Eric J. Beh

In the correspondence analysis literature, advocates over the past 45 years (or so) have discussed the advantages of studying the square root of the cell counts of a contingency table, rather than the cell counts themselves. Doing so means that the Freeman-Tukey statistic is used as an alternative measure of association to Pearson’s chi-squared statistic. While there are advantages to using the classic Freeman-Tukey statistic, it is not always beneficial since it does not always behave like a chi-squared random variable. To address this issue, various improvements to the classic Freeman-Tukey statistic have been proposed, including a class of Freeman-Tukey statistics introduced in the early 1990s. This paper shows how correspondence analysis can be performed using two special cases of this class. By using these two special cases, we discuss the construction and properties of a low-dimensional space that visually represents the structure of a statistically significant association. We also quantify and interpret the distance between two row (or column) points in this low-dimensional space in terms of the link between their Hellinger and chi-squared differences.

One of the first serious attempts to study the indices that assess the association between the variables of a 2 × 2 contingency table was undertaken in the mid-1880s. Central to this study is the 1884 tornado observation/prediction data collected by Seargent John Park Finley (1854–1943), while working for the US Army Signal Service, and the controversial index he proposed to evaluate the success of his tornado predictions, which he denoted i. Subsequent improvements to Finley’s index were proposed, all of which pre-date the development of association measures made by pioneers such as Sir Francis Galton and Karl Pearson. This paper discusses Finley’s data, his index i, and the improvements made to this index. We also give historical context to Finley and his successors and their place in the early development of contingency table analysis.

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.

When analysing the association between the ordered categorical variables of a contingency table, orthogonal polynomials derived from the recurrence formulae of Emerson (1968, Biometrics , 24: 695 - 701) have been extensively used. The calculation of such polynomials is somewhat limited because they reflect only the univariate structure of each variable. This paper proposes a new generalisation of Emerson’s recurrence formulae that reflects bivariate and, more generally, multivariate, association structures for the construction of orthogonal polynomials for multi-way contingency tables. We shall demonstrate the utility of this generalisation by giving special attention to the Gray-Williams index, a three-way variant of the Goodman-Kruskal tau index.

The analysis of aggregate data has received increasing attention in the statistical discipline over the past 20 years, with the ongoing development of a suite of techniques that are classified as ecological inference. Much of its development has been focused solely on estimating the cell frequencies in a 2 × 2 contingency table where only the marginal totals are given; an approach that has been received with mixed reviews. More recently, the focus has shifted toward analyzing the overall association structure, rather than on the estimation of cell frequencies. This article provides some insight into how informative the aggregate data in a single 2 × 2 contingency table are for assessing the association between the variables. This is achieved through the development of a new index, the aggregate informative index. This new index quantifies how much information, on a [0, 100] scale, is needed in the marginal information in a 2 × 2 contingency table to conclude that a statistically significant association exists between the variables. It is established that, unlike Pearson’s (and other forms of the) chi-squared statistic, this new index is immune to changes in the sample size. It is also shown that the new index remains stable when the 2 × 2 contingency table consists of extreme marginal information.

The analysis of aggregate data has received increasing attention in the statistical discipline over the past 20 years. Much of this attention has been focused on estimating the cells frequencies of a 2×2 contingency table given only the marginal totals; the analyses proposed have been received with missed reviews. More recently, the focus has shifted toward analysing the overall association structure rather than on the estimation of the cell frequencies. This article provides some insight int how informative the aggregate data of a single 2×2 contingency table is for assessing the association. The information contained in the margins of the table is quantified using the squared deviation of the expected cell frequencies over the possible range of values, where the row and column totals are known and fixed. The new measure we discuss to quantify this association is referred to as the aggregate informative index (AII) and is shown to be the standardized area under the squared deviation curve.

Using data from the Swedish National Diabetes Register, this study examines the gender disparity among patients with type 1 diabetes who have experienced a specific cardiovascular complication, while exploring the association between their weight variability, age group, and gender. Fourteen cardiovascular complications have been considered. This analysis is conducted using three-way correspondence analysis (CA), which allows for the partitioning and decomposition of Pearson’s three-way chi-squared statistic. The dataset comprises information organized in a data cube, detailing how weight variability among these patients correlates with a cardiovascular complication, age group, and gender. The three-way CA method presented in this paper allows one to assess the statistical significance of the association between these variables and to visualize this association, highlighting the gender gap among these patients. From this analysis, we find that the association between weight variability, age group, and gender varies among different types of cardiovascular complications.

Recently, Beh and Lombardo (2022, Symmetry, 14, 1103) showed how to perform a correspondence analysis on a two-way contingency table where Bowker’s statistic lies at the numerical heart of this analysis. Thus, we showed how this statistic could be used to visually identify departures from perfect symmetry. Interestingly, Bowker’s statistic is a special case of the symmetry version of the Cressie–Read family of divergence statistics. Therefore, this paper presents a new framework for visually assessing departures from perfect symmetry using a second-order Taylor series approximation of the Cressie–Read family of divergence statistics.

Recently, there appeared in this journal (Beh and Lombardo \textbf{2022}, {\it Symmetry}, 14, 1103) a paper that showed how to perform a correspondence analysis on a two-way contingency table where Bowker's statistic lies at the numerical heart of this analysis. Thus, we showed how this statistic can be used to visually identify departures from perfect symmetry. Interestingly, Bowker's statistic is a special case of the symmetry-version of the Cressie-Read family of divergence statistics. Therefore, this paper presents a new framework for visually assessing departures from perfect symmetry using a second-order Taylor series approximation of the Cressie-Read family of divergence statistics.