Timeline for Argument on Interactions in The Book of Why
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 20, 2020 at 11:59 | vote | accept | T.E.G. | ||
| Jun 23, 2019 at 11:03 | vote | accept | T.E.G. | ||
| Nov 20, 2020 at 11:59 | |||||
| Jun 19, 2019 at 6:21 | comment | added | Carlos Cinelli | @T.E.G. that’s right, we would say this is not a linear structural equation | |
| Jun 19, 2019 at 5:39 | comment | added | T.E.G. | "A structural model is said to be linear if all functions are linear" in variables, right? So, a model like $Y= \beta_0 +\beta_1X_1 + \beta_2X_2 + \beta_3(X_1 \times X_2) + \varepsilon$ is not linear in causal modeling framework. | |
| Jun 19, 2019 at 5:03 | comment | added | Carlos Cinelli | Hi @T.E.G. the answer you cite is talking about regression models. Here we are talking about causal (structural) models. The structural equation y = f(x, z) is linear if f(x,z) is a linear function of x and z. You may be able to estimate f(x,z) with OLS and variable transformations, but f(x,z) is still not linear. A structural model is said to be linear if all functions are linear. This is not just a difference in semantics--if the structural model is not linear then, as Pearl says: (1) backdoor adjustment differs from regression adjustment; and (2) the decomposition TE = DE + IE does not hold | |
| Jun 18, 2019 at 14:59 | comment | added | T.E.G. | It seems to me that I am not conflating anything, only asking for clarification. Both answers boil down to this: Pearl means something else by “linear model.” But I have no reason to adopt the use of term Pearl prefers. And linear models (as in linear in parameters) do allow interactions. If what I overlook is only the different way “linear model“ used here, then I will accept this answer. | |
| Jun 18, 2019 at 14:58 | comment | added | T.E.G. | As stated in this answer (stats.stackexchange.com/a/8706/109647), “linear refers to the relationship between the parameters that you are estimating and the outcome.” That’s one of the first things I learned in a regression course; you can model nonlinear relations (e.g., polynomial terms) in linear regression. We preserve the term nonlinear for those models which are not linear in parameters (e.g., $y= e^{\beta} + \varepsilon$)… | |
| Jun 18, 2019 at 14:58 | comment | added | T.E.G. | Thank you, @CarlosCinelli. I know your interest in Pearl’s work from this thread (stats.stackexchange.com/a/376925/109647) and I am glad you had time to write an answer here. It is more detailed compared to previous answer, but basically agrees with it. So by linear model, Pearl means linear in variables, but not in parameters. But, here is my issue: the term “linear” in linear model does not refer to being linear in variables. As far as I know, it never does… | |
| Jun 18, 2019 at 0:50 | history | edited | Carlos Cinelli | CC BY-SA 4.0 |
added 41 characters in body
|
| Jun 18, 2019 at 0:24 | history | edited | Carlos Cinelli | CC BY-SA 4.0 |
deleted 4 characters in body
|
| Jun 18, 2019 at 0:14 | history | edited | Carlos Cinelli | CC BY-SA 4.0 |
added 207 characters in body
|
| Jun 18, 2019 at 0:09 | history | answered | Carlos Cinelli | CC BY-SA 4.0 |