There is currently much interest in conducting spatial analyses of health outcomes at the small-area scale. This requires sophisticated statistical techniques, usually involving Bayesian models, to smooth the underlying risk estimates because the data are typically sparse. However, questions have been raised about the performance of these models for recovering the “true” risk surface, about the influence of the prior structure specified, and about the amount of smoothing of the risks that is actually performed. We describe a comprehensive simulation study designed to address these questions. Our results show that Bayesian disease-mapping models are essentially conservative, with high specificity even in situations with very sparse data but low sensitivity if the raised-risk areas have only a moderate (< 2-fold) excess or are not based on substantial expected counts (> 50 per area). Semiparametric spatial mixture models typically produce less smoothing than their conditional autoregressive counterpart when there is sufficient information in the data (moderate-size expected count and/or high true excess risk). Sensitivity may be improved by exploiting the whole posterior distribution to try to detect true raised-risk areas rather than just reporting and mapping the mean posterior relative risk. For the widely used conditional autoregressive model, we show that a decision rule based on computing the probability that the relative risk is above 1 with a cutoff between 70 and 80% gives a specific rule with reasonable sensitivity for a range of scenarios having moderate expected counts (~ 20) and excess risks (~1.5- to 2-fold). Larger (3-fold) excess risks are detected almost certainly using this rule, even when based on small expected counts, although the mean of the posterior distribution is typically smoothed to about half the true value.

With the advent of routine health data indexed at a fine geographical resolution, small area disease mapping studies have become an established technique in geographical epidemiology. The specific issues posed by the sparseness of the data and possibility for local spatial dependence belong to a generic class of statistical problems involving an underlying (latent) spatial process of interest corrupted by observational noise. These are naturally formulated within the framework of hierarchical models, and over the past decade, a variety of spatial models have been proposed for the latent level(s) of the hierarchy. In this article, we provide a comprehensive review of the main classes of such models that have been used for disease mapping within a Bayesian estimation paradigm, and report a performance comparison between representative models in these classes, using a set of simulated data to help illustrate their respective properties. We also consider recent extensions to model the joint spatial distribution of multiple disease or health indicators. The aim is to help the reader choose an appropriate structural prior for the second level of the hierarchical model and to discuss issues of sensitivity to this choice.

We analyze the perturbative massive open string spectrum of even-dimensional superstring compactifications with four, eight and sixteen supercharges. In each of such cases, we focus on universal states that exist independently on the internal geometry and other compatification details. We analytically compute refined partition functions that count these states at each mass level. Such refined partition functions are written in a super-Poincaré covariant form, providing information on how supermultiplets transform under the little group and the R symmetry. Various asymptotic limits of the partition functions and their associated quantities, such as the leading and subleading Regge trajectories, are studied empirically and analytically. In the phenomenologically relevant case of four supercharges, the partition function can be cast into the most compact form and the asymptotic formula in the large spin limit is derived explicitly. Crown Lüst et al. / Nuclear Physics B 876 (2013) Fig. 1. Classes of superstring compactifications for which we will discuss the universal particle content. The arrows within the columns represent dimensional reduction on a T 2 torus.

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