I appreciate all the comments here on my article on ANOVA for the New Palgrave Dictionary of Economics. The main suggestion was that I clarify what ANOVA gives you that regression doesn’t. I hope this revision does the trick. Here’s the revised article and here’s the added section:

The analysis of variance is often understood by economists in relation to linear regression (e.g., Goldberger, 1964). From the perspective of linear (or generalized linear) models, we identify ANOVA with the structuring of coefficients into batches, with each batch corresponding to a “source of variation” (in ANOVA terminology).

As discussed by Gelman (2005), the relevant inferences from ANOVA can be reproduced using regression—but not always least-squares regression. Multilevel models are needed for analyzing hierarchical data structures such as the split-plot design in Figure 2, where between-plot effects are compared to the main-plot errors, and within-plot effects are compared to sub-plot errors.

Given that we can already fit regression models, what do we gain by thinking about ANOVA? To start with, there are settings in which the sources of variation are of more interest than the individual effects. For example, the analysis of internet connect times in Figure 3 represents thousands of coefficients—but without taking the model too seriously, we can use the ANOVA display as a helpful exploratory summary of data. For another example, the two plots in Figure 4 allow us to quickly understand and compare two multilevel logistic regressions, again without getting overwhelmed with dozens of coefficient estimates.

More generally, we think of the analysis of variance as a way of understanding and structuring multilevel models—not as an alternative to regression but as a tool for summarizing complex high-dimensional inferences, as can be seen, for example, in Figure 5 (finite-population and superpopulation standard deviations) and Figures 6-7 (group-level coefficients and trends).

Also, I thanked the commenters who gave their full names. If anyone else has suggestions and also wants to be achknowledged in the article, just let me know your name. Thanks again.