Gelman A, Price P N
Department of Statistics, Columbia University, 618 Mathematics Building, New York, New York 10027, USA.
Stat Med. 1999 Dec 15;18(23):3221-34. doi: 10.1002/(sici)1097-0258(19991215)18:23<3221::aid-sim312>3.0.co;2-m.
Maps are frequently used to display spatial distributions of parameters of interest, such as cancer rates or average pollutant concentrations by county. It is well known that plotting observed rates can have serious drawbacks when sample sizes vary by area, since very high (and low) observed rates are found disproportionately in poorly-sampled areas. Unfortunately, adjusting the observed rates to account for the effects of small-sample noise can introduce an opposite effect, in which the highest adjusted rates tend to be found disproportionately in well-sampled areas. In either case, the maps can be difficult to interpret because the display of spatial variation in the underlying parameters of interest is confounded with spatial variation in sample sizes. As a result, spatial patterns occur in adjusted rates even if there is no spatial structure in the underlying parameters of interest, and adjusted rates tend to look too uniform in areas with little data. We introduce two models (normal and Poisson) in which parameters of interest have no spatial patterns, and demonstrate the existence of spatial artefacts in inference from these models. We also discuss spatial models and the extent to which they are subject to the same artefacts. We present examples from Bayesian modelling, but, as we explain, the artefacts occur generally.
地图经常被用于展示感兴趣参数的空间分布，比如各县的癌症发病率或平均污染物浓度。众所周知，当样本量因地区而异时，绘制观测率可能会有严重缺陷，因为在抽样不足的地区会不成比例地出现非常高（和低）的观测率。不幸的是，调整观测率以考虑小样本噪声的影响可能会产生相反的效果，即调整后最高的率往往不成比例地出现在抽样良好的地区。在这两种情况下，地图都可能难以解读，因为感兴趣的潜在参数的空间变化显示与样本量的空间变化相互混淆。结果，即使在感兴趣的潜在参数中不存在空间结构，调整后的率中也会出现空间模式，而且在数据少的地区调整后的率往往看起来过于均匀。我们引入了两个模型（正态和泊松），其中感兴趣的参数没有空间模式，并证明了从这些模型进行推断时空间伪像的存在。我们还讨论了空间模型以及它们在多大程度上会受到同样的伪像影响。我们给出了贝叶斯建模的例子，但正如我们所解释的，这些伪像普遍存在。
