Analysis of recurrent event data with spatial random effects using a Bayesian approach.

The conventional Markov chain Monte Carlo (MCMC) method is limited to the selected shape and size of proposal distribution and is not easy to start when the initial proposal distribution is far away from the target distribution. To overcome these drawbacks of the conventional MCMC method, two useful improvements in MCMC method, adaptive Metropolis (AM) algorithm and delayed rejection (DR) algorithm, are attempted to be combined. The AM algorithm aims at adapting the proposal distribution by using the generated estimators, and the DR algorithm aims at enhancing the efficiency of the improved MCMC method. Based on the improved MCMC method, a Bayesian amplitude versus offset (AVO) inversion method on the basis of the exact Zoeppritz equation has been developed, with which the P- and S-wave velocities and the density can be obtained directly, and the uncertainty of AVO inversion results has been estimated as well. The study based on the logging data and the seismic data demonstrates the feasibility and robustness of the method and shows that all three parameters are well retrieved. So the exact Zoeppritz-based nonlinear inversion method by using the improved MCMC is not only suitable for reservoirs with strong-contrast interfaces and long-offset ranges but also it is more stable, accurate and anti-noise.

A proper choice of a proposal distribution for Markov chain Monte Carlo methods, for example for the Metropolis-Hastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis (AM) algorithm, where the Gaussian proposal distribution is updated along the process using the full information cumulated so far. Due to the adaptive nature of the process, the AM algorithm is non-Markovian, but we establish here that it has the correct ergodic properties. We also include the results of our numerical tests, which indicate that the AM algorithm competes well with traditional Metropolis-Hastings algorithms, and demonstrate that the AM algorithm is easy to use in practical computation.

Recurrent events are frequently encountered in biomedical studies. Evaluating the covariates effects on the marginal recurrent event rate is of practical interest. There are mainly two types of rate models for the recurrent event data: the multiplicative rates model and the additive rates model. We consider a more flexible additive-multiplicative rates model for analysis of recurrent event data, wherein some covariate effects are additive while others are multiplicative. We formulate estimating equations for estimating the regression parameters. The estimators for these regression parameters are shown to be consistent and asymptotically normally distributed under appropriate regularity conditions. Moreover, the estimator of the baseline mean function is proposed and its large sample properties are investigated. We also conduct simulation studies to evaluate the finite sample behavior of the proposed estimators. A medical study of patients with cystic fibrosis suffered from recurrent pulmonary exacerbations is provided for illustration of the proposed method.

Recurrent events, including cardiovascular events, are commonly observed in biomedical studies. Understanding the effects of various treatments on recurrent events and investigating the underlying mediation mechanisms by which treatments may reduce the frequency of recurrent events are crucial tasks for researchers. Although causal inference methods for recurrent event data have been proposed, they cannot be used to assess mediation. This study proposed a novel methodology of causal mediation analysis that accommodates recurrent outcomes of interest in a given individual. A formal definition of causal estimands (direct and indirect effects) within a counterfactual framework is given, and empirical expressions for these effects are identified. To estimate these effects, a semiparametric estimator with triple robustness against model misspecification was developed. The proposed methodology was demonstrated in a real-world application. The method was applied to measure the effects of two diabetes drugs on the recurrence of cardiovascular disease and to examine the mediating role of kidney function in this process.

This paper adopts a Bayesian spatial random effect modelling approach to analyse the risk of domestic burglary in Cambridgeshire, England, at the census output area level (OA). The model, in the form of Binomial spatial logistic regression, integrates offence and offender based theories and takes into account unknown local risk factors (represented as unexplained spatial autocorrelation in the model). A score of ‘proximity to offenders’ was calibrated for each OA based on the number of likely offenders in the county, the OAs they reside, and their proximities. Our results indicate that areas that have a score higher than the average score were at higher risks of being burgled. Household occupied by non-couple and economically inactivity are positively associated confounders. Household occupied by owner is a negatively associated confounder. These confounders diminish the effect of high score of proximity to offenders, which, however, remains positively associated with the risk of burglary. Bayesian spatial random effect modelling, which adds to the traditional (non-spatial) regression model a spatial random effect term, stabilizes estimated risks and remarkably improves model fit and causation inference. Mapping the results of spatial random effect reveals locations of high risk of burglary after controlling for offender and socioeconomic factors. Limitations of the study and strategies to deter burglaries based on the results of spatial random effect modelling are discussed.

Comparison of Multivariate Poisson lognormal spatial and temporal crash models to identify hot spots of intersections based on crash types

Grapham: Graphical models with adaptive random walk Metropolis algorithms

Chagas disease, caused by Trypanosoma cruzi, is still a public health problem in Latin America and in the Southern Cone countries, where Triatoma infestans is the main vector. We evaluated the relationships among the density of green vegetation around rural houses, sociodemographic characteristics, and domestic (re)infestation with T. infestans while accounting for their spatial dependence in the municipality of Pampa del Indio between 2007 and 2016. The study comprised sociodemographic and ecological variables from 734 rural houses with no missing data. Green vegetation density surrounding houses was estimated by the normalized difference vegetation index (NDVI). We used a hierarchical Bayesian logistic regression composed of fixed effects and spatial random effects to estimate domestic infestation risk and quantile regressions to evaluate the association between surrounding NDVI and selected sociodemographic variables. Qom ethnicity and the number of poultry were negatively associated with surrounding NDVI, whereas overcrowding was positively associated with surrounding NDVI. Hierarchical Bayesian models identified that domestic infestation was positively associated with surrounding NDVI, suitable walls for triatomines, and overcrowding over both intervention periods. Preintervention domestic infestation also was positively associated with Qom ethnicity. Models with spatial random effects performed better than models without spatial effects. The former identified geographic areas with a domestic infestation risk not accounted for by fixed-effect variables. Domestic infestation with T. infestans was associated with the density of green vegetation surrounding rural houses and social vulnerability over a decade of sustained vector control interventions. High density of green vegetation surrounding rural houses was associated with households with more vulnerable social conditions. Evaluation of domestic infestation risk should simultaneously consider social, landscape and spatial effects to control for their mutual dependency. Hierarchical Bayesian models provided a proficient methodology to identify areas for targeted triatomine and disease surveillance and control.

Environmental health studies are of great interest in research to evaluate the mortality-temperature relationship by adjusting spatially correlated random effects as well as identifying significant change points in temperature. However, this relationship is often not expressed using parametric models, which makes identifying change points an even more challenging problem. This paper proposes a unified semiparametric approach to simultaneously identify the nonlinear mortality-temperature relationship and detect spatially-dependent change points. A unified method is proposed for the model estimation, spatially dependent change points detection, and testing whether they are significant simultaneously by a permutation-based test. We operate under the assumption that change points remain constant, yet acknowledge the uncertainty regarding their precise number. These change points are influenced by the smoothing of an unknown function, which in turn relies on a smoothing variable and spatial random effects. Consequently, the detection of change points may be influenced by spatial effects. In this paper, several simulation studies are conducted to evaluate the performance of our proposed approach. The advantages of this unified approach are demonstrated using epidemiological data on mortality and temperature.

Malaria is a leading cause of morbidity and mortality in developing countries especially in rural areas where local resources are limited. Accurate disease forecasts can provide information to public and clinical health services to design targeted interventions for malaria control that make effective use of limited resources. Using verbal autopsy data, space-time model was used to forecast mortality due to malaria. The study used longitudinal data which were collected from Rufiji and Ifakara Health Demographic Surveillance System (HDSS) sites for the period of 1999 to 2011 and 2002 to 2012 respectively to assess models. The models included environmental factors and mosquito net ownership as predictor variables for mortality due to malaria. Deviance information criteria (DIC), logarithm score and root mean square error (RMSE) were used to assess the goodness of fit and forecasting accuracy of the models. The results indicate that the model included spatial and temporal random effect terms had small deviance information criteria, logarithm score and root mean square error. This model was the best model for forecasting and prediction of mortality due to malaria in both HDSS sites. In addition, mosquito net ownership and rainfall were significantly associated with mortality due to malaria. The model with spatial and temporal random effect terms is useful tool to provide reasonably reliable forecasts for mortality due to malaria. This might help to design appropriate strategies for targeting malaria control. On the other hand, including spatially and temporal varying random terms in the model is necessary and good strategy for modelling mortality due to malaria.

Few methods are currently available for group sequential analysis of recurrent events data subject to a terminal event in the clinical trial setting. This research helps fill this gap by developing a completely nonparametric group sequential monitoring procedure for use with the two-sample Tayob and Murray statistic. Advantages of the Tayob and Murray statistic include high power to detect treatment differences when there is correlation between recurrent event times or between recurrent and terminal events in an individual. This statistic does not suffer bias from dependent censoring, regardless of the correlation between event times in an individual. This manuscript briefly reviews the Tayob and Murray statistic, develops and describes how to use methods for its group sequential analysis, and through simulation, compares its operating characteristics with those of Cook and Lawless, which is currently in use as the only available nonparametric method for group sequential analysis of recurrent event data. The merits of our proposed approach are most clearly demonstrated when gap times between recurrent events are correlated; when gap times between events are independent, the Cook and Lawless method is difficult to beat. Simulations demonstrate that as correlation between recurrent event times grows, the reduction in power using the Cook and Lawless approach is substantial when compared to our method. Finally, we use our method to analyze recurrent acute exacerbation outcomes from the azithromycin in chronic obstructive pulmonary disease trial.

<p>The actual temperature profile is a determinant of ice rheology, which controls ice deformation and flow, and sliding over the underlying bedrock. Importantly, the ice flow in turn affects its temperature profile through strain heating, which makes observed temperature profiles a powerful input for ice sheet model validation.</p><p><span>Up to now temperature profile was available in few boreholes or from glaciological models. Recently, </span><span>Macelloni et al. (2016)</span><span> opened up new opportunities for probing ice </span><span>temperature</span><span> from space with the low-frequency passive sensors. </span><span>Indeed</span><span>, at L-band frequency, the very low absorption of ice and the low scattering by particles (grain size, bubbles in ice) allow a large penetration in the dry snow and ice (several hundreds of meters). Macelloni et al. (2019) performed the first retrieval of the ice sheet temperature in Antarctica by using the European Space Agency (ESA)’s Soil Moisture and Ocean Salinity (SMOS) L-band observations. The</span><span>y used </span><span>the</span><span> minimization of the difference between SMOS brightness temperature and microwave emission model simulations that include</span><span>s</span><span> a glaciological model.</span></p><p><span>Here, in the framework of the ESA 4D-Antarctica project, we propose a new method based on a Bayesian approach in order to improve the accuracy of the retrieved ice temperature and to provide an uncertainty estimation along the profiles. As a first step, a one-dimensional ice temperature profile model (Robin 1955) is used, which limits the retrieval to the Antarctic Plateau. Then, the new temperature emulator based on the three-dimensional glaciological GRISLI (Quiquet et al., 2018) will be used to enable retrievals over the entire continent (cf. Ritz’s presentation in this session for the GRISLI emulator </span><span>description</span><span>).</span></p><p><span>The Bayesian inference takes as free parameters: </span><span>ice thickness, </span><span>surface ice temperature, snow accumulation and geothermal heat flux (GHF). Their prior probability distribution is defined as normal, centered around a priori values taken from literature, and truncated to stay in a realistic range. The observed brightness temperature distribution is normal and a normal likelihood function is used to quantify the matching between the observed and simulated brightness temperature. The parameter space investigation is achieved through a Markov Chain Monte Carlo (MCMC) method. Here, the differential evolution adaptive Metropolis (DREAM) algorithm is used, which runs multiple different Markov chains in parallel and uses a discrete proposal distribution to evolve the sampler to the posterior distribution </span><span>(</span><span>Laloy and Vrugt, 2012).</span></p><p>For each SMOS brightness temperature observation, 1000 iterations are run on 5 parallel chains. The 2500 first iterations are discarded (aka. burn-in) and only the last 2500 are used for the final ice temperature profile estimation. The posterior probability distribution captures the most likely parameter set (i.e. a surface temperature, snow accumulation and GHF combination), and so, the most likely ice temperature profiles associated to this SMOS observation. It also provides the standard deviation which is an accurate estimate of the temperature uncertainty along the depth obtained with the method.</p>

Markov Chain Monte Carlo (MCMC) is widely applied to fit complex, practical models in various disciplines, such as engineering, physics, agriculture, etc. Indeed, when the target density $f$ is not available in closed form, a Markov chain is constructed with stationary density $f$ for forming Monte Carlo estimators of means with respect to $f$. Although a Harris ergodic Markov chain will converge to the target distribution as the Monte Carlo sample size tends to infinity, and the population means can be consistently estimated by the corresponding MCMC estimators, in practice, it is important to estimate the standard errors associated with these Monte Carlo estimates due to the availability of only a finite sample. Therefore, the geometric ergodicity of a Markov Chain is highly desired as it is the most standard method for guaranteeing a central limit theorem (CLT) for MCMC estimators and finding its standard errors. Among the different MCMC algorithms, Metropolis-Hastings (MH) algorithms are predominant. This thesis establishes conditions under which MH algorithms with a position-dependent proposal covariance matrix will or will not have the geometric convergence rate. Indeed, we provide these results for MH algorithms where the normal proposal density has a general mean function $c(x)$ and a covariance matrix $G(x)$ depending on the current position $x$ of the Markov chain. For modern variants of the popular Metropolis adjusted Langevin algorithms (MALA), like the manifold MALA (MMALA) that adapt to the geometry of the target distributions, the proposal covariance matrix changes in every iteration. Thus, we provide conditions for the geometric ergodicity of different variations of the Langevin algorithms, e.g., MMALA, position-dependent MALA (PMALA), and the pre-conditioned MALA (PCMALA). We also provide conditions guaranteeing geometric ergodicity of the pre-conditioned unadjusted Langevin algorithms (PCULA). Generalized linear mixed models (GLMMs), where the response variable is allowed to follow a non-Gaussian distribution, are popular statistical models, although their likelihood functions are not available in closed form. GLMMs can handle the over-dispersion often present in real-world data, and MCMC methods are generally used to fit frequentist and Bayesian GLMMs. Since the likelihood function for GLMMs is not available in closed form, Monte Carlo EM or Monte Carlo maximum likelihood methods are often used for making inferences from GLMMs. Simulation from the conditional (on the data) distribution of the random effects is, in turn, needed for implementing the Monte Carlo EM or Monte Carlo maximum likelihood methods. In this thesis, our general conditions for geometric ergodicity of variants of Langevin algorithms are also verified in the context of conditional simulation from the two most popular GLMMs, namely the binomial GLMM with the logit link and the Poisson GLMM with the log link. These results have important practical implications as users can now confidently fit these models by attaching asymptotically valid standard errors with their Markov chain-based Monte Carlo estimates. Spatial generalized linear mixed models (SGLMMs) are GLMMs where the random effects form a spatial process and are useful for modeling spatially correlated data. We compare the performance of different variants of MALA mentioned above as well as other popular MH algorithms like Random Walk Metropolis in the context of conditional simulation from SGLMMs. Next, we consider MCMC algorithms for full Bayesian analysis of GLMMs. Blocking and reparameterization are techniques used to improve the efficiency and convergence properties of MCMC algorithms. We combine these strategies with pre-conditioning to develop efficient Langevin algorithms for Bayesian GLMMs and SGLMMs. We illustrate our algorithms using extensive real and simulated data analyses.

The near-shore ocean ecosystem is influenced by catchment runoff. The Great Barrier Reef has been experiencing significant water quality deterioration over the past 150 years, due in part to agricultural intensification and urban settlement in adjacent catchments (Thorburn et al., 2013). There is a need for us to understand the influences of catchment characteristics on stream water quality, with an aim to mitigate and manage the water quality issue in the terrestrial stream runoff derived from the adjacent Great Barrier Reef catchments. The event-based water quality monitoring data set from the Paddock to Reef Integrated Monitoring Program across six Natural Management Regions provides a potential opportunity to develop a data-driven understanding of catchment characteristics affecting water quality at the catchment scale. This requires a robust and reliable modelling tool to relate the monitoring data to anthropogenic and natural processes. In this study, monitoring data of Total Suspended Solids (TSS) and dissolved Oxidised Nitrogen (NOX) from 32 sites across the Great Barrier Reef catchments are selected as case study constituents due to the high risk they pose to reef health when exported from catchments to the receiving marine environment. Also, these two constituents have distinct biogeochemical processes in catchments. Specifically, TSS is typically conserved (although mobilized and deposited) while travelling through river systems; while NOX can be potentially processed and removed from the system altogether. A Bayesian hierarchical linear modelling framework is adopted, due to its ability to borrow strength among sites, allowing information to be transferred across space, and due to its ability to provide uncertainty of the predictions. The Bayesian hierarchical linear regression model in this study is developed to evaluate the significance of various catchment characteristics (e.g., land uses, catchment topography and geology) on spatial variation in water quality. The main findings of this study are listed as follows, • Sites located in the Burdekin and Fitzroy Natural Management Regions tend to have greater TSS concentrations, illustrated by the modelled site-specific spatial random effects (deviation from the overall average concentration). Additionally, grazing and dry land agriculture land uses are positively correlated with the spatial random effect on TSS. The complexity of interaction between different catchment characteristics (e.g., land use and topography) can potentially result in a negative spatial random effect on TSS, which is reduced in the relatively steeper Great Barrier Reef catchments with a denser stream network. • Sugar cane is one of the most significant NOX contributors according to the modelling results, likely partially due to the excessive application of fertilizers; however, conservation land use has limited effect on NOX removal, indicating denitrification process alone may not be sufficient to remove NOX. The modelling results demonstrate different land management strategies are required for the purpose of reducing different constituents. This work will provide scientific insights for water quality management at the catchment scale.

My dissertation is motivated by the need for improved statistical approaches for understanding differences in cancer risk in rural vs. urban areas. To accomplish this goal, we develop three statistical methods that can be used to model and explain differences in age-adjusted rates between small areas with differing characteristics. These methods allow researchers to draw meaningful inferences from sparse spatial or spatio-temporal datasets at the ZIP code or county levels. Furthermore, they can be used to identify regions that might require public health interventions and to motivate costly data collection efforts at the individual level. The statistical methods we propose in this dissertation can be classified into two categories: estimation and mediation. Chapter 2 focuses on precisely estimating age-adjusted cancer mortality rates for small areas from spatio-temporal datasets with an excessive number of zeros. A large proportion of zeros are often present in datasets involving low-prevalence diseases and in datasets that include rural regions with small population sizes. When stratifying counts for each region and year by age group in the process of estimating age-adjusted rates, the proportion of zeros in the dataset is further inflated. We propose a novel Bayesian hierarchical hurdle model with spatial and temporal random effects for estimating age-adjusted rates in these settings. Via a simulation study and two data examples, we study the performance of this model and make recommendations as to when it should be used over a Bayesian hierarchical Poisson regression model for age-adjusted rates. Chapters 3 and 4 address the goal of mediation. Specifically, Chapter 3 focuses on a method for performing a mediation analysis and Chapter 4 introduces a method for performing a decomposition analysis. Both chapters aim to explain a difference in age-adjusted rates at the ZIP code level based on either a ZIP code-level exposure (mediation) or a fixed characteristic (decomposition). We develop flexible statistical methods that use Bayesian hierarchical models with spatial random effects and a Bayesian version of the g-computation technique from causal inference. Both methods were designed to create counterfactual small area estimates of age-adjusted rates within the analysis and consequently stable estimates of the effects of interest. Through a simulation study, we illustrate a high level of precision and minimal bias in the total, direct, and indirect effects when using our proposed mediation analysis method. We also compare our proposed mediation method to two methods used in the medical literature and show that our method exhibits the best trade-off of bias and variance. To demonstrate the performance of our decomposition method, we perform a data analysis to understand whether park access explains the disparity in age-adjusted colorectal cancer incidence rates in rural vs. urban ZIP codes in Iowa. We illustrate that precise inferences can be made from sparse ZIP code-level datasets when incorporating small area estimation techniques into each analysis.