Repeated Measures ANOVA with Huynh and Feldt Epsilon Correction

The test statistics is:

[math]\displaystyle{ F=\frac{\sum^s_{j=1} \sum^{n_j}_{i = 1} w_{ij}(\bar{x}_j - \bar{x})^2 / (j-1)} {\sum^s_{j=1} \sum^{n_j}_{i=1} w_{ij}(\bar{x}_j - x_{ij})^2 / ((\sum^s_{j=1} \sum^{n_j}_{i = 1} w_{ij}-1) (j - 1))} }[/math]

where:

[math]\displaystyle{ x_{ij} }[/math] is the value of the [math]\displaystyle{ i }[/math]th of [math]\displaystyle{ n_j }[/math] observations in the [math]\displaystyle{ j }[/math]th of [math]\displaystyle{ s }[/math] groups, where [math]\displaystyle{ x_{ij} }[/math] has been 'centered' such that [math]\displaystyle{ \sum^s_{j=1} x_{ij} = 0\forall i }[/math],

[math]\displaystyle{ \bar{x}_j }[/math] is the average in the [math]\displaystyle{ j }[/math]th group,

[math]\displaystyle{ \bar{x} }[/math] is the overall average,

[math]\displaystyle{ w_{ij} }[/math] is the calibrated weight,

[math]\displaystyle{ p \approx \Pr(F_{(s-1)\epsilon,(\sum^s_{j=1} \sum^{n_j}_{i = 1} w_{ij}-1) (s - 1))\epsilon} \ge F ) }[/math], and

[math]\displaystyle{ \epsilon }[/math] is computed using the Greenhouse-Geisser method.

and [math]\displaystyle{ F }[/math] is evaulated using the F-distribution with [math]\displaystyle{ (j-1)\epsilon }[/math] and [math]\displaystyle{ ((\sum^s_{j=1} \sum^{n_j}_{i = 1} w_{ij}-1) (j - 1))\epsilon }[/math] degrees of freedom, where [math]\displaystyle{ \epsilon = 1 / (j - 1) }[/math].

See also

• ANOVA-Type Tests - Comparing Three or More Groups
• Planned ANOVA-Type Test

References

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