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It seems to me, that with today's computing power, tools such as spearman's rank/correlation are completely useless. They uncover the exact same information as a regression, except they can't make predictions, can't establish cause and effect and don't fit with curvilinear data.

A regression on the other hand can do everything a Spearman's correlation can do.

So my question is. Why would anyone ever use Spearman's correlation?

I realize the question may sound provocative but I am here to learn, not blindly criticize the model. I just explained it the way I currently see it.

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    $\begingroup$ It is a descriptive, unit-free measure of association strength, thus complementing information gain from regression. Correlation does not fight against regression, it adds insights - so does a scatter plot. $\endgroup$
    – Michael M
    Commented Jan 11, 2020 at 13:18
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    $\begingroup$ I see almost no connection between regression and Spearman's rank correlation coefficient. An analogous premise on a cooking site might be "tools like eggbeaters are completely useless because we have automatic hot pots now. A hot pot can do everything an eggbeater can." I hope that this facetious comparison indicates what is unconstructive about the question and at least hints at how it might be reformulated. $\endgroup$
    – whuber ♦
    Commented Jan 11, 2020 at 13:29
  • $\begingroup$ Thank you for your answers, can you explain how it complements information from regression? You can do so in an "answer" if you'd like. That's essentially what I am asking. $\endgroup$
    – Paze
    Commented Jan 11, 2020 at 13:47

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Regression can't do everything rank correlation does. If you are talking about simple linear regression on the raw data then

  • Regression makes assumptions that Spearman's does not.
  • Regression results are in terms of the units, Spearman's is not.
  • Regression posits that one variable is dependent and the other is independent. Spearman's does not.
  • Regression includes an intercept term, Spearman's does not.
  • Spearman's is based on ranks, regression is based on actual values.
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You are not the only one to have thought correlation useless! John Tukey also had such ideas, see the paper John Tukey and the correlation coefficient by David Brillinger. The paper has many quotes (with refs), but trying to copy quotes here only results in chinese ... so not. Have a look at the paper!

Tukey's reason for disliking correlations did not have anything to do with with today's computing power, as he did most of his calculations by pencil and paper.

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As a matter of fact, Spearman's rank correlation coefficient may for some data be more informative (and less misleading) than a simple linear regression. The crucial difference between Spearman's rank correlation coefficient and linear regression is that the former can detect non-linear associations, while linear regression cannot and may even lead to false conclusions.

First of all, you confused some statements about linear regression and causality here. Linear regression can explain the association between a dependent variable and some independent explanatory variables, it says nothing about causality. Causality is much harder to measure and can only rigorously be established by conducting a controlled experiment (it is however possible to measure a weaker concept of causality through time-based effect observations, based on the assumption that the cause must be observed before its effect, or through methods such as non-parametric regression discontinuity design).

More in detail, while it is true that the linear regression function is based on linear correlation (similar to Pearson's correlation coefficient), they are not the same. A (scalar-valued) linear regression model assumes that:

$y_i = \alpha + \beta * x_i + \epsilon_i$

where $y_i$, the dependent variable to be explained or predicted, may for example be the (numerical) GPA of student $i$. Then, $x_i$ is the independent or explanatory variable, e.g., the number of hours of studying the student spent during last semester, which helps explain the GPA. $\alpha$ is the intercept term, giving the average GPA level without using $x_i$ as an explanatory variable and $\epsilon_i$ is the error term. Now the assumption in plain-vanilla linear regression is that this error term $\epsilon_i$ is independently and identically distributed (i.i.d) with mean zero. Often it is wrongly assumed that the dependent variable $y_i$ should be normally distributed, which is not required in linear regression.

So, if your data is actually non-linearly related, i.e. if $\beta$ (the increase in average GPA per additional hour of studying) in the regression does not stay constant but depends on the level of $x_i$ (say $\beta = 0.5$ for $x < 10$, but $\beta = 2$ for $x >= 10$), then linear regression analysis is misleading. In our example this could, e.g., be the case if an effort of less than 10 hours of studying does not result in a good enough understanding of the subject at hand to raise a student's GPA by more than 0.5 per additional hour of studying on average. However, an effort of a total of more than 10 hours put in is associated with a significant increase of GPA per additional hour of studying on average. This is exactly where Spearman's rank correlation coefficient would give more accurate results than Pearson's correlation coefficient or linear regression, since Spearman's rank correlation coefficient is non-parametric and does not assume a linear relation.

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    $\begingroup$ "Now the crucial assumption in plain-vanilla linear regression is that this error term $\epsilon_i$ is normally distributed." What makes this so crucial to your argument? The Gauss-Markov theorem, giving the conditions where the OLS estimator is the minimum variance estimator among all linear estimators of the regression coefficients, does not make any particular distribution assumption about the error term, for instance. (You absolutely correct that the assumption about normality, when we make it, if for the error, not for the marginal/pooled distribution of the response variable.) $\endgroup$
    – Dave
    Commented Nov 12, 2020 at 20:33
  • $\begingroup$ Re "linear regression cannot:" This seems to compare the two approaches unfairly. If we conceive of Spearman's coefficient as estimating the slope of a regression of transformed variables (with the transformation depending on the data), then a better comparison would be against linear regression procedures that permit arbitrary (nonlinear) univariate transformations of the data. Those procedures are quite capable of detecting and accurately modeling nonlinear relations. $\endgroup$
    – whuber ♦
    Commented Nov 12, 2020 at 20:39
  • $\begingroup$ @Dave You're right, my formulation here is not a hundred percent accurate and to the point, I'll edit this part. Thanks. The point was rather that even the optimal minimum variance estimator for a linear function still only works well if the underlying relationship between the variables in the data is actually linear. $\endgroup$
    – This_is_it
    Commented Nov 12, 2020 at 21:24
  • $\begingroup$ @whuber Probably a really fair comparison can only be made between Spearman's rho and Pearson's linear correlation coefficient, since they are meant to measure more or less the same statistic. However, the question asked for something else… I get your point, but combining both the non-linear transformation and the linear regression would amount to something similar as just directly modelling the relation as a non-linear regression problem, no? $\endgroup$
    – This_is_it
    Commented Nov 12, 2020 at 21:31
  • $\begingroup$ A fair comparison could be made by computing Peason's $\rho$ for the predicted vs. actual responses in a linear regression. You might be trying to make too much of Spearman's coefficient. Its claims to fame, and the bases of any virtues it might have, are (1) its robustness and (2) simplicity of calculation. $\endgroup$
    – whuber ♦
    Commented Nov 12, 2020 at 21:34

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