Residual confounding is a common source of bias in observational studies. In this article, we build upon a series of sensitivity analyses methods for residual confounding developed by Brumback et al. and Chiba whose sensitivity parameters are constructed to quantify deviation from conditional exchangeability, given measured confounders. These sensitivity parameters are combined with the observed data to produce a “bias-corrected” estimate of the causal effect of interest. We provide important generalizations of these sensitivity analyses, by allowing for arbitrary exposures and a wide range of different causal effect measures, through the specification of the target causal effect as a parameter in a generalized linear model with the arbitrary link function. We show how our generalized sensitivity analysis can be easily implemented with standard software, and how its sensitivity parameters can be calibrated against measured confounders. We demonstrate our sensitivity analysis with an application to publicly available data from a cohort study of behavior patterns and coronary heart disease.