We have determined quasi-geometric distances to the Magellanic Clouds, M31 and M33. Our analysis uses a Bayesian statistical method to provide mathematically rigorous and objective solutions for individual Cepheids. We combine the individual distances with a hierarchical Bayesian model to determine the galactic distances. We obtain distance moduli 18.87 ± 0.07 mag (LMC, 12 stars), 19.14 ± 0.10 (SMC, 8 stars), 23.83 ± 0.35 mag (M33, 1 star) and 25.2 ± 0.6 mag (M31, 1 star) – all uncorrected for metallicity. The M31 and M33 distances are very preliminary. If the PL relations of the LMC, SMC and Galaxy are identical, our results exclude the metallicity effect in the V, (V – R) surface brightness method predicted by Hindsley & Bell (1989) at the 5σ level. Alternately, if Hindsley & Bell’s prediction is adopted as true, we find a metallicity effect intrinsic to the Cepheid PL relation requiring a correction Δ(V – Mv
) = (0.36 ± 0.07)Δ[A/H] mag. The latter has the opposite sign to other observational estimates of the Cepheid metallicity effect.