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Compare the various measures of correlation that are used to assess dependence, noting how they are alike and how they differ.

FRM® Part I · Quantitative Analysis · Measuring Returns, Volatility, and Correlation · FRM1.QA.12.h

The idea

Pearson correlation measures linear dependence and is sensitive to outliers and to the marginal distributions. Rank correlations use only the ordering of the observations. Spearman's rho is the Pearson correlation of the ranks. With no ties it has a simple form. Spearman = 1 − 6 × Σd² ÷ (n × (n² − 1)). Here d is the difference between the two ranks of an observation, and n is the number of observations. Kendall's tau compares pairs of observations. Kendall = (C − D) ÷ (n × (n − 1) ÷ 2). Here C and D are the numbers of concordant and discordant pairs. Both rank measures are unchanged by any strictly increasing transformation of either variable, so an exact monotone but nonlinear relation gives Spearman and Kendall values of 1 while Pearson is below 1. They are also more robust to outliers. All three lie between −1 and +1 and are 0 for independent variables, but none captures the full dependence structure. For bivariate normal data the two are linked. Kendall = (2 ÷ π) × arcsin(ρ). Here ρ is the Pearson correlation.

Formulas

Spearman's rank correlation with no ties

ρS = 1 − 6 × Σd² ÷ [n × (n² − 1)]

d
the difference between an observation's rank on X and its rank on Y
n
the number of paired observations

Rank first, then compute; outliers affect only their rank. With ties, compute the Pearson correlation of the ranks instead.

Kendall's tau

τ = (C − D) ÷ [n × (n − 1) ÷ 2]

C
the number of concordant pairs
D
the number of discordant pairs
n
the number of observations

The denominator is the number of pairs, not the number of observations. Unchanged by any strictly increasing transformation of either variable.

A practice question

From the question bank, with the reason each wrong option is wrong.

Kendall's tau for a sample of paired observations is computed as:

  1. the number of concordant pairs divided by the number of discordant pairs.Why not: Takes a ratio, which is not bounded between -1 and +1.
  2. (number of concordant pairs minus number of discordant pairs) divided by the number of observations.Why not: Divides by n instead of the number of pairs, n(n - 1) / 2.
  3. (number of concordant pairs minus number of discordant pairs) divided by the total number of pairs. - correct
  4. the Pearson correlation of the ranks of the observations.Why not: This is Spearman's rho, not Kendall's tau.

tau = (C - D) / (n(n - 1) / 2). A pair is concordant if both variables order the two observations the same way, discordant if not.

A harder one

Six paired observations contain no ties and give 11 concordant pairs and 4 discordant pairs. Kendall's tau is closest to:

  1. 0.47 - correct
  2. 0.73Why not: Uses the share of concordant pairs, 11 / 15, without subtracting the discordant ones.
  3. 0.27Why not: Uses the share of discordant pairs, 4 / 15.
  4. 0.19Why not: Divides (C - D) by n² (36) instead of by the number of pairs.

Pairs = 6 x 5 / 2 = 15. tau = (11 - 4) / 15 = 0.467.

The hardest tier

Paired observations of X are 1, 2, 3, 4, 5, 50 and of Y are 3, 1, 2, 5, 4, 100. Spearman's rank correlation is closest to:

  1. 0.77 - correct
  2. 1.00Why not: Reports the Pearson correlation of the raw values, which the outlying pair drives to about 1.
  3. 0.23Why not: Computes 6 x sum(d²) / (n(n² - 1)) and omits the '1 minus' at the start of the formula.
  4. 0.96Why not: Leaves the factor 6 out of the numerator.

Ranks of X: 1-6. Ranks of Y: 3, 1, 2, 5, 4, 6. Rank differences: -2, 1, 1, -1, 1, 0, so sum(d²) = 8. rho = 1 - 6 x 8 / (6 x 35) = 0.771. The outlier does not affect ranks.

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