Bayesian quantile regression‐based nonlinear mixed‐effects joint models for time‐to‐event and longitudinal data with multiple features

This article explores Bayesian joint models for a quantile of longitudinal response, mismeasured covariate and event time outcome with an attempt to (i) characterize the entire conditional distribution of the response variable based on quantile regression that may be more robust to outliers and misspecification of error distribution; (ii) tailor accuracy from measurement error, evaluate non‐ignorable missing observations, and adjust departures from normality in covariate; and (iii) overcome shortages of confidence in specifying a time‐to‐event model. When statistical inference is carried out for a longitudinal data set with non‐central location, non‐linearity, non‐normality, measurement error, and missing values as well as event time with being interval censored, it is important to account for the simultaneous treatment of these data features in order to obtain more reliable and robust inferential results. Toward this end, we develop Bayesian joint modeling approach to simultaneously estimating all parameters in the three models: quantile regression‐based nonlinear mixed‐effects model for response using asymmetric Laplace distribution, linear mixed‐effects model with skew‐t distribution for mismeasured covariate in the presence of informative missingness and accelerated failure time model with unspecified nonparametric distribution for event time. We apply the proposed modeling approach to analyzing an AIDS clinical data set and conduct simulation studies to assess the performance of the proposed joint models and method. Copyright © 2016 John Wiley & Sons, Ltd.     

A spatiotemporal quantile regression model for emergency department expenditures

Motivated by a recent study of geographic and temporal trends in emergency department care, we develop a spatiotemporal quantile regression model for the analysis of emergency department‐related medical expenditures. The model yields distinct spatial patterns across time for each quantile of the response distribution, which is important in the spatial analysis of expenditures, as there is often little spatiotemporal variation in mean expenditures but more pronounced variation in the extremes. The model has a hierarchical structure incorporating patient‐level and region‐level predictors as well as spatiotemporal random effects. We model the random effects via intrinsic conditionally autoregressive priors, improving small‐area estimation through maximum spatiotemporal smoothing. We adopt a Bayesian modeling approach based on an asymmetric Laplace distribution and develop an efficient posterior sampling scheme that relies solely on conjugate full conditionals. We apply our model to data from the Duke support repository, a large georeferenced database containing health and financial data for Duke Health System patients residing in Durham County, North Carolina. Copyright © 2015 John Wiley & Sons, Ltd.     

Longitudinal quantile regression in the presence of informative dropout through longitudinal–survival joint modeling

We propose a joint model for a time‐to‐event outcome and a quantile of a continuous response repeatedly measured over time. The quantile and survival processes are associated via shared latent and manifest variables. Our joint model provides a flexible approach to handle informative dropout in quantile regression. A Monte Carlo expectation maximization strategy based on importance sampling is proposed, which is directly applicable under any distributional assumption for the longitudinal outcome and random effects. We consider both parametric and nonparametric assumptions for the baseline hazard. We illustrate through a simulation study and an application to an original data set about dilated cardiomyopathies. Copyright © 2014 John Wiley & Sons, Ltd.     

Bayesian epidemic models for spatially aggregated count data

Epidemic data often possess certain characteristics, such as the presence of many zeros, the spatial nature of the disease spread mechanism, environmental noise, serial correlation and dependence on time‐varying factors. This paper addresses these issues via suitable Bayesian modelling. In doing so, we utilize a general class of stochastic regression models appropriate for spatio‐temporal count data with an excess number of zeros. The developed regression framework does incorporate serial correlation and time‐varying covariates through an Ornstein–Uhlenbeck process formulation. In addition, we explore the effect of different priors, including default options and variations of mixtures of g‐priors. The effect of different distance kernels for the epidemic model component is investigated. We proceed by developing branching process‐based methods for testing scenarios for disease control, thus linking traditional epidemiological models with stochastic epidemic processes, useful in policy‐focused decision making. The approach is illustrated with an application to a sheep pox dataset from the Evros region, Greece. Copyright © 2017 John Wiley & Sons, Ltd.     

Classification of array CGH data using smoothed logistic regression model

Array comparative genomic hybridization (aCGH) provides a genome‐wide information of DNA copy number that is potentially useful for disease classification. One immediate problem is that the data contain many features (probes) but only a few samples. Existing approaches to overcome this problem include features selection, ridge regression and partial least squares. However, these methods typically ignore the spatial characteristic of aCGH data. To explicitly make use of this spatial information we develop a procedure called smoothed logistic regression (SLR) model. The procedure is based on a mixed logistic regression model, where the random component is a mixture distribution that controls smoothness and sparseness. Conceptually such a procedure is straightforward, but its implementation is complicated due to computational problems. We develop a fast and reliable iterative weighted least‐squares algorithm based on the singular value decomposition. Simulated data and two real data sets are used to illustrate the procedure. For real data sets, error rates are calculated using the leave‐one‐out cross validation procedure. For both simulated and real data examples, SLR achieves better misclassification error rates compared with previous methods. Copyright © 2009 John Wiley & Sons, Ltd.     

A simple significance test for quantile regression

Where OLS regression seeks to model the mean of a random variable as a function of observed variables, quantile regression seeks to model the quantiles of a random variable as functions of observed variables. Tests for the dependence of the quantiles of a random variable upon observed variables have only been developed through the use of computer resampling or based on asymptotic approximations resting on distributional assumptions. We propose an exceedingly simple but heretofore undocumented likelihood ratio test within a logistic regression framework to test the dependence of a quantile of a random variable upon observed variables. Simulated data sets are used to illustrate the rationale, ease, and utility of the hypothesis test. Simulations have been performed over a variety of situations to estimate the type I error rates and statistical power of the procedure. Results from this procedure are compared to (1) previously proposed asymptotic tests for quantile regression and (2) bootstrap techniques commonly used for quantile regression inference. Results show that this less computationally intense method has appropriate type I error control, which is not true for all competing procedures, comparable power to both previously proposed asymptotic tests and bootstrap techniques, and greater computational ease. We illustrate the approach using data from 779 adolescent boys age 12-18 from the Third National Health and Nutrition Examination Survey (NHANES III) to test hypotheses regarding age, ethnicity, and their interaction upon quantiles of waist circumference.     

Weighted quantile regression for analyzing health care cost data with missing covariates

Analysis of health care cost data is often complicated by a high level of skewness, heteroscedastic variances and the presence of missing data. Most of the existing literature on cost data analysis have been focused on modeling the conditional mean. In this paper, we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates. The weighted quantile regression estimator is consistent, unlike the naive estimator, and asymptotically normal. Furthermore, we propose a modified BIC for variable selection in quantile regression when the covariates are missing at random. The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data. The method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates. We investigate the weighted quantile regression procedure and the modified BIC via extensive simulations. We illustrate the application by analyzing a real data set from a health care cost study. Copyright © 2013 John Wiley & Sons, Ltd.     

Linear models of coregionalization for multivariate lattice data: a general framework for coregionalized multivariate CAR models

We present a general coregionalization framework for developing coregionalized multivariate Gaussian conditional autoregressive (cMCAR) models for Bayesian analysis of multivariate lattice data in general and multivariate disease mapping data in particular. This framework is inclusive of cMCARs that facilitate flexible modelling of spatially structured symmetric or asymmetric cross‐variable local interactions, allowing a wide range of separable or non‐separable covariance structures, and symmetric or asymmetric cross‐covariances, to be modelled. We present a brief overview of established univariate Gaussian conditional autoregressive (CAR) models for univariate lattice data and develop coregionalized multivariate extensions. Classes of cMCARs are presented by formulating precision structures. The resulting conditional properties of the multivariate spatial models are established, which cast new light on cMCARs with richly structured covariances and cross‐covariances of different spatial ranges. The related methods are illustrated via an in‐depth Bayesian analysis of a Minnesota county‐level cancer data set. We also bring a new dimension to the traditional enterprize of Bayesian disease mapping: estimating and mapping covariances and cross‐covariances of the underlying disease risks. Maps of covariances and cross‐covariances bring to light spatial characterizations of the cMCARs and inform on spatial risk associations between areas and diseases. Copyright © 2016 John Wiley & Sons, Ltd.     

The bivariate combined model for spatial data analysis

To describe the spatial distribution of diseases, a number of methods have been proposed to model relative risks within areas. Most models use Bayesian hierarchical methods, in which one models both spatially structured and unstructured extra‐Poisson variance present in the data. For modelling a single disease, the conditional autoregressive (CAR) convolution model has been very popular. More recently, a combined model was proposed that ‘combines’ ideas from the CAR convolution model and the well‐known Poisson‐gamma model. The combined model was shown to be a good alternative to the CAR convolution model when there was a large amount of uncorrelated extra‐variance in the data. Less solutions exist for modelling two diseases simultaneously or modelling a disease in two sub‐populations simultaneously. Furthermore, existing models are typically based on the CAR convolution model. In this paper, a bivariate version of the combined model is proposed in which the unstructured heterogeneity term is split up into terms that are shared and terms that are specific to the disease or subpopulation, while spatial dependency is introduced via a univariate or multivariate Markov random field. The proposed method is illustrated by analysis of disease data in Georgia (USA) and Limburg (Belgium) and in a simulation study. We conclude that the bivariate combined model constitutes an interesting model when two diseases are possibly correlated. As the choice of the preferred model differs between data sets, we suggest to use the new and existing modelling approaches together and to choose the best model via goodness‐of‐fit statistics. Copyright © 2016 John Wiley & Sons, Ltd.     

Quantile regression methods for reference growth charts

Estimation of reference growth curves for children's height and weight has traditionally relied on normal theory to construct families of quantile curves based on samples from the reference population. Age-specific parametric transformation has been used to significantly broaden the applicability of these normal theory methods. Non-parametric quantile regression methods offer a complementary strategy for estimating conditional quantile functions. We compare estimated reference curves for height using the penalized likelihood approach of Cole and Green with quantile regression curves based on data used for modern Finnish reference charts. An advantage of the quantile regression approach is that it is relatively easy to incorporate prior growth and other covariates into the analysis of longitudinal growth data. Quantile specific autoregressive models for unequally spaced measurements are introduced and their application to diagnostic screening is illustrated.     

On Bayesian shared component disease mapping and ecological regression with errors in covariates

Recent literature on Bayesian disease mapping presents shared component models (SCMs) for joint spatial modeling of two or more diseases with common risk factors. In this study, Bayesian hierarchical formulations of shared component disease mapping and ecological models are explored and developed in the context of ecological regression, taking into consideration errors in covariates. A review of multivariate disease mapping models (MultiVMs) such as the multivariate conditional autoregressive models that are also part of the more recent Bayesian disease mapping literature is presented. Some insights into the connections and distinctions between the SCM and MultiVM procedures are communicated. Important issues surrounding (appropriate) formulation of shared‐ and disease‐specific components, consideration/choice of spatial or non‐spatial random effects priors, and identification of model parameters in SCMs are explored and discussed in the context of spatial and ecological analysis of small area multivariate disease or health outcome rates and associated ecological risk factors. The methods are illustrated through an in‐depth analysis of four‐variate road traffic accident injury (RTAI) data: gender‐specific fatal and non‐fatal RTAI rates in 84 local health areas in British Columbia (Canada). Fully Bayesian inference via Markov chain Monte Carlo simulations is presented. Copyright © 2010 John Wiley & Sons, Ltd.     

Identifying significant gene‐environment interactions using a combination of screening testing and hierarchical false discovery rate control

Although gene‐environment (G× E) interactions play an important role in many biological systems, detecting these interactions within genome‐wide data can be challenging due to the loss in statistical power incurred by multiple hypothesis correction. To address the challenge of poor power and the limitations of existing multistage methods, we recently developed a screening‐testing approach for G× E interaction detection that combines elastic net penalized regression with joint estimation to support a single omnibus test for the presence of G× E interactions. In our original work on this technique, however, we did not assess type I error control or power and evaluated the method using just a single, small bladder cancer data set. In this paper, we extend the original method in two important directions and provide a more rigorous performance evaluation. First, we introduce a hierarchical false discovery rate approach to formally assess the significance of individual G× E interactions. Second, to support the analysis of truly genome‐wide data sets, we incorporate a score statistic‐based prescreening step to reduce the number of single nucleotide polymorphisms prior to fitting the first stage penalized regression model. To assess the statistical properties of our method, we compare the type I error rate and statistical power of our approach with competing techniques using both simple simulation designs as well as designs based on real disease architectures. Finally, we demonstrate the ability of our approach to identify biologically plausible SNP‐education interactions relative to Alzheimer's disease status using genome‐wide association study data from the Alzheimer's Disease Neuroimaging Initiative (ADNI).     

Mapping disability-adjusted life years: a Bayesian hierarchical model framework for burden of disease and injury assessment

This paper presents a Bayesian disability-adjusted life year (DALY) methodology for spatial and spatiotemporal analyses of disease and/or injury burden. A Bayesian disease mapping model framework, which blends together spatial modelling, shared-component modelling (SCM), temporal modelling, ecological modelling, and non-linear modelling, is developed for small-area DALY estimation and inference. In particular, we develop a model framework that enables SCM as well as multivariate CAR modelling of non-fatal and fatal disease or injury rates and facilitates spline smoothing for non-linear modelling of temporal rate and risk trends. Using British Columbia (Canada) hospital admission-separation data and vital statistics mortality data on non-fatal and fatal road traffic injuries to male population age 20-39 for year 1991-2000 and for 84 local health areas and 16 health service delivery areas, spatial and spatiotemporal estimation and inference on years of life lost due to premature death, years lived with disability, and DALYs are presented. Fully Bayesian estimation and inference, with Markov chain Monte Carlo implementation, are illustrated. We present a methodological framework within which the DALY and the Bayesian disease mapping methodologies interface and intersect. Its development brings the relative importance of premature mortality and disability into the assessment of community health and health needs in order to provide reliable information and evidence for community-based public health surveillance and evaluation, disease and injury prevention, and resource provision.     

Power and type I error results for a bias‐correction approach recently shown to provide accurate odds ratios of genetic variants for the secondary phenotypes associated with primary diseases

We recently proposed a bias correction approach to evaluate accurate estimation of the odds ratio (OR) of genetic variants associated with a secondary phenotype, in which the secondary phenotype is associated with the primary disease, based on the original case‐control data collected for the purpose of studying the primary disease. As reported in this communication, we further investigated the type I error probabilities and powers of the proposed approach, and compared the results to those obtained from logistic regression analysis (with or without adjustment for the primary disease status). We performed a simulation study based on a frequency‐matching case‐control study with respect to the secondary phenotype of interest. We examined the empirical distribution of the natural logarithm of the corrected OR obtained from the bias correction approach and found it to be normally distributed under the null hypothesis. On the basis of the simulation study results, we found that the logistic regression approaches that adjust or do not adjust for the primary disease status had low power for detecting secondary phenotype associated variants and highly inflated type I error probabilities, whereas our approach was more powerful for identifying the SNP‐secondary phenotype associations and had better‐controlled type I error probabilities. Genet. Epidemiol. 2011. © 2011 Wiley Periodicals, Inc. 35:739‐743, 2011     

The role of spatial representation in the development of a LUR model for Ottawa, Canada

A land use regression (LUR) model for the mapping of NO(2) concentrations in Ottawa, Canada was created based on data from 29 passive air quality samplers from the City of Ottawa's National Capital Air Quality Mapping Project and two permanent stations. Model sensitivity was assessed against three spatial representations of population: population at the dissemination area level, population at the dissemination block level and a dasymetrically derived population representation. A spatial database with land use, roads, population, zoning, greenspaces and elevation was created. Polycategorical zoning data were used in dasymetric mapping to spatially focus population data derived from the dissemination blocks to a sub-block level for comparison purposes. Dasymetric population mapping provided no significant LUR model improvement in explained variance when compared to block level population; however, both the former were significantly better than the dissemination area level population representations. However, where block level population is not available or too costly to acquire, our method using polycategorical zoning data provides a viable alternative in LUR modelling endeavours.     

Modelling the geographical distribution of co-infection risk from single-disease surveys

Background: The need to deliver interventions targeting multiple diseases in a cost‐effective manner calls for integrated disease control efforts. Consequently, maps are required that show where the risk of co‐infection is particularly high. Co‐infection risk is preferably estimated via Bayesian geostatistical multinomial modelling, using data from surveys screening for multiple infections simultaneously. However, only few surveys have collected this type of data. Methods: Bayesian geostatistical shared component models (allowing for covariates, disease‐specific and shared spatial and non‐spatial random effects) are proposed to model the geographical distribution and burden of co‐infection risk from single‐disease surveys. The ability of the models to capture co‐infection risk is assessed on simulated data sets based on multinomial distributions assuming light‐ and heavy‐dependent diseases, and a real data set of Schistosoma mansoni –hookworm co‐infection in the region of Man, Côte d'Ivoire. The data were restructured as if obtained from single‐disease surveys. The estimated results of co‐infection risk, together with independent and multinomial model results, were compared via different validation techniques. Results: The results showed that shared component models result in more accurate estimates of co‐infection risk than models assuming independence in settings of heavy‐dependent diseases. The shared spatial random effects are similar to the spatial co‐infection random effects of the multinomial model for heavy‐dependent data. Conclusions: In the absence of true co‐infection data geostatistical shared component models are able to estimate the spatial patterns and burden of co‐infection risk from single‐disease survey data, especially in settings of heavy‐dependent diseases. Copyright © 2011 John Wiley & Sons, Ltd.     

The quantile regression approach to efficiency measurement: insights from Monte Carlo simulations

In the health economics literature there is an ongoing debate over approaches used to estimate the efficiency of health systems at various levels, from the level of the individual hospital - or nursing home - up to that of the health system as a whole. The two most widely used approaches to evaluating the efficiency with which various units deliver care are non-parametric data envelopment analysis (DEA) and parametric stochastic frontier analysis (SFA). Productivity researchers tend to have very strong preferences over which methodology to use for efficiency estimation. In this paper, we use Monte Carlo simulation to compare the performance of DEA and SFA in terms of their ability to accurately estimate efficiency. We also evaluate quantile regression as a potential alternative approach. A Cobb-Douglas production function, random error terms and a technical inefficiency term with different distributions are used to calculate the observed output. The results, based on these experiments, suggest that neither DEA nor SFA can be regarded as clearly dominant, and that, depending on the quantile estimated, the quantile regression approach may be a useful addition to the armamentarium of methods for estimating technical efficiency.     

Meta‐regression with partial information on summary trial or patient characteristics

We present a model for meta‐regression in the presence of missing information on some of the study level covariates, obtaining inferences using Bayesian methods. In practice, when confronted with missing covariate data in a meta‐regression, it is common to carry out a complete case or available case analysis. We propose to use the full observed data, modelling the joint density as a factorization of a meta‐regression model and a conditional factorization of the density for the covariates. With the inclusion of several covariates, inter‐relations between these covariates are modelled. Under this joint likelihood‐based approach, it is shown that the lesser assumption of the covariates being Missing At Random is imposed, instead of the more usual Missing Completely At Random (MCAR) assumption. The model is easily programmable in WinBUGS, and we examine, through the analysis of two real data sets, sensitivity and robustness of results to the MCAR assumption. Copyright © 2010 John Wiley & Sons, Ltd.     

Clustered spatio‐temporal varying coefficient regression model

In regression analysis for spatio‐temporal data, identifying clusters of spatial units over time in a regression coefficient could provide insight into the unique relationship between a response and covariates in certain subdomains of space and time windows relative to the background in other parts of the spatial domain and the time period of interest. In this article, we propose a varying coefficient regression method for spatial data repeatedly sampled over time, with heterogeneity in regression coefficients across both space and over time. In particular, we extend a varying coefficient regression model for spatial‐only data to spatio‐temporal data with flexible temporal patterns. We consider the detection of a potential cylindrical cluster of regression coefficients based on testing whether the regression coefficient is the same or not over the entire spatial domain for each time point. For multiple clusters, we develop a sequential identification approach. We assess the power and identification of known clusters via a simulation study. Our proposed methodology is illustrated by the analysis of a cancer mortality dataset in the Southeast of the U.S.     

A conditional likelihood approach for regression analysis using biomarkers measured with batch‐specific error

Measurement error is common in epidemiological and biomedical studies. When biomarkers are measured in batches or groups, measurement error is potentially correlated within each batch or group. In regression analysis, most existing methods are not applicable in the presence of batch‐specific measurement error in predictors. We propose a robust conditional likelihood approach to account for batch‐specific error in predictors when batch effect is additive and the predominant source of error, which requires no assumptions on the distribution of measurement error. Although a regression model with batch as a categorical covariable yields the same parameter estimates as the proposed conditional likelihood approach for linear regression, this result does not hold in general for all generalized linear models, in particular, logistic regression. Our simulation studies show that the conditional likelihood approach achieves better finite sample performance than the regression calibration approach or a naive approach without adjustment for measurement error. In the case of logistic regression, our proposed approach is shown to also outperform the regression approach with batch as a categorical covariate. In addition, we also examine a ‘hybrid’ approach combining the conditional likelihood method and the regression calibration method, which is shown in simulations to achieve good performance in the presence of both batch‐specific and measurement‐specific errors. We illustrate our method by using data from a colorectal adenoma study. Copyright © 2012 John Wiley & Sons, Ltd.