  We apply the Projected Gross-Pitaevskii equation (PGPE) formalism to the
experimental problem of the shift in critical temperature $T_c$ of a
harmonically confined Bose gas as reported in Gerbier \emph{et al.} [Phys. Rev.
Lett. \textbf{92}, 030405 (2004)]. The PGPE method includes critical
fluctuations and we find the results differ from various mean-field theories,
and are in best agreement with experimental data. To unequivocally observe
beyond mean-field effects, however, the experimental precision must either
improve by an order of magnitude, or consider more strongly interacting
systems. This is the first application of a classical field method to make
quantitative comparison with experiment.
