## 1 Introduction

Generally speaking, epidemiological surveillance consists of continuously gathering and analyzing data for changes in disease occurrence (Last, 2001). Surveillance may be based on time or space or a combination of time-space, with an active or passive approach. Disease mapping, *i.e.* the study of the variability of disease occurrence on space, is a cornerstone of epidemiologic surveillance. Currently, the availability of data on a small scale makes it popular to scan for abnormal disease rates potentially associated with widespread environmental exposures or to search for a localized cluster of cases in proximity of putative sources of pollution (Elliott *et al.*, 2000). In disease mapping, a moderate to large number of area-level relative risks are considered. However, the large heterogeneity of population density among small areas leads to smaller *p*-values paradoxically associated with relative risk estimates closer to the null. Such inconsistency justified the development of shrinkage estimators (Clayton and Kaldor, 1987). Shrinkage estimators, as empirical Bayes or full Bayes, are now accepted as standard tools in spatial epidemiology, but they leave unresolved the multiple comparison problem.

Control of Family Wise Error Rate (FWER) that is a global control of type I error is generally pursued in the Surveillance framework (Frisén, 2003; Kulldorff, 2001). In his article of 2007, Rolka discussed the cost in sensitivity of adopting a FWER control procedure and he mentioned control of the False Discovery Rate (FDR). FDR is the rate of false positives among all rejected hypotheses and was introduced with examples in the context of clinical trials by Benjamini and Hochberg (1995). FDR has a Bayesian interpretation and it is connected to the *q*-value, a Bayesian alternative to the *p*-value (Storey, 2003).

In the disease mapping literature, posterior probability for each area having a risk higher than a predefined threshold after having specified an appropriate hierarchical Bayesian model, was suggested as a way to screen areas at higher/lower risk (Bernardinelli and Montomoli, 1992; Richardson *et al.*, 2004). This is not sufficient to assure that the posterior inference adjusts for multiple testing. To accomplish this task, the probability model needs to include a null prior and related hyperparameters that define the prior probability mass for non-*divergent* areas (Scott and Berger, 2006). In the following article, we consider a two-sided alternative hypothesis and use the term *divergent* to denote areas at risk different from the null. This meaning of the word divergent was used by Olhssen *et al.* (2007).

### 1.1 Aim of the study

This article aims to develop a hierarchical Bayesian modeling approach to multiple testing in the context of disease mapping. The idea to use an FDR approach instead of an FWER control is based upon the fact that the erroneous rejection of the null hypothesis for some municipalities does not challenge the result of the whole descriptive analysis whose aim is to assess heterogeneity of risk in the entire study region. Therefore, the FWER control is too strict for the application's needs (Benjamini, 2009).

In the following analysis a tri-level hierarchical Bayesian model is proposed to estimate for each area the probability of belonging to the null, to be used to explore areas at divergent risk (higher or lower then the reference disease rate) while controlling for multiple testing. We took advantage of real data regarding the mortality rate due to lung cancer in males at the municipal level in the Tuscan region (Italy) during the period 1995–1999.

In Section 2, we describe the mortality data. In Section 3, we briefly introduce the problem of multiple comparisons; we then describe the proposed hierarchical Bayesian models for disease mapping and how to estimate posterior classification probabilities. The results are presented in Section 4. The conclusion and discussion follow in Section 5.