ANOVA: Fixed effects, omnibus, one-way

The fixed effects one-way ANOVA tests whether there are any differences between the means µ_i of k ≥ 2 normally distributed random variables with equal variance σ. The random variables represent measurements of a variable X in k fixed populations. The one-way ANOVA can be viewed as an extension of the two group t test for a difference of means to more than two groups. The null hypothesis is that all k means are identical, H₀: µ₁ = µ₂ = . . . = µ_k. The alternative hypothesis states that at least two of the k means differ, H₁: µ_i ≠ µ_j , for at least one pair i, j with 1 ≤ i, j ≤ k.

Effect size index

The effect size f is defined as: f = σ_m/σ. In this equation σ_m is the standard deviation of the group means µ_i and σ the common standard deviation within each of the k groups. The total variance is then σ²_t = σ²_m + σ². A different but equivalent way to specify the effect size is in terms of η², which is defined as η² = σ²_m/σ²_t. That is, η² is the ratio of the between-groups variance σ²_m and the total variance σ²_t and can be interpreted as the proportion of variance explained by the group membership. The relationship between η² and f is:

η² = f²/(1 + f²)

or, if solved for f,

f = √(η² /(1 − η²)).

Cohen (1969, p.348) defined the following effect size conventions:

small f = 0.10
medium f = 0.25
large f = 0.40

If the mean µ_i and size n_i of all k groups are known then the standard deviation σ_m can be calculated in the following way:
where w_i = n_i /(n₁ + n₂ + · · · + n_k ) stands for the relative size of group i. Pressing the Determine button to the left of the effect size label opens the effect size drawer. You can use this drawer to calculate the effect size f from variances, from η², or from the group means and group sizes. The drawer essentially contains two different dialogs, and you can use the Select procedure selection field to choose one of them.

Effect size from means

In this dialog you normally start by setting the number of groups. G*Power  then provides you with a mean and group size table of appropriate size. Insert the standard deviation σ common to all groups in the SD σ within each group field. Then you need to specify the mean µ_i and size n_i for each group. If all group sizes are equal then you may insert the common group size in the input field to the right of the Equal n button. Clicking on this button fills the Size column of the table with the chosen value.

Clicking on the Calculate button provides a preview of the effect size that results from your inputs. If you click on the Calculate and transfer to main window button then G*Power calculates the effect size and transfers the result into the effect size field in the main window. If the number of groups or the total sample size given in the effect size drawer differ from the corresponding values in the main window, you will be asked whether you want to change the values in the main window to the corresponding values in the effect size drawer.

Effect size from variance

This dialog offers two ways to specify f. If you choose From Variances then you need to insert the variance of the group means, that is, σ²_m into the Variance explained by special effect field, and the square of the common standard deviation within each group, that is, σ², into the Variance within groups field. Alternatively, you may choose the Direct option and then specify the effect size f via η².

Options

This test has no options.

Examples

We compare 10 groups, and we have reason to expect a "medium" effect size (f = .25). How many subjects do we need in a test with α = 0.05 to achieve a power of 0.95?.

Select

Type of power analysis: A priori

Input

Effect size f : 0.25
α err prob: 0.05
Power (1-β err prob): 0.95
Number of groups: 10

Output

Noncentrality parameter λ: 24.375000
Critical F: 1.904538
Numerator df: 9
Denominator df: 380
Total sample size: 390
Actual Power: 0.952363

Thus, we need 39 subjects in each of the 10 groups. What if we had only 200 subjects available? Assuming that both α and β error are equally costly (i.e., the ratio q := beta/alpha = 1), which probably is the default in basic research, we can compute the following compromise power analysis:

Select

Type of power analysis: Compromise

Input

Effect size f: 0.25
β /α ratio: 1
Total sample size: 200
Number of groups: 10

Output

Noncentrality parameter λ: 12.500000
Critical F: 1.476210
Numerator df: 9
Denominator df: 190
α err prob: 0.159194
β err prob: 0.159194
Power (1-β err prob): 0.840806

Related tests

ANOVA: Fixed effects, special, main effects and interactions
ANOVA: Repeated measures, between factors

Implementation notes

The distribution under H₀ is the central F(k - 1, N - k) distribution with numerator df₁ = k - 1 and denominator df₂ = N − k. The distribution under H₁ is the noncentral F(k - 1, N - k, λ) distribution with the same df 's and noncentrality parameter λ = f² N. Note: k is the number of groups, N is the total sample size.

Validation

The results were checked against the values produced by GPower 2.0.