Six rate decay functions have been fit to 14 aftershock sequences, mostly from southern California. The best fit was usually by the modified Omori function, but five of the sequences were best fit by new variants of the stretched exponential model, and three more were best fit by the modified Omori with a steady background rate added. Background rate terms in the modified Omori model never improved the fit to sequences best fit by exponential models. The variety of decay laws observed can be explained if the underlying aftershock generation process follows stretched exponential relations with spatial inhomogeneities in q value. Tests of a large number of synthetic sequences show that if different parts of a fault zone produced aftershocks decaying with stretched exponentials having a wide variety of q values, the combined sequence will be best fit by a variant of the modified Omori relation. The sequences best fit by modified exponential may be associated with more homogeneous material properties or simpler geometry.