scipy.stats.pearsonr

scipy.stats.pearsonr(x, y)

Pearson correlation coefficient and p-value for testing non-correlation.

The Pearson correlation coefficient [1] measures the linear relationship between two datasets. The calculation of the p-value relies on the assumption that each dataset is normally distributed. (See Kowalski [3] for a discussion of the effects of non-normality of the input on the distribution of the correlation coefficient.) Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply an exact linear relationship. Positive correlations imply that as x increases, so does y. Negative correlations imply that as x increases, y decreases.

The p-value roughly indicates the probability of an uncorrelated system producing datasets that have a Pearson correlation at least as extreme as the one computed from these datasets.

Parameters

x(N,) array_like

Input

y(N,) array_like

Input

Returns

rfloat

Pearson’s correlation coefficient

p-valuefloat

two-tailed p-value

Warns

PearsonRConstantInputWarning

Raised if an input is a constant array. The correlation coefficient is not defined in this case, so np.nan is returned.

PearsonRNearConstantInputWarning

Raised if an input is “nearly” constant. The array x is considered nearly constant if norm(x - mean(x)) < 1e-13 * abs(mean(x)). Numerical errors in the calculation x - mean(x) in this case might result in an inaccurate calculation of r.

See also

spearmanr

Spearman rank-order correlation coefficient.

kendalltau

Kendall’s tau, a correlation measure for ordinal data.

Notes

The correlation coefficient is calculated as follows:

\[r = \frac{\sum (x - m_x) (y - m_y)} {\sqrt{\sum (x - m_x)^2 \sum (y - m_y)^2}}\]

where \(m_x\) is the mean of the vector \(x\) and \(m_y\) is the mean of the vector \(y\).

Under the assumption that x and y are drawn from independent normal distributions (so the population correlation coefficient is 0), the probability density function of the sample correlation coefficient r is ([1], [2]):

       (1 - r**2)**(n/2 - 2)
f(r) = ---------------------
          B(1/2, n/2 - 1)

where n is the number of samples, and B is the beta function. This is sometimes referred to as the exact distribution of r. This is the distribution that is used in pearsonr to compute the p-value. The distribution is a beta distribution on the interval [-1, 1], with equal shape parameters a = b = n/2 - 1. In terms of SciPy’s implementation of the beta distribution, the distribution of r is:

dist = scipy.stats.beta(n/2 - 1, n/2 - 1, loc=-1, scale=2)

The p-value returned by pearsonr is a two-sided p-value. For a given sample with correlation coefficient r, the p-value is the probability that abs(r’) of a random sample x’ and y’ drawn from the population with zero correlation would be greater than or equal to abs(r). In terms of the object dist shown above, the p-value for a given r and length n can be computed as:

p = 2*dist.cdf(-abs(r))

When n is 2, the above continuous distribution is not well-defined. One can interpret the limit of the beta distribution as the shape parameters a and b approach a = b = 0 as a discrete distribution with equal probability masses at r = 1 and r = -1. More directly, one can observe that, given the data x = [x1, x2] and y = [y1, y2], and assuming x1 != x2 and y1 != y2, the only possible values for r are 1 and -1. Because abs(r’) for any sample x’ and y’ with length 2 will be 1, the two-sided p-value for a sample of length 2 is always 1.

References

1(1,2,3)

“Pearson correlation coefficient”, Wikipedia, https://en.wikipedia.org/wiki/Pearson_correlation_coefficient

2(1,2)

Student, “Probable error of a correlation coefficient”, Biometrika, Volume 6, Issue 2-3, 1 September 1908, pp. 302-310.

3(1,2)

C. J. Kowalski, “On the Effects of Non-Normality on the Distribution of the Sample Product-Moment Correlation Coefficient” Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 21, No. 1 (1972), pp. 1-12.

Examples

>>> from scipy import stats
>>> a = np.array([0, 0, 0, 1, 1, 1, 1])
>>> b = np.arange(7)
>>> stats.pearsonr(a, b)
(0.8660254037844386, 0.011724811003954649)

>>> stats.pearsonr([1, 2, 3, 4, 5], [10, 9, 2.5, 6, 4])
(-0.7426106572325057, 0.1505558088534455)