Solved – What does it indicate when the Spearman correlation is a definite amount less than Pearson

correlationpearson-rspearman-rho

I have a bunch of related datasets. The pearson correlations between pairs of them are typically definitely larger than the spearman correlations. That suggests any correlation is linear, but one might expect that even if the pearson and spearman were the same. What does it mean when there is a definite gap between the pearson and the spearman correlation and the pearson is larger?
This seems to be a consistent feature across my datasets.

Best Answer

The Spearman correlation is just the Pearson correlation using the ranks (order statistics) instead of the actual numeric values. The answer to your question is that they're not measuring the same thing. Pearson: linear trend, Spearman: monotonic trend. That the Pearson correlation is higher just means the linear correlation is larger than the rank correlation. This is probably due to influential observations in the tails of the distribution that have large influence relative to their ranked values. Tests of association using the Pearson correlation are of higher power when the linearity holds in the data.

Related Solutions

Solved – Is it a reasonable rule of thumb to examine Pearson and Spearman correlations, and use the latter if they are very different

Pearson and Spearman correlation coefficients do not have the same meaning. Pearson gives information on the linear dependency between the two variables while Spearman give information on the generic relationship between the two variables.

For example in R,

x = rnorm(1000) # simulate 1000 samples of a standard Gaussian variable
cor(x, exp(100 * x), method = "spearman") # calculate Spearman correlation
# [1] 1 
cor(x, exp(100 * x), method = "pearson") # calculate Pearson correlation
# [1] 0.101593

You should report the correlation coefficient that makes more sense for your needs.

Solved – Explaining the difference between Pearson correlation and distance correlation

Distance correlation or distance covariance is a measure of dependence between two paired random vectors of arbitrary, not necessarily equal dimension.

The population distance correlation coefficient is zero if and only if the random vectors are independent. Thus, distance correlation measures both linear and nonlinear association between two random variables or random vectors.

This is in contrast to Pearson's correlation, which can only detect linear association between two random variables.

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Related Solutions

Solved – Is it a reasonable rule of thumb to examine Pearson and Spearman correlations, and use the latter if they are very different

Solved – Explaining the difference between Pearson correlation and distance correlation

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