Confounding is a major concern in observational studies. To adjust for confounding bias, the potential confounder(s) for a study must first be identified and measured. But this is not always possible. The unmeasured factors may also exhibit effect modification, and this further complicates the situation. In this paper, the author derives bounding formulas for the bias of unmeasured factors with confounding and effect-modifying potentials. Based on these formulas, the author derives two conditions (for the unmeasured factors) to explain away an observed positive finding: the low-threshold (for the minimum of two parameters related to the unmeasured factors) and the high-threshold (for the maximum) conditions. All these should help researchers make more prudent interpretations of their (potentially biased) results. Copyright © 2011 John Wiley & Sons, Ltd.