent normal deviates. Having established the theoretical foundations of the class of models and validated the DIC as a means of comparing models, we examine preterm birth and low birth weight counts in the state of Arkansas from 1994 to 2005. We find that preterm birth and low birth weight have different spatial patterns of risk, and that rates of low birth weight can be fit with a strikingly simple model that includes a constant spatial effect for all periods, a linear trend, and three covariates. It is also found that the risks of each outcome are increasing over time, even with adjustment for covariates. x CHAPTER 1 INTRODUCTION With the Markov-Chain Monte Carlo