For many classification problems, in addition to providing accurate classification results, very often it is equally important to determine which predictors are contributing to the results, and by how much. This motivates this article, where a new sensitivity analysis framework is developed for classification problems to address this issue. The effectiveness of this framework is illustrated through simulation studies and application to real data. Notice that this framework can be coupled with different classification methods. However, in practice it is recommended to pair it with a classification method called Bayesian smoothing spline ANOVA probit regression (BSSANOVA). When compared with other existing methods through numerical experiments, BSSANOVA performs extremely well for both classification and sensitivity analysis.