An Evaluation of Non‐iterative Methods for Estimating the Linear‐by‐linear Parameter of Ordinal Log‐linear Models

Summary Parameter estimation for association and log‐linear models is an important aspect of the analysis of cross‐classified categorical data. Classically, iterative procedures, including Newton's method and iterative scaling, have typically been used to calculate the maximum likelihood estimates of these parameters. An important special case occurs when the categorical variables are ordinal and this has received a considerable amount of attention for more than 20 years. This is because models for such cases involve the estimation of a parameter that quantifies the linear‐by‐linear association and is directly linked with the natural logarithm of the common odds ratio. The past five years has seen the development of non‐iterative procedures for estimating the linear‐by‐linear parameter for ordinal log‐linear models. Such procedures have been shown to lead to numerically equivalent estimates when compared with iterative, maximum likelihood estimates. Such procedures also enable the researcher to avoid some of the computational difficulties that commonly arise with iterative algorithms. This paper investigates and evaluates the performance of three non‐iterative procedures for estimating this parameter by considering 14 contingency tables that have appeared in the statistical and allied literature. The estimation of the standard error of the association parameter is also considered.