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The "Introductory Statistics with R" book contains a section that deals with correlations (section 6.4 in the second edition). The book shows Pearson, Spearman and Kendall correlation coefficients computed on the blood.glucose and short.velocity columns of the thuesen data set. The p-values associated with these coefficients are 0.048, 0.139 and 0.119, correspondingly. The book then says the following:

Notice that neither of the two nonparametric correlations is significant at the 5% level, which the Pearson correlation is, albeit only borderline significant.

I have several problems with this paragraph.

First of all, my naive guess would be that since the non-parametric coefficients do not imply linearity, they will tend to be "significant" more frequently than Pearson's r. Am I right?

Secondly, and more importantly, is such a comparison between p-values of different tests applied on the same data legit? (I'm talking about real-life comparisons and not about trivial examples in a text book) If it is, how one need to interpret the notion that linear correlation is "significant", while rank or concordance correlation isn't?

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One explanation is that outliers, even mild ones can affect the results in a pearson correlation. If the outlier is a legitimate point (not a typo or other error) then it should increase the significance of the correlation (as you see), but will not change much in the other 2, so it is easy for the pearson correlation to be larger and more significant. In real data analysis seeing this would suggest looking for outliers (you should be plotting the data anyways) that are influencing the results. What to do next depends on what question you are asking and what assumptions are reasonable given the science.

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@Greg Snow is on the money about your first question.

In regard to your second, comparing the two tests is misleading since two hypotheses are different even though the scientific question is (ostensibly) the same. This is a case where it's really important to be explicit about what hypothesis test you're using.

To be explicit, the test using $r$ is testing something like $H_0: r=0$ vs $H_1: r \neq 0$. For Spearman's rho, you're testing $H_0: \rho=0$ vs $H_1: \rho \neq 0$. Using $r$ presumes a linear relationship, while using $\rho$ presumes a more general monotonic relationship since it's based on the observed ranks (which is also where it gets its robustness). The two hypotheses are actually quite different.

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