Annals of Statistics

Analysis of variance—why it is more important than ever

Andrew Gelman

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Analysis of variance (ANOVA) is an extremely important method in exploratory and confirmatory data analysis. Unfortunately, in complex problems (e.g., split-plot designs), it is not always easy to set up an appropriate ANOVA. We propose a hierarchical analysis that automatically gives the correct ANOVA comparisons even in complex scenarios. The inferences for all means and variances are performed under a model with a separate batch of effects for each row of the ANOVA table.

We connect to classical ANOVA by working with finite-sample variance components: fixed and random effects models are characterized by inferences about existing levels of a factor and new levels, respectively. We also introduce a new graphical display showing inferences about the standard deviations of each batch of effects.

We illustrate with two examples from our applied data analysis, first illustrating the usefulness of our hierarchical computations and displays, and second showing how the ideas of ANOVA are helpful in understanding a previously fit hierarchical model.

Article information

Ann. Statist., Volume 33, Number 1 (2005), 1-53.

First available in Project Euclid: 8 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62J10: Analysis of variance and covariance 62J07: Ridge regression; shrinkage estimators 62F15: Bayesian inference 62J05: Linear regression 62J12: Generalized linear models

ANOVA Bayesian inference fixed effects hierarchical model linear regression multilevel model random effects variance components


Gelman, Andrew. Analysis of variance—why it is more important than ever. Ann. Statist. 33 (2005), no. 1, 1--53. doi:10.1214/009053604000001048.

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