## 21.3 Sample Size Planning for ANOVA

### 21.3.1 Balanced Designs

#### 21.3.1.1 Single Factor Studies

##### 21.3.1.1.1 Fixed cell means

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

where $$\phi$$ is the non-centrality 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 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

$\Delta = \max(\mu_i) - \min(\mu_i)$

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.

#### 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:

\begin{aligned} \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) \end{aligned}

Test for Factor $$A$$ main effects:

\begin{aligned} \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) \end{aligned}

Test for Factor $$B$$ main effects:

\begin{aligned} \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) \end{aligned}

Procedure:

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:

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

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

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