A Guide on Data Analysis

To decide on the power probabilities we use the non-central F distribution.

We could use the power table directly when effects are fixed and design is balanced by using minimum range of factor level means for your desired differences

\[ \Delta = \max(\mu_i) - \min(\mu_i) \]

• \(\alpha\) level
• \(\Delta\)
• \(\sigma\)
• \(\beta\)

• When \(\Delta/\sigma\) is small greatly affects sample size, but if \(\Delta/\sigma\) is large.
• Reducing \(\alpha\) or \(\beta\) increases the required sample sizes.
• Error in estimating \(\sigma\) can make a large difference.

21.3.1.2 Multi-factor Studies

The same noncentral \(F\) tables can be used here

For two-factor fixed effect model

Test for interactions:

Test for Factor \(A\) main effects:

Test for Factor \(B\) main effects:

1. Specify the minimum range of Factor \(A\) means
2. Obtain sample sizes with \(r = a\) . The resulting sample size is \(bn\) , from which \(n\) can be obtained.
3. Repeat the first 2 steps for Factor \(B\) minimum range.
4. Choose the greater number of sample size between \(A\) and \(B\) .

21.3.2 Randomized Block Experiments

Analogous to completely randomized designs . The power of the F-test for treatment effects for randomized block design uses the same non-centrality parameter as completely randomized design:

However, the power level is different from the randomized block design because

• error variance \(\sigma^2\) is different
• df(MSE) is different.