pearsonr#
- scipy.stats.mstats.pearsonr(x, y)[source]#
- 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. - Parameters:
- x(N,) array_like
- Input array. 
- y(N,) array_like
- Input array. 
 
- Returns:
- rfloat
- Pearson’s correlation coefficient. 
- p-valuefloat
- Two-tailed p-value. 
 
- Warns:
- ConstantInputWarning
- Raised if an input is a constant array. The correlation coefficient is not defined in this case, so - np.nanis returned.
- NearConstantInputWarning
- Raised if an input is “nearly” constant. The array - xis 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 - pearsonrto 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 - pearsonris a two-sided p-value. 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. More precisely, 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- distshown 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)- “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 >>> from scipy.stats import mstats >>> mstats.pearsonr([1, 2, 3, 4, 5], [10, 9, 2.5, 6, 4]) (-0.7426106572325057, 0.1505558088534455) - 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. - >>> s = 0.5 >>> x = stats.norm.rvs(size=500) >>> e = stats.norm.rvs(scale=s, size=500) >>> y = x + e >>> mstats.pearsonr(x, y) (0.9029601878969703, 8.428978827629898e-185) # may vary - 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) >>> mstats.pearsonr(x, y) (-0.016172891856853524, 0.7182823678751942) # may vary - 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) >>> mstats.pearsonr(x, y) (0.8537091583771509, 3.183461621422181e-143) # may vary - 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.