📉Graphical Representation: Multivariate Data (Just the Quotes)

"An especially effective device for enhancing the explanatory power of time-series displays is to add spatial dimensions to the design of the graphic, so that the data are moving over space" (in two or three dimensions) as well as over time. […] Occasionally graphics are belligerently multivariate, advertising the technique rather than the data." (Edward R Tufte, "The Visual Display of Quantitative Information", 1983)

"Graphical excellence is the well-designed presentation of interesting data - a matter of substance, of statistics, and of design. Graphical excellence consists of complex ideas communicated with clarity, precision, and efficiency. Graphical excellence is that which gives to the viewer the greatest number of ideas in the shortest time with the least ink in the smallest space. Graphical excellence is nearly always multivariate. And graphical excellence requires telling the truth about the data." (Edward R Tufte, "The Visual Display of Quantitative Information", 1983)

"At the heart of quantitative reasoning is a single question: Compared to what? Small multiple designs, multivariate and data bountiful, answer directly by visually enforcing comparisons of changes, of the differences among objects, of the scope of alternatives. For a wide range of problems in data presentation, small multiples are the best design solution." (Edward R Tufte, "Envisioning Information", 1990)

"Show multivariate data; that is, show more than 1 or 2 variables." (Edward R Tufte, "Beautiful Evidence", 2006)

"The purpose of an evidence presentation is to assist thinking. Thus presentations should be constructed so as to assist with the fundamental intellectual tasks in reasoning about evidence: describing the data, making multivariate comparisons, understanding causality, integrating a diversity of evidence, and documenting the analysis. Thus the Grand Principle of analytical design: 'The principles of analytical design are derived from the principles of analytical thinking.' Cognitive tasks are turned into principles of evidence presentation and design." (Edward R Tufte, "Beautiful Evidence", 2006)

"Multivariate techniques often summarize or classify many variables to only a few groups or factors (e.g., cluster analysis or multi-dimensional scaling). Parallel coordinate plots can help to investigate the influence of a single variable or a group of variables on the result of a multivariate procedure. Plotting the input variables in a parallel coordinate plot and selecting the features of interest of the multivariate procedure will show the influence of different input variables." (Martin Theus & Simon Urbanek, "Interactive Graphics for Data Analysis: Principles and Examples", 2009)

"Parallel coordinate plots are often overrated concerning their ability to depict multivariate features. Scatterplots are clearly superior in investigating the relationship between two continuous variables and multivariate outliers do not necessarily stick out in a parallel coordinate plot. Nonetheless, parallel coordinate plots can help to find and understand features such as groups/clusters, outliers and multivariate structures in their multivariate context. The key feature is the ability to select and highlight individual cases or groups in the data, and compare them to other groups or the rest of the data." (Martin Theus & Simon Urbanek, "Interactive Graphics for Data Analysis: Principles and Examples", 2009)

"Given the important role that correlation plays in structural equation modeling, we need to understand the factors that affect establishing relationships among multivariable data points. The key factors are the level of measurement, restriction of range in data values" (variability, skewness, kurtosis), missing data, nonlinearity, outliers, correction for attenuation, and issues related to sampling variation, confidence intervals, effect size, significance, sample size, and power." (Randall E Schumacker & Richard G Lomax, "A Beginner’s Guide to Structural Equation Modeling" 3rd Ed., 2010)

"A heatmap is a visualization where values contained in a matrix are represented as colors or color saturation. Heatmaps are great for visualizing multivariate data" (data in which analysis is based on more than two variables per observation), where categorical variables are placed in the rows and columns and a numerical or categorical variable is represented as colors or color saturation." (Mario Döbler & Tim Großmann, "The Data Visualization Workshop", 2nd Ed., 2020)

📉Graphical Representation: Univariate vs Multivariate Data (Just the Quotes)

"Fitting data means finding mathematical descriptions of structure in the data. An additive shift is a structural property of univariate data in which distributions differ only in location and not in spread or shape. […] The process of identifying a structure in data and then fitting the structure to produce residuals that have the same distribution lies at the heart of statistical analysis. Such homogeneous residuals can be pooled, which increases the power of the description of the variation in the data." (William S Cleveland, "Visualizing Data", 1993)

"The logarithm is one of many transformations that we can apply to univariate measurements. The square root is another. Transformation is a critical tool for visualization or for any other mode of data analysis because it can substantially simplify the structure of a set of data. For example, transformation can remove skewness toward large values, and it can remove monotone increasing spread. And often, it is the logarithm that achieves this removal." (William S Cleveland, "Visualizing Data", 1993)

"When the distributions of two or more groups of univariate data are skewed, it is common to have the spread increase monotonically with location. This behavior is monotone spread. Strictly speaking, monotone spread includes the case where the spread decreases monotonically with location, but such a decrease is much less common for raw data. Monotone spread, as with skewness, adds to the difficulty of data analysis. For example, it means that we cannot fit just location estimates to produce homogeneous residuals; we must fit spread estimates as well. Furthermore, the distributions cannot be compared by a number of standard methods of probabilistic inference that are based on an assumption of equal spreads; the standard t-test is one example. Fortunately, remedies for skewness can cure monotone spread as well." (William S Cleveland, "Visualizing Data", 1993)

"The only thing that is 2-dimensional about evidence is the physical flatland of paper and computer screen. Flatlandy technologies of display encourage flatlandy thinking. Reasoning about evidence should not be stuck in 2 dimensions, for the world seek to understand is profoundly multivariate. Strategies of design should make multivariateness routine, nothing out of the ordinary. To think multivariate, show multivariate; the Third Principle for the analysis and presentation of data: 'Show multivariate data; that is, show more than 1 or 2 variables.'" (Edward R Tufte, "Beautiful Evidence", 2006)

"The simplest way to plot univariate continuous data is a dotplot. Because the points are distributed along only one axis, overplotting is a serious problem, no matter how small the sample is. The usual technique to avoid overplotting is jittering, i.e., the data are randomly spread along a virtual second axis." (Antony Unwin et al [in "Graphics of Large Datasets: Visualizing a Million"], 2006)

"Multivariate techniques often summarize or classify many variables to only a few groups or factors (e.g., cluster analysis or multi-dimensional scaling). Parallel coordinate plots can help to investigate the influence of a single variable or a group of variables on the result of a multivariate procedure. Plotting the input variables in a parallel coordinate plot and selecting the features of interest of the multivariate procedure will show the influence of different input variables." (Martin Theus & Simon Urbanek, "Interactive Graphics for Data Analysis: Principles and Examples", 2009)

"Parallel coordinate plots are often overrated concerning their ability to depict multivariate features. Scatterplots are clearly superior in investigating the relationship between two continuous variables and multivariate outliers do not necessarily stick out in a parallel coordinate plot. Nonetheless, parallel coordinate plots can help to find and understand features such as groups/clusters, outliers and multivariate structures in their multivariate context. The key feature is the ability to select and highlight individual cases or groups in the data, and compare them to other groups or the rest of the data." (Martin Theus & Simon Urbanek, "Interactive Graphics for Data Analysis: Principles and Examples", 2009)

"Eye-catching data graphics tend to use designs that are unique (or nearly so) without being strongly focused on the data being displayed. In the world of Infovis, design goals can be pursued at the expense of statistical goals. In contrast, default statistical graphics are to a large extent determined by the structure of the data (line plots for time series, histograms for univariate data, scatterplots for bivariate nontime-series data, and so forth), with various conventions such as putting predictors on the horizontal axis and outcomes on the vertical axis. Most statistical graphs look like other graphs, and statisticians often think this is a good thing." (Andrew Gelman & Antony Unwin, "Infovis and Statistical Graphics: Different Goals, Different Looks" , Journal of Computational and Graphical Statistics Vol. 22(1), 2013)

"Multivariate analysis refers to incorporation of multiple exploratory variables to understand the behavior of a response variable. This seems to be the most feasible and realistic approach considering the fact that entities within this world are usually interconnected. Thus the variability in response variable might be affected by the variability in the interconnected exploratory variables." (Danish Haroon, "Python Machine Learning Case Studies", 2017)

"A heatmap is a visualization where values contained in a matrix are represented as colors or color saturation. Heatmaps are great for visualizing multivariate data (data in which analysis is based on more than two variables per observation), where categorical variables are placed in the rows and columns and a numerical or categorical variable is represented as colors or color saturation." (Mario Döbler & Tim Großmann, "The Data Visualization Workshop", 2nd Ed., 2020)

See also: bivariate data