In observational studies, when estimating the causal effect of X on Y based on a limited sample and using regression, in addition to controlling for confounders, is it necessary to control for variables that have a significant effect on the dependent variable Y but no effect on the independent variable X? What is the purpose of doing so? (disregarding selection bias and measurement error)
A simple example that may clarify this for you is when both X and Y are caused by a third variable (confounder) Z. For example, there is a (negative) relationship between individuals' hair length (X) and their height (Y). However, hair length does not cause height. The relationship exists only because of the confounder gender (Z). Women are (on average) smaller than men and tend to have longer hair than men (on average). The relationship between hair length and height is thus a spurious relationship. It should disappear (go to zero) once you control for the confounder gender, for example, through multiple regression or partial correlation. Notice that the confounder is correlated with both X and Y. The confounder so to speak "explains" why X and Y are correlated.
Thank you for your kindly reply! Of course, as in the example you gave, the common cause Z (gender) of X (hair length) and Y (height) would lead to a non-causal correlation between the two. Only after adjusting for these confounding factors can the correlation between X and Y be interpreted as causal.
However, my confusion is that if there is a variable U which is the direct cause of Y but is unrelated to X. Obviously, U is not a confounder because it is not a common cause of X and Y, then is an adjustment for U necessary at this point? I have often seen in the literature the practice of including similar variables as control variables in regression models. Am I to understand that the reason for doing this is to control for random error due to a finite sample (the correlation of U with X due to a finite random sample, although in theory they are uncorrelated)?@Christian Geiser
Random measurement error does not typically lead to spurious correlations (although this is not completely impossible). Instead, random measurement error would typically make correlations smaller ("attenuate" correlation coefficients) because random measurement error adds random noise (error variance). There is even a formula in classical test theory to correct for this (called "correction for attenuation"; developed by Charles Spearman). Correlations may be different from zero due to random sampling error, but this is not typically so much of a concern and it would not be corrected for by adding other variables U to the model.
Adding variables U that are causes of Y (but that are uncorrelated with X) is useful to have a complete specification of the (regression) model (as complete as possible). That is, you want to include all relevant causes of Y (if possible) to avoid underspecification/underfitting of the model and to reduce the error/residual/unexplained variance in Y.
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