As empathy is important for the emotional interaction between a human and a robot, the design factors which induce human empathy toward robots need to be explored. Human empathy toward a robot can be affected by the presence of a robot. Thus, we focused on the levels of agency and the physical embodiment of a robot, which are influential factors pertaining to social presence, by executing two experiments. In the first experiment, in a 2 (levels of agency: mediated vs. simulated) between-participants experiment, participants interacted with either a mediated robot which delivers the emotional state of a remote user or a simulated robot which expresses its own emotion. Participants empathized more with the mediated robot than with the simulated robot, demonstrating that the proper form of an emotional robot is as a mediator during emotional interaction between people. In the second study, in a 2 (physical embodiment: physically embodied vs. physically disembodied) between-participants experiment design, participants interacted with either a physically embodied robot or a physically disembodied robot. The results showed that participants empathized more with a physically embodied robot than with a physically disembodied robot, indicating the impact of physical embodiment on human empathy. Implications for the design of human-robot interactions are discussed.