Many of the results in modern astrophysics rest on the notion that the initial mass function (IMF) is universal. Our observations of a sample of H I selected galaxies in the light of Ha and the far-ultraviolet (FUV) challenge this result. The extinction-corrected flux ratio FH(alpha/)f(FUV) from these two tracers of star formation shows strong correlations with the surface brightness in Ha and the R band: low surface brightness (LSB) galaxies have lower FH(alpha/)f(FUV) ratios compared to high surface brightness galaxies as well as compared to expectations from equilibrium models of constant star formation rate (SFR) using commonly favored IMF parameters. Weaker but significant correlations of FH(alpha/)f(FUV) with luminosity, rotational velocity, and dynamical mass as well as a systematic trend with morphology, are found. The correlated variations of FH(alpha/)f(FUV) with other global parameters are thus part of the larger family of galaxy scaling relations. The FH(alpha/)f(FUV) correlations cannot be due to residual extinction correction errors, while systematic variations in the star formation history (SFH) cannot explain the trends with both Ha and R surface brightness nor with other global properties. The possibility that LSB galaxies have a higher escape fraction of ionizing photons seems inconsistent with their high gas fraction, and observations of color-magnitude diagrams (CMDs) of a few systems which indicate a real deficit of O stars. The most plausible explanation for the correlations is the systematic variations of the upper mass limit M(u) and/or the slope. which define the upper end of the IMF. We outline a scenario of pressure driving the correlations by setting the efficiency of the formation of the dense star clusters where the highest mass stars preferentially form. Our results imply that the SFR measured in a galaxy is highly sensitive to the tracer used in the measurement. A nonuniversal IMF would also call into question the interpretation of metal abundance patterns in dwarf galaxies as well as SFHs derived from CMDs.