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We apply the Projected Gross-Pitaevskii equation (PGPE) formalism to theexperimental problem of the shift in critical temperature $T_c$ of aharmonically confined Bose gas as reported in Gerbier \emph{et al.} [Phys. Rev.Lett. \textbf{92}, 030405 (2004)]. The PGPE method includes criticalfluctuations and we find the results differ from various mean-field theories,and are in best agreement with experimental data. To unequivocally observebeyond mean-field effects, however, the experimental precision must eitherimprove by an order of magnitude, or consider more strongly interactingsystems. This is the first application of a classical field method to makequantitative comparison with experiment.Comment: revtex4, four pages, three figures. v2: updated to published version.  Several additions to figures, and better explanations in text in response to  referee comment

Critical Temperature of a Trapped Bose Gas: Comparison of Theory and Experiment