17.9.4.2 Algorithm ((PSS) One-Way ANOVA)

Generally, when performing an ANOVA test, we want to know the power of the test with specific sample sizes, or the sample sizes at specified power levels. The PSS ANOVA1 X-Function can be used to calculate the power as well as sample size.

The algorithm for Power calculation.

We assume that each group has equal size. The formula of the Power calculation is given by

$Power=Prob(F> F_0,\nu _{1,}\nu _2,\lambda )\,\!$

where $F\,\!$ is distributed as the non-central $F(\lambda ,\nu _{1,}\nu _2) \,\!$

$F_0=F_{(1-\alpha ,\nu _{1,}\nu _2)}$ , the $1-\alpha \,\!$ quartile of the F-distribution with $\nu _1 \,\!$ and $\nu _2 \,\!$ degrees of freedom

$\nu _1=r-1 \,\!$ is the degrees of freedom of the numerator

$\nu _2=n*r-r \,\!$ is the degrees of freedom of the denominator

$n \,\!$ is the number per groupi

$r \,\!$ is the number of groups

$\lambda =\frac{n*CSS}{S^2}$ is the noncentrality parameter

If we specify the group means difference by CSS of Means:

$CSS=\sum _{g=1}^r(\mu _g-\mu )^2$

$\mu _g \,\!$ is the mean of the ith group

$\mu \,\!$ is the overall mean

$S^2 \,\!$ is estimated by the mean square error

If we specify the group means difference by Maximum Difference between Means:

$CSS=D^2/2$

$D=\mu_{max}-\mu_{min}$ is the difference between the smallest group mean and the largest group mean.

$\mu_{max}$ is the largest group mean.

$\mu_{min}$ is the smallest group mean.