By default, when this test is selected, standard errors are computed 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 \(\bar x_1\) and \(\bar x_2\) are the two means, the test statistic is:

\[\begin{align} =\frac{\bar x_1-\bar x_2}{\sqrt{\frac{e_1 max(1, e_1-b)s^2_{\bar x_1} + e_2 max(e_2-b, 1)s^2_{\bar x_2}}{n_{1} + n_{1} - 2b}}(\frac{1}{e_1}+\frac{1}{e_2}-r \frac{2e_o}{e_1 e_2})} \end{align},\]

Otherwise, the test statistic is:

\[\begin{align} =t=\frac{\bar x_1-\bar x_2}{\sqrt{\frac{e_1 max(1, \pi_1-b)s^2_{\bar x_1} + e_2 max(\pi_2-b, 1)s^2_{\bar x_2}}{\pi_1 - \frac{\sum^{n_{1}}_{i=1} w^2_{i1}}{\pi_2} + \pi_2 - \frac{\sum^{n_{2}}_{i=1} w^2_{i2}}{\pi_2}}(\frac{1}{e_1}+\frac{1}{e_2}-r \frac{2e_o}{e_1 e_2})}} \end{align},\]

where:

\(1\) and \(2\) refer to the two variables and \(o\) to the overlap,

\(n_j\) is the sample size,

\(e_j\) is the effective sample size,

\(\pi_j\) is the weighted sample size,

\(w_{ji} \) is the weight for the \(i\)th observation in the \(j\)th group,

\(b=1\) if **Bessel's correction** is selected and 0 otherwise,

\(p \approx 2 \Pr(t_{round( e_1 + e_2 - e_0 - 2)} \ge |t|),\)

\(r\) is Pearson's Product Moment Correlation for the overlapping sample,

\(s_{\bar x_j}\) is the Standard Error of the mean,

the Statistics - p is returned as NaN if the denominator is less than 0.00001.

Also note that if \(n_o = 0\) the *Independent* version of the test is conducted instead.