andrew_gelman_stats andrew_gelman_stats-2013 andrew_gelman_stats-2013-1656 knowledge-graph by maker-knowledge-mining
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Introduction: David Hoaglin writes: After seeing it cited, I just read your paper in Technometrics. The home radon levels provide an interesting and instructive example. I [Hoaglin] have a different take on the difficulty of interpreting the estimated coefficient of the county-level basement proportion (gamma-sub-2) on page 434. An important part of the difficulty involves “other things being equal.” That sounds like the widespread interpretation of a regression coefficient as telling how the dependent variable responds to change in that predictor when the other predictors are held constant. Unfortunately, as a general interpretation, that language is oversimplified; it doesn’t reflect how regression actually works. The appropriate general interpretation is that the coefficient tells how the dependent variable responds to change in that predictor after allowing for simultaneous change in the other predictors in the data at hand. Thus, in the county-level regression gamma-sub-2 summarize
sentIndex sentText sentNum sentScore
1 I [Hoaglin] have a different take on the difficulty of interpreting the estimated coefficient of the county-level basement proportion (gamma-sub-2) on page 434. [sent-3, score-0.361]
2 ” That sounds like the widespread interpretation of a regression coefficient as telling how the dependent variable responds to change in that predictor when the other predictors are held constant. [sent-5, score-1.439]
3 The appropriate general interpretation is that the coefficient tells how the dependent variable responds to change in that predictor after allowing for simultaneous change in the other predictors in the data at hand. [sent-7, score-1.717]
4 Thus, in the county-level regression gamma-sub-2 summarizes the relation of alpha to x-bar after allowing for the contribution of u (the log of the uranium level in the county). [sent-8, score-0.675]
5 What was the relation between the basement proportion and the uranium level? [sent-9, score-0.406]
6 I continue to be surprised at the number of textbooks that shortchange students by teaching the “held constant” interpretation of coefficients in multiple regression. [sent-18, score-0.363]
7 ” My reply: As Jennifer and I discuss in our book, regression coefficients can be interpreted in more than one way. [sent-23, score-0.364]
8 Hoaglin writes, “The appropriate general interpretation is that the coefficient tells how the dependent variable responds to change in that predictor after allowing for simultaneous change in the other predictors in the data at hand,” and Speed says something similar. [sent-24, score-1.717]
9 But I don’t actually find that description very helpful because I don’t really know how to interpret the phrase, “allowing for simultaneous change in the other predictors. [sent-25, score-0.472]
10 ” If I’m in purely descriptive mode, I prefer to say that, if you’re regressing y on u and v, the coefficient of u is the average difference in y per difference in u, comparing pairs of items that differ in u but are identical in v. [sent-26, score-0.77]
11 (See my paper with Pardoe on average predictive comparisons for more on this idea, including how to define this averaging so that, in a simple linear model, you end up with the usual regression coefficient. [sent-27, score-0.478]
12 Because, in its most basic form, regression tells you nothing at all about change. [sent-31, score-0.409]
13 For sparse or continuous data, you can’t really find these comparisons where v is identical, so it’s clear that regression coefficients are model-based. [sent-37, score-0.513]
14 In that sense, I don’t mind vague statements such as “allowing for simultaneous change in the other predictors. [sent-38, score-0.36]
15 ” I’d prefer the term “comparison” rather than “change,” but the real point is that regression coefficients represent averages in a sort of smoothed comparison, a particular smoothing based on a linear model. [sent-39, score-0.439]
16 More in a bit, but first another quote from Terry: Think of the world of difference between using a regression model for prediction and using one for estimating a parameter with a causal interpretation, for example, the effect of class size on school children’s test scores. [sent-53, score-0.603]
17 With prediction, we don’t need our relationship to be causal, but we do need to be concerned with the relation between our training and our test set. [sent-54, score-0.415]
18 When estimating the causal parameter, we do need to ask whether the children were randomly assigned to classes of different sizes, and if not, we need to find a way to deal with possible selection bias. [sent-56, score-0.408]
19 Terry seems unaware of the potential-outcome framing of causal inference, in which causal estimands are defined in terms of various hypothetical scenarios. [sent-58, score-0.314]
20 Terry continues: I would like to see multiple regression taught as a series of case studies, each study addressing a sharp question, and focussing on those aspects of the topic that are relevant to that question. [sent-61, score-0.376]
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