A geometric examination of linear model assumptions

Graham R. Wood*, David J. Saville

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    7 Citations (Scopus)

    Abstract

    Summary: From a geometric perspective, linear model theory relies on a single assumption, that ('corrected') data vector directions are uniformly distributed in Euclidean space. We use this perspective to explore pictorially the effects of violations of the traditional assumptions (normality, independence and homogeneity of variance) on the Type I error rate. First, for several non-normal distributions we draw geometric pictures and carry out simulations to show how the effects of non-normality diminish with increased parent distribution symmetry and continuity, and increased sample size. Second, we explore the effects of dependencies on Type I error rate. Third, we use simulation and geometry to investigate the effect of heterogeneity of variance on Type I error rate. We conclude, in a fresh way, that independence and homogeneity of variance are more important assumptions than normality. The practical implication is that statisticians and authors of statistical computing packages need to pay more attention to the correctness of these assumptions than to normality.

    Original languageEnglish
    Pages (from-to)285-303
    Number of pages19
    JournalAustralian and New Zealand Journal of Statistics
    Volume55
    Issue number3
    DOIs
    Publication statusPublished - Sep 2013

    Keywords

    • Homogeneity of variance
    • Independence
    • Linear algebra
    • Normality
    • Spherical symmetry
    • Vector geometry

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