Since the seminal introduction of the propensity score (PS) by Rosenbaum and Rubin, PS-based methods have been widely used for drawing causal inferences in the behavioral and social sciences. However, the PS approach depends on the ignorability assumption: there are no unobserved confounders once observed covariates are taken into account. For situations where this assumption may be violated, Heckman and his associates have recently developed a novel approach based on marginal treatment effects (MTEs). In this article, we (1) explicate the consequences for PS-based methods when aspects of the ignorability assumption are violated, (2) compare PS-based methods and MTE-based methods by making a close examination of their identification assumptions and estimation performances, (3) apply these two approaches in estimating the economic return to college using data from the National Longitudinal Survey of Youth (NLSY) of 1979 and discuss their discrepancies in results. When there is a sorting gain but no systematic baseline difference between treated and untreated units given observed covariates, PS-based methods can identify the treatment effect of the treated (TT). The MTE approach performs best when there is a valid and strong instrumental variable (IV). In addition, this article introduces the "smoothing-difference PS-based method," which enables us to uncover heterogeneity across people of different PSs in both counterfactual outcomes and treatment effects.