Generalisations of the Shapley value decomposition of goodness-of-fit measures in regression models

拟合优度测量用于指示回归模型的质量。虽然这样的度量指的是整个模型，但单个回归变量的重要性通常不会超出其各自的统计意义。作为替代诊断工具，也可以将模型的拟合优度分解为各个回归变量。Shapley值和合作博弈论的相关概念构成了在参与者之间分配联合生产所创造的价值的解决方案。合作对策论的一个主要成就是其解概念的公理化基础。它已被证明，Shapley值基本上是唯一的解决方案的概念，满足一个理想的单调性。因此，它可以最好地识别每个参与者对联合产生的价值的贡献-或者每个回归变量对模型拟合优度的贡献。这个公理化的基础已经被转化为一个特殊的统计应用：线性回归分析中R平方的分解。与这些概念相关的计算--尽管是NP困难的--可以通过现代计算机相当快速地完成。该项目旨在为其他统计应用提供这样的公理化。一方面，这涉及进一步拟合优度度量的分解。这对于非线性模型是更需要的，例如，用最大似然法估计。另一方面，在统计应用的背景下，有相关的Shapley值的概括。例如，这允许适当处理变量组（如解码定性变量的虚拟变量）和基本变量。虽然各自的解决方案的概念已经建立在合作博弈论文献中，它们的公理化的统计应用的背景下仍然失踪。弥合这一差距是该项目的任务。此外，分解拟合优度的一般概念将通过实证例子来证明。

Goodness-of-fit measures are used to indicate the quality of regression models. While such a measure refers to the model as a whole, the importance of individual regressor variables is often not discussed beyond their respective statistical significance. As an alternative diagnostic tool, one may also decompose the goodness-of-fit of a model among the individual regressor variables. The Shapley value and related concepts from cooperative game theory constitute solutions to distribute the value created from joint production among participating players. A major achievement of cooperative game theory is the axiomatic foundation of its solution concepts. It has been proven that the Shapley value is basically the only solution concept that satisfies a desirable monotonicity property. Thus, it identifies best the contribution of each player to jointly produced values - or the contribution of each regressor variable to the goodness of fit of the model. This axiomatic foundation has already been translated to a particular statistical application: the decomposition of R-squared in linear regression analysis.The computations associated with these concepts - though NP-hard - can be accomplished by modern computers quite rapidly.The project aims to provide such axiomatisations for additional statistical applications. On the one hand, this refers to the decomposition of further goodness-of-fit measures. This is the more a desideratum for nonlinear models that are, e.g., estimated by with the maximum likelihood method. On the other hand, there are generalisations of the Shapley value that are relevant in the context of statistical applications. For instance, this allows for the appropriate treatment of variable groups (such as dummies decoding a qualitative variable) and of essential variables. While the respective solution concepts are already established in the cooperative game theory literature, their axiomatisations in the context of statistical applications are still missing. Bridging this gap is the task of the project. Moreover, the generalisedconcepts of decomposing goodness-of-fit will be demonstrated with empirical examples.