A Simple Lack-of-Fit Test for Regression Models

TL;DR

This paper introduces a formal statistical test for assessing the correctness of a specified regression function, providing a more rigorous alternative to residual plots, capable of handling heteroscedastic errors without replicates.

Contribution

The paper presents a novel, simple test for regression model fit that complements graphical diagnostics and can handle heteroscedasticity and non-replicated data.

Findings

Test statistic based on maximum consecutive overestimation or underestimation sequences.

Recursive formulas for exact distribution under null and alternative hypotheses.

Applicable to heteroscedastic errors without requiring replicates.

Abstract

A simple test is proposed for examining the correctness of a given completely specified response function against unspecified general alternatives in the context of univariate regression. The usual diagnostic tools based on residuals plots are useful but heuristic. We introduce a formal statistical test supplementing the graphical analysis. Technically, the test statistic is the maximum length of the sequences of ordered (with respect to the covariate) observations that are consecutively overestimated or underestimated by the candidate regression function. Note that the testing procedure can cope with heteroscedastic errors and no replicates. Recursive formulae allowing to calculate the exact distribution of the test statistic under the null hypothesis and under a class of alternative hypotheses are given.