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

• Published in: Communications in statistics. Simulation and computation Vol. 41; no. 2; pp. 141 - 154
• Main Authors: Schützenmeister, A., Jensen, U., Piepho, H.-P.
• Format: Journal Article
• Language: English
• Published: Colchester Taylor & Francis Group 01.02.2012; Taylor & Francis; Taylor & Francis Ltd
• Subjects: 62-09; 62J5; Approximation; Computer simulation; Constant residual variance; Correlation; Diagnostic systems; Distribution theory; Exact sciences and technology; Homogeneity; Linear inference, regression; Mathematics; Model checking; Monte Carlo; Multiplicity; Multivariate analysis; Normality; Numerical analysis; Numerical analysis. Scientific computation; Numerical methods in probability and statistics; Probability and statistics; Residual plot; Sciences and techniques of general use; Simulation; Simultaneous tolerance band; Statistics; Tolerance interval; Tolerance region; Tolerances; Transformations; Correlation; Generalized linear model; Statistical distribution; Error estimation; Data transformation; Gaussian distribution; Non parametric estimation; Multivariate analysis; Stochastic method; Constant residual variance; Linear model; Variance; Tolerance interval; Outlier; Statistical test; Monte Carlo; Multiplicity; Model checking; Approximation theory; 62F25; Monte Carlo method; Discriminant analysis; Covariance analysis; Independence; Probability; Tolerance; Statistical association; Statistical estimation; Algorithm; Variance analysis; Heteroscedasticity; Confidence interval; Statistical method; Statistical regression; Diagnostic techniques; Residual plot; Numerical analysis; Non normality; Simulation; Simultaneous tolerance band; Correlation analysis; Tolerance region; Normality test; Numerical simulation; Computing method
• ISSN: 0361-0918; 1532-4141
• DOI: 10.1080/03610918.2011.582560

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.]