To date, we have seen the emergence of a large literature on multivariate disease mapping. That is, incidence of (or mortality from) multiple diseases is recorded at the scale of areal units where incidence (mortality) across the diseases is expected to manifest dependence. The modeling involves a hierarchical structure: a Poisson model for disease counts (conditioning on the rates) at the first stage, and a specification of a function of the rates using spatial random effects at the second stage. These random effects are specified as a prior and introduce spatial smoothing to the rate (or risk) estimates. What we see in the literature is the amount of smoothing induced under a given prior across areal units compared with the observed/empirical risks. Our contribution here extends previous research on smoothing in univariate areal data models. Specifically, for three different choices of multivariate prior, we investigate both within prior smoothing according to hyperparameters and across prior smoothing. Its benefit to the user is to illuminate the expected nature of departure from perfect fit associated with these priors since model performance is not a question of goodness of fit. We propose both theoretical and empirical metrics for our investigation and illustrate with both simulated and real data.

翻译：迄今为止，关于多元疾病制图的研究已形成大量文献。该方法记录多个疾病在区域单元尺度上的发病率（或死亡率），预期不同疾病间的发病率（死亡率）会呈现相关性。建模采用分层结构：第一阶段建立疾病计数的泊松模型（以发病率为条件），第二阶段利用空间随机效应构建发病率的函数表达式。这些随机效应通过先验分布进行设定，并对发病率（或风险）估计引入空间平滑。现有文献主要关注特定先验分布下各区域单元获得的平滑程度与观测/经验风险的比较。本文的贡献在于拓展了单变量区域数据模型中的平滑研究。具体而言，针对三种不同的多元先验选择，我们同时考察了超参数决定的先验内部平滑与先验间平滑。该方法的价值在于：由于模型性能并非拟合优度问题，它能帮助使用者明晰这些先验分布导致理想拟合偏离的预期性质。我们提出了理论与实证相结合的评估指标，并通过模拟数据与真实数据进行了实证分析。