\frac{T_{\rm S} - T_{\gamma}}{1+z}\left(1-{\rm e}^{-\tau_{\nu_0}}\right) \nonumber \\
&\approx& 27x_{{\ifmmode\mathrm{H\,{\scriptscriptstyle I}}\else{}H\,{\scriptsize I}\fi}{}}(1+\delta_{\rm nl})\left(\frac{H}{{\rm d}v_{\rm r}/{\rm d}r+H}\right)
\left(1 - \frac{T_{\gamma}}{T_{\rm S}}\right) \nonumber \\
& & \times \left(\frac{1+z}{10}\frac{0.15}{\Omega_{\rm m}h^{2}}\right)^{1/2}
\left(\frac{\Omega_{\rm b}h^{2}}{0.023}\right)~{\rm mK},\end{aligned}$$ where $T_{\rm S}$ is the gas spin temperature, $\tau_{\nu_0}$ is the optical depth at the 21 cm frequency, $\nu_0$, $\delta_{\rm nl}(\boldsymbol{x},z)$ is the evolved (Eularian) overdensity, $H(z)$ is the Hubble parameter, ${\rm d}v_{\rm r}/{\rm d}r$ is the gradient of the line of sight component of the velocity and all quantities are evaluated at redshift $z = \nu_{0}/\nu - 1$. Throughout this work, we include the effects of peculiar velocities and assume $T_{\rm S} \gg T_{\gamma}$ (likely appropriate for the advanced stages of reionization, corresponding to an average neutral fraction of $\bar{x}_{{\ifmmode\mathrm{H\,{\scriptscriptstyle I}}\else{}H\,{\scriptsize I}\fi}} \lesssim 0.8$, e.g. @Mesinger:2013p1835) as the explicit computation of the spin temperature within [<span style="font-variant:small-caps;">21CMFAST</span>]{} is computationally expensive; however, in future we will remove this assumption.

Although [<span style="font-variant:small-caps;">21CMFAST</span>]{} produces 3D maps of the 21 cm brightness temperature, we require a goodness of fit statistic in order to quantitatively compare an [EoR]{} realization with the mock observation. We choose the common spherically averaged 21 cm PS, $\Delta^{2}_{21}(k,z) \equiv k^{3}/(2\upi^{2}V)\,\delta \bar{T}^{2}_{\rm b}(z)\,\langle |\delta_{21}(\boldsymbol{k},z)|^{2}\rangle_{k}$ where $\delta_{21}(\boldsymbol{x},z) \equiv \delta T_{\rm b}(\boldsymbol{x},z)/\delta 