An Exact Test of Fit for the Gaussian Linear Model Using Optimal Nonbipartite Matching

ABSTRACT

Fisher tested the fit of Gaussian linear models using replicated observations. We refine this method by (1) constructing near-replicates using an optimal nonbipartite matching and (2) defining a distance that focuses on predictors important to the model’s predictions. Near-replicates may not exist unless the predictor set is low-dimensional; the test addresses dimensionality by betting that model failures involve a subset of predictors important in the old fit. Despite using the old fit to pair observations, the test has exactly its stated level under the null hypothesis. Simulations show the test has reasonable power even when many spurious predictors are present.

KEYWORDS:

• Combinatorial optimization
• Network optimization
• Tukey’s test for nonadditivity.

Acknowledgments

Supported by Fellowship FA9550-11-C-0028 from the Department of Defense, Army Research Office, National Defense Science and Engineering Graduate (NDSEG), Fellowship Program, 32 CFR 168a4, 2013--2016.