Non-iterative estimation methods for ordinal log-linear models

Log-linear modelling has proven to be a rich and diverse area of research in categorical data analysis since the 1950s. However, until the mid-1970s, the log-linear model has only considered nominal variables and did not take into consideration the structure of ordered categories. So, it was during this time that ordinal loglinear models became popular. While such models were not commonly applied at the time, they grew in use in most fields of research, especially in the social sciences. Of all of the ordinal categorical data analytic techniques that are now available, ordinal log-linear models are amid the most widely used. Traditionally, the parameters from such models have been estimated using iterative algorithms including the Newton-Raphson method and iterative proportional fitting. However, issues such as the choice of poor initial values, poor (or no) rates of convergence, excessively high number of iterations and the size of the contingency table all can hamper the estimation of the parameters using these iterative techniques. For the analysis of the association between the categorical variables of a two-way and three-way contingency table, more recent advances have been proposed for parameter estimation for ordinal log-linear models. These advances include non-iterative estimation techniques that give numerically similar estimates to those obtained using conventional iterative methods. These recently introduced non-iterative estimation methods provide an alternative, and interestingly, closed-form estimates which do not require any iteration to estimate the association parameters of the ordinal log-linear models.