Propensity score matching (PSM) is one of the most widely-used causal inference methods to estimate the causal estimands such as average treatment effect or average treatment effect on the treated from observational studies. To implement PSM, a researcher first selects an appropriate set of confounders, estimates the propensity score, and matches the treated group with the control group using a matching algorithm such as nearest neighborhood or optimal matching. In this paper, we highlight the importance of investigating the assumptions employed in the PSM procedure thoroughly because they strongly affect the analysis result, but are not testable using observational data. We explain how to exploit the domain knowledge to avoid the potential risks from the violation of the untestable assumptions, and show how the research purpose is linked to selecting the matching algorithm and downstream analysis after PSM. In addition, to examine the vulnerability of the causal result, we highlight the use of sensitivity analysis for the analysis after PSM. These points are demonstrated in detail using National Supported Work data.