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

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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.