Within the field of hierarchical modelling, little attention is paid to micro-macro models: those in which group-level outcomes are dependent on covariates measured at the level of individuals within groups. Although such models are perhaps underrepresented in the literature, they have applications in economics, epidemiology, and the social sciences. Despite the strong mathematical similarities between micro-macro and measurement error models, few efforts have been made to apply the much better-developed methodology of the latter to the former. Here, we present a new empirical Bayesian technique for micro-macro data, called FRODO (Functional Regression On Densities of Observations). The method jointly infers group-specific densities for multilevel covariates and uses them as functional predictors in a functional linear regression, resulting in a model that is analogous to a generalized additive model (GAM). In doing so, it achieves a level of generality comparable to more sophisticated methods developed for errors-in-variables models, while further leveraging the larger group sizes characteristic of multilevel data to provide richer information about the within-group covariate distributions. After explaining the hierarchical structure of FRODO, its power and versatility are demonstrated on several simulated datasets, showcasing its ability to accommodate a wide variety of covariate distributions and regression models.

翻译：在层次建模领域，微观-宏观模型受到的关注较少：这类模型中，群体层面的结果依赖于群体内个体层面测量的协变量。尽管此类模型在文献中可能代表性不足，但它们在经济学、流行病学和社会科学中具有应用价值。尽管微观-宏观模型与测量误差模型在数学上具有高度相似性，但很少有人尝试将后者更为成熟的方法应用于前者。本文提出了一种用于微观-宏观数据的新经验贝叶斯技术，称为FRODO（观测密度函数回归）。该方法联合推断多层次协变量的群体特定密度，并将其用作函数线性回归中的函数预测变量，从而得到一个类似于广义加性模型（GAM）的模型。通过这种方式，它实现了与为变量误差模型开发的更复杂方法相媲美的泛化能力，同时进一步利用多层次数据中较大的群体规模特征，为组内协变量分布提供更丰富的信息。在解释FRODO的层次结构后，我们在多个模拟数据集上展示了其效能和多功能性，证明了其能够适应各种协变量分布和回归模型。