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

拟合度被用来表示回归模型的质量。虽然这样的衡量标准指的是整个模型，但除了各自的统计意义外，往往不会讨论单个回归变量的重要性。作为另一种诊断工具，人们还可以在各个回归变量中分解模型的拟合优度。合作博弈论中的Shapley值和相关概念构成了在参与者之间分配联合生产创造的价值的解决方案。合作博弈论的一个重要成果是其解概念的公理化基础。已证明Shapley值基本上是唯一满足理想单调性的解概念。因此，它最好地确定了每个参与者对联合产生的值的贡献-或者每个回归变量对模型拟合优度的贡献。这个公理基础已经被转化为一个特殊的统计学应用：线性回归分析中的R平方分解。与这些概念相关的计算-尽管是NP-难的-可以由现代计算机相当快地完成。该项目旨在为更多的统计应用提供这样的公理。一方面，这指的是进一步分解拟合优度指标。这对于例如用最大似然法估计的非线性模型来说是更理想的。另一方面，在统计应用的背景下，存在Shapley值的一般化。例如，这允许适当地处理变量组(如对定性变量进行解码的假人)和基本变量。虽然各自的解概念已经在合作博弈论文献中建立，但它们在统计应用背景下的公理化仍然缺乏。弥合这一差距是该项目的任务。此外，还将用实证实例说明分解拟合度的一般概念。

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.