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.

If you do not mind the time predictor, it is hard to believe that this is a light curve, time dependent data. At a glance, this data set represents a simple block design for the one-way ANOVA. ANOVA stands for Analysis of Variance, which is not a familiar nomenclature for astronomers.

Consider a case that you have a very long strip of land that experienced FIVE different geological phenomena. What you want to prove is that crop productivity of each piece of land is different. So, you make FIVE beds and plant same kind seeds. You measure the location of each seed from the origin. Each bed has some dozens of seeds, which are very close to each other but their distances are different. On the other hand, the distance between planting beds are quite far unable to say that plants in the test bed A affects plants in B. In other words, A and B are independent suiting for my statistical inference procedure by the F-test. All you need is after a few months, measuring the total weight of crop yield from each plant (with measurement errors).

Now, let’s look at the plot above. If you replace distance to time and weight to flux, the pattern in data collection and its statistical inference procedure matches with the one-way ANOVA. It’s hard to say this data set is designed for time series analysis apart from the complication in statistical inference due to measurement errors. How to design the statistical study with measurement errors, huge gaps in time, and unequal time intervals is complex and unexplored. It depends highly on the choice of inference methods, assumptions on error i.e. likelihood function construction, prior selection, and distribution family properties.

Speaking of ANOVA, using the F-test means that we assume residuals are Gaussian from which one can comfortably modify the model with additive measurement errors. Here I assume there’s no correlation in measurement errors and plant beds. How to parameterize the measurement errors into model depends on such assumptions as well as how to assess sampling distribution and test statistics.

Although I know this Crab nebula data set is not for the one-way ANOVA, the pattern in the scatter plot drove me to test the data set. The output said to reject the null hypothesis of statistically equivalent flux in FIVE time blocks. The following is R output without measurement errors.

Df Sum Sq Mean Sq F value Pr(>F)
factor 4 4041.8 1010.4 143.53 < 2.2e-16 ***
Residuals 329 2316.2 7.0

If the gaps are minor, I would consider time series with missing data next. However, the missing pattern does not agree with my knowledge in missing data analysis. I wonder how astronomers handle such big gaps in time series data, what assumptions they would take to get a best fit and its error bar, how the measurement errors are incorporated into statistical model, what is the objective of statistical inference, how to relate physical meanings to statistical significant parameter estimates, how to assess the model choice is proper, and more questions. When the contest is over, if available, I’d like to check out any statistical treatments to answer these questions. I hope there are scientists who consider similar statistical issues in these data sets by the INTEGRAL team.

I was browsing posters; there were so many but the acronym ANOVA couldn’t miss my eyes. I decided to take a look. The poster was made by an education graduate student who designed experiments on children and performed a statistical analysis (ANOVA) with collected data to find a factor that affects children (testing the significance of a factor via the F-test). Frankly, I didn’t read it through but was desperate to know the occasion in astronomy where ANOVA can be utilized. I was half excited because ANOVA appeared in the AAS meeting and half disappointed because ANOVA was not performed as a part of astronomical research. A train of thoughts, nonetheless, came along.

ANOVA or statistical methods from design of experiments may be of no use in astronomical experiments (collecting data by observations, i.e. collecting random incidents seems to not allow experiment designing stages). However, these methods can be used in evaluating proposals, summaries of projects’ quality, standardizing decision making processes, improving usages of instrument times, and so on. These statistical experiment design tools could be a servant to improve the procedures of collecting data and allocating time slots, and could amplify already tremendous efforts of renovating/creating expensive instruments.

Experiment design has never been of my interest because I could not see any chance of using them in astronomy. Upon finding ANOVA at the AAS, my second thought combined with my experience at CfA is now quite opposite to my first thought. There are plenty of rooms where well sought experiment design can be adopted in the astronomical society.