The relationship between CUB and loglinear models with latent variables

The "combination of uniform and shifted binomial" (CUB) model is a distribution for ordinal variables that has received considerable recent attention and specialized development. This article notes that the CUB model is a special case of the well-known loglinear latent class model, an observation that is useful for two reasons. First, we show how it can be used to estimate the CUB model in familiar standard software such as Mplus or Latent GOLD. Second, the mathematical equivalence of CUB with this well-known model and its correspondingly long history allows well-known results to be applied straightforwardly, subsuming a wide range of specialized recent developments of CUB and suggesting several possibly useful future ones. Thus, the observation that CUB and its extensions are restricted loglinear latent class models should be useful to both applied practitioners and methodologists.