Correlation Coefficients – Is Spearman Correlation Never Greater Than Pearson Correlation

correlationpearson-rspearman-rho

If Spearman correlation is Pearson correlation, but on ranks of the data – does it mean that absolute Spearman will always be lower or equal to absolute Pearson correlation, but never greater? Is it possible to have variables that show greater absolute Spearman than absolute Pearson?

Example:

# Spearman == Pearson
# We're correlating ranked variables from the beginning 
foo <- 1:1e3
bar <- 1:1e3
sapply(c("pearson", "spearman"), function(x) abs(cor(foo, bar, method = x)))

# Spearman < Pearson
foo <- rnorm(1e3)
bar <- rnorm(1e3)
sapply(c("pearson", "spearman"), function(x) abs(cor(foo, bar, method = x)))

Best Answer

Simple example in which Spearman correlation is greater than Pearson correlation:

x = 1:10;  y = x^2
cor(x,y, meth = "p")
[1] 0.9745586
cor(x,y, meth = "s")
[1] 1

Related Solutions

Solved – How to choose between Pearson and Spearman correlation

If you want to explore your data it is best to compute both, since the relation between the Spearman (S) and Pearson (P) correlations will give some information. Briefly, S is computed on ranks and so depicts monotonic relationships while P is on true values and depicts linear relationships.

As an example, if you set:

x=(1:100);  
y=exp(x);                         % then,
corr(x,y,'type','Spearman');      % will equal 1, and 
corr(x,y,'type','Pearson');       % will be about equal to 0.25

This is because $y$ increases monotonically with $x$ so the Spearman correlation is perfect, but not linearly, so the Pearson correlation is imperfect.

corr(x,log(y),'type','Pearson');  % will equal 1

Doing both is interesting because if you have S > P, that means that you have a correlation that is monotonic but not linear. Since it is good to have linearity in statistics (it is easier) you can try to apply a transformation on $y$ (such a log).

I hope this helps to make the differences between the types of correlations easier to understand.

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

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.

Best Answer

Related Solutions

Solved – How to choose between Pearson and Spearman correlation

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

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