There is an expanding literature on different representations of spatial random effects for different types of spatial correlation structure within the conditional autoregressive class of priors for Bayesian spatial models. However, little is known about the impact of these different priors when the number of areas is small. This paper aimed to investigate this problem both in the context of a case study of spatial analysis of dengue fever and more generally through a simulation study. Both the simulation study and the case study considered count data aggregated to a small area level in a region. Five different conditional autoregressive priors for a simple Bayesian Poisson model were considered: independent, Besag-York-Mollié, Leroux, and two variants of a localised clustering model. Data were simulated with eight different sizes of areal grids, ranging from 4 to 2500 areas, and two different levels of both spatial autocorrelation and disease counts. Model goodness-of-fit measures and model estimates were compared. A case study involving dengue fever cases in 14 local areas in Makassar, Indonesia, was also considered. The simulation study showed that model performance varied under different scenarios. When areas had low autocorrelation and high counts, and the number of areas was at most 25, the BYM, Leroux and localised $$G = 2$$ models performed similarly and better than the independent and localised $$G = 3$$ models. However, when the number of areas were at least 100, all models performed differently, and the Leroux model performed the best. Overall, the Leroux model performed the best for every scenario especially when there were at least 16 areas. Based on the case study, the comparative performance of spatial models may also vary for a small number of areas, especially when the data have a relatively large mean and variance over areas. In this case, the localised model with G = 3 was a better choice. Detecting spatial patterns can be difficult when there are very few areas. Understanding the characteristics of the data and the relative influence of alternative conditional autoregressive priors is essential in selecting an appropriate Bayesian spatial model.

中文翻译：

---

评估少量区域对空间估计的影响

在关于贝叶斯空间模型的先验条件自回归类中，对于不同类型的空间相关结构的空间随机效应的不同表示形式，已有越来越多的文献报道。但是，当区域数较少时，对于这些不同先验的影响知之甚少。本文旨在通过对登革热空间分析的案例研究以及更广泛的模拟研究来研究这个问题。模拟研究和案例研究都考虑了将计数数据汇总到一个区域中的小区域级别。考虑了简单贝叶斯泊松模型的五个不同的条件自回归先验：独立，Besag-York-Mollié，Leroux和局部聚类模型的两个变体。使用八个不同大小的面网格对数据进行了模拟，范围从4到2500个区域，以及空间自相关和疾病计数的两个不同级别。比较了模型拟合优度度量和模型估计。还考虑了一个涉及印度尼西亚望加锡14个地区的登革热病例的案例研究。仿真研究表明，模型性能在不同情况下会有所不同。当区域的自相关性较低且计数很高时，区域的数量最多为25个，则BYM，Leroux和本地化$$ G = 2 $$模型的性能类似，并且优于独立和本地化的$$ G = 3 $$模型。但是，当区域数至少为100时，所有模型的性能都不同，而Leroux模型的性能最佳。总体而言，Leroux模型在每种情况下都表现最佳，尤其是当至少有16个区域时。根据案例研究 空间模型的比较性能也可能在少数区域内发生变化，尤其是当数据在区域内具有相对较大的均值和方差时。在这种情况下，G = 3的本地化模型是更好的选择。当区域很少时，检测空间模式可能很困难。在选择适当的贝叶斯空间模型时，了解数据的特征以及替代条件自回归先验的相对影响至关重要。