Interpreting posterior relative risk estimates in disease-mapping studies

Environmental Health Perspectives, June 15, 2004 by Sylvia Richardson, Andrew Thomson, Nicky Best, Paul Elliott

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 me 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. Key words: Bayesian hierarchical models, cancer mapping, environmental epidemiology, sensitivity, small-area studies, spatial smoothing, specificity. Environ Health Perspect 112:1016-1025 (2004). doi:10.1289/ehp.6740 available via http://dx.doi.org/[Online 15 April 2004]

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Spatial analyses of health outcomes have long been recognized in the epidemiologic literature as playing a specific and important role in description and analysis. In particular, they can highlight sources of heterogeneity underlying spatial patterns in the health outcomes and consequently are able to suggest important public health determinants or etiologic clues. A good example of geographic epidemiology is the seminal monograph by Doll (1980), which described some of the first hypotheses concerning the influence of environment and lifestyle characteristics on cancer mortality and discussed how these arose from studying the geographic distribution of various cancers. These early studies were usually performed on a large geographic scale, using mostly international or regional comparisons.

Recently, the availability of local geographically indexed health and population data, together with advances in computing and geographic information systems, has encouraged the analysis of health data on a small geographic scale (Elliott et al. 2000). The motivation is the increased interpretability of small-scale studies, as they are in principle less susceptible to the component of ecologic bias created by the within-area heterogeneity of exposure or other determinants. They are also better able to detect highly localized effects such as those related to industrial pollution in the vicinity. Conversely, small-scale studies require more sophisticated statistical analysis techniques than, for example, an analysis between countries, because the data are typically sparse with low (even zero) counts of events in many of the small areas. Further, frequently there is evidence of overdispersion of the counts with respect to the Poisson model as well as spatial patterns indicating some dependence between the counts in neighboring areas.

Faced with these nonstandard characteristics, statistical models have been developed to address these issues and make best use of small-area health data. In connection with generic developments in a flexible modeling strategy using the paradigm of Bayesian hierarchical models, hierarchical disease-mapping models based on conditional autoregressions (CAR) were proposed in the 1990s through the work of Besag et al. (1991), Clayton and Bernardinelli (1992), and Clayton et al. (1993). These CAR models are now commonly used both by statisticians and epidemiologists, and their implementation is facilitated by existing software such as WinBUGS (Spiegdhalter et al. 2002). Alternative semiparametric formulations to CAR have also been proposed recently (Denison and Holmes 2001; Green and Richardson 2002; Knorr-Held and Rasser 2000) to model more heterogeneous risk surfaces and particularly to allow for potential discontinuities in the risk. The main characteristic of all these models is to provide some shrinkage and spatial smoothing of the raw relative risk estimates that otherwise would be computed separately in each area. Such shrinkage gives a more stable estimate of the pattern of underlying risk of disease than that provided by the raw estimates. The pattern of the raw risks, strongly influenced by the size of the population at risk, leads to a noisy and blurred picture of the true unobserved risks.