correlation

Calculate confidence intervals for correlation coefficients, including Pearson's R, Kendall's tau, Spearman's rho, and customized correlation measures.

Methodology

Two approaches are offered to calculate the confidence intervals, one parametric approach based on normal approximation, and one non-parametric approach based on bootstrapping.

Parametric Approach

Say r_hat is the correlation we obtained, then with a transformation

z = ln((1+r)/(1-r))/2,

z would approximately follow a normal distribution,
with a mean equals to z(r_hat),
and a variance sigma^2 that equals to 1/(n-3), 0.437/(n-4), (1+r_hat^2/2)/(n-3) for the Pearson's r, Kendall's tau, and Spearman's rho, respectively (read Ref. [1, 2] for more details). n is the array length.

The (1-alpha) CI for r would be

(T(z_lower), T(z_upper))

where T is the inverse of the transformation mentioned earlier

T(x) = (exp(2x) - 1) / (exp(2x) + 1),

z_lower = z - z_(1-alpha/2) sigma,

z_upper = z + z_(1-alpha/2) sigma.

This normal approximation works when the absolute values of the Pearson's r, Kendall's tau, and Spearman's rho are less than 1, 0.8, and 0.95, respectively.

Nonparametric Approach

For the nonparametric approach, we simply adopt a naive bootstrap method.

• We sample a pair (x_i, y_i) with replacement from the original (paired) samples until we have a sample size that equals to n, and calculate a correlation coefficient from the new samples.
• Repeat this process for a large number of times (by default we use 5000),
• then we could obtain the (1-alpha) CI for r by taking the alpha/2 and (1-alpha/2) quantiles of the obtained correlation coefficients.

References

[1] Bonett, Douglas G., and Thomas A. Wright. "Sample size requirements for estimating Pearson, Kendall and Spearman correlations." Psychometrika 65, no. 1 (2000): 23-28.
[2] Bishara, Anthony J., and James B. Hittner. "Confidence intervals for correlations when data are not normal." Behavior research methods 49, no. 1 (2017): 294-309.

Installation:

pip install correlation

or

conda install -c wangxiangwen correlation

Example Usage:

>>> import correlation
>>> a, b = list(range(2000)), list(range(200, 0, -1)) * 10
>>> correlation.corr(a, b, method='spearman_rho')
(-0.0999987624920335,          # correlation coefficient
 -0.14330929583811683,         # lower endpoint of CI
 -0.056305939127336606,        # upper endpoint of CI
 7.446171861744971e-06)        # p-value