Checking Normality and Homoscedasticity in the General Linear Model Using Diagnostic Plots

Abstract

Inference for the general linear model makes several assumptions, including independence of errors, normality, and homogeneity of variance. Departure from the latter two of these assumptions may indicate the need for data transformation or removal of outlying observations. Informal procedures such as diagnostic plots of residuals are frequently used to assess the validity of these assumptions or to identify possible outliers. A simulation-based approach is proposed, which facilitates the interpretation of various diagnostic plots by adding simultaneous tolerance bounds. Several tests exist for normality or homoscedasticity in simple random samples. These tests are often applied to residuals from a linear model fit. The resulting procedures are approximate in that correlation among residuals is ignored. The simulation-based approach accounts for the correlation structure of residuals in the linear model and allows simultaneously checking for possible outliers, non normality, and heteroscedasticity, and it does not rely on formal testing.

[Supplementary materials are available for this article. Go to the publisher's online edition of Communications in Statistics—Simulation and Computation® for the following three supplemental resource: a word file containing figures illustrating the mode of operation for the bisectional algorithm, QQ-plots, and a residual plot for the mussels data.]

Keywords:

• Constant residual variance
• Model checking
• Monte Carlo
• Multiplicity
• Residual plot
• Simulation
• Simultaneous tolerance band
• Tolerance interval
• Tolerance region

Mathematics Subject Classification:

• 62-09
• 62F25
• 62J5
• 62J10

Acknowledgments

We thank the anonymous reviewers for helpful comments. The last author acknowledges support within the DFG project PI 377/12-1.