Point-biserial correlation coefficient

To calculate r_pb, assume that the dichotomous variable Y has the two values 0 and 1. If we divide the data set into two groups, group 1 which received the value "1" on Y and group 2 which received the value "0" on Y, then the point-biserial correlation coefficient is calculated as follows:

${\displaystyle r_{pb}={\frac {M_{1}-M_{0}}{s_{n}}}{\sqrt {\frac {n_{1}n_{0}}{n^{2}}}},}$

where s_n is the standard deviation used when data are available for every member of the population:

${\displaystyle s_{n}={\sqrt {{\frac {1}{n}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}}}\,,}$

M₁ being the mean value on the continuous variable X for all data points in group 1, and M₀ the mean value on the continuous variable X for all data points in group 2. Further, n₁ is the number of data points in group 1, n₀ is the number of data points in group 2 and n is the total sample size. This formula is a computational formula that has been derived from the formula for r_XY in order to reduce steps in the calculation; it is easier to compute than r_XY.

There is an equivalent formula that uses s_n−1:

${\displaystyle r_{pb}={\frac {M_{1}-M_{0}}{s_{n-1}}}{\sqrt {\frac {n_{1}n_{0}}{n(n-1)}}},}$

where s_n−1 is the standard deviation used when data are available only for a sample of the population:

${\displaystyle s_{n-1}={\sqrt {{\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}}}.}$

The version of the formula using s_n−1 is useful if one is calculating point-biserial correlation coefficients in a programming language or other development environment where there is a function available for calculating s_n−1, but no function available for calculating s_n.

Glass and Hopkins' book Statistical Methods in Education and Psychology, (3rd Edition)[1] contains a correct version of point biserial formula.

Also the square of the point biserial correlation coefficient can be written:

${\displaystyle {\frac {(M_{1}-M_{0})^{2}}{\sum _{i=1}^{n}(X_{i}-{\overline {X}})^{2}}}\left({\frac {n_{1}n_{0}}{n}}\right)\,.}$

We can test the null hypothesis that the correlation is zero in the population. A little algebra shows that the usual formula for assessing the significance of a correlation coefficient, when applied to r_pb, is the same as the formula for an unpaired t-test and so

${\displaystyle r_{pb}{\sqrt {\frac {n_{1}+n_{0}-2}{1-r_{pb}^{2}}}}}$

follows Student's t-distribution with (n₁+n₀ − 2) degrees of freedom when the null hypothesis is true.

One disadvantage of the point biserial coefficient is that the further the distribution of Y is from 50/50, the more constrained will be the range of values which the coefficient can take. If X can be assumed to be normally distributed, a better descriptive index is given by the biserial coefficient

${\displaystyle r_{b}={\frac {M_{1}-M_{0}}{s_{n}}}{\frac {n_{1}n_{0}}{n^{2}u}},}$

where u is the ordinate of the normal distribution with zero mean and unit variance at the point which divides the distribution into proportions n₀/n and n₁/n. This is not easy to calculate, and the biserial coefficient is not widely used in practice.

A specific case of biserial correlation occurs where X is the sum of a number of dichotomous variables of which Y is one. An example of this is where X is a person's total score on a test composed of n dichotomously scored items. A statistic of interest (which is a discrimination index) is the correlation between responses to a given item and the corresponding total test scores. There are three computations in wide use,[2] all called the point-biserial correlation: (i) the Pearson correlation between item scores and total test scores including the item scores, (ii) the Pearson correlation between item scores and total test scores excluding the item scores, and (iii) a correlation adjusted for the bias caused by the inclusion of item scores in the test scores. Correlation (iii) is

${\displaystyle r_{upb}={\frac {M_{1}-M_{0}-1}{\sqrt {{\frac {n^{2}s_{n}^{2}}{n_{1}n_{0}}}-2(M_{1}-M_{0})+1}}}.}$

A slightly different version of the point biserial coefficient is the rank biserial which occurs where the variable X consists of ranks while Y is dichotomous. We could calculate the coefficient in the same way as where X is continuous but it would have the same disadvantage that the range of values it can take on becomes more constrained as the distribution of Y becomes more unequal. To get round this, we note that the coefficient will have its largest value where the smallest ranks are all opposite the 0s and the largest ranks are opposite the 1s. Its smallest value occurs where the reverse is the case. These values are respectively plus and minus (n₁ + n₀)/2. We can therefore use the reciprocal of this value to rescale the difference between the observed mean ranks on to the interval from plus one to minus one. The result is

${\displaystyle r_{rb}=2{\frac {M_{1}-M_{0}}{n_{1}+n_{0}}},}$

where M₁ and M₀ are respectively the means of the ranks corresponding to the 1 and 0 scores of the dichotomous variable. This formula, which simplifies the calculation from the counting of agreements and inversions, is due to Gene V Glass (1966).

It is possible to use this to test the null hypothesis of zero correlation in the population from which the sample was drawn. If r_rb is calculated as above then the smaller of

${\displaystyle (1+r_{rb}){\frac {n_{1}n_{0}}{2}}}$

and

${\displaystyle (1-r_{rb}){\frac {n_{1}n_{0}}{2}}}$

is distributed as Mann–Whitney U with sample sizes n₁ and n₀ when the null hypothesis is true.

• Point Biserial Coefficient (Keith Calkins, 2005)