# Multiple Comparisons t-Test with False Discovery Rate Correction

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The test statistic is:

$\displaystyle{ t=\frac{\bar x_1-\bar x_2}{\sqrt{\frac{\sum^{J}_{j=1}\sum^{n_j}_{i=1} w_{ij}(x_{ij} - \bar x_j)^2}{v}(\frac{1}{e_1}+\frac{1}{e_1})}} }$

where:

$\displaystyle{ \bar x_1 }$ and $\displaystyle{ \bar x_2 }$ are the means of the two groups being compared and $\displaystyle{ \bar x_j }$ is the mean of the $\displaystyle{ j }$ of $\displaystyle{ J }$ groups,
when applying the test to Repeated Measures, each respondent’s average is initially subtracted from their data and it is this corrected data that constitutes of $\displaystyle{ x_{ij} }$,
$\displaystyle{ n_j }$ is the number of observations in the $\displaystyle{ j }$th of $\displaystyle{ J }$ groups,
$\displaystyle{ w_{ij} }$ is the Calibrated Weight for the $\displaystyle{ i }$th observation in the $\displaystyle{ j }$ group,
$\displaystyle{ e_j }$ is the Effective Sample Size for the $\displaystyle{ j }$ group,
$\displaystyle{ v = (J - 1)(\sum^J_{j=1} e_j - 1) }$ for Repeated Measures and $\displaystyle{ v = \sum^J_{j=1} e_j - J }$ otherwise.
$\displaystyle{ p \approx 2\Pr(t_v \ge |t|) }$,
the p-value is then corrected using the False Discovery Rate Correction.