A Bridge Between Regression and ANOVA Thinking

In dose-response studies, the dose level can be treated either as a classification variable in an ANOVA-type model or as a continuous variable in a regression model. There is a fun little bridge between these two seemingly different approaches, for example, when the underlying dose-response relationship can be represented using generalized linear models (GLMs). This post illustrates that bridge in the specific context of Gaussian linear models.

It should be noted that scaling the contrast in Eq (1) affects only the conclusion in the second bullet point. In such as case, the two estimators are no longer identical but remain directly proportional.

It's also worth noting that the contrasts derived from Eq (1) are the same as the usual orthogonal polynomial contrasts (or any proportional scaling thereof) for testing a linear trend. When dose levels are equally spaced, for example, the contrast coefficients for ℓ = 3, 4, and 5 are (-1, 0, 1), (-3, -1, 1, 3) and (-2, -1, 0, 1, 2), respectively.

Finally, similar relationships and the corresponding equivalence between regression and ANOVA-type testing naturally extend to all GLMs when evaluated on the link-function scale, because the GLM log-likelihood depends on the data only through the linear predictor.