Spatial autoregressive models for scan statistic

Abstract

Spatial scan statistics are well-known methods for cluster detection and are widely used in epidemiology and medical studies for detecting and evaluating the statistical significance of disease hotspots. For the sake of simplicity, the classical spatial scan statistic assumes that the observations of the outcome variable in different locations are independent, while in practice the data may exhibit a spatial correlation. In this article, we use spatial autoregressive (SAR) models to account the spatial correlation in parametric/non-parametric scan statistic. Firstly, the correlation parameter is estimated in the SAR model to transform the outcome into a new independent outcome over all locations. Secondly, we propose an adapted spatial scan statistic based on this independent outcome for cluster detection. A simulation study highlights the better performance of the proposed methods than the classical one in presence of spatial correlation in the data. The latter shows a sharp increase in Type I error and false-positive rate but also decreases the true-positive rate when spatial correlation increases. Besides, our methods retain the Type I error and have stable true and false positive rates with respect to the spatial correlation. The proposed methods are illustrated using a spatial economic dataset of the median income in Paris city. In this application, we show that taking spatial correlation into account leads to the identification of more concentrated clusters than those identified by the classical spatial scan statistic.

Appendix 1

Explicit expressions of the parameters estimators for the Gaussian spatial scan statistics

Under ${\mathcal {H}}_0$, the MLEs of $\alpha$ and $\sigma ^2$ have the following explicit expressions:

$$\begin{aligned} {\widehat{\alpha }}=\frac{1}{n}\sum _{i=1}^{n}Y_i\qquad \text{ and } \qquad \widehat{\sigma ^2}=\frac{1}{n}\sum _{i=1}^{n}\left( Y_i-{\widehat{\alpha }}\right) ^{2}. \end{aligned}$$

Under ${\mathcal {H}}_1$, the MLEs of $\alpha$, $\sigma ^2$ and $\delta _k$ have the following explicit expressions:

$$\begin{aligned} {\widehat{\alpha }}_{k}=\frac{1}{n-n_k}\sum _{i=1}^{n}\left( 1-\xi ^{(k)}_i\right) Y_i,\qquad {\widehat{\delta }}_{k}=\frac{1}{n-n_k}\sum _{i=1}^{n}\left( \frac{n}{n_k}\xi ^{(k)}_i-1\right) {Y}_i \end{aligned}$$

and

$$\begin{aligned} \widehat{\sigma ^2}_{k}= & {} \frac{1}{n}\sum _{i=1}^{n}\left( Y_i-{\widehat{\alpha }}_{k}-{\widehat{\delta }}_{k}\xi _{i}^{(k)}\right) ^2\\= & {} \frac{1}{n}\left\{ \sum _{i \notin C_k}\left( Y_i-\frac{1}{n-n_k}\sum _{j \notin C_k }Y_j\right) ^2 +\sum _{i\in C_k}\left( Y_i-\frac{1}{n_k}\sum _{j \in C_k}Y_j\right) ^2\right\} \end{aligned}$$

where $n_k$ is the number of locations inside $C_k$. Thus, the last decomposition is equal to the estimator of $\sigma ^2$ under ${\mathcal {H}}_1$ given in Kulldorff et al. (2009).