Comment: Fisher Lecture: Dimension Reduction in Regression

I am pleased to participate in this well-deserved recognition of Dennis Cook’s remarkable career. Cook points out Fisher’s insistence that predictor variables in regression be chosen without reference to the dependent variable. Reduction by principal components clearly satisfies that dictum. One of my primary objections to partial least squares regression when I first encountered it as an alternative to principal components was that the predictor variables were being chosen with reference to the dependent variable. (I now have other objections to partial least squares.) Yet on the other hand, variable selection in regression is well accepted and it clearly chooses variables based on their relationship to the dependent variable. Perhaps variable selection is better thought of as a form of shrinkage estimation rather than as a process for choosing predictor variables. Cook also reiterates something that I think is difficult to overemphasize: Fisher’s point that “More or less elaborate forms [for models] will be suitable according to the volume of the data.” We see this now on a regular basis as modern technology provides larger data sets to which elaborate models are regularly fitted. With regard to Cook’s work, it seems to me that the key issue in the development of Cook’s models (2), (5), (10) and (13) is whether they are broadly reasonable. The question did not seem to be extensively addressed but Cook shows that much can be gained if we can reasonably use them. When they are appropriate, the results in the corresponding propositions are rather stunning. It has long been