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What a summary of a scatter is a property of

R-squared is read as a property of a relationship and it is a property of a design. Five studies differing only in how far apart they placed their readings report it from 0.021 to 0.849, with the same line and an estimated residual spread of 0.993 in every one. For a simple regression the squared t statistic is (n − 2) times R-squared over (1 − R-squared), exactly, checked to sixteen significant figures across five hundred fits — so a report carrying both has carried one number twice. A coefficient that is zero only under independence does exist: distance correlation separates the four datasets built to share a correlation by 0.10, and the rank coefficient that guarantees nothing separates them by 0.49.

One relationship at five designs, residual spread 1.00. Every panel has the same slope of 1, the same intercept of 0 and the same residual standard deviation of 1.00. Only the range of x differs. R-squared runs from 0.021 to 0.849, and the estimated residual spread is 0.9932 in all five.

R² is a property of the design

One line, one residual spread, five studies that differ only in how far apart they placed their x values. R² runs from 0.021 to 0.849 and the estimated residual spread is 0.993 in every one of them. Nothing about the relationship changed.

Two studies with an R-squared of 0.85. The left study's points sit 0.50 from the line and the right study's 2.00 — a factor of 4.0. Both report an R-squared of 0.85 and, at the same sample size, the same standard error for the slope. The design was chosen to make it so, and it can always be chosen.

The t statistic wearing different clothes

For a simple regression, t² = (n − 2)R²/(1 − R²), exactly, on every dataset — checked to sixteen significant figures over five hundred fits. So a paper reporting R² and a p-value has reported one number twice, and two studies with the same R² have points four times further from the line.

How far apart four summaries put the quartet. Each bar is the largest value minus the smallest across the four datasets. Pearson spans 0.0018, which is the construction working. Distance correlation — the measure that is zero if and only if the variables are independent — spans 0.101, and Spearman spans 0.491.

The summary that was meant to work

Distance correlation is zero if and only if two variables are independent, which is exactly the guarantee a correlation coefficient lacks. Run on the four datasets that share a correlation, it spreads them by 0.10 — and Spearman, which guarantees nothing, spreads them by 0.49.

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