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Looking at Person's, Spearman's and Kendall's correlation coefficients for the same data, we can see that both Spearman's Rho and Kendall's Tau misrepresent the acutal correlation, if the data is higher than ordinally scaled and ranks therefore don't represent the actual data well.

Let's look at some examples first and my own data last. Here is some R code (please scroll the code window up to see all of the code) and a plot for different data:

### perfectly correlated:

p <- c(1, 1000)
q <- c(1, 1000)
cor(p, q, method = "pearson")
# 1
cor(p, q, method = "spearman")
# 1
cor(p, q, method = "kendall")
# 1
par(mfrow = c(2, 2))
plot(p, q, main = "correlated", xlab = "", ylab = "", axes = FALSE)
box()

### perfectly uncorrelated:

p <- c(1, 2, 1, 2)
q <- c(1, 1, 2, 2)
cor(p, q, method = "pearson")
# 0
cor(p, q, method = "spearman")
# 0
cor(p, q, method = "kendall")
# 0
plot(p, q, main = "uncorrelated", xlab = "", ylab = "", axes = FALSE)
box()

### almost perfectly correlated:

p <- c(1, 2, 999, 1000)
q <- c(2, 1, 1000, 999)
cor(p, q, method = "pearson")
# 0.999998
cor(p, q, method = "spearman")
# 0.6
cor(p, q, method = "kendall")
# 0.3333333
plot(jitter(p, 100), jitter(q, 100), main = "almost", xlab = "", ylab = "", axes = FALSE)
box()

### my data

p <- c(1.139434, 1.901322, 1.461096, 2.459053, 4.643259, 2.397895, 1.99243, 3.013225, 1.654558, NA, 1.529395, 3.861899, 1.07881, 2.942148, 3.791436, 3.349904, NA, 2.34857, 2.944439, 3.251079, 3.766229, 3.94266, 2.125251, 1.934076, 2.238047, 1.731135, 1.511458, 3.311585, 2.66921, NA, 0.4700036, 1.751754, 1.548813, 4.01228, 0.7503056, 3.430397, 3.718977, 3.154634, 0.8873032, 1.824549, 2.837728, 3.057768, 3.709399, 2.674149, 1.832581, NA, 2.710713, 1.738219, 0.8754687, NA, 3.272417, 2.89395, 1.386294, 1.814749, 2.1366, 4.857225, 0.8043728, 3.531694, 4.75359, 1.791759, 1.754019, 2.367124, 2.736221, 4.004119, 4.39834, 3.745575)
q <- c(0.9162907, 1.332227, 0.415127, 1.765906, 1.523495, 1.722767, 1.622683, 2.455054, 0.6931472, NA, 1.495494, 2.890372, 0.05715841, 2.221092, 3.326474, 2.732743, NA, 1.791759, 2.273598, 2.524516, 2.803572, 3.028522, 1.252763, 1.538763, 1.558145, 1.386294, 1.029619, 2.655252, 2.397895, NA, 0.9808293, 1.32567, 1.548813, 2.585711, 0.6931472, 2.914763, 2.86537, 2.654806, 0.6931472, 1.386294, 2.135531, 2.95491, 2.632064, 2.564949, 1.098612, NA, 1.99606, 0.4770875, 0.4054651, 1.213682, 3.107944, 2.383124, 1.072637, 1.249435, 1.644123, 3.628776, 0.1625189, 2.008824, 3.590034, 1.920377, 0.7985077, 1.813738, 2.436116, 3.754337, 3.335957, 2.908721)
cor(p, q, use = "pairwise.complete.obs", method = "pearson")
# 0.8890321
cor(p, q, use = "pairwise.complete.obs", method = "spearman")
# 0.9087856
cor(p, q, use = "pairwise.complete.obs", method = "kendall")
# 0.7669589
plot(p, q, main = "my data", xlab = "", ylab = "", axes = FALSE)
box()

plot of the correlations

Even if eyeballing is a bad way to assess data, I'm sure you agree with me that in my almost perfectly correlating third example some of the three calculated correlation coefficients must be off the mark: rP = 0.999998, rS = 0.6, rK = 0.3333333.

Considering the fact that my data is metric and ratio-scaled (the data are seconds), but not normally distributed, what is the best way to calculate its correlation?

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  • 3
    $\begingroup$ Reading another question might help you. $\endgroup$ – ttnphns Sep 8 '15 at 7:59
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    $\begingroup$ I can simply look at the plot and answer ... how correlated the data is. Unfortunately, it is not simple to answer. Just try and reflect on, and find that you are basing your description of the correlation plot on some visual "good" prototype or premise. If all people in any time interpreted such plots identically there would be no reason to invent different correlation measures. $\endgroup$ – ttnphns Sep 8 '15 at 8:58
  • 1
    $\begingroup$ Kendall and Spearman correlations are computed on the ranked data. 1 2 3 4 and 2 1 4 3, respectively. Did you just forget it? $\endgroup$ – ttnphns Sep 8 '15 at 9:43
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    $\begingroup$ Your "almost" plot is only "almost" when viewed in terms of the raw data, as PMCC would do. After rank transformation it doesn't look so "almost" after all. Apply a monotonic transformation so that the X and Y values don't tend to cluster up around two values (something visually meaningful to you, but not at all detected by rank-based correlation measures) and the correlation would not look so strong. If you use visual intuition based on raw data, don't be surprised if this intuition is not reflected in rank-based measures, especially when you picked an arrangement where they differ so much! $\endgroup$ – Silverfish Sep 8 '15 at 9:44
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    $\begingroup$ Looking at Person's, Spearman's and Kendall's correlation coefficients for the same data, we can see that both Spearman's Rho and Kendall's Tau misrepresent the acutal correlation, if the data is higher than ordinally scaled and ranks therefore don't represent the actual data well. I'm returning to my 2nd comment. What is "actual correlation"? There exist no one. Not single. We always imply some this or that clear or dim model of relationship. Models can be plenty, coefficients of correlation can be infinite number of. $\endgroup$ – ttnphns Sep 8 '15 at 10:42