# Dependent Samples - Quantum Column Proportions Test

Where ${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$ are the two proportions, ${\displaystyle e_{1}}$ and ${\displaystyle e_{2}}$ are the effective sample sizes, ${\displaystyle \pi _{1}}$ and ${\displaystyle \pi _{2}}$ are the weighted sample sizes, ${\displaystyle r}$ is Pearson's Product Moment Correlation for the overlapping sample and ${\displaystyle e_{o}}$ is the effective sample size that overlaps between the two samples, the test statistic is:
${\displaystyle t={\frac {p_{1}-p_{2}}{\sqrt {p_{12}(1-p_{12})(e_{1}^{-1}+e_{2}^{-1}-r{\frac {2e_{o}}{e_{1}e_{2}}})({\frac {e_{1}+e_{2}}{e_{1}+e_{2}-1}})}}}}$,
where: ${\displaystyle p_{12}={\frac {\pi _{1}p_{1}+\pi _{2}p_{2}}{\pi _{1}+\pi _{2}}}}$, and
${\displaystyle p\approx 2\Pr(t_{e_{1}+e_{2}-1}\geq |t|)}$.
Also note that if ${\displaystyle e_{o}=0}$ the Independent version of the test is conducted instead.