Table 2.

Correlated confounders X and U: Omitted variable bias (OVB) and imbalance before and after adjusting for X*.

	Initial OVB and Imbalance	OVB and Imbalance after adjusting for X*
Omitted variable bias	OVB(τ̂ | {}) = αXβX + αUβU + αXρβU + αUρβX	$\mathit{OVB}(\widehat{\tau }\mid X^{\ast })=\frac{{\alpha }_U{\beta }_U(1-{\stackrel{\sim }{\rho }}^2)+({\alpha }_X{\beta }_X+{\alpha }_X\rho {\beta }_U+{\alpha }_U\rho {\beta }_X)(1-\gamma )}{1-{({\alpha }_X+{\alpha }_U\rho )}^2\gamma }$
Imbalance in U	Imbalance(U | {}) = αU + αXρ	$\mathit{Imbalance}(U\mid X^{\ast })=\frac{{\alpha }_U(1-{\stackrel{\sim }{\rho }}^2)+{\alpha }_X\rho (1-\gamma )}{1-{({\alpha }_X+{\alpha }_U\rho )}^2\gamma }$
Imbalance in X	Imbalance(X|{}) = αX + αUρ	$\mathit{Imbalance}(X\mid X^{\ast })=\frac{({\alpha }_X+{\alpha }_U\rho )(1-\gamma )}{1-{({\alpha }_X+{\alpha }_U\rho )}^2\gamma }$
	Effect of conditioning on X* when …
	biases are in same the direction	biases offset each other
Absolute omitted variable bias	Increase in OVB is most likely if the bias induced by the unobserved confounder U is much larger than the bias induced by confounder X and the correlation between X and U is low, or confounder X strongly affects Z –a high correlation between X and U strongly boosts bias amplification.	Whether OVB increases strongly depends. on the signs and magnitudes of all five parameters. If the biases induced by X and U strongly offset each other, an increase in OVB almost surely results – unless the correlation between X and U is close to 1.
Absolute imbalance	Imbalance in U may increase or decrease. Imbalance in X always decreases.	Imbalance in U may increase or decrease. Imbalance in X always decreases.
Effect of measurement error	Attenuates any increase in OVB and attenuates any decrease in OVB.	Attenuates any increase in OVB (and might even turn an increase into a decrease) but may attenuate or strengthen any decrease in OVB.