  This paper revisits the classical Kennicutt method for inferring the stellar
initial mass function (IMF) from the integrated light properties of galaxies.
The large size, uniform high quality data set from the Sloan Digital Sky Survey
DR4 is combined with more in depth modeling and quantitative statistical
analysis to search for systematic IMF variations as a function of galaxy
luminosity. Galaxy H alpha equivalent widths are compared to a broadband color
index to constrain the IMF. This parameter space is useful for breaking
degeneracies which are traditionally problematic. Age and dust corrections are
largely orthogonal to IMF variations. In addition the effects of metallicity
and smooth star formation history e-folding times are small compared to IMF
variations. We find that for the sample as a whole the best fitting IMF slope
above 0.5 M_sun is Gamma = 1.4535 with a negligible random error of +/- 0.0004
and a systematic error of +/- 0.1. Galaxies brighter than around M_r,0.1 = -20
(including galaxies like the Milky Way which has M_r,0.1 ~ -21) are well fit by
a universal Gamma ~ 1.4 IMF, similar to Salpeter, and smooth, exponential star
formation histories (SFH). Fainter galaxies prefer steeper IMFs and the quality
of the fits reveal that for these galaxies a universal IMF with smooth SFHs is
actually a poor assumption. Several sources of sample bias are ruled out as the
cause of these luminosity dependent IMF variations. Analysis of bursting SFH
models shows that an implausible coordination of burst times is required to fit
a universal IMF to the M_r,0.1 = -17 galaxies. This leads to the conclusions
that the IMF in low luminosity galaxies has fewer massive stars, either by
steeper slope or lower upper mass cutoff, and is not universal.
