We consider an analytic model of cosmic star formation which incorporates supernova feedback, gas accretion and enriched outflows, reproducing the history of cosmic star formation, metallicity, Type II supernova rates and the fraction of baryons allocated to structures. We present a new statistical treatment of the available observational data on the star formation rate and metallicity that accounts for the presence of possible systematics. We then employ a Bayesian Markov Chain Monte Carlo method to compare the predictions of our model with observations and derive constraints on the seven free parameters of the model. We find that the dust-correction scheme one chooses to adopt for the star formation data is critical in determining which scenario is favoured between a hierarchical star formation model, where star formation is prolonged by accretion, infall and merging, and a monolithic scenario, where star formation is rapid and efficient. We distinguish between these modes by defining a characteristic minimum mass, M≳ 1011M⊙, in our fiducial model, for early-type galaxies where star formation occurs efficiently. Our results indicate that the hierarchical star formation model can achieve better agreement with the data, but that this requires a high efficiency of supernova-driven outflows. In a monolithic model, our analysis points to the need for a mechanism that drives metal-poor winds, perhaps in the form of supermassive black hole induced outflows. Furthermore, the relative absence of star formation beyond z∼ 5 in the monolithic scenario requires an alternative mechanism to dwarf galaxies for re-ionizing the universe at z∼ 11, as required by observations of the microwave background. While the monolithic scenario is less favoured in terms of its quality-of-fit, it cannot yet be excluded.