Lack of fit tests for linear regression models with many predictor variables using minimal weighted maximal matchings

We develop lack of fit tests for linear regression models with many predictor variables. General alternatives for model comparison are constructed using minimal weighted maximal matchings consistent with graphs on the predictor vectors. The weighted graphs we employ have edges based on model-driven distance thresholds in predictor space, thereby making our testing procedure implementable and computationally efficient in higher dimensional settings. In addition, it is shown that the testing procedure adapts to efficacious maximal matchings. An asymptotic analysis, along with simulation results, demonstrate that our tests are effective against a broad class of lack of fit.