Cochran's Q
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This test is a non-parametric equivalent of Repeated Measures ANOVA which is applicable with binary data (i.e., proportions). Note that, given this is used on repeated measures data, the overlap of respondents between categories is taken into account ([math]\displaystyle{ u_i }[/math] below), and thus cells with similar counts and sample sizes can have different significant results. The test statistic is:
[math]\displaystyle{ Q = \frac{c(c-1)\sum^c_{j=1}(\sum^n_{i=1}w_i x_{ij} - \frac{1}{c}\sum^n_{i=1}w_i u_i)^2}{c\sum^n_{i=1}w_i u_i - \sum^n_{i=1} w_i u^2_i} }[/math]
where:
- [math]\displaystyle{ x_{ij} }[/math] is the data for [math]\displaystyle{ j }[/math]th of [math]\displaystyle{ c }[/math] categories of the [math]\displaystyle{ i }[/math]th of [math]\displaystyle{ n }[/math] observations,
- [math]\displaystyle{ x_{ij}\in {0,1} }[/math],
- [math]\displaystyle{ w_i }[/math] is the Calibrated Weight, and
- [math]\displaystyle{ u_i = \sum^c_{j=1} x_{ij} }[/math]
- [math]\displaystyle{ p \approx \Pr(\chi^2_{c-1} \ge Q) }[/math]