scipy.stats.pearsonr#

scipy.stats.pearsonr(x, y, *, alternative='two-sided', method=None)[source]#

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

The Pearson correlation coefficient [1] measures the linear relationship between two datasets. 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.

This function also performs a test of the null hypothesis that the distributions underlying the samples are uncorrelated and normally distributed. (See Kowalski [3] for a discussion of the effects of non-normality of the input on the distribution of the correlation coefficient.) 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 array.

y(N,) array_like

Input array.

alternative{‘two-sided’, ‘greater’, ‘less’}, optional

Defines the alternative hypothesis. Default is ‘two-sided’. The following options are available:

‘two-sided’: the correlation is nonzero
‘less’: the correlation is negative (less than zero)
‘greater’: the correlation is positive (greater than zero)

New in version 1.9.0.

methodResamplingMethod, optional

Defines the method used to compute the p-value. If method is an instance of PermutationMethod/MonteCarloMethod, the p-value is computed using scipy.stats.permutation_test/scipy.stats.monte_carlo_test with the provided configuration options and other appropriate settings. Otherwise, the p-value is computed as documented in the notes.

New in version 1.11.0.

Returns:

resultPearsonRResult

An object with the following attributes:

statisticfloat: Pearson product-moment correlation coefficient.
pvaluefloat: The p-value associated with the chosen alternative.

The object has the following method:

confidence_interval(confidence_level, method): This computes the confidence interval of the correlation coefficient statistic for the given confidence level. The confidence interval is returned in a namedtuple with fields low and high. If method is not provided, the confidence interval is computed using the Fisher transformation [1]. If method is an instance of BootstrapMethod, the confidence interval is computed using scipy.stats.bootstrap with the provided configuration options and other appropriate settings. In some cases, confidence limits may be NaN due to a degenerate resample, and this is typical for very small samples (~6 observations).

Warns:

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

\[f(r) = \frac{{(1-r^2)}^{n/2-2}}{\mathrm{B}(\frac{1}{2},\frac{n}{2}-1)}\]

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 when the method parameter is left at its default value (None). 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 default 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.

For backwards compatibility, the object that is returned also behaves like a tuple of length two that holds the statistic and the p-value.

References

[1] (1,2,3)

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

[2]

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

[3]

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

>>> import numpy as np
>>> from scipy import stats
>>> x, y = [1, 2, 3, 4, 5, 6, 7], [10, 9, 2.5, 6, 4, 3, 2]
>>> res = stats.pearsonr(x, y)
>>> res
PearsonRResult(statistic=-0.828503883588428, pvalue=0.021280260007523286)

To perform an exact permutation version of the test:

>>> rng = np.random.default_rng()
>>> method = stats.PermutationMethod(n_resamples=np.inf, random_state=rng)
>>> stats.pearsonr(x, y, method=method)
PearsonRResult(statistic=-0.828503883588428, pvalue=0.028174603174603175)

To perform the test under the null hypothesis that the data were drawn from uniform distributions:

>>> method = stats.MonteCarloMethod(rvs=(rng.uniform, rng.uniform))
>>> stats.pearsonr(x, y, method=method)
PearsonRResult(statistic=-0.828503883588428, pvalue=0.0188)

To produce an asymptotic 90% confidence interval:

>>> res.confidence_interval(confidence_level=0.9)
ConfidenceInterval(low=-0.9644331982722841, high=-0.3460237473272273)

And for a bootstrap confidence interval:

>>> method = stats.BootstrapMethod(method='BCa', random_state=rng)
>>> res.confidence_interval(confidence_level=0.9, method=method)
ConfidenceInterval(low=-0.9983163756488651, high=-0.22771001702132443)  # may vary

There is a linear dependence between x and y if y = a + b*x + e, where a,b are constants and e is a random error term, assumed to be independent of x. For simplicity, assume that x is standard normal, a=0, b=1 and let e follow a normal distribution with mean zero and standard deviation s>0.

>>> rng = np.random.default_rng()
>>> s = 0.5
>>> x = stats.norm.rvs(size=500, random_state=rng)
>>> e = stats.norm.rvs(scale=s, size=500, random_state=rng)
>>> y = x + e
>>> stats.pearsonr(x, y).statistic
0.9001942438244763

This should be close to the exact value given by

>>> 1/np.sqrt(1 + s**2)
0.8944271909999159

For s=0.5, we observe a high level of correlation. In general, a large variance of the noise reduces the correlation, while the correlation approaches one as the variance of the error goes to zero.

It is important to keep in mind that no correlation does not imply independence unless (x, y) is jointly normal. Correlation can even be zero when there is a very simple dependence structure: if X follows a standard normal distribution, let y = abs(x). Note that the correlation between x and y is zero. Indeed, since the expectation of x is zero, cov(x, y) = E[x*y]. By definition, this equals E[x*abs(x)] which is zero by symmetry. The following lines of code illustrate this observation:

>>> y = np.abs(x)
>>> stats.pearsonr(x, y)
PearsonRResult(statistic=-0.05444919272687482, pvalue=0.22422294836207743)

A non-zero correlation coefficient can be misleading. For example, if X has a standard normal distribution, define y = x if x < 0 and y = 0 otherwise. A simple calculation shows that corr(x, y) = sqrt(2/Pi) = 0.797…, implying a high level of correlation:

>>> y = np.where(x < 0, x, 0)
>>> stats.pearsonr(x, y)
PearsonRResult(statistic=0.861985781588, pvalue=4.813432002751103e-149)

This is unintuitive since there is no dependence of x and y if x is larger than zero which happens in about half of the cases if we sample x and y.