Kernel dimension reduction in regression

We present a new methodology for sufficient dimension reduction (SDR). Our methodology derives directly from the formulation of SDR in terms of the conditional independence of the covariate $X$ from the response $Y$, given the projection of $X$ on the central subspace [cf. J. Amer. Statist. Assoc. 86 (1991) 316--342 and Regression Graphics (1998) Wiley]. We show that this conditional independence assertion can be characterized in terms of conditional covariance operators on reproducing kernel Hilbert spaces and we show how this characterization leads to an $M$-estimator for the central subspace. The resulting estimator is shown to be consistent under weak conditions; in particular, we do not have to impose linearity or ellipticity conditions of the kinds that are generally invoked for SDR methods. We also present empirical results showing that the new methodology is competitive in practice.Comment: Published in at http://dx.doi.org/10.1214/08-AOS637 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Developing and evaluating interventions to reduce inappropriate prescribing by general practitioners of antibiotics for upper respiratory tract infections: a randomised controlled trial to compare paper-based and web-based modelling experiments

Background: Much implementation research is focused on full-scale trials with little evidence of preceding modelling work. The Medical Research Council Framework for developing and evaluating complex interventions has argued for more and better theoretical and exploratory work prior to a trial as a means of improving intervention development. Intervention modelling experiments (IMEs) are a way of exploring and refining an intervention before moving to a full-scale trial. They do this by delivering key elements of the intervention in a simulation that approximates clinical practice by, for example, presenting general practitioners (GPs) with a clinical scenario about making a treatment decision. Methods: The current proposal will run a full, web-based IME involving 250 GPs that will advance the methodology of IMEs by directly comparing results with an earlier paper-based IME. Moreover, the web-based IME will evaluate an intervention that can be put into a full-scale trial that aims to reduce antibiotic prescribing for upper respiratory tract infections in primary care. The study will also include a trial of email versus postal invitations to participate. Discussion: More effective behaviour change interventions are needed and this study will develop one such intervention and a system to model and test future interventions. This system will be applicable to any situation in the National Health Service where behaviour needs to be modified, including interventions aimed directly at the public. Trial registration: ClinicalTrials (NCT): NCT0120673