We report a systematic study of nonlinearity in the ferromagnetic resonance of a series of submicron Permalloy ellipses with varying aspect ratios. At high excitation powers, the resonances are found to shift to higher or lower applied field. We focus here on the sign of the shift and its dependence on the applied field and shape-induced anisotropy of the ellipses. Using ferromagnetic resonance force microscopy, we find that the measured nonlinear coefficient changes sign as a function of anisotropy field and applied field in qualitative agreement with a macrospin analysis. This macrospin analysis also points to origins of the nonlinearity in a combination of hard-axis in-plane anisotropy and precession ellipticity. In comparison of the macrospin predictions with both experimental and micromagnetic modeling results, we measure/model values of the nonlinear coefficient that are more positive than predicted by the macrospin model. The results are useful in understanding nonlinear physics in nanomagnets and applications of spin-torque oscillators.