Adaptive functional principal components analysis

Functional data analysis (FDA) almost always involves smoothing discrete observations into curves, because they are never observed in continuous time and rarely without error. Although smoothing parameters affect the subsequent inference, data-driven methods for selecting these parameters are not well-developed, frustrated by the difficulty of using all the information shared by curves while being computationally efficient. On the one hand, smoothing individual curves in an isolated, albeit sophisticated way, ignores useful signals present in other curves. On the other hand, bandwidth selection by automatic procedures such as cross-validation after pooling all the curves together quickly become computationally unfeasible due to the large number of data points. In this paper we propose a new data-driven, adaptive kernel smoothing, specifically tailored for functional principal components analysis (FPCA) through the derivation of sharp, explicit risk bounds for the eigen-elements. The minimization of these quadratic risk bounds provide refined, yet computationally efficient bandwidth rules for each eigen-element separately. Both common and independent design cases are allowed. Rates of convergence for the adaptive eigen-elements estimators are derived. An extensive simulation study, designed in a versatile manner to closely mimic characteristics of real data sets, support our methodological contribution, which is available for use in the R package FDAdapt

Ticagrelor in patients with diabetes and stable coronary artery disease with a history of previous percutaneous coronary intervention (THEMIS-PCI) : a phase 3, placebo-controlled, randomised trial

Background: Patients with stable coronary artery disease and diabetes with previous percutaneous coronary intervention (PCI), particularly those with previous stenting, are at high risk of ischaemic events. These patients are generally treated with aspirin. In this trial, we aimed to investigate if these patients would benefit from treatment with aspirin plus ticagrelor. Methods: The Effect of Ticagrelor on Health Outcomes in diabEtes Mellitus patients Intervention Study (THEMIS) was a phase 3 randomised, double-blinded, placebo-controlled trial, done in 1315 sites in 42 countries. Patients were eligible if 50 years or older, with type 2 diabetes, receiving anti-hyperglycaemic drugs for at least 6 months, with stable coronary artery disease, and one of three other mutually non-exclusive criteria: a history of previous PCI or of coronary artery bypass grafting, or documentation of angiographic stenosis of 50% or more in at least one coronary artery. Eligible patients were randomly assigned (1:1) to either ticagrelor or placebo, by use of an interactive voice-response or web-response system. The THEMIS-PCI trial comprised a prespecified subgroup of patients with previous PCI. The primary efficacy outcome was a composite of cardiovascular death, myocardial infarction, or stroke (measured in the intention-to-treat population). Findings: Between Feb 17, 2014, and May 24, 2016, 11 154 patients (58% of the overall THEMIS trial) with a history of previous PCI were enrolled in the THEMIS-PCI trial. Median follow-up was 3·3 years (IQR 2·8–3·8). In the previous PCI group, fewer patients receiving ticagrelor had a primary efficacy outcome event than in the placebo group (404 [7·3%] of 5558 vs 480 [8·6%] of 5596; HR 0·85 [95% CI 0·74–0·97], p=0·013). The same effect was not observed in patients without PCI (p=0·76, p interaction=0·16). The proportion of patients with cardiovascular death was similar in both treatment groups (174 [3·1%] with ticagrelor vs 183 (3·3%) with placebo; HR 0·96 [95% CI 0·78–1·18], p=0·68), as well as all-cause death (282 [5·1%] vs 323 [5·8%]; 0·88 [0·75–1·03], p=0·11). TIMI major bleeding occurred in 111 (2·0%) of 5536 patients receiving ticagrelor and 62 (1·1%) of 5564 patients receiving placebo (HR 2·03 [95% CI 1·48–2·76], p<0·0001), and fatal bleeding in 6 (0·1%) of 5536 patients with ticagrelor and 6 (0·1%) of 5564 with placebo (1·13 [0·36–3·50], p=0·83). Intracranial haemorrhage occurred in 33 (0·6%) and 31 (0·6%) patients (1·21 [0·74–1·97], p=0·45). Ticagrelor improved net clinical benefit: 519/5558 (9·3%) versus 617/5596 (11·0%), HR=0·85, 95% CI 0·75–0·95, p=0·005, in contrast to patients without PCI where it did not, p interaction=0·012. Benefit was present irrespective of time from most recent PCI. Interpretation: In patients with diabetes, stable coronary artery disease, and previous PCI, ticagrelor added to aspirin reduced cardiovascular death, myocardial infarction, and stroke, although with increased major bleeding. In that large, easily identified population, ticagrelor provided a favourable net clinical benefit (more than in patients without history of PCI). This effect shows that long-term therapy with ticagrelor in addition to aspirin should be considered in patients with diabetes and a history of PCI who have tolerated antiplatelet therapy, have high ischaemic risk, and low bleeding risk