Summary: When data composed of several categorical responses together with categorical or continuous predictors are observed, it is often useful to relate the response probabilities to the predictors via a generalised linear model with a composite link function. This paper discusses a class of link functions that lie between the two extremes of the multivariate logistic transform of P. McCullagh and J.A. Nelder [Generalized linear models. 2. ed. (1989; Zbl 0744.62098)] and the log-linear decomposition of contingency table analysis. The models derived from these link functions are shown to inherit various desirable properties of both the multivariate logistic regression models and the log-linear regression models. A computational scheme for implementing these models is derived and they are demonstrated to be computationally more tractable than the multivariate logistic regression models. Their application is illustrated in a numerical example.