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Suppose I have a study with a response variable $y$ and two explanatory variables $x_1, x_2$. I do a regression such as lm(y~x1) and get a p-value of $p_1'$ for $x_1$. Then I control for $x_2$ and run lm(y~x1 + x2) and get $p_1''$ for $x_1$ and $p_2''$.

Question 1 How would I interpret the following cases in terms of if my model benefits from adding the covariate?

  1. $p_1'>p_1''$
  2. $p_1'<p_1''$
  3. $p_1'=p_1''$

Question 2 Will the answer extend to more explanatory covariates?

Question 3 What role does $p_2''$ play in my analysis/interpretation?

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You don't interpret whether some p-values are smaller or larger than each other, that's a meaningless comparison.

You would usually focus on one of the p-values (assuming both are meaningful analyses) and pre-specify that in your protocol before seeing the data. The other analysis might be a supportive analysis that you also look at or something that you look at, because it answers a different question, which you are also interested in.

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  • $\begingroup$ Is my model better or worse with adding the covariate? $\endgroup$ – abalter Mar 6 '19 at 21:04
  • $\begingroup$ It may be better, it might be worse, one of them might be totally inappropriate for three question you are asking and so on, there is no generic answer to that question, it depends on the context. $\endgroup$ – Björn Mar 7 '19 at 5:21
  • $\begingroup$ The goal is to assess whether or not the additional variables are confounders. That is, do I need to control for them, or are they irrelevant. How do I determine this? $\endgroup$ – abalter Mar 7 '19 at 20:53
  • $\begingroup$ By logically considering for situation, looking at p-values would be a terrible way of doing that. In fact, in an observational study adjusting for just one potential confounder would be a strong signal that the model and/or the collected data are inadequate. $\endgroup$ – Björn Mar 8 '19 at 5:29
  • $\begingroup$ I think I need to formulate a new question asking specifically what I'm trying to get at. $\endgroup$ – abalter Mar 8 '19 at 18:08

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