Discussion of “ Analysis of Variance — Why It Is More Important than Ever ”

Andrew Gelman’s contribution shifts the focus of “Analysis of Variance” (ANOVA) from the limited sense in which it has been commonly used in classical statistics, as a method of testing, to the broader framework of estimation and inference. The term more commonly used in this sense, “variance components modeling,” also captures the same spirit. The essential idea is that of constructing distributions by using ideas of exchangeability; each variance component corresponds to a collection of exchangeable effects. This extremely powerful approach to linking the scientific structure of a dataset with a model has been and will continue to be widely applicable. The transition from ANOVA to variance components modeling shifts attention from decomposition of the variance of the sample as in classical ANOVA to decomposition of the variance of the population. This shift in focus is appropriate to a world in which scientific questions become increasingly complex and are less frequently answerable through simple designed experiments. As Gelman notes, in a Bayesian framework the estimation of variance components is relatively automatic; attention can be focused on defining sensible models rather than on constructing designs that can be analyzed easily. Finally, Gelman’s discussion of the manifold definitions of “fixed and random effects” is itself worth the price of admission.