# scipy.stats.pointbiserialr¶

scipy.stats.pointbiserialr(x, y)[source]

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

Parameters
xarray_like of bools

Input array.

yarray_like

Input array.

Returns
correlationfloat

R value.

pvaluefloat

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:

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

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

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

>>> 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.       ]])