When having many pairs of variables that demands numerous scatter plots, one possibility is using parallel coordinates and a matrix of correlation coefficients. If Gaussian distribution is assumed, which seems to be almost all cases, particularly when parametrizing measurement errors or fitting models of physics, then error bars of these coefficients also can be reported in a matrix form. If one considers more complex relationships with multiple tiers of data sets, then one might want to check ANCOVA (ANalysis of COVAriance) to find out how statisticians structure observations and their uncertainties into a model to extract useful information.

I’m not saying those simple examples from wikipedia, wikiversity, or publicly available tutorials on ANCOVA are directly applicable to statistical modeling for astronomical data. Most likely not. Astrophysics generally handles complicated nonlinear models of physics. However, identifying dependent variables, independent variables, latent variables, covariates, response variables, predictors, to name some jargon in statistical model, and defining their relationships in a rather comprehensive way as used in ANCOVA, instead of pairing variables for scatter plots, would help to quantify relationships appropriately and to remove artificial correlations. Those spurious correlations appear frequently because of data projection. For example, datum points on a circle on the XY plane of the 3D space centered at zero, when seen horizontally, look like that they form a bar, not a circle, producing a perfect correlation.

As a matter of fact, astronomers are aware of removing these unnecessary correlations via some corrections. For example, fitting a straight line or a 2nd order polynomial for extinction correction. However, I rarely satisfy with such linear shifts of data with uncertainty because of changes in the uncertainty structure. Consider what happens when subtracting background leading negative values, a unrealistic consequence. Unless probabilistically guaranteed, linear operation requires lots of care. We do not know whether residuals y-E(Y|X=x) are perfectly normal only if μ and σs in the gaussian density function can be operated linearly (about Gaussian distribution, please see the post why Gaussianity? and the reference therein). An alternative to the subtraction is linear approximation or nonparametric model fitting as we saw through applications of principle component analysis (PCA). PCA is used for whitening and approximating nonlinear functional data (curves and images). Taking the sources of uncertainty and their hierarchical structure properly is not an easy problem both astronomically and statistically. Nevertheless, identifying properties of the observed both from physics and statistics and putting into a comprehensive and structured model could help to find out the causality^[2] and the significance of correlation, better than throwing numerous scatter plots with lines from simple regression analysis.

In order to understand why statisticians studied ANCOVA or, in general, ANOVA (ANalysis Of VAriance) in addition to the material in wiki:ANCOVA, you might want to check this page^[3] and to utilize your search engine with keywords of interest on top of ANCOVA to narrow down results.

From the linear model perspective, if a response is considered to be a function of redshift (z), then z becomes a covariate. The significance of this covariate in addition to other factors in the model, can be tested later when one fully fit/analyze the statistical model. If one wants to design a model, say rotation speed (indicator of dark matter occupation) as a function of redshift, the degrees of spirality, and the number of companions – this is a very hypothetical proposal due to my lack of knowledge in observational cosmology. I only want to point that the model fitting problem can be seen from statistical modeling like ANCOVA by identifying covariates and relationships – because the covariate z is continuous, and the degrees are fixed effect (0 to 7, 8 tuples), and the number of companions are random effect (cluster size is random), the comprehensive model could be described by ANCOVA. To my knowledge, scatter plots and simple linear regression are marginalizing all additional contributing factors and information which can be the main contributors of correlations, although it may seem Y and X are highly correlated in the scatter plot. At some points, we must marginalize over unknowns. Nonetheless, we still have some nuisance parameters and latent variables that can be factored into the model, different from ignoring them, to obtain advanced insights and knowledge from observations in many measures/dimensions.

Something, I think, can be done with a small/ergonomic chart/table via hypothesis testing, multivariate regression, model selection, variable selection, dimension reduction, projection pursuit, or names of some state of the art statistical methods, is done in astronomy with numerous scatter plots, with colors, symbols, and lines to account all possible relationships within pairs whose correlation can be artificial. I also feel that trees, electricity, or efforts can be saved from producing nice looking scatter plots. Fitting/Analyzing more comprehensive models put into a statistical fashion helps to identify independent variables or covariates causing strong correlation, to find collinear variables, and to drop redundant or uncorrelated predictors. Bayes factors or p-values can be assessed for comparing models, testing significance their variables, and computing error bars appropriately, not the way that the null hypothesis probability is interpreted.

Lastly, ANCOVA is a complete [MADS].

1. This is not an assuring absolute statement but a personal impression after reading articles of various fields in addition to astronomy. My readings of other fields tell that many rely on correlation statistics but less scatter plots by adding straight lines going through data sets for the purpose of imposing relationships within variable pairs
2. the way that chi-square fitting is done and the goodness-of-fit test is carried out is understood by the notion that X causes Y and by the practice that the objective function, the sum of (Y-E[Y|X])^2/σ^2 is minimized
3. It’s a website of Vassar college, that had a first female faculty in astronomy, Maria Mitchell. It is said that the first building constructed is the Vassar College Observatory, which is now a national historic landmark. This historical factor is the only reason of pointing this website to drag some astronomers attentions beyond statistics.

Both good and bad reviews exist but I don’t believe there’s a book this extensive to cover the grammar of graphics. Not many statisticians are handling images compared to computer vision engineers but at some points, all engineers and scientists must present their work into graphs and tables. By the same token, tongs are different, although alphabets are common. Often times, plots from scientist A cannot talk to scientist B (A \ne B). This communication discrepancy seems prevalent between astronomy and statistics.

Almost all chapters begin with the Greek or Latin origins of chapter names to reflect the common origins of lexicons in graphics regardless of subjects. Some chapters, on the contrary, tend to illuminate different practices/perspectives/interests in graphics between astronomers and statisticians:

• Chap. 6 [Scale]: Scaling by log transformation is meant to stabilize errors (Box-Cox transformation) in statistics; in contrast, in astronomy to impose a linear relationship between predictor and response which is manifested better in log scale.
• Chap. 7 [Statistics]: Discussion on error bars, bins, and histogram; although graphical tools are same but the objectives seem different (statistics – optimal binning: astronomy – enhancing signals in each bin).
• Chap 15. [Uncertainty]: Concepts of uncertainty; many words are associated with uncertainty, for example, variability, noise, incompleteness, indeterminacy, bias, error, accuracy, precision, reliability, validity, quality, and integrity.

Overall, the ideas are implored to be included adaptively in the astronomical data analysis packages for visualizing the analyzed products. Perhaps, it may inspire some astronomers to transform the ways of visualization. For instance, instead of histograms, in my opinion, box-plots, qq-plots, and scatter plots would shed improved information while maintaining the simplicity but except scatter plots, other summary plots are not commonly used in astronomy. A benefit from box plot and qq plot is checking gaussianity without sacrificing information from binning. However, there’s no golden rule which type or grammar of graphics is correct and shall be used . Only exists user preference.

Different disciplines maintain their ways of presenting graphics and expect that they can talk to viewers of other disciplines. No one fully reached that point, disappointingly. Extensive discussion and persuasion is required to deliver stories behind graphics to others.

As Felice Frankel pointed out the way of visualization could enhance recognition and understanding of deliberate delivering of information. To the purpose, a few interesting quotes from the book is replaced the conclusion of this post.

• The first ed. of this book, and Part 1 of the current ed., explicitly cautioned that the grammar of graphics is not a visualization system.
• We are surprised, nevertheless, to discover how little some visualization researchers in various fields know about the origins of many the of techniques that are routinely applied in visualization.
• The grammar of graphics determined how algebra, geometry, aesthetics, statistics, scales, and coordinates interact. In the world of statistical graphics, we cannot confuse aesthetics with geometry by picking a tree graphics to represent a continuous flow of migrating insects across a geographic field simply because we like the impression in conveys.
• If we must choose a single word to characterize the focus of modern statistics, it would be uncertainty (Stigler, 1983)
• … decision-makers need statistical tools to formalize the scenarios they encounter and they need graphical aids to keep them from making irrational decisions. … the use of graphics for decision-making under uncertainty is a relatively recent field. … We need to go beyond the use of error bars to incorporate other aesthetics in the representation of error. And we need research to assess the effectiveness of decision-making based on these graphics using a Bayesian yardstick.