Generalized Geographically Weighted Regression Model within a Modularized Bayesian Framework.

Geographically weighted regression (GWR) models handle geographical dependence through a spatially varying coefficient model and have been widely used in applied science, but its general Bayesian extension is unclear because it involves a weighted log-likelihood which does not imply a probability distribution on data. We present a Bayesian GWR model and show that its essence is dealing with partial misspecification of the model. Current modularized Bayesian inference models accommodate partial misspecification from a single component of the model. We extend these models to handle partial misspecification in more than one component of the model, as required for our Bayesian GWR model. Information from the various spatial locations is manipulated via a geographically weighted kernel and the optimal manipulation is chosen according to a Kullback-Leibler (KL) divergence. We justify the model via an information risk minimization approach and show the consistency of the proposed estimator in terms of a geographically weighted KL divergence.

Factors Associated with Significant Weight Loss in Hospitalised Patients with COVID-19: A Retrospective Cohort Study in a Large Teaching Hospital.

SARS-CoV-2 infection (COVID-19) is associated with malnutrition risk in hospitalised individuals. COVID-19 and malnutrition studies in large European cohorts are limited, and post-discharge dietary characteristics are understudied. This study aimed to assess the rates of and risk factors for ≥10% weight loss in inpatients with COVID-19, and the need for post-discharge dietetic support and the General Practitioner (GP) prescription of oral nutritional supplements, during the first COVID-19 wave in a large teaching hospital in the UK. Hospitalised adult patients admitted between March and June 2020 with a confirmed COVID-19 diagnosis were included in this retrospective cohort study. Demographic, anthropometric, clinical, biochemical, and nutritional parameters associated with ≥10% weight loss and post-discharge characteristics were described. Logistic regression models were used to identify risk factors for ≥10% weight loss and post-discharge requirements for ongoing dietetic input and oral nutritional supplement prescription. From the total 288 patients analysed (40% females, 72 years median age), 19% lost ≥ 10% of their admission weight. The length of hospital stay was a significant risk factor for ≥10% weight loss in multivariable analysis (OR 1.22; 95% CI 1.08-1.38; p = 0.001). In addition, ≥10% weight loss was positively associated with higher admission weight and malnutrition screening scores, dysphagia, ICU admission, and artificial nutrition needs. The need for more than one dietetic input after discharge was associated with older age and ≥10% weight loss during admission. A large proportion of patients admitted to the hospital with COVID-19 experienced significant weight loss during admission. Longer hospital stay is a risk factor for ≥10% weight loss, independent of disease severity, reinforcing the importance of repeated malnutrition screening and timely referral to dietetics.

The Prognostic and Discriminatory Utility of the Clinical Frailty Scale and Modified Frailty Index Compared to Age.

Background: There is no consensus on the optimal method for the assessment of frailty. We compared the prognostic utility of two approaches (modified Frailty Index [mFI], Clinical Frailty Scale [CFS]) in older adults (≥65 years) hospitalised with COVID-19 versus age. Methods: We used a test and validation cohort that enrolled participants hospitalised with COVID-19 between 27 February and 30 June 2020. Multivariable mixed-effects logistic modelling was undertaken, with 28-day mortality as the primary outcome. Nested models were compared between a base model, age and frailty assessments using likelihood ratio testing (LRT) and an area under the receiver operating curves (AUROC). Results: The primary cohort enrolled 998 participants from 13 centres. The median age was 80 (range:65−101), 453 (45%) were female, and 377 (37.8%) died within 28 days. The sample was replicated in a validation cohort of two additional centres (n = 672) with similar characteristics. In the primary cohort, both mFI and CFS were associated with mortality in the base models. There was improved precision when fitting CFS to the base model +mFI (LRT = 25.87, p < 0.001); however, there was no improvement when fitting mFI to the base model +CFS (LRT = 1.99, p = 0.16). AUROC suggested increased discrimination when fitting CFS compared to age (p = 0.02) and age +mFI (p = 0.03). In contrast, the mFI offered no improved discrimination in any comparison (p > 0.05). Similar findings were seen in the validation cohort. Conclusions: These observations suggest the CFS has superior prognostic value to mFI in predicting mortality following COVID-19. Our data do not support the use of the mFI as a tool to aid clinical decision-making and prognosis.

Unquantifiably low aldosterone concentrations are prevalent in hospitalised COVID-19 patients but may not be revealed by chemiluminescent immunoassay.

Objective: Previous studies have reported conflicting findings regarding aldosterone levels in patients hospitalised with COVID-19. We therefore used the gold-standard technique of liquid chromatography-tandem mass spectrometry (LCMSMS) to address this uncertainty. Design: All patients admitted to Cambridge University Hospitals with COVID-19 between 10 March 2020 and 13 May 2021, and in whom a stored blood sample was available for analysis, were eligible for inclusion. Methods: Aldosterone was measured by LCMSMS and by immunoassay; cortisol and renin were determined by immunoassay. Results: Using LCMSMS, aldosterone was below the limit of detection (<70 pmol/L) in 74 (58.7%) patients. Importantly, this finding was discordant with results obtained using a commonly employed clinical immunoassay (Diasorin LIAISON®), which over-estimated aldosterone compared to the LCMSMS assay (intercept 14.1 (95% CI -34.4 to 54.1) + slope 3.16 (95% CI 2.09-4.15) pmol/L). The magnitude of this discrepancy did not clearly correlate with markers of kidney or liver function. Solvent extraction prior to immunoassay improved the agreement between methods (intercept -14.9 (95% CI -31.9 to -4.3) and slope 1.0 (95% CI 0.89-1.02) pmol/L) suggesting the presence of a water-soluble metabolite causing interference in the direct immunoassay. We also replicated a previous finding that blood cortisol concentrations were often increased, with increased mortality in the group with serum cortisol levels > 744 nmol/L (P = 0.005). Conclusion: When measured by LCMSMS, aldosterone was found to be profoundly low in a significant proportion of patients with COVID-19 at the time of hospital admission. This has likely not been detected previously due to high levels of interference with immunoassays in patients with COVID-19, and this merits further prospective investigation.

Interoperability of statistical models in pandemic preparedness: principles and reality.

We present interoperability as a guiding framework for statistical modelling to assist policy makers asking multiple questions using diverse datasets in the face of an evolving pandemic response. Interoperability provides an important set of principles for future pandemic preparedness, through the joint design and deployment of adaptable systems of statistical models for disease surveillance using probabilistic reasoning. We illustrate this through case studies for inferring and characterising spatial-temporal prevalence and reproduction numbers of SARS-CoV-2 infections in England.

A numerically stable algorithm for integrating Bayesian models using Markov melding.

When statistical analyses consider multiple data sources, Markov melding provides a method for combining the source-specific Bayesian models. Markov melding joins together submodels that have a common quantity. One challenge is that the prior for this quantity can be implicit, and its prior density must be estimated. We show that error in this density estimate makes the two-stage Markov chain Monte Carlo sampler employed by Markov melding unstable and unreliable. We propose a robust two-stage algorithm that estimates the required prior marginal self-density ratios using weighted samples, dramatically improving accuracy in the tails of the distribution. The stabilised version of the algorithm is pragmatic and provides reliable inference. We demonstrate our approach using an evidence synthesis for inferring HIV prevalence, and an evidence synthesis of A/H1N1 influenza.

Combining chains of Bayesian models with Markov melding.

A challenge for practitioners of Bayesian inference is specifying a model that incorporates multiple relevant, heterogeneous data sets. It may be easier to instead specify distinct submodels for each source of data, then join the submodels together. We consider chains of submodels, where submodels directly relate to their neighbours via common quantities which may be parameters or deterministic functions thereof. We propose chained Markov melding, an extension of Markov melding, a generic method to combine chains of submodels into a joint model. One challenge we address is appropriately capturing the prior dependence between common quantities within a submodel, whilst also reconciling differences in priors for the same common quantity between two adjacent submodels. Estimating the posterior of the resulting overall joint model is also challenging, so we describe a sampler that uses the chain structure to incorporate information contained in the submodels in multiple stages, possibly in parallel. We demonstrate our methodology using two examples. The first example considers an ecological integrated population model, where multiple data sets are required to accurately estimate population immigration and reproduction rates. We also consider a joint longitudinal and time-to-event model with uncertain, submodel-derived event times. Chained Markov melding is a conceptually appealing approach to integrating submodels in these settings.

Development and validation of a dynamic 48-hour in-hospital mortality risk stratification for COVID-19 in a UK teaching hospital: a retrospective cohort study.

OBJECTIVES: To develop a disease stratification model for COVID-19 that updates according to changes in a patient's condition while in hospital to facilitate patient management and resource allocation. DESIGN: In this retrospective cohort study, we adopted a landmarking approach to dynamic prediction of all-cause in-hospital mortality over the next 48 hours. We accounted for informative predictor missingness and selected predictors using penalised regression. SETTING: All data used in this study were obtained from a single UK teaching hospital. PARTICIPANTS: We developed the model using 473 consecutive patients with COVID-19 presenting to a UK hospital between 1 March 2020 and 12 September 2020; and temporally validated using data on 1119 patients presenting between 13 September 2020 and 17 March 2021. PRIMARY AND SECONDARY OUTCOME MEASURES: The primary outcome is all-cause in-hospital mortality within 48 hours of the prediction time. We accounted for the competing risks of discharge from hospital alive and transfer to a tertiary intensive care unit for extracorporeal membrane oxygenation. RESULTS: Our final model includes age, Clinical Frailty Scale score, heart rate, respiratory rate, oxygen saturation/fractional inspired oxygen ratio, white cell count, presence of acidosis (pH <7.35) and interleukin-6. Internal validation achieved an area under the receiver operating characteristic (AUROC) of 0.90 (95% CI 0.87 to 0.93) and temporal validation gave an AUROC of 0.86 (95% CI 0.83 to 0.88). CONCLUSIONS: Our model incorporates both static risk factors (eg, age) and evolving clinical and laboratory data, to provide a dynamic risk prediction model that adapts to both sudden and gradual changes in an individual patient's clinical condition. On successful external validation, the model has the potential to be a powerful clinical risk assessment tool. TRIAL REGISTRATION: The study is registered as 'researchregistry5464' on the Research Registry (www.researchregistry.com).

Clinical Features, Inpatient Trajectories and Frailty in Older Inpatients with COVID-19: A Retrospective Observational Study.

INTRODUCTION: We describe the clinical features and inpatient trajectories of older adults hospitalized with COVID-19 and explore relationships with frailty. METHODS: This retrospective observational study included older adults admitted as an emergency to a University Hospital who were diagnosed with COVID-19. Patient characteristics and hospital outcomes, primarily inpatient death or death within 14 days of discharge, were described for the whole cohort and by frailty status. Associations with mortality were further evaluated using Cox Proportional Hazards Regression (Hazard Ratio (HR), 95% Confidence Interval). RESULTS: 214 patients (94 women) were included of whom 142 (66.4%) were frail with a median Clinical Frailty Scale (CFS) score of 6. Frail compared to nonfrail patients were more likely to present with atypical symptoms including new or worsening confusion (45.1% vs. 20.8%, p < 0.001) and were more likely to die (66% vs. 16%, p = 0.001). Older age, being male, presenting with high illness acuity and high frailty were independent predictors of death and a dose-response association between frailty and mortality was observed (CFS 1-4: reference; CFS 5-6: HR 1.78, 95% CI 0.90, 3.53; CFS 7-8: HR 2.57, 95% CI 1.26, 5.24). CONCLUSIONS: Clinicians should have a low threshold for testing for COVID-19 in older and frail patients during periods of community viral transmission, and diagnosis should prompt early advanced care planning.

Stochastic approximation cut algorithm for inference in modularized Bayesian models.

Bayesian modelling enables us to accommodate complex forms of data and make a comprehensive inference, but the effect of partial misspecification of the model is a concern. One approach in this setting is to modularize the model and prevent feedback from suspect modules, using a cut model. After observing data, this leads to the cut distribution which normally does not have a closed form. Previous studies have proposed algorithms to sample from this distribution, but these algorithms have unclear theoretical convergence properties. To address this, we propose a new algorithm called the stochastic approximation cut (SACut) algorithm as an alternative. The algorithm is divided into two parallel chains. The main chain targets an approximation to the cut distribution; the auxiliary chain is used to form an adaptive proposal distribution for the main chain. We prove convergence of the samples drawn by the proposed algorithm and present the exact limit. Although SACut is biased, since the main chain does not target the exact cut distribution, we prove this bias can be reduced geometrically by increasing a user-chosen tuning parameter. In addition, parallel computing can be easily adopted for SACut, which greatly reduces computation time.

MultiBUGS: A Parallel Implementation of the BUGS Modelling Framework for Faster Bayesian Inference.

MultiBUGS is a new version of the general-purpose Bayesian modelling software BUGS that implements a generic algorithm for parallelising Markov chain Monte Carlo (MCMC) algorithms to speed up posterior inference of Bayesian models. The algorithm parallelises evaluation of the product-form likelihoods formed when a parameter has many children in the directed acyclic graph (DAG) representation; and parallelises sampling of conditionally-independent sets of parameters. A heuristic algorithm is used to decide which approach to use for each parameter and to apportion computation across computational cores. This enables MultiBUGS to automatically parallelise the broad range of statistical models that can be fitted using BUGS-language software, making the dramatic speed-ups of modern multi-core computing accessible to applied statisticians, without requiring any experience of parallel programming. We demonstrate the use of MultiBUGS on simulated data designed to mimic a hierarchical e-health linked-data study of methadone prescriptions including 425,112 observations and 20,426 random effects. Posterior inference for the e-health model takes several hours in existing software, but MultiBUGS can perform inference in only 28 minutes using 48 computational cores.

Joining and splitting models with Markov melding.

Analysing multiple evidence sources is often feasible only via a modular approach, with separate submodels specified for smaller components of the available evidence. Here we introduce a generic framework that enables fully Bayesian analysis in this setting. We propose a generic method for forming a suitable joint model when joining submodels, and a convenient computational algorithm for fitting this joint model in stages, rather than as a single, monolithic model. The approach also enables splitting of large joint models into smaller submodels, allowing inference for the original joint model to be conducted via our multi-stage algorithm. We motivate and demonstrate our approach through two examples: joining components of an evidence synthesis of A/H1N1 influenza, and splitting a large ecology model.

A Gibbs Sampler for Learning DAGs.

We propose a Gibbs sampler for structure learning in directed acyclic graph (DAG) models. The standard Markov chain Monte Carlo algorithms used for learning DAGs are random-walk Metropolis-Hastings samplers. These samplers are guaranteed to converge asymptotically but often mix slowly when exploring the large graph spaces that arise in structure learning. In each step, the sampler we propose draws entire sets of parents for multiple nodes from the appropriate conditional distribution. This provides an efficient way to make large moves in graph space, permitting faster mixing whilst retaining asymptotic guarantees of convergence. The conditional distribution is related to variable selection with candidate parents playing the role of covariates or inputs. We empirically examine the performance of the sampler using several simulated and real data examples. The proposed method gives robust results in diverse settings, outperforming several existing Bayesian and frequentist methods. In addition, our empirical results shed some light on the relative merits of Bayesian and constraint-based methods for structure learning.

Rapid model exploration for complex hierarchical data: application to pharmacokinetics of insulin aspart.

We consider situations, which are common in medical statistics, where we have a number of sets of response data, from different individuals, say, potentially under different conditions. A parametric model is defined for each set of data, giving rise to a set of random effects. Our goal here is to efficiently explore a range of possible 'population' models for the random effects, to select the most appropriate model. The range of possible models is potentially vast, because the random effects may depend on observed covariates, and there may be multiple credible ways of partitioning their variability. Here, we consider pharmacokinetic (PK) data on insulin aspart, a fast acting insulin analogue used in the treatment of diabetes. PK models are typically nonlinear (in their parameters), often complex and sometimes only available as a set of differential equations, with no closed-form solution. Fitting such a model for just a single individual can be a challenging task. Fitting a joint model for all individuals can be even harder, even without the complication of an overarching model selection objective. We describe a two-stage approach that decouples the population model for the random effects from the PK model applied to the response data but nevertheless fits the full, joint, hierarchical model, accounting fully for uncertainty. This allows us to repeatedly reuse results from a single analysis of the response data to explore various population models for the random effects. This greatly expedites not only model exploration but also cross-validation for the purposes of model criticism. © 2015 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd.

Modelling the dynamics of an experimental host-pathogen microcosm within a hierarchical Bayesian framework.

The advantages of Bayesian statistical approaches, such as flexibility and the ability to acknowledge uncertainty in all parameters, have made them the prevailing method for analysing the spread of infectious diseases in human or animal populations. We introduce a Bayesian approach to experimental host-pathogen systems that shares these attractive features. Since uncertainty in all parameters is acknowledged, existing information can be accounted for through prior distributions, rather than through fixing some parameter values. The non-linear dynamics, multi-factorial design, multiple measurements of responses over time and sampling error that are typical features of experimental host-pathogen systems can also be naturally incorporated. We analyse the dynamics of the free-living protozoan Paramecium caudatum and its specialist bacterial parasite Holospora undulata. Our analysis provides strong evidence for a saturable infection function, and we were able to reproduce the two waves of infection apparent in the data by separating the initial inoculum from the parasites released after the first cycle of infection. In addition, the parameter estimates from the hierarchical model can be combined to infer variations in the parasite's basic reproductive ratio across experimental groups, enabling us to make predictions about the effect of resources and host genotype on the ability of the parasite to spread. Even though the high level of variability between replicates limited the resolution of the results, this Bayesian framework has strong potential to be used more widely in experimental ecology.