Aftershock decay is often correlated with the modified Omori's law: dN/ dt = tau(-1)( 1 + t/c)(-p), where dN/ dt is the occurrence rate of aftershocks with magnitudes greater than a lower cutoff m, t is time since a mainshock, t and c are characteristic times, and p is an exponent. Extending this approach, we derive two possibilities: ( 1) c is a constant independent of m and t scales with m and ( 2) c scales with m and t is a constant independent of m. These two are tested by using aftershock sequences of four relatively recent and large earthquakes in Japan. We first determine for each sequence the threshold magnitude above which all aftershocks are completely recorded and use only events above this magnitude. Then, visual inspection of the decay curves and statistical analysis shows that the second possibility is the better approximation for our sequences. This means that the power law decay of smaller aftershocks starts after larger times from the mainshock. We find a close association of our second result with a solution obtained for a damage mechanics model of aftershock decay. The time delays associated with aftershocks, according to the second possibility, can be understood as the times needed to nucleate microcracks ( aftershocks). Our result supports the idea that the c value is a real consequence of aftershock dynamics associated with damage evolution.