Dependent Samples - Quantum Column Proportions Test

Where [math]\displaystyle{ p_1 }[/math] and [math]\displaystyle{ p_2 }[/math] are the two proportions, [math]\displaystyle{ e_1 }[/math] and [math]\displaystyle{ e_2 }[/math] are the effective sample sizes, [math]\displaystyle{ \pi_1 }[/math] and [math]\displaystyle{ \pi_2 }[/math] are the weighted sample sizes, [math]\displaystyle{ r }[/math] is Pearson's Product Moment Correlation for the overlapping sample and [math]\displaystyle{ e_o }[/math] is the effective sample size that overlaps between the two samples, the test statistic is:

[math]\displaystyle{ t=\frac{p_1-p_2}{\sqrt{p_{12}(1-p_{12})(e^{-1}_1+e^{-1}_2 - r \frac{2e_o}{e_1 e_2})(\frac{e_1+e_2}{e_1+e_2-1})}} }[/math],

where: [math]\displaystyle{ p_{12}=\frac{\pi_1 p_1 + \pi_2 p_2}{\pi_1+\pi_2} }[/math], and

[math]\displaystyle{ p \approx 2 \Pr(t_{e_1+e_2-1} \ge |t|) }[/math].

Note that if wishing to replicate results from IBM SPSS Data Collection products (including Survey Reporter and Quantum) additional settings need to be modified (see How to Replicate Survey Reporter Significance Tests).

Also note that if [math]\displaystyle{ e_o = 0 }[/math] the Independent version of the test is conducted instead.