We analyze isothermal aging of a four-dimensional Edwards-Anderson model in detail by Monte Carlo simulations. We analyze the data in the view of an extended version of the droplet theory proposed recently which is based on the original droplet theory plus conjectures on the anomalously soft droplets in the presence of domain walls. We found that the scaling laws including some fundamental predictions of the original droplet theory explain our results well. The results of our simulation strongly suggest the separation of the breaking of the time translational invariance and the fluctuation dissipation theorem in agreement with our scenario.