Bravais-Pearson correlation coefficient and the Phi coefficient

Question

Given the $\Phi$ coefficient $(h_{11}h_{22}-h_{12}h_{21})/\sqrt{h_{1\cdot}h_{2\cdot}h_{\cdot 1}h_{\cdot 2}}$ of a dichotomic table $\begin{pmatrix}h_{11} & h_{12}\\ h_{21}& h_{22}\end{pmatrix}$ I would like to understand how to connect it with the calculation of the Bravais-Pearson correlation coefficient: How to map the 2D data $(i,j)\mapsto h_{i,j}$ to something of the form $x_i \mapsto y_i$ so that the formula for $r$ can be used?

Answer 1

The trick is to reinterpret the $h_{ij}$ entries as frequencies of the cases $(X=0,Y=0),(0,1),(1,0),(1,1)$. Therefore one can write down the Bravais-Pearson correlation coefficient $r$ as \begin{align} r ={}& \frac{\bar{x}\bar{y}h_{11} + (0 - \bar{x})(1-\bar{y})h_{12} + (1 - \bar{x})(0-\bar{y})h_{21} + (1 - \bar{x})(1-\bar{y})h_{22}}{\sqrt{[\bar{x}^2(h_{11} + h_{12}) + (1 - \bar{x})^2(h_{21} + h_{22})][\bar{y}^2(h_{11} + h_{21}) + (1 - \bar{y})^2(h_{12} + h_{22})]}}\;.\label{r2} \end{align} With $N= h_{11} + h_{12} + h_{21} + h_{22}$ and \begin{align} \bar{x} ={}& \frac{h_{21} + h_{22}}{N}\;,\\ \bar{y} ={}& \frac{h_{12} + h_{22}}{N}\;, \end{align} we find \begin{align} r ={}& \frac{h_{11}h_{22} - h_{12}h_{21}}{\sqrt{h_{1\cdot}h_{2\cdot}h_{\cdot 1}h_{\cdot 2}}} = \Phi\;. \end{align}