Least‐squares estimators of the linear‐by‐linear association parameter from an ordinal log‐linear model

Summary When modelling the association between the ordinal categorical variables of a contingency table, ordinal log‐linear models are typically used; these models are a variation of the more popular log‐linear model, which has attracted considerable attention in the statistics and allied literature since the late 1960s. Estimating the parameters of an ordinal log‐linear model usually involves the use of iterative techniques, typically Newton's method and iterative proportional fitting. However, the early 2000s brought with it more direct estimation methods that do not require the use of iterative techniques. When the focus is on the parameter that reflects the linear‐by‐linear association between the variables, these methods have proven to provide unbiased, consistent and normally distributed estimates. Despite this new work, no attention has been given to the estimation of the least‐squares estimator. Therefore, this article derives the least‐squares estimator of the linear‐by‐linear association parameter and shows it to be equivalent to one of the existing non‐iterative estimators recently described. We also derive two further least‐squares estimators based on the Box‐Cox transformation and derive their variance.