Table 1.

Uncorrelated 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	$\mathit{OVB}(\widehat{\tau }\mid X^{\ast })=\frac{{\alpha }_U{\beta }_U+{\alpha }_X{\beta }_X(1-\gamma )}{1-{\alpha }_X^2\gamma }$
Imbalance in U	Imbalance(U | {}) = αU	$\mathit{Imbalance}(U\mid X^{\ast })=\frac{{\alpha }_U}{1-{\alpha }_X^2\gamma }$
Imbalance in X	Imbalance(X|{}) = αX	$\mathit{Imbalance}(X\mid X^{\ast })=\frac{{\alpha }_X(1-\gamma )}{1-{\alpha }_X^2\gamma }$
	Effect of conditioning on X* when …
	biases are in the same 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 or confounder X strongly affects Z.	If the bias induced by the unobserved confounder U exceeds half of the bias induced by X, OVB always increases (this case also includes almost perfectly offsetting biases). If the bias induced by the unobserved confounder U is less than half of the bias induced by X, OVB most likely increases if X strongly affects Z (provided X is reliably measured).
Absolute imbalance	Imbalance in U always increases. Imbalance in X always decreases.	Imbalance in U always increases. Imbalance in X always decreases.
Effect of measurement error	Attenuates any increase in OVB and attenuates any decrease in OVB.	If the bias induced by the unobserved confounder U exceeds half of the bias induced by X, measurement error attenuates any increase in OVB. If the bias induced by the unobserved confounder U is less than half of the bias induced by X, measurement error attenuates any increase in OVB (and might even turn an increase into a decrease) but may attenuate or strengthen any decrease in OVB.