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Article Dans Une Revue Journal of Computational and Graphical Statistics Année : 2018

Goodness of fit of logistic regression models for random graphs

Résumé

The logistic regression model constitutes a natural and simple tool to understand how covariates (when available) contribute to explain the topology of a binary network. After estimating the logistic parameters, one of the main questions which arises in practice is to assess the goodness of fit of the corresponding model. To address this problem, we add a general term, related to the graphon function of W-graph models, to the logistic function. Such an extra term aims at characterizing the residual structure of the network, that is not explained by the covariates. We approximate this new generic logistic model using a class of models with blockwise constant residual structure. This framework allows to derive a Bayesian procedure from a model based selection context using goodness-of-fit criteria. All these criteria depend on marginal likelihood terms for which we do provide estimates relying on two series of variational approximations. Experiments on toy data are carried out to assess the inference procedure. Finally, six networks from social sciences and ecology are studied to illustrate the proposed methodology. Logistic regression is a natural and simple tool to understand how covariates contribute to explain the topology of a binary network. Once the model is fitted, the practitioner is interested in the goodness-of-fit of the regression in order to check if the covariates are sufficient to explain the whole topology of the network and, if they are not, to analyze the residual structure. To address this problem, we introduce a generic model that combines logistic regression with a network-oriented residual term. This residual term takes the form of the graphon function of a W-graph. Using a variational Bayes framework, we infer the residual graphon by averaging over a series of blockwise constant functions. This approach allows us to define a generic goodness-of-fit criterion, which corresponds to the posterior probability for the residual graphon to be constant. Experiments on toy data are carried out to assess the accuracy of the procedure. Several networks from social sciences and ecology are studied to illustrate the proposed methodology.
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Dates et versions

hal-01583801 , version 1 (07-09-2017)

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Pierre Latouche, Stephane Robin, Sarah Ouadah. Goodness of fit of logistic regression models for random graphs. Journal of Computational and Graphical Statistics, 2018, 27 (1), pp.98-109. ⟨10.1080/10618600.2017.1349663⟩. ⟨hal-01583801⟩
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