In many papers linear regressions of data are reported: A recent example: "we find a significant
negative relationship (r 2 = 0.21; p = 4 × 10 -8)." The data points scatter strongly around the regression line such that by eye this line cannot be envisaged. This is consistent with low r2.
But what does the extremely low p-value mean in such a situation?
Dear Wolfgang,
in the context of a linear regression, the p-values gives you - as always - the probability of the R² value being produced by mere chance. In you'r example, the p-value indicates that the R² is significant, even though the correlation is poor.
That being said: If you'd run a linear regression on a number of random variables, you'd find that the R² is normally distributed around 0 and the p-values are uniformally distributed from 0 to 1 - like seen in this figure: https://tinyurl.com/y7cegsrg
The take-home message is that with poor correlations, the p-value itself becomes essentially random. You'd have a 2% chance that radom values produce a combination of R² > 0.2 and p < 0.05.
However, in your case, there's a VERY low p-value, so the chance of having R² 0.21 p 10^-8 coming from pure chance in close to zero.
... i should maybe also add, that I assume the p-value refers to the R² in your example. It might also be that they calculated it for the slope of the regression line - should be indicated somewhere maybe?
The p-value is presumably for the test that the expected r² is 0. The observed r² from the sample is different, and the test should help to judge if the sample size is large enough to call this "significant" (i.e., having such sample sizes and an expected r² of zero will very unlikely result in observed r² values of 0.21 or larger).
For any given non-zero value of the expected r², the p-value is a function of the sample size, larger samples will more likely give smaller p-values. Any trivial difference between the tested and the actual expected r² can produce p-values arbitrarily small - it's just a matter of the sample size. For such a small p-value of 4E-8 and r² = 0.21, the sample size should be somwehere around 1300 (assuming a bivariate normal distribution).
Paul Toechterle , "R² is normally distributed around 0" - nope. R² > 0, so it cannot be normally distributed. However, r (so that R² = r²) is (approximately) normal distributed around zero; and from this is follows that R² is approximately Chi²-distributed, with some appropriate scaling.
Another note: if R² is Pearson's coefficient of determination (and r would be Pearson's product-moment coefficient), than testing r or R² or the slope of a linear regression is all the same and should give the very same p-value.
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