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It has long been understood that there exists a strong association between a student’s belief in the future utility of mathematics and their self-efficacy in mathematics. This study re-examines this association by studying these variables based on data collected from a sample of 21,444 ninth-grade students who participated in the 2009 High School Longitudinal Study (HSLS09). The nature of the association between future utility beliefs of students in mathematics and self-efficacy of students in mathematics is explored visually using the simple correspondence analysis technique. The main feature that will be utilised from this technique is a two-dimensional graphical display, referred to as a correspondence plot. By studying the HSLS09 data, the first two axes of such a plot summarised nearly 99% of the statistically significant association that exists between a student’s beliefs in the future utility of mathematics and their mathematics self-efficacy. It is shown visually that students who strongly believe in the future importance of studying mathematics also perform strongly in the subject, while those who do not believe that there is any future utility from studying mathematics do not perform well at it. This study, therefore, suggests that mathematics ability is associated with a student’s perception of its future importance.

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.

A comprehensive overview of the internationalisation of correspondence analysis Correspondence Analysis: Theory, Practice and New Strategies examines the key issues of correspondence analysis, and discusses the new advances that have been made over the last 20 years. The main focus of this book is to provide a comprehensive discussion of some of the key technical and practical aspects of correspondence analysis, and to demonstrate how they may be put to use.  Particular attention is given to the history and mathematical links of the developments made. These links include not just those major contributions made by researchers in Europe (which is where much of the attention surrounding correspondence analysis has focused) but also the important contributions made by researchers in other parts of the world. Key features include: * A comprehensive international perspective on the key developments of correspondence analysis. * Discussion of correspondence analysis for nominal and ordinal categorical data. * Discussion of correspondence analysis of contingency tables with varying association structures (symmetric and non-symmetric relationship between two or more categorical variables). * Extensive treatment of many of the members of the correspondence analysis family for two-way, three-way and multiple contingency tables. Correspondence Analysis offers a comprehensive and detailed overview of this topic which will be of value to academics, postgraduate students and researchers wanting a better understanding of correspondence analysis. Readers interested in the historical development, internationalisation and diverse applicability of correspondence analysis will also find much to enjoy in this book.

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.

Master the fundamentals of correspondence analysis with this illuminating resource An Introduction to Correspondence Analysis assists researchers in improving their familiarity with the concepts, terminology, and application of several variants of correspondence analysis.

Over the past decade, a series of procedures has been introduced to estimate, using a non-iterative method, the linear-by-linear association parameter of an ordinal log-linear model. This paper will examine the two key non-iteratively determined estimates of the parameter for the analysis of the association between the two categorical variables that form a contingency table; these are the log and the Beh-Davy non-iterative estimates, referred to simply as the LogNI and the BDNI estimates, respectively. Such an examination will focus on determining their asymptotic characteristics. To do so, a computational study was undertaken for tables of varying sizes to show that these two estimates are asymptotically unbiased. It is also shown that both estimates are asymptotically normally distributed. On the basis of the standard errors, their relative efficiency was established for the 13 commonly analysed contingency tables that appear throughout the literature.

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.

In the framework of multi-way data analysis, this paper presents symmetrical and non-symmetrical variants of three-way correspondence analysis that are suitable when a three-way contingency table is constructed from ordinal variables. In particular, such variables may be modelled using general recurrence formulae to generate orthogonal polynomial vectors instead of singular vectors coming from one of the possible three-way extensions of the singular value decomposition. As we shall see, these polynomials, that until now have been used to decompose two-way contingency tables with ordered variables, also constitute an alternative orthogonal basis for modelling symmetrical, non-symmetrical associations and predictabilities in three-way contingency tables. Consequences with respect to modelling and graphing will be highlighted.

This paper provides a composite indicator for comparing the perceived service of satisfaction of public transport by residents of a southern Italian city across three time periods spanning 2008–2012. Data were collected from 400 respondents that rated their agreement with 15 attribute-related statements regarding local public transport services. This study identifies passenger satisfaction in terms of the various quality aspects of public transport services using features of ordered multiple correspondence analysis. Such a method combines dimension reduction and cluster analysis for categorical data by objectively assigning individuals to clusters and identifying optimal scaling values to each of the categories. The main findings of our study indicate that there are differences in how public transport is perceived during the period of time studied.