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# scipy.stats.pointbiserialr

`scipy.stats.``pointbiserialr`(x, y)

  Calculates a point biserial correlation coefficient and its p-value.

  The point biserial correlation is used to measure the relationship between a binary variable, x, and a continuous variable, y. Like other correlation coefficients, this one varies between -1 and +1 with 0 implying no correlation. Correlations of -1 or +1 imply a determinative relationship.

  This function uses a shortcut formula but produces the same result as `pearsonr`.

  Parameters: | x : array_like of bools Input array. y : array_like Input array.
  Returns: | correlation : float R value pvalue : float 2-tailed p-value

  Notes

  `pointbiserialr` uses a t-test with `n-1` degrees of freedom. It is equivalent to pearsonr.

  The value of the point-biserial correlation can be calculated from:

  \[r_{pb} = \frac{\overline{Y_{1}} - \overline{Y_{0}}}{s_{y}}\sqrt{\frac{N_{1} N_{2}}{N (N - 1))}}\]

  Where \(Y_{0}\) and \(Y_{1}\) are means of the metric observations coded 0 and 1 respectively; \(N_{0}\) and \(N_{1}\) are number of observations coded 0 and 1 respectively; \(N\) is the total number of observations and \(s_{y}\) is the standard deviation of all the metric observations.

  A value of \(r_{pb}\) that is significantly different from zero is completely equivalent to a significant difference in means between the two groups. Thus, an independent groups t Test with \(N-2\) degrees of freedom may be used to test whether \(r_{pb}\) is nonzero. The relation between the t-statistic for comparing two independent groups and \(r_{pb}\) is given by:

  \[t = \sqrt{N - 2}\frac{r_{pb}}{\sqrt{1 - r^{2}_{pb}}}\]

  References

  [R618] | J. Lev, “The Point Biserial Coefficient of Correlation”, Ann. Math. Statist., Vol. 20, no.1, pp. 125-126, 1949.

  [R619] | R.F. Tate, “Correlation Between a Discrete and a Continuous Variable. Point-Biserial Correlation.”, Ann. Math. Statist., Vol. 25, np. 3, pp. 603-607, 1954.

  [R620] | http://onlinelibrary.wiley.com/doi/10.1002/9781118445112.stat06227/full

  Examples

      >>> from scipy import stats
      >>> a = np.array([0, 0, 0, 1, 1, 1, 1])
      >>> b = np.arange(7)
      >>> stats.pointbiserialr(a, b)
      (0.8660254037844386, 0.011724811003954652)
      >>> stats.pearsonr(a, b)
      (0.86602540378443871, 0.011724811003954626)
      >>> np.corrcoef(a, b)
      array([[ 1.       ,  0.8660254],
             [ 0.8660254,  1.       ]])