Bayesian regression discontinuity designs: Incorporating clinical knowledge in the causal analysis of primary care data

The regression discontinuity (RD) design is a quasi-experimental design that estimates the causal effects of a treatment by exploiting naturally occurring treatment rules. It can be applied in any context where a particular treatment or intervention is administered according to a pre-specified rule linked to a continuous variable. Such thresholds are common in primary care drug prescription where the RD design can be used to estimate the causal effect of medication in the general population. Such results can then be contrasted to those obtained from randomised controlled trials (RCTs) and inform prescription policy and guidelines based on a more realistic and less expensive context. In this paper we focus on statins, a class of cholesterol-lowering drugs, however, the methodology can be applied to many other drugs provided these are prescribed in accordance to pre-determined guidelines. NHS guidelines state that statins should be prescribed to patients with 10 year cardiovascular disease risk scores in excess of 20%. If we consider patients whose scores are close to this threshold we find that there is an element of random variation in both the risk score itself and its measurement. We can thus consider the threshold a randomising device assigning the prescription to units just above the threshold and withholds it from those just below. Thus we are effectively replicating the conditions of an RCT in the area around the threshold, removing or at least mitigating confounding. We frame the RD design in the language of conditional independence which clarifies the assumptions necessary to apply it to data, and which makes the links with instrumental variables clear. We also have context specific knowledge about the expected sizes of the effects of statin prescription and are thus able to incorporate this into Bayesian models by formulating informative priors on our causal parameters.Comment: 21 pages, 5 figures, 2 table

Can ethnic disparities in sentencing be taken as evidence of judicial discrimination?

Large research efforts have been directed at the exploration of ethnic disparities in the criminal justice system, documenting harsher treatment of minority ethnic defendants, across offence types, criminal justice decisions, and jurisdictions. However, most studies on the topic have relied on observational data, which can only approximate ‘like with like’ comparisons. We use causal diagrams to lay out explicitly the different ways estimates of ethnic disparities in sentencing derived from observational data could be biased. Beyond the commonly acknowledged problem of unobserved case characteristics, we also discuss other less well-known, yet likely more consequential problems: measurement error in the form of racially-determined case characteristics or as a result of disparities within the ‘Whites’ reference group, and selection bias from non-response and missing offenders’ ethnicity data. We apply such causal framework to review findings from two recent studies showing ethnic disparities in custodial sentences imposed at the Crown Court (England and Wales). We also use simulations to recreate the most comprehensive of those studies, and demonstrate how the reported ethnic disparities appear robust to a problem of unobserved case characteristics. We conclude that ethnic disparities observed in the Crown Court are likely reflecting evidence of direct discrimination in sentencing

Missing data: a unified taxonomy guided by conditional independence

Recent work (Seaman et al., 2013; Mealli & Rubin, 2015) attempts to clarify the not always well-understood difference between realised and everywhere definitions of missing at random (MAR) and missing completely at random. Another branch of the literature (Mohan et al., 2013; Pearl & Mohan, 2013) exploits always-observed covariates to give variable-based definitions of MAR and missing completely at random. In this paper, we develop a unified taxonomy encompassing all approaches. In this taxonomy, the new concept of ‘complementary MAR’ is introduced, and its relationship with the concept of data observed at random is discussed. All relationships among these definitions are analysed and represented graphically. Conditional independence, both at the random variable and at the event level, is the formal language we adopt to connect all these definitions. Our paper covers both the univariate and the multivariate case, where attention is paid to monotone missingness and to the concept of sequential MAR. Specifically, for monotone missingness, we propose a sequential MAR definition that might be more appropriate than both everywhere and variable-based MAR to model dropout in certain contexts

Bayesian Interrupted Time Series for evaluating policy change on mental well-being: an application to England's welfare reform

Factors contributing to social inequalities are also associated with negative mental health outcomes leading to disparities in mental well-being. We propose a Bayesian hierarchical model which can evaluate the impact of policies on population well-being, accounting for spatial/temporal dependencies. Building on an interrupted time series framework, our approach can evaluate how different profiles of individuals are affected in different ways, whilst accounting for their uncertainty. We apply the framework to assess the impact of the United Kingdoms welfare reform, which took place throughout the 2010s, on mental well-being using data from the UK Household Longitudinal Study. The additional depth of knowledge is essential for effective evaluation of current policy and implementation of future policy.Comment: 13 pages, 5 figures, 2 table

Bayesian modelling for binary outcomes in the regression discontinuity design

The regression discontinuity (RD) design is a quasi-experimental design which emulates a randomized study by exploiting situations where treatment is assigned according to a continuous variable as is common in many drug treatment guidelines. The RD design literature focuses principally on continuous outcomes. We exploit the link between the RD design and instrumental variables to obtain an estimate for the causal risk ratio for the treated when the outcome is binary. Occasionally this risk ratio for the treated estimator can give negative lower confidence bounds. In the Bayesian framework we impose prior constraints that prevent this from happening. This is novel and cannot be easily reproduced in a frequentist framework. We compare our estimators with those based on estimating equation and generalized methods-of-moments methods. On the basis of extensive simulations our methods compare favourably with both methods and we apply our method to a real example to estimate the effect of statins on the probability of low density lipoprotein cholesterol levels reaching recommended levels

Ecosystem services mapping and assessment for policy- and decision-making: Lessons learned from a comparative analysis of European case studies

This paper analyses and compares a set of case studies on ecosystem services (ES) mapping and assessment with the purpose of formulating lessons learned and recommendations. Fourteen case studies were selected during the EU Horizon 2020 “Coordination and Support Action” ESMERALDA to represent different policy- and decision-making processes throughout the European Union, across a wide range of themes, biomes and scales. The analysis is based on a framework that addresses the key steps of an ES mapping and assessment process, namely policy questions, stakeholder identification and involvement, application of mapping and assessment methods, dissemination and communication and implementation. The analysis revealed that most case studies were policy-orientated or gave explicit suggestions for policy implementation in different contexts, including urban, rural and natural areas. Amongst the findings, the importance of starting stakeholder engagement early in the process was confirmed in order to generate interest and confidence in the project and to increase their willingness to cooperate. Concerning mapping and assessment methods, it was found that the integration of methods and results is essential for providing a comprehensive overview from different perspectives (e.g. social, economic). Finally, lessons learned for effective implementation of ES mapping and assessment results are presented and discussed