A case-crossover analysis is used as a simple but powerful tool for estimating the effect of short-term environmental factors such as extreme temperatures or poor air quality on mortality. The environment on the day of each death is compared to the one or more "control days" in previous weeks, and higher levels of exposure on death days than control days provide evidence of an effect. Current stateof-the-art methodology and software (integrated nested Laplace approximation [INLA]) cannot be used to fit the most flexible case-crossover models to large datasets, because the likelihood for case-crossover models cannot be expressed in a manner compatible with this methodology. In this paper, we develop a flexible and scalable modeling framework for case-crossover models with linear and semiparametric effects which retains the flexibility and computational advantages of INLA. We apply our method to quantify nonlinear associations between mortality and extreme temperatures in India. An R package implementing our methods is publicly available.
In many applications that involve the inference of an unknown smooth function, the inference of its derivatives will often be just as important as that of the function itself. To make joint inferences of the function and its derivatives, a class of Gaussian processes called p th order Integrated Wiener's Process (IWP), is considered. Methods for constructing a finite element (FEM) approximation of an IWP exist but have focused only on the order p = 2 case which does not allow appropriate inference for derivatives, and their computational feasibility relies on additional approximation to the FEM itself. In this article, we propose an alternative FEM approximation, called overlapping splines (O-spline), which pursues computational feasibility directly through the choice of test functions, and mirrors the construction of an IWP as the Ospline results from the multiple integrations of these same test functions. The Ospline approximation applies for any order p ∈ Z + , is computationally efficient and provides consistent inference for all derivatives up to order p − 1. It is shown both theoretically, and empirically through simulation, that the O-spline approximation converges to the true IWP as the number of knots increases. We further provide a unified and interpretable way to define priors for the smoothing parameter based on
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