Bayesian P-splines Research Articles

A Bayesian Joint Model of Longitudinal Kidney Disease Progression, Recurrent Cardiovascular Events, and Terminal Event in Patients with Chronic Kidney Disease

AbstractNearly 15% (37 million) of adults in the United States (US) have chronic kidney disease (CKD). The longitudinal decline of kidney function is intricately related to the development of cardiovascular disease (CVD) and eventual “terminal” event (kidney failure and mortality) in patients with CKD. Understanding the mechanism and risk factors underlying the three key outcome processes, (1) CKD progression, (2) CVD, and (3) subsequent terminal event in the CKD patient population remains incomplete. Thus, in this work, we develop a novel trivariate joint model to study the risk factors associated with the interdependent outcomes of kidney function (as measured by longitudinal estimated glomerular filtration rate), recurrent cardiovascular events, and the terminal event. Efficient estimation and inference is proposed within a Bayesian framework using Markov Chain Monte Carlo and Bayesian P-splines for hazard functions. The proposed Bayesian framework is directly generalizable beyond trivariate outcome processes to accommodate other potential modeling of complex multi-disease processes. The method is applied to study the aforementioned trivariate processes using data from the Chronic Renal Insufficiency Cohort Study, an ongoing prospective cohort study, established by the National Institute of Diabetes and Digestive and Kidney Diseases to address the rising epidemic of CKD in the US.

Applying parametric modelling to generate the ecosystem path of innovation and entrepreneurship education in applied undergraduate colleges and universities

Abstract This paper firstly researches the generation path of innovation and entrepreneurship education ecosystem in applied undergraduate universities from the four aspects of education ecosystem, teacher team, conceptual change and system mechanism. Then, it constructs the ecological index system of innovation and entrepreneurship education and transforms the model into a parametric model by modeling the mean value parameter and dispersion parameter and approximating the smooth function of the model by using the Bayesian P-spline method. Finally, University H is taken as an example to evaluate and test the innovation and entrepreneurship education system based on the parametric model. The results show that education and teaching occupy a large proportion of the ecosystem, indicating that by actively initiating or participating in the innovation and entrepreneurship education ecology, applied undergraduate colleges and universities can rely on social capital endeavors to prompt a constant input of resources into the ecosystem, thus maintaining the benign and sustainable development of the innovation and entrepreneurship education ecosystem.

Functional and variables selection in extreme value models for regional flood frequency analysis

The problem of estimating return levels of river discharge, relevant in flood frequency analysis, is tackled by relying on the extreme value theory. The Generalized Extreme Value (GEV) distribution is assumed to model annual maxima values of river discharge registered at multiple gauging stations belonging to the same river basin. The specific features of the data from the Upper Danube basin drive the definition of the proposed statistical model. Firstly, Bayesian P-splines are considered to account for the non-linear effects of station-specific covariates on the GEV parameters. Secondly, the problem of functional and variable selection is addressed by imposing a grouped horseshoe prior to the coefficients to encourage the shrinkage of non-relevant components to zero. A cross-validation study is organized to compare the proposed modeling solution to other models, showing its potential to reduce the uncertainty of the ungauged predictions without affecting their calibration.

Penalty parameter selection and asymmetry corrections to Laplace approximations in Bayesian P-splines models

Laplace P-splines (LPS) combine the P-splines smoother and the Laplace approximation in a unifying framework for fast and flexible inference under the Bayesian paradigm. The Gaussian Markov random field prior assumed for penalized parameters and the Bernstein-von Mises theorem typically ensure a razor-sharp accuracy of the Laplace approximation to the posterior distribution of these quantities. This accuracy can be seriously compromised for some unpenalized parameters, especially when the information synthesized by the prior and the likelihood is sparse. Therefore, we propose a refined version of the LPS methodology by splitting the parameter space in two subsets. The first set involves parameters for which the joint posterior distribution is approached from a non-Gaussian perspective with an approximation scheme tailored to capture asymmetric patterns, while the posterior distribution for the penalized parameters in the complementary set undergoes the LPS treatment with Laplace approximations. As such, the dichotomization of the parameter space provides the necessary structure for a separate treatment of model parameters, yielding improved estimation accuracy as compared to a setting where posterior quantities are uniformly handled with Laplace. In addition, the proposed enriched version of LPS remains entirely sampling-free, so that it operates at a computing speed that is far from reach to any existing Markov chain Monte Carlo approach. The methodology is illustrated on the additive proportional odds model with an application on ordinal survey data.

Regularised B-splines Projected Gaussian Process Priors to Estimate Time-trends in Age-specific COVID-19 Deaths

The COVID-19 pandemic has caused severe public health consequences in the United States. In this study, we use a hierarchical Bayesian model to estimate the age-specific COVID-19 attributable deaths over time in the United States. The model is specified by a novel non-parametric spatial approach over time and age, a low-rank Gaussian Process (GP) projected by regularised B-splines. We show that this projection defines a GP with attractive smoothness and computational efficiency properties, derive its kernel function, and discuss the penalty terms induced by the projected GP. Simulation analyses and benchmark results show that the B-splines projected GP may perform better than standard B-splines and Bayesian P-splines, and equivalently well as a standard GP at considerably lower runtimes. We apply the model to weekly, age-stratified COVID-19 attributable deaths reported by the US Centers for Disease Control, which are subject to censoring and reporting biases. Using the B-splines projected GP, we can estimate longitudinal trends in COVID-19 associated deaths across the US by 1-year age bands. These estimates are instrumental to calculate age-specific mortality rates, describe variation in age-specific deaths across the US, and for fitting epidemic models. Here, we couple the model with age-specific vaccination rates to show that vaccination rates were significantly associated with the magnitude of resurgences in COVID-19 deaths during the summer 2021. With counterfactual analyses, we quantify the avoided COVID-19 deaths under lower vaccination rates and avoidable COVID-19 deaths under higher vaccination rates. The B-splines projected GP priors that we develop are likely an appealing addition to the arsenal of Bayesian regularising priors.

Dynamics of SARS-CoV-2 infection hospitalisation and infection fatality ratios over 23 months in England.

The relationship between prevalence of infection and severe outcomes such as hospitalisation and death changed over the course of the COVID-19 pandemic. Reliable estimates of the infection fatality ratio (IFR) and infection hospitalisation ratio (IHR) along with the time-delay between infection and hospitalisation/death can inform forecasts of the numbers/timing of severe outcomes and allow healthcare services to better prepare for periods of increased demand. The REal-time Assessment of Community Transmission-1 (REACT-1) study estimated swab positivity for Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) infection in England approximately monthly from May 2020 to March 2022. Here, we analyse the changing relationship between prevalence of swab positivity and the IFR and IHR over this period in England, using publicly available data for the daily number of deaths and hospitalisations, REACT-1 swab positivity data, time-delay models, and Bayesian P-spline models. We analyse data for all age groups together, as well as in 2 subgroups: those aged 65 and over and those aged 64 and under. Additionally, we analysed the relationship between swab positivity and daily case numbers to estimate the case ascertainment rate of England's mass testing programme. During 2020, we estimated the IFR to be 0.67% and the IHR to be 2.6%. By late 2021/early 2022, the IFR and IHR had both decreased to 0.097% and 0.76%, respectively. The average case ascertainment rate over the entire duration of the study was estimated to be 36.1%, but there was some significant variation in continuous estimates of the case ascertainment rate. Continuous estimates of the IFR and IHR of the virus were observed to increase during the periods of Alpha and Delta's emergence. During periods of vaccination rollout, and the emergence of the Omicron variant, the IFR and IHR decreased. During 2020, we estimated a time-lag of 19 days between hospitalisation and swab positivity, and 26 days between deaths and swab positivity. By late 2021/early 2022, these time-lags had decreased to 7 days for hospitalisations and 18 days for deaths. Even though many populations have high levels of immunity to SARS-CoV-2 from vaccination and natural infection, waning of immunity and variant emergence will continue to be an upwards pressure on the IHR and IFR. As investments in community surveillance of SARS-CoV-2 infection are scaled back, alternative methods are required to accurately track the ever-changing relationship between infection, hospitalisation, and death and hence provide vital information for healthcare provision and utilisation.

Trends in SARS-CoV-2 infection prevalence during England's roadmap out of lockdown, January to July 2021.

Following rapidly rising COVID-19 case numbers, England entered a national lockdown on 6 January 2021, with staged relaxations of restrictions from 8 March 2021 onwards. We characterise how the lockdown and subsequent easing of restrictions affected trends in SARS-CoV-2 infection prevalence. On average, risk of infection is proportional to infection prevalence. The REal-time Assessment of Community Transmission-1 (REACT-1) study is a repeat cross-sectional study of over 98,000 people every round (rounds approximately monthly) that estimates infection prevalence in England. We used Bayesian P-splines to estimate prevalence and the time-varying reproduction number (Rt) nationally, regionally and by age group from round 8 (beginning 6 January 2021) to round 13 (ending 12 July 2021) of REACT-1. As a comparator, a separate segmented-exponential model was used to quantify the impact on Rt of each relaxation of restrictions. Following an initial plateau of 1.54% until mid-January, infection prevalence decreased until 13 May when it reached a minimum of 0.09%, before increasing until the end of the study to 0.76%. Following the first easing of restrictions, which included schools reopening, the reproduction number Rt increased by 82% (55%, 108%), but then decreased by 61% (82%, 53%) at the second easing of restrictions, which was timed to match the Easter school holidays. Following further relaxations of restrictions, the observed Rt increased steadily, though the increase due to these restrictions being relaxed was offset by the effects of vaccination and also affected by the rapid rise of Delta. There was a high degree of synchrony in the temporal patterns of prevalence between regions and age groups. High-resolution prevalence data fitted to P-splines allowed us to show that the lockdown was effective at reducing risk of infection with school holidays/closures playing a significant part.

Semiparametric finite mixture of regression models with Bayesian P-splines

AbstractMixture models provide a useful tool to account for unobserved heterogeneity and are at the basis of many model-based clustering methods. To gain additional flexibility, some model parameters can be expressed as functions of concomitant covariates. In this Paper, a semiparametric finite mixture of regression models is defined, with concomitant information assumed to influence both the component weights and the conditional means. In particular, linear predictors are replaced with smooth functions of the covariate considered by resorting to cubic splines. An estimation procedure within the Bayesian paradigm is suggested, where smoothness of the covariate effects is controlled by suitable choices for the prior distributions of the spline coefficients. A data augmentation scheme based on difference random utility models is exploited to describe the mixture weights as functions of the covariate. The performance of the proposed methodology is investigated via simulation experiments and two real-world datasets, one about baseball salaries and the other concerning nitrogen oxide in engine exhaust.

EpiLPS: A fast and flexible Bayesian tool for estimation of the time-varying reproduction number.

In infectious disease epidemiology, the instantaneous reproduction number [Formula: see text] is a time-varying parameter defined as the average number of secondary infections generated by an infected individual at time t. It is therefore a crucial epidemiological statistic that assists public health decision makers in the management of an epidemic. We present a new Bayesian tool (EpiLPS) for robust estimation of the time-varying reproduction number. The proposed methodology smooths the epidemic curve and allows to obtain (approximate) point estimates and credible intervals of [Formula: see text] by employing the renewal equation, using Bayesian P-splines coupled with Laplace approximations of the conditional posterior of the spline vector. Two alternative approaches for inference are presented: (1) an approach based on a maximum a posteriori argument for the model hyperparameters, delivering estimates of [Formula: see text] in only a few seconds; and (2) an approach based on a Markov chain Monte Carlo (MCMC) scheme with underlying Langevin dynamics for efficient sampling of the posterior target distribution. Case counts per unit of time are assumed to follow a negative binomial distribution to account for potential overdispersion in the data that would not be captured by a classic Poisson model. Furthermore, after smoothing the epidemic curve, a "plug-in'' estimate of the reproduction number can be obtained from the renewal equation yielding a closed form expression of [Formula: see text] as a function of the spline parameters. The approach is extremely fast and free of arbitrary smoothing assumptions. EpiLPS is applied on data of SARS-CoV-1 in Hong-Kong (2003), influenza A H1N1 (2009) in the USA and on the SARS-CoV-2 pandemic (2020-2021) for Belgium, Portugal, Denmark and France.

A Bayesian multilevel time-varying framework for joint modeling of hospitalization and survival in patients on dialysis.

Over 782 000 individuals in the United States have end-stage kidney disease with about 72% of patients on dialysis, a life-sustaining treatment. Dialysis patients experience high mortality and frequent hospitalizations, at about twice per year. These poor outcomes are exacerbated at key time periods, such as the fragile period after transition to dialysis. In order to study the time-varying effects of modifiable patient and dialysis facility risk factors on hospitalization and mortality, we propose a novel Bayesian multilevel time-varying joint model. Efficient estimation and inference is achieved within the Bayesian framework using Markov chain Monte Carlo, where multilevel (patient- and dialysis facility-level) varying coefficient functions are targeted via Bayesian P-splines. Applications to the United States Renal Data System, a national database which contains data on nearly all patients on dialysis in the United States, highlight significant time-varying effects of patient- and facility-level risk factors on hospitalization risk and mortality. Finite sample performance of the proposed methodology is studied through simulations.

Bayesian P-Splines Applied to Semiparametric Models with Errors Following a Scale Mixture of Normals

Bayesian P-Splines Applied to Semiparametric Models with Errors Following a Scale Mixture of Normals

Appropriately smoothing prevalence data to inform estimates of growth rate and reproduction number

The time-varying reproduction number (Rt) can change rapidly over the course of a pandemic due to changing restrictions, behaviours, and levels of population immunity. Many methods exist that allow the estimation of Rt from case data. However, these are not easily adapted to point prevalence data nor can they infer Rt across periods of missing data. We developed a Bayesian P-spline model suitable for fitting to a wide range of epidemic time-series, including point-prevalence data. We demonstrate the utility of the model by fitting to periodic daily SARS-CoV-2 swab-positivity data in England from the first 7 rounds (May 2020–December 2020) of the REal-time Assessment of Community Transmission-1 (REACT-1) study. Estimates of Rt over the period of two subsequent rounds (6–8 weeks) and single rounds (2–3 weeks) inferred using the Bayesian P-spline model were broadly consistent with estimates from a simple exponential model, with overlapping credible intervals. However, there were sometimes substantial differences in point estimates. The Bayesian P-spline model was further able to infer changes in Rt over shorter periods tracking a temporary increase above one during late-May 2020, a gradual increase in Rt over the summer of 2020 as restrictions were eased, and a reduction in Rt during England’s second national lockdown followed by an increase as the Alpha variant surged. The model is robust against both under-fitting and over-fitting and is able to interpolate between periods of available data; it is a particularly versatile model when growth rate can change over small timescales, as in the current SARS-CoV-2 pandemic. This work highlights the importance of pairing robust methods with representative samples to track pandemics.

Bayesian P-Splines Quantile Regression of Partially Linear Varying Coefficient Spatial Autoregressive Models

This paper deals with spatial data that can be modelled by partially linear varying coefficient spatial autoregressive models with Bayesian P-splines quantile regression. We evaluate the linear and nonlinear effects of covariates on the response and use quantile regression to present comprehensive information at different quantiles. We not only propose an empirical Bayesian approach of quantile regression using the asymmetric Laplace error distribution and employ P-splines to approximate nonparametric components but also develop an efficient Markov chain Monte Carlo technique to explore the joint posterior distributions of unknown parameters. Monte Carlo simulations show that our estimators not only have robustness for different spatial weight matrices but also perform better compared with quantile regression and instrumental variable quantile regression estimators in finite samples at different quantiles. Finally, a set of Sydney real estate data applications is analysed to illustrate the performance of the proposed method.

Bayesian Estimation of Partially Linear Additive Spatial Autoregressive Models with P-Splines

In this paper, we aim to develop a partially linear additive spatial autoregressive model (PLASARM), which is a generalization of the partially linear additive model and spatial autoregressive model. It can be used to simultaneously evaluate the linear and nonlinear effects of the covariates on the response for spatial data. To estimate the unknown parameters and approximate nonparametric functions by Bayesian P-splines, we develop a Bayesian Markov Chain Monte Carlo approach to estimate the PLASARM and design a Gibbs sampler to explore the joint posterior distributions of unknown parameters. Furthermore, we illustrate the performance of the proposed model and estimation method by a simulation study and analysis of Chinese housing price data.

Variational Bayesian partially linear mean shift models for high-dimensional Alzheimer's disease neuroimaging data.

Alzheimer's disease can be diagnosed by analyzing brain images (eg, magnetic resonance imaging, MRI) and neuropsychological tests (eg, mini-mental state examination, MMSE). A partially linear mean shift model (PLMSM) is here proposed to investigate the relationship between MMSE score and high-dimensional regions of interest in MRI, and detect the outliers. In the presence of high-dimensional data, existing Bayesian approaches (eg, Markov chain Monte Carlo) to analyze a PLMSM take intensive computational cost and require huge memory, and have low convergence rate. To address these issues, a variational Bayesian inference is developed to simultaneously estimate parameters and nonparametric functions and identify outliers in a PLMSM. A Bayesian P-splines method is presented to approximate nonparametric functions, a Bayesian adaptive Lasso approach is employed to select predictors, and outliers are detected by the classification variable. Two simulation studies are conducted to assess the finite sample performance of the proposed method. An MRI dataset with elderly cognitive ability is provided to corroborate the proposed method.

Spatial distribution of various forms of malnutrition among reproductive age women in Nepal: A Bayesian geoadditive quantile regression approach.

Addressing both the under-and over-nutritional status of women is an eminent challenge for developing countries like Nepal. This paper examined a critical analysis of factors associated with various forms of malnutrition using Bayesian geoadditive quantile regression approach and assessed spatial variations of malnutrition among Nepalese women using Asian cut-off values. Data drawn from the 2016 Nepal Demographic and Health Survey was utilized to assess the spatial distributions of underweight, overweight and obesity at the provincial level. Spatial and nonlinear components were estimated using Markov random fields and Bayesian P-splines, respectively. The analysis of 4,338 women confirmed that women living in extremely urbanized areas and in Province 1, Province 3, and Province 4 were more likely to be overweight/obese. Similarly, the likelihood of being underweight was prominently high among women residing in rural municipality and women residing in Province 2 and Province 7. Women from the richest and richer quintiles, and with primary education were more likely to be obese. Furthermore, currently-working women and women having access to protected water source were less likely to be obese while improved toilet and access to electricity facility were associated with obesity. Women with access to newspaper and radio were less prone to obesity. Inconsistent distribution of under- and over-nutrition existed in Nepal, given that the high prevalence of overweight/obesity among women living in metropolitan and undernutrition among rural women. Specific intervention measures, addressing location-specific nutrition issues are urgent. Rigorous implementation of strategies incorporated in the national nutrition plan is called for to curb the burden of overweight/obesity. Involving mass media to promote healthier lifestyle and nutritious food could be advantageous at the population level, especially in rural municipalities.

Laplace approximations for fast Bayesian inference in generalized additive models based on P-splines

Generalized additive models (GAMs) are a well-established statistical tool for modeling complex nonlinear relationships between covariates and a response assumed to have a conditional distribution in the exponential family. To make inference in this model class, a fast and flexible approach is considered based on Bayesian P-splines and the Laplace approximation. The proposed Laplace-P-spline model contributes to the development of a new methodology to explore the posterior penalty space by considering a deterministic grid-based strategy or a Markov chain sampler, depending on the number of smooth additive terms in the predictor. The approach has the merit of relying on a simple Gaussian approximation to the conditional posterior of latent variables with closed form analytical expressions available for the gradient and Hessian of the approximate posterior penalty vector. This enables to construct accurate posterior pointwise and credible set estimators for (functions of) regression and spline parameters at a relatively low computational budget even for a large number of smooth additive components. The performance of the Laplace-P-spline model is confirmed through different simulation scenarios and the method is illustrated on two real datasets.

Analysis of the spatial patterns of malnutrition among women in Nigeria with a Bayesian structured additive model

Malnutrition among women has severe consequences on their health, households and national development. The different forms of malnutrition among adult individuals are severe thinness, underweight, overweight and obesity which are usually assessed using height and weight indices calculated as body mass index (BMI). In this study, the spatial distributions of the various forms of malnutrition measured at the upper and lower tails of the BMI were accessed among women of reproductive age in Nigeria after accounting for the influence of socio-economic factors. Bayesian quantile regression that permits for inference on conditional quantiles, yielding a complete description of the functional changes among covariates at different quantiles of the response variable was adopted. Markov random field and Bayesian P-splines were used as prior distributions for the spatial and nonlinear effects respectively, while computation was based on MCMC approach. Data were derived from the 2013 Nigeria Demographic and Health Survey. Findings indicate the existence of spatial structure in the various forms of malnutrition with neighbouring states sharing similar patterns and that the socio-economic variables exact dissimilar influence on the various forms of malnutrition. The study provides valuable insights for policy makers in the quest for halting all forms of malnutrition among Nigerian women.

Nonlinear association structures in flexible Bayesian additive joint models.

Joint models of longitudinal and survival data have become an important tool for modeling associations between longitudinal biomarkers and event processes. The association between marker and log hazard is assumed to be linear in existing shared random effects models, with this assumption usually remaining unchecked. We present an extended framework of flexible additive joint models that allows the estimation of nonlinear covariate specific associations by making use of Bayesian P-splines. Our joint models are estimated in a Bayesian framework using structured additive predictors for all model components, allowing for great flexibility in the specification of smooth nonlinear, time-varying, and random effects terms for longitudinal submodel, survival submodel, and their association. The ability to capture truly linear and nonlinear associations is assessed in simulations and illustrated on the widely studied biomedical data on the rare fatal liver disease primary biliary cirrhosis. All methods are implemented in the R package bamlss to facilitate the application of this flexible joint model in practice.

Binge drinking is associated with reduced quality of life in young students: A pan-European study

BackgroundBinge drinking (BD) is frequently observed in youth, with psychological and cognitive consequences, but its link with quality of life has been scarcely explored. MethodsSociodemographic and alcohol consumption characteristics were collected in a cross-sectional survey including 15,020 European students. Health-related quality of life was assessed using the Alcohol Quality of Life Scale (AQoLS) measuring the self-reported negative impact of alcohol consumption. A flexible link function, using Bayesian P-splines, was used to study the relationship between alcohol-related quality of life and alcohol consumption. ResultsA non-linear relationship between BD and AQoLS scores was identified, showing that: (1) For students presenting moderate BD pattern, alcohol consumption is related to a robust reduction in quality of life, this link remaining stable for students with more intense BD patterns; (2) BD are not strongly associated with social, personal, and work activities, but are linked to an increase in perceived loss of control over consumption; (3) Harmful or hazardous consumption is also related with a massive decrease in quality of life; (4) The strongest relationship between BD and impacted quality of life is found among males and Eastern European students. ConclusionsThese results demonstrate the importance of measuring the perceived relation between alcohol and quality of life, beyond the classically assessed consequences, as this relation is strong among young students. Prevention programs should take this into account, notably regarding the perceived loss of control over alcohol consumption, which constitutes a key factor for the emergence of severe alcohol-use disorders.