When I'm using linear regression as an estimation method to infer the average treatment effect (between treatment and outcome), am I assuming there is a linear relation only between treatment and outcome? Or am I assuming a linear relation between treatment, outcome and confounding variable(s)?
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2 One assumption is that the regression model is correctly specified. This assumption implies that (1) treatment and outcome are linear and (2) confounding variables and outcome are linear if only main effect terms are included.
For (2), you could include quadratic, spline, or other nonlinear terms. You just need to assume you have correctly modeled the relationship for all those confounding variables. See this paper for a full discussion of the assumptions, including correct model specification.
- $\begingroup$ I'm sorry, English is not my main language. I'm having trouble understanding (2). Do I just only need to assume any type of causal relationship between confounding variables and outcome? In other words: when using linear regression as an estimation method to infer the average treatment effect, I'm assuming a linear relationship between the treatment and the outcome variable only (the rest could be any type of causal relationship)? $\endgroup$Rui Lima– Rui Lima2023-05-01 22:57:16 +00:00Commented May 1, 2023 at 22:57
- 1$\begingroup$ No problem. The rest of the variables (the confounding variables) can be any type of relationship (linear, non-linear, step, etc.). The important part for (2) is that our model includes those variables correctly. Say W is a continuous confounding variable and Y is our outcome. If the true relationship between W and Y is quadratic but we model it as linear, we might not get the right answer (i.e., we will have bias for the ATE). Correct model specification doesn't say anything about the specific functional form between W and Y, just that our model needs to get that relationship correct. $\endgroup$pzivich– pzivich2023-05-02 13:35:19 +00:00Commented May 2, 2023 at 13:35