We develop factor copula models for analysing the dependence among mixed continuous and discrete responses. Factor copula models are canonical vine copulas that involve both observed and latent variables, hence they allow tail, asymmetric and non-linear dependence. They can be explained as conditional independence models with latent variables that don't necessarily have an additive latent structure. We focus on important issues that would interest the social data analyst, such as model selection and goodness-of-fit. Our general methodology is demonstrated with an extensive simulation study and illustrated by re-analysing three mixed response datasets. Our study suggests that there can be a substantial improvement over the standard factor model for mixed data and makes the argument for moving to factor copula models.