We discuss a new condition based on the odds ratios for sensitivity analysis on two by two contingency tables, regarding the average effect of a treatment or exposure on a response or outcome, with estimates adjusted for and conditional on a single, unmeasured, dichotomous covariate. We derive the odds ratio condition with a mathematical proof showing the condition to be necessary for Simpson's paradox. In addition we will present the results of simulations showing the odds ratio condition to be as reliable as other commonly used conditions. While other conditions involve quantities naturally ascertained from a mediating covariate, the odds ratio condition can be easily applied when the covariate is a confounding variable. In an example application we demonstrate the utility of the odds ratio condition in data analysis.