# Dependent Samples - Quantum Column Means Test

By default, when this test is selected Q computes standard errors using Taylor series linearization. To replicate the results of IBM SPSS Data Collection Model programs (e.g., Survey Reporter), you need to also change the Weights and significance to Kish approximation in Statistical Assumptions.

Where Taylor series linearization is performed, and ${\displaystyle {\bar {x}}_{1}}$ and ${\displaystyle {\bar {x}}_{2}}$ are the two means, the test statistic is:

${\displaystyle t={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{{\sqrt {\frac {e_{1}max(1,e_{1}-b)s_{{\bar {x}}_{1}}^{2}+e_{2}max(e_{2}-b,1)s_{{\bar {x}}_{2}}^{2}}{n_{1}+n_{1}-2b}}}({\frac {1}{e_{1}}}+{\frac {1}{e_{2}}}-r{\frac {2e_{o}}{e_{1}e_{2}}})}}}$

Otherwise, the test statistic is:

${\displaystyle t={\frac {{\bar {x}}_{1}-{\bar {x}}_{2}}{\sqrt {{\frac {e_{1}max(1,\pi _{1}-b)s_{{\bar {x}}_{1}}^{2}+e_{2}max(\pi _{2}-b,1)s_{{\bar {x}}_{2}}^{2}}{\pi _{1}-{\frac {\sum _{i=1}^{n_{1}}w_{i1}^{2}}{\pi _{2}}}+\pi _{2}-{\frac {\sum _{i=1}^{n_{2}}w_{i2}^{2}}{\pi _{2}}}}}({\frac {1}{e_{1}}}+{\frac {1}{e_{2}}}-r{\frac {2e_{o}}{e_{1}e_{2}}})}}}}$

where:

${\displaystyle 1}$ and ${\displaystyle 2}$ refer to the two variables and ${\displaystyle o}$ to the overlap,
${\displaystyle n_{j}}$ is the sample size,
${\displaystyle e_{j}}$ is the effective sample size,
${\displaystyle \pi _{j}}$ is the weighted sample size,
${\displaystyle w_{ji}}$ is the weight for the ${\displaystyle i}$th observation in the ${\displaystyle j}$th group,
${\displaystyle b=1}$ if Bessel's correction is selected and 0 otherwise,
${\displaystyle p\approx 2\Pr(t_{round(e_{1}+e_{2}-e_{0}-2)}\geq |t|)}$,
${\displaystyle r}$ is Pearson's Product Moment Correlation for the overlapping sample,
${\displaystyle s_{{\bar {x}}_{j}}}$ is the Standard Error of the mean,
the p is returned as NaN if the denominator is less than 0.00001.

Also note that if ${\displaystyle n_{o}=0}$ the Independent version of the test is conducted instead.