Rejoinder to “analysis of Variance—why It Is More Important than Ever” by A. Gelman

ANOVA is more important than ever because we are fitting models with many parameters, and these parameters can often usefully be structured into batches. The essence of “ANOVA” (as we see it) is to compare the importance of the batches and to provide a framework for efficient estimation of the individual parameters and related summaries such as comparisons and contrasts. Classical ANOVA is associated with many things, including linear models, F-tests of nested and nonnested interactions, decompositions of sums of squares and hypothesis tests. Our paper focuses on generalizing the assessment, with uncertainty bounds, of the importance of each row of the “ANOVA table.” This falls in the general category of data reduction or model summary, and presupposes an existing model (most simply, a linear regression) and an existing batching of coefficients (or more generally “effects,” as noted by McCullagh) into named batches. We thank the discussants for pointing out that more work needs to be done to generalize these ideas beyond classical regression settings with exchangeable batches of parameters. In this rejoinder, we review the essentials of our approach and then address some specific issues raised in the discussions.