We propose a family of tests to assess the goodness of fit of a high dimensional generalized linear model. Our framework is flexible and may be used to construct an omnibus test or directed against testing specific non‐linearities and interaction effects, or for testing the significance of groups of variables. The methodology is based on extracting left‐over signal in the residuals from an initial fit of a generalized linear model. This can be achieved by predicting this signal from the residuals by using modern powerful regression or machine learning methods such as random forests or boosted trees. Under the null hypothesis that the generalized linear model is correct, no signal is left in the residuals and our test statistic has a Gaussian limiting distribution, translating to asymptotic control of type I error. Under a local alternative, we establish a guarantee on the power of the test. We illustrate the effectiveness of the methodology on simulated and real data examples by testing goodness of fit in logistic regression models. Software implementing the methodology is available in the R package GRPtests.

中文翻译：

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高维广义线性模型的拟合优度测试

我们提出了一系列测试，以评估高维广义线性模型的拟合优度。我们的框架很灵活，可用于构建综合测试或针对测试特定的非线性和交互作用，或用于测试变量组的重要性。该方法基于从广义线性模型的初始拟合中提取残差中的剩余信号。这可以通过使用现代强大的回归或机器学习方法（例如随机森林或茂密的树木）从残差预测此信号来实现。在广义线性模型正确的零假设下，残差中不留任何信号，并且我们的检验统计量具有高斯极限分布，转化为I型误差的渐近控制。在本地替代方案下，我们建立了对测试力量的保证。我们通过测试逻辑回归模型的拟合优度来说明该方法在模拟和真实数据示例中的有效性。实现该方法的软件可在R包GRPtests。