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

Abstract

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.

A Computational Study Assessing Maximum Likelihood and Noniterative Methods for Estimating the Linear-by-Linear Parameter for Ordinal Log-Linear Models

“…Refer to Goodman [3], Haberman [4,10], and Agresti [5] for further details on the method. Here a brief description of Newton's method will be provided although a more detailed account on the algorithm used in this paper is provided in Beh and Farver [9].…”

“…Beh and Farver [9] continued to examine ISRN Computational Mathematics 3 these issues and proposed the following two noniterative estimates:…”

“…When this stopping criterion is satisfied, the MLE of φ using Newton's method, φ N , is the final iterative estimate φ (t+1) N . In the study undertaken by Beh and Farver [9], an initial value of φ (0) NR = 0 was chosen and, as (2) indicates, coincides with independence between the two variables. The standard error of the MLE of φ is…”

“…They considered a moderately reliable and relatively mathematically simple means of estimating φ noniteratively. Beh and Farver [9] greatly improved upon this approach. By applying these estimation techniques to fourteen commonly analysed two-way contingency tables considered in the statistics literature, Beh and Farver [9] showed that noniterative procedures provide estimates of φ that are comparable, and in most cases statistically indistinguishable, to ML estimates using Newton's unidimensional method.…”

“…Beh and Farver [9] greatly improved upon this approach. By applying these estimation techniques to fourteen commonly analysed two-way contingency tables considered in the statistics literature, Beh and Farver [9] showed that noniterative procedures provide estimates of φ that are comparable, and in most cases statistically indistinguishable, to ML estimates using Newton's unidimensional method. Here we further elaborate on this empirical study by examining the relative stability of estimating φ noniteratively and by using Newton's unidimensional method.…”