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Michael Friendly and Howard Wainer recount the tale of Francis Galton and his discovery of weather patterns, which led to modern weather maps – providing a glimpse into the history of visual thinking and graphic communication, the subject of their new book

The sinking of the Titanic has inspired books, movies and documentaries. But it has also motivated data visualisation designers to tell the story of the tragedy in new ways. Michael Friendly, Jürgen Symanzik and Ortac Onder review the first graph of the disaster and some recent history

This paper is designed as a tutorial to highlight some recent developments for visualizing the relationships among response and predictor variables in multivariate linear models (MLMs), and implemented in convenient packages for R. These models include multivariate multiple regression analysis (MMRA), multivariate analysis of variance (MANOVA) and multivariate analysis of covariance (MANCOVA). The methods we describe go well beyond what can be understood and explained from simple univariate graphical methods for the separate response variables. We describe extensions of these methods for the case of more than just a few response variables, where the important relationships can be readily seen in the low-dimensional (2D) space that accounts for most of the relevant information. As befits the tutorial nature of this paper, we analyze some sample psychological research studies utilizing these multivariate designs, showing examples in R. In the process, we also take up several practical problems related to the assumptions of MLMs, and how these can be dealt with using graphical methods. Finally, we provide guidelines to aid researchers in conducting multivariate research, pertaining to the analysis, visualization, and reporting of such designs. The graphical and statistical methods described here are all freely available and implemented in the R packages candisc, car, heplots, and mvinfluence.

The graphic representation of scientific phenomena serves at least four purposes. One of these is standardizing phenomena in visual form; a second is communicating such phenomena more broadly, thus...

Visualizing both conditional and marginal associations in contingency tables by using simple diagrams is discussed

This paper explores a variety of topics related to the question of testing the equality of covariance matrices in multivariate linear models, particularly in the MANOVA setting. The main focus is on graphical methods that can be used to address the evaluation of this assumption. We introduce some extensions of data ellipsoids, hypothesis-error (HE) plots and canonical discriminant plots and demonstrate how they can be applied to the testing of equality of covariance matrices. Further, a simple plot of the components of Box's M test is proposed that shows _how_ groups differ in covariance and also suggests other visualizations and alternative test statistics. These methods are implemented and freely available in the **heplots** and **candisc** packages for R. Examples from the paper are available in supplementary materials., The American Statistician, in press (2018)

A document retrieved from the archives of the Conservatoire National des Arts et Metiers (CNAM) in Paris sheds new light on the invention by Andre-Michel Guerry of a mechanical device for obtaining statistical summaries and for examining the relationship between different variables, well before general purpose statistical calculators and the idea of correlation had even been conceived. Guerry's ordonnateur statistique may arguably be considered as the first example of a mechanical device devoted to statistical calculations. This article describes what is now known about this machine and illustrates how Guerry probably used it in his program of statistique analytique to reason about the relationship of types of crimes to various potential causes or associations. Supplementary materials for this article are available online.

In ridge regression and related shrinkage methods, the ridge trace plot, a plot of estimated coefficients against a shrinkage parameter, is a common graphical adjunct to help determine a favorable trade-off of bias against precision (inverse variance) of the estimates. However, standard unidimensional versions of this plot are ill-suited for this purpose because they show only bias directly and ignore the multidimensional nature of the problem. A generalized version of the ridge trace plot is introduced, showing covariance ellipsoids in parameter space, whose centers show bias and whose size and shape show variance and covariance, respectively, in relation to the criteria for which these methods were developed. These provide a direct visualization of both bias and precision. Even two-dimensional bivariate versions of this plot show interesting features not revealed in the standard univariate version. Low-rank versions of this plot, based on an orthogonal transformation of predictor space extend these idea...

A 1644 diagram by Michael Florent van Langren, showing estimates of the difference in longitude between Toledo and Rome, is sometimes considered to be the first known instance of a graph of statistical data. Some recently discovered documents help to date the genesis of this graphic to before March 1628, and shed some light on why van Langren chose to display this information in this form. In the process, we discovered three earlier versions of the 1644 graph and one slightly later reproduction. This article describes these early attempts on the solution of “the problem of longitude” from the perspective of a history of data visualization. This article has supplementary material online.

Collinearity diagnostics are widely used, but the typical tabular output used in almost all software makes it hard to tell what to look for and how to understand the results. We describe a simple improvement to the standard tabular display, a graphic rendition of the salient information as a which is simultaneously a biplot of the smallest dimensions of the correlation matrix of the predictors, RXX, and the largest dimensions of R–1xx, on which the standard collinearity diagnostics are based.

In the debate over null hypothesis significance testing, Paul Meehl strongly advocated appraising theories through the generation and evaluation of precise predictions (e.g., Meehl, 1978 ). The study of personality structure through the five-factor model (FFM; McCrae & John, 1992 ) is an important area of research where one encounters many precise predictions. Extant methods of assessing such predictions, however, do not allow researchers to examine the outcome of the predictions in great detail. That is, it may be difficult to determine how estimates fail to match predicted values. As Meehl argued, one must examine how a theory fails to predict in order to refine and improve the theory. To promote better theory appraisal in FFM research, we present a powerful new tool, called a tableplot ( Kwan, 2008a ), that can summarize and clarify factor analytic results. Specifically, we illustrate how the tableplot enables detailed appraisal of precise predictions in the FFM.

The cluster heat map is an ingenious display that simultaneously reveals row and column hierarchical cluster structure in a data matrix. It consists of a rectangular tiling, with each tile shaded on a color scale to represent the value of the corresponding element of the data matrix. The rows (columns) of the tiling are ordered such that similar rows (columns) are near each other. On the vertical and horizontal margins of the tiling are hierarchical cluster trees. This cluster heat map is a synthesis of several different graphic displays developed by statisticians over more than a century. We locate the earliest sources of this display in late 19th century publications, and trace a diverse 20th century statistical literature that provided a foundation for this most widely used of all bioinformatics displays.

Hypothesis-error (or “HE”) plots, introduced by Friendly (J Stat Softw 17(6):1–42, 2006a; J Comput Graph Stat 16:421–444, 2006b), permit the visualization of hypothesis tests in multivariate linear models by representing hypothesis and error matrices of sums of squares and cross-products as ellipses. This paper describes the implementation of these methods in the heplots package for R, as well as their extension, for example from two to three dimensions and by scaling hypothesis ellipses and ellipsoids in a natural manner relative to error.

Multivariate analysis of variance (MANOVA) extends the ideas and methods of univariate ANOVA in simple and straightforward ways. But the familiar graphical methods typically used for univariate ANOVA are inadequate for showing how measures in a multivariate response vary with each other, and how their means vary with explanatory factors. Similarly, the graphical methods commonly used in multiple regression are not widely available or used in multivariate multiple regression (MMRA). We describe a variety of graphical methods for multiple-response (MANOVA and MMRA) data aimed at understanding what is being tested in a multivariate test, and how factor/predictor effects are expressed across multiple response measures.In particular, we describe and illustrate: (a) Data ellipses and biplots for multivariate data; (b) HE plots, showing the hypothesis and error covariance matrices for a given pair of responses, and a given effect; (c) HE plot matrices, showing all pairwise HE plots; and (d) reduced-rank analogs ...

This paper outlines a general framework for ordering information in visual displays (tables and graphs) according to the effects or trends which we desire to see. This idea, termed effect-ordered data displays, applies principally to the arrangement of unordered factors for quantitative data and frequency data, and to the arrangement of variables and observations in multivariate displays (star plots, parallel coordinate plots, and so forth).As examples of this principle, we present several techniques for ordering items, levels or variables "optimally", based on some desired criterion. All of these may be based on eigenvalue or singular-value decompositions.Along the way, we tell some stories about data display, illustrated by graphs--some surprisingly bad, and some surprisingly good--for showing patterns, trends, and anomalies in data. We hope to raise more questions than we can provide answers for.

The modules in the statistical package ViSta related to categorical data analysis are presented These modules are: visualization of frequency data with mosaic and bar plots, correspondence analysis, multiple correspondence analysis and loglinear analysis. All these methods are implemented in ViSta with a big emphasis on plots and graphical representations of data, as well as interactivity for the user with the system. These provide a system that has shown to be easy, useful, and powerful, both for novice and experienced users.

Correlation and covariance matrices provide the basis for all classical multivariate techniques. Many statistical tools exist for analyzing their structure, but, surprisingly, there are few techniques for exploratory visual display, and for depicting the patterns of relations among variables in such matrices directly, particularly when the number of variables is moderately large. This paper describes a set of techniques we subsume under the name “corrgram”, based on two main schemes: (a) rendering the value of a correlation to depict its sign and magnitude. We consider some of the properties of several iconic representations, in relation to the kind of task to be performed. (b) re-ordering the variables in a correlation matrix so that “similar” variables are positioned adjacently, facilitating perception. In addition, the extension of this visualization to matrices for conditional independence and partial independence is described and illustrated, and we provide an easily-used SAS implementation of these methods.

This article provides an illustrated history of the visual and conceptual ideas leading to the development of mosaic displays. We trace the origins of the use of rectangles and area to depict data quantities and their relations, of early forms of mosaic displays including subdivided bar-like charts and various cartograms, to the modern forms used in log-linear analysis and in space-filling tree maps.

Charles Joseph Minard is most widely known for a single work—his poignant flow-map depiction of the fate of Napoleon’s Grand Army in the disastrous Russian campaign of 1812. In fact, Minard was a true pioneer in thematic cartography and in statistical graphics; he developed many novel graphic forms to depict data, always with the goal to let the data “speak to the eyes.” This article reviews Minard’s contributions to statistical graphics, the time course of his work, and some background behind the famous March on Moscow graphic. This article also looks at some modern revisions of this graph from an information visualization perspective and examines some lessons this graphic provides as a test case for the power and expressiveness of computer systems or languages for graphic information display and visualization.

Background Multiple sclerosis is a polysymptomatic disease. Little is known about relative contributions of the different multiple sclerosis symptoms to self-perception of health. Objectives To investigate the relationship between symptom severity in 11 domains affected by multiple sclerosis and self-rated health. Methods Multiple sclerosis patients in two multiple sclerosis centers assessed self-rated health with a validated instrument and symptom burden with symptoMScreen, a validated battery of Likert scales for 11 domains commonly affected by multiple sclerosis. Pearson correlations and multivariate linear regressions were used to investigate the relationship between symptoMScreen scores and self-rated health. Results Among 1865 multiple sclerosis outpatients (68% women, 78% with relapsing–remitting multiple sclerosis, mean age 46.38 ± 12.47 years, disease duration 13.43 ± 10.04 years), average self-rated health score was 2.30 (‘moderate to good’). Symptom burden (composite symptoMScreen score) highly correlated with self-rated health ( r = 0.68, P Conclusions Pain contributed the most to multiple sclerosis outpatients’ perception of health, followed by gait dysfunction and fatigue. These findings suggest that ‘invisible disability’ may be more important to patients’ sense of wellbeing than physical disability, and challenge the notion that physical disability should be the primary outcome measure in multiple sclerosis.

Visual insights into a wide variety of statistical methods, for both didactic and data analytic purposes, can often be achieved through geometric diagrams and geometrically based statistical graphs. This paper extols and illustrates the virtues of the ellipse and her higher-dimensional cousins for both these purposes in a variety of contexts, including linear models, multivariate linear models and mixed-effect models. We emphasize the strong relationships among statistical methods, matrix-algebraic solutions and geometry that can often be easily understood in terms of ellipses., Comment: Published in at http://dx.doi.org/10.1214/12-STS402 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org)

As has often been recognized, several characteristics of APL make it especially well suited for programming statistical applications: APL treats arrays—vectors, matrices, and higher dimensional arrays—as data structures that can be processed without reference to their elements; it contains many powerful primitive functions and operators for manipulating data; and it is an extensible language, so that user-written programs are employed in the same manner as the primitives. APL2 incorporates several significant extensions of the language, the most important of which are generalized arrays—arrays whose elements may themselves be arrays. Using generalized arrays along with other features of APL2, the authors build an object-oriented programming environment for statistics and data analysis called APL2STAT.

Since its introduction, APL has frequently been touted as an ideal programming language for statistical applications. Among the attractive features of APL for statistics are its extensibility, the presence of primitives for operations such as sorting, matrix inversion, and arranging data, and powerful facilities for handling matrices and other arrays. Newer programming languages incorporating some of the features of APL—notably New S and XLisp-Stat—have been designed specifically for statistical applications. Is there still a niche for APL? We believe that APL2 continues to offer some significant benefits for statistical computation, including user-defined operators, nested arrays, and convenient implementation of arrays of any dimension. We use these characteristics of the language in the course of designing an extensible computing environment for data analysis and programming based on APL2 that incorporates some of the features of modern statistical programming languages, such as data objects, ...