Complex Samples Dependent T-Test - Comparing a Sub-Group Proportion to Total

Where $g_{t}$ is the proportion in the total sample and $g_{s}$ is the proportion in the sub-group, $n_{t}$ and $n_{s}$ are their respective effective sample sizes, and $w_{s}$ is the proportion of the total population in sub-group $s$ :

$t = \frac{g_{t} - g_{s}}{(1 - w_{s}) \sqrt{d_{e f f} (s e_{s}^{2} + s e_{k}^{2})}}$ ,

where:

p = 2 \Pr (t_{v} \geq | t |)

,

g_{k} = (g_{t} - g_{s} w_{s}) / (1 - w_{s})

,

s e_{k} = \sqrt{\frac{g_{k} (1 - g_{k})}{n_{t} - n_{s} - b}}

,

s e_{g} = \sqrt{\frac{g_{g} (1 - g_{g})}{n_{s} - b}}

,

v = \frac{(\frac{s e_{s}^{2}}{n_{s}} + \frac{s e_{k}^{2}}{n_{k}})^{2}}{\frac{(\frac{s e_{s}^{2}}{n_{s}})^{2}}{n_{s} - b} + \frac{(\frac{s e_{k}^{2}}{n_{k}})^{2}}{n_{k} - b}}

, or, if Weights and significance is set to Un-weighted sample size in tests (see Weights, Effective Sample Size and Design Effects),

v = n_{t} - 2 b

b

is 1 if Bessel's correction is selected for Proportions in Statistical Assumptions and 0 otherwise,

d_{e f f}

is Extra Deff, and

w_{s}

is the proportion of the Population represents by the sub-group.

Complex Samples Dependent T-Test - Comparing a Sub-Group Proportion to Total

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