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I want to identify confounding factors with multiple regression. I have two models.

Y ~ X     # model 1, Y = outcome, X = exposure
Y ~ X + Z # model 2, Z = a potential confounder

Can I tell whether Z is a confounder in X→Y by:

  1. comparing the coefficient of X between model 1 & model 2
  2. comparing the 95% CI or significance of the coefficient of X between model 1 & model 2

The reason I'm having this question is, as far as I know, the existance of a confounder masked the true association between X and Y. However, in a multiple regression, the coefficient of X does not represent the level of association between independent variable and dependent variable. Also, some example showed that by including a confounder into the regression model, the coefficient of exposure changed from biased to unbiased. So I don't know if I can identify a confounder by comparing the coefficient or properties of the coefficient (SE, Var, p-value etc.) before and after adjusting for the confounder. Thank you!

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You cannot identify confounders in this way. This is because the same pattern of results could appear if $Z$ was a confounder or not. If $Z$ was a mediator or a collider, for example, you would see the coefficient on $X$ change between the two models. Also, if there is confounding by other unmeasured variables, the coefficient on $X$ may change regardless of the status of $Z$.

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  • $\begingroup$ Thanks for the answer! By saying "the coefficient on X change between the two models" do you mean change of significance or change of value? "if there is confounding by other unmeasured variables" Could you be more specific? Does that mean if there is any uncontroled confonuder of X→Y, the coefficient of X changed when I included a covariate Z? I thought the coefficient of X changes more or less when I included a new independent variable regardless the existance of an uncontroled confounder? $\endgroup$
    – Ian Wang
    Nov 17, 2021 at 3:15
  • $\begingroup$ Significance is a function of the sample size, and the change in the sample estimates could reflect sampling variability rather than a legitimate change due to removing bias due to confounding. I'm saying that even if you had population data, you could not determine whether $Z$ is a confounder from this method (or any statistical method that does not rely on causal assumptions). $\endgroup$
    – Noah
    Nov 17, 2021 at 3:19
  • $\begingroup$ If there is unmeasured confounding, including an instrumental variable (which is not a confounder and which would not change the coefficient on $X$ is the absence of unmeasured confounding), would change the coefficient on $X$ (in particular, it would induce bias). See Steiner & Kim (2016) for example. I recommend reading a text on causal inference to learn more about confounding. Pearl's Book of Why and Hernan and Robins' What If are excellent introductions. $\endgroup$
    – Noah
    Nov 17, 2021 at 3:22
  • $\begingroup$ I have a question about this "the change in the sample estimates could reflect sampling variability". To my understanding, sampling variability happens when I resampled from the same population. But in the context of adjusting for a potential confounder, I'm still using the same sample data with the same value of exposure and outcome. $\endgroup$
    – Ian Wang
    Nov 17, 2021 at 11:05
  • $\begingroup$ What I meant was that events that occur in analyzing your sample may not perfectly reflect patterns in the population because your sample is an imperfect representation of the population. The coefficients in your sample will not equal the population coefficients, and the magnitude of the change in coefficients in your sample when you perform your procedure will not equal the change in coefficients were you to perform the same procedure using population data. $\endgroup$
    – Noah
    Nov 17, 2021 at 15:02

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