Usually, in spatial generalised linear models, covariates that are spatially smooth are collinear with spatial random effects. This affects the bias and precision of the regression coefficients. This is known in the spatial statistics literature as spatial confounding. We discuss the problem of confounding in the case of multilevel spatial models wherein there are multiple observations within clusters. We show that even under the standard multilevel model, which allows for independent (i.e. not spatially correlated) cluster effects, the cluster-level fixed effects might be biased depending on the structure of the ‘true’ generating mechanism of the processes. We provide simulation studies in order to investigate the effects of confounding in the estimation of fixed effects present in random intercept models under different scenarios of confounding. One remedy to spatial confounding is restricted spatial regression wherein the spatial random effects are constrained to be orthogonal to the fixed effects of the model. We propose one way to fit a restricted spatial regression model for multilevel data and illustrate it with artificial data analyses. We also briefly describe the issue of confounding in random intercept and slope models.

中文翻译：

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分层模型中空间混杂的影响

通常，在空间广义线性模型中，空间平滑的协变量与空间随机效应共线。这会影响回归系数的偏差和精度。这在空间统计文献中称为空间混杂。我们讨论了在多级空间模型的情况下的混杂问题，其中集群内有多个观察。我们表明，即使在允许独立（即非空间相关）集群效应的标准多级模型下，集群级固定效应可能会根据过程的“真实”生成机制的结构而产生偏差。我们提供模拟研究，以研究在不同混杂情况下随机截距模型中存在的固定效应估计中混杂的影响。空间混杂的一种补救措施是受限空间回归，其中空间随机效应被约束为与模型的固定效应正交。我们提出了一种方法来拟合多级数据的受限空间回归模型，并用人工数据分析来说明它。我们还简要描述了随机截距和斜率模型中的混杂问题。