Finding confounder in multiple regression 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:

*

*comparing the coefficient of X between model 1 & model 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!

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 $Z$.
