Starburst galaxies and star-forming regions in the Milkyway, with high rate of supernova activities, are candidate sources of high-energy neutrinos. Using a gamma-ray selected sample of these sources we perform statistical analysis of their angular correlation with the four-year sample of high-energy starting events (HESE), detected by the IceCube Neutrino Observatory. We find that the two samples (starburst galaxies and local star-forming regions) are correlated with cosmic neutrinos at $\sim (2-3)\sigma$ (pre-trial) significance level, when the full HESE sample with deposited energy $\gtrsim 20$~TeV is considered. However when we consider the HESE sample with deposited energy $\gtrsim 60$~TeV, which is almost free of atmospheric neutrino and muon backgrounds, the significance of correlation decreased drastically. We perform a similar study for Galactic sources in the 2FHL catalog as well, obtaining $\sim (2-3)\sigma$ (pre-trial) correlation, however the significance of correlation increases with higher cutoff energy in the HESE sample for this case. We also fit available gamma-ray data from these sources using a $pp$ interaction model and calculate expected neutrino fluxes. We find that the expected neutrino fluxes for most of the sources are at least an order of magnitude lower than the fluxes required to produce the HESE events from these sources. This puts the starburst sources being the origin of the IceCube HESE events in question.