Premium
Goodness‐of‐fit testing in high dimensional generalized linear models
Author(s) -
Janková Jana,
Shah Rajen D.,
Bühlmann Peter,
Samworth Richard J.
Publication year - 2020
Publication title -
journal of the royal statistical society: series b (statistical methodology)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 6.523
H-Index - 137
eISSN - 1467-9868
pISSN - 1369-7412
DOI - 10.1111/rssb.12371
Subject(s) - goodness of fit , test statistic , generalized linear model , null distribution , mathematics , statistical hypothesis testing , linear model , logistic regression , computer science , statistics , statistic , null hypothesis , artificial intelligence
Summary 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.