24.5 Sample Size Planning for ANOVA

24.5.1 Balanced Designs

Choosing an appropriate sample size for an ANOVA study requires ensuring sufficient power while balancing practical constraints.

24.5.2 Single Factor Studies

24.5.2.1 Fixed Cell Means Model

The probability of rejecting $H_0$ when it is false (power) is given by:

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

where:

• $\phi$ is the non-centrality parameter (measuring the inequality among treatment means $\mu_i$):

  $$\phi = \frac{1}{\sigma} \sqrt{\frac{n}{a} \sum_{i} (\mu_i - \mu_.)^2}, \quad (n_i \equiv n)$$

• $\mu_.$ is the overall mean:

  $$\mu_. = \frac{\sum \mu_i}{a}$$

To determine power, we use the non-central F distribution.

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Using Power Tables

Power tables can be used directly when:

1. The effects are fixed.

2. The design is balanced.

3. The minimum range of factor level means $\Delta$ is known:

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

Thus, the required inputs are:

• Significance level ($\alpha$)
• Minimum range of means ($\Delta$)
• Error standard deviation ($\sigma$)
• Power ($1 - \beta$)

Notes on Sample Size Sensitivity

• When $\Delta/\sigma$ is small, sample size requirements increase dramatically.
• Lowering $\alpha$ or $\beta$ increases required sample sizes.
• Errors in estimating $\sigma$ can significantly impact sample size calculations.

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24.5.3 Multi-Factor Studies

The same noncentral $F$ tables apply for multi-factor models.

24.5.3.1 Two-Factor Fixed Effects Model

24.5.3.1.1 Test for Interaction Effects

The non-centrality parameter:

$$\phi = \frac{1}{\sigma} \sqrt{\frac{n \sum_i \sum_j (\alpha \beta)_{ij}^2}{(a-1)(b-1)+1}}$$

or equivalently:

$$\phi = \frac{1}{\sigma} \sqrt{\frac{n \sum_i \sum_j (\mu_{ij} - \mu_{i.} - \mu_{.j} + \mu_{..})^2}{(a-1)(b-1)+1}}$$

where degrees of freedom are:

$$\begin{aligned} \upsilon_1 &= (a-1)(b-1) \\ \upsilon_2 &= ab(n-1) \end{aligned}$$

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24.5.3.1.2 Test for Factor $A$ Main Effects

The non-centrality parameter:

$$\phi = \frac{1}{\sigma} \sqrt{\frac{nb \sum \alpha_i^2}{a}}$$

or equivalently:

$$\phi = \frac{1}{\sigma} \sqrt{\frac{nb \sum (\mu_{i.} - \mu_{..})^2}{a}}$$

where degrees of freedom are:

$$\begin{aligned} \upsilon_1 &= a-1 \\ \upsilon_2 &= ab(n-1) \end{aligned}$$

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24.5.3.1.3 Test for Factor $B$ Main Effects

The non-centrality parameter:

$$\phi = \frac{1}{\sigma} \sqrt{\frac{na \sum \beta_j^2}{b}}$$

or equivalently:

$$\phi = \frac{1}{\sigma} \sqrt{\frac{na \sum (\mu_{.j} - \mu_{..})^2}{b}}$$

where degrees of freedom are:

$$\begin{aligned} \upsilon_1 &= b-1 \\ \upsilon_2 &= ab(n-1) \end{aligned}$$

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24.5.4 Procedure for Sample Size Selection

1. Specify the minimum range of Factor $A$ means.
2. Obtain sample size from power tables using $r = a$.
   • The resulting sample size is $bn$, from which $n$ can be derived.
3. Repeat steps 1-2 for Factor $B$.
4. Choose the larger sample size from the calculations for Factors $A$ and $B$.

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24.5.5 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.

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