On some models for multivariate binary variables parallel in complexity with the multivariate Gaussian distribution

It is shown that both the simple form of the Rasch model for binary data and a generalisation are essentially equivalent to special dichotomised Gaussian models.In these the underlying Gaussian structure is of single factor form; that is, the correlations between the binary variables arise via a single underlying variable, called in psychometrics a latent trait. The implications for scoring of the binary variables are discussed, in particular regarding the scoring system as in effect estimating the latent trait. In particular, the role of the simple sum score, in effect the total number of ‘successes’, is examined. Relations with the principal component analysis of binary data are outlined and some connections with the quadratic exponential binary model are sketched.