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，多变量疾病制图领域已涌现大量文献。即，在区域单元尺度上记录多种疾病的发病率（或死亡率），且预期这些疾病之间的发病（死亡）存在依赖性。该建模采用分层结构：第一阶段对疾病计数（基于发病率条件）采用泊松模型，第二阶段利用空间随机效应指定发病率的函数。这些随机效应被设定为先验，并对发病率（或风险）估计引入空间平滑。文献中常见的做法是将给定先验下跨区域单元的平滑量与观测/经验风险进行比较。我们的贡献在于拓展了先前关于单变量区域数据模型平滑的研究。具体而言，针对三种不同的多变量先验选择，我们既根据超参数研究了先验内部平滑，也研究了跨先验平滑。这对用户的益处在于阐明这些先验下与完美拟合的预期偏离性质——因为模型性能并非拟合优度问题。我们为研究提出了理论与经验指标，并通过模拟数据与实际数据加以验证。