# scipy.stats.pointbiserialr

scipy.stats.pointbiserialr(x, y)

  Calculate 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 may be computed using a shortcut formula but produces the same result as `pearsonr`.

  Parameters:
    x array_like of bools

      Input array.

    y array_like

      Input array.

  Returns:
    res: SignificanceResult

      An object containing attributes:

      statisticfloat

        The R value.

      pvaluefloat

        The two-sided 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:

  \[r_{pb} = \frac{\overline{Y_1} - \overline{Y_0}} {s_y} \sqrt{\frac{N_0 N_1} {N (N - 1)}}\]

  Where \(\overline{Y_{0}}\) and \(\overline{Y_{1}}\) 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

  [1]

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

  [2]

  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.

  [3]

  D. Kornbrot “Point Biserial Correlation”, In Wiley StatsRef: Statistics Reference Online (eds N. Balakrishnan, et al.), 2014. DOI:10.1002/9781118445112.stat06227

  Examples

      >>> import numpy as np
      >>> 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.       ]])