Point bisiserial correlation

As a point-biserial correlation coefficient is correlation coefficient of the relationship between an interval scaled characteristic and a dichotomous ( Bernoulli distributed ) feature called. It is not an independent measure, but a special case of the usual Pearson correlation coefficient, which in this case can be calculated as ${\displaystyle I}$ ${\displaystyle D}$

${\displaystyle \rho ={\frac {{\overline {I}}_{D=1}-{\overline {I}}_{D=0}}{\sqrt {\mathrm {QS} (I)}}}\cdot {\sqrt {n\cdot p\cdot q}}}$,

where the sum of squares , the sample size, the proportion of the examination units with the property recorded in D and the proportion of the examination units without the characteristic recorded in D denotes. ${\displaystyle \mathrm {QS} }$ ${\displaystyle n}$ ${\displaystyle p}$ ${\displaystyle q}$

Derivation from the Pearson correlation

For the sake of simplicity, it is assumed that the dichotomous feature takes on the values ​​0 and 1, so that the mean value in is equal to . The correlation between and over is calculated according to the general formula ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle p}$ ${\displaystyle I}$ ${\displaystyle D}$

${\displaystyle \rho ={\frac {\sum _{i=1}^{n}(I_{i}-{\bar {I}})(D_{i}-{\bar {D}})}{\sqrt {\mathrm {QS} (I)\cdot \mathrm {QS} (D)}}}}$.

A distinction can now be made between cases: units of investigation are D = 1 and are above the mean value in D, the other units of investigation are D = 0 and are below the mean value in D. This applies ${\displaystyle n\cdot p}$ ${\displaystyle 1-p=q}$ ${\displaystyle n\cdot q}$ ${\displaystyle 0-p=-p}$

${\displaystyle \rho ={\frac {n\cdot p\cdot ({\bar {I}}_{D=1}-{\bar {I}})\cdot q+n\cdot q\cdot ({\bar {I}}_{D=0}-{\bar {I}})\cdot (-p)}{\sqrt {\mathrm {QS} (I)\cdot (n\cdot p\cdot q^{2}+n\cdot q\cdot (-p)^{2})}}}}$,

what is about

${\displaystyle \rho ={\frac {n\cdot p\cdot q\cdot ({\bar {I}}_{D=1}-{\bar {I}}_{D=0})}{\sqrt {\mathrm {QS} (I)\cdot (n\cdot p\cdot q)}}}}$

can be simplified to the above equation.

Use in common statistics software

SPSS and R automatically use the point-to-point calculation method if the commands CORRELATEor cor, are cor.testrequested and one of the variables has only two characteristics (e.g. the values ​​0 and 1) that are also considered relevant to the calculation (−7 or 99 e.g. B. can be marked as missing values ​​in SPSS and thus ignored).

literature

• Jürgen Bortz: Statistics for human and social scientists. 6th edition. Springer, Berlin a. a. 2005, ISBN 3-540-21271-X .
• J. Cohen, P. Cohen, SG West, LS Aiken: Applied Multiple Regression / Correlation Analysis For The Behavioral Sciences. London 2003, ISBN 0-8058-2223-2 .