BACKGROUND All analyses of spatially aggregated data are vulnerable to the modifiable areal unit problem (MAUP), which describes the sensitivity of analytical results to the arbitrary choice of spatial aggregation unit at which data are measured. The MAUP is a serious problem endemic to analyses of spatially aggregated data in all scientific disciplines. However, the impact of the MAUP is rarely considered, perhaps partly because it is still widely considered to be unsolvable. RESULTS It was originally understood that a solution to the MAUP should constitute a comprehensive statistical framework describing the regularities in estimates of association observed at different combinations of spatial scale and zonation. Additionally, it has been debated how such a solution should incorporate the geographical characteristics of areal units (e.g. shape, size, and configuration), and in particular whether this can be achieved in a purely mathematical framework (i.e. independent of areal units). We argue that the consideration of areal units must form part of a solution to the MAUP, since the MAUP only manifests in their presence. Thus, we present a theoretical and statistical framework that incorporates the characteristics of areal units by combining estimates obtained from different scales and zonations. We show that associations estimated at scales larger than a minimal geographical unit of analysis are systematically biased from a true minimal-level effect, with different zonations generating uniquely biased estimates. Therefore, it is fundamentally erroneous to infer conclusions based on data that are spatially aggregated beyond the minimal level. Instead, researchers should measure and display information, estimate effects, and infer conclusions at the smallest possible meaningful geographical scale. The framework we develop facilitates this. CONCLUSIONS The proposed framework represents a new minimum standard in the estimation of associations using spatially aggregated data, and a reference point against which previous findings and misconceptions related to the MAUP can be understood.

中文翻译：

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将地理学纳入一个新的广义的理论和统计框架，以解决可修改的面积单位问题。

背景技术所有对空间聚集数据的分析都容易受到可修改的面积单位问题（MAUP）的影响，该问题描述了分析结果对测量数据的空间聚集单位的任意选择的敏感性。MAUP是所有科学学科中空间聚集数据分析所特有的严重问题。但是，很少考虑MAUP的影响，部分原因是它仍然被广泛认为是无法解决的。结果最初理解为MAUP的解决方案应构成一个全面的统计框架，该框架应描述在空间尺度和分区的不同组合下所观察到的关联估计的规律性。此外，关于这种解决方案应如何结合区域单位的地理特征（例如形状，大小和配置），特别是是否可以在纯数学框架中实现（即与面积单位无关）。我们认为，考虑面积单位必须构成MAUP解决方案的一部分，因为MAUP仅在其存在时才表现出来。因此，我们提出了一个理论和统计框架，通过组合从不同规模和分区获得的估算值，结合了单位面积的特征。我们显示，在大于最小分析地理单位的范围内估计的关联会从真正的最小水平效应中系统性地产生偏差，不同的区域会产生唯一的偏差估计。因此，从空间上汇总的数据超出最小水平推断结论，从根本上来说是错误的。代替，研究人员应在尽可能小的有意义的地理范围内测量和显示信息，估计效果并推断结论。我们开发的框架对此起到了促进作用。结论所提出的框架代表了一种使用空间聚合数据进行关联估计的新的最低标准，也是一个可以理解与MAUP相关的先前发现和误解的参考点。