20.3 Sample Size Planning for ANOVA

20.3.1 Balanced Designs

20.3.1.1 Single Factor Studies

20.3.1.1.1 Fixed cell means

\[ P(F>f_{(1-\alpha;a-1,N-a)}|\phi) = 1 - \beta \]

where \(\phi\) is the noncentrality parameter (measures how unequal the treatment means \(\mu_i\) are)

\[ \phi = \frac{1}{\sigma}\sqrt{\frac{n}{a}\sum_i (\mu_i - \mu_.)^2} , (n_i \equiv n) \]

and

\[ \mu_. = \frac{\sum \mu_i}{a} \]

To decide on the power probabilities we use the noncetral 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

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

Hence, we need

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

Notes:

  • 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.

20.3.1.2 Multi-factor Studies

The same noncentral F tables can be used here

For two-factor fixed effect model

Test for interactions:

\[ \phi = \frac{1}{\sigma} \sqrt{\frac{n \sum \sum (\alpha \beta_{ij})^2}{(a-1)(b-1)+1}} = \frac{1}{\sigma} \sqrt{\frac{n \sum \sum (\mu_{ij}- \mu_{i.} - \mu_{.j} + \mu_{..})^2}{(a-1)(b-1)+1}} \\ \upsilon_1 = (a-1)(b-1) \\ \upsilon_2 = ab(n-1) \]

Test for Factor A main effects:

\[ \phi = \frac{1}{\sigma} \sqrt{\frac{nb \sum \alpha_i^2}{a}} = \frac{1}{\sigma}\sqrt{\frac{nb \sum (\mu_{i.}- \mu_{..})^2}{a}} \\ \upsilon_1 = a-1 \\ \upsilon_2 = ab(n-1) \]

Test for Factor B main effects:

\[ \phi = \frac{1}{\sigma} \sqrt{\frac{na \sum \beta_j^2}{b}} = \frac{1}{\sigma}\sqrt{\frac{na \sum (\mu_{.j}- \mu_{..})^2}{b}} \\ \upsilon_1 = b-1 \\ \upsilon_2 = ab(n-1) \]

Procedure:

  1. Specify the minimu 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.

20.3.2 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:

\[ \phi = \frac{1}{\sigma} \sqrt{\frac{n}{r} \sum (\mu_i - \mu_.)^2} \]

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

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