Calculating Power for Multi-Way Anovas Using Gpower

Biostatistics 11/9/2006

Calculating Power for Multi-way ANOVAs using GPower

In the lecture notes, formulae are provided for calculating the non-central parameter (φ) for terms in a multi-way ANOVA. Power may then be obtained from the power tables, such as those in Zar.

However, GPower does not allow you to use φ to calculate power. Instead, the effect size (f) is calculated based on the sums of squares as follows:

1. Under Tests Menu select F-Test (ANOVA) (the same as for one-way ANOVA) and change the Hypothesis type to Special. This will allow you to calculate the effect size (f) from the Sums of Squares (see # 2 below) and to specify the proper numerator degrees of freedoms and the number of groups (see # 3 below).

Note: Under the Global option, GPower assumes you are testing the effect in an one-way ANOVA and calculates the df as number of group (k) – 1.

2. Use the Calc Effectsize option and in the dialog box, toggle the Eta – Var button so that it allows you to enter the “Variance explained by special effect” and “Error variance”. The former is the Sums of Squares (SS) for the model term (i.e. main effect or interaction) for which you wish to calculate power and the later is the Error SS (called the Residual SS in SPlus). GPower uses the values to calculate a Partial eta2 and an Effect size (f). Calc & Copy the effect size into the previous screen.

Example: Calculating the effect size of the Interaction A x B using GPOWER

SPLUS Output:

Type III Sums of Square

Df / Sum of Sq / Mean Sq / F Value / Pr(F)
Factor A / 2 / 1356.302 / 678.151 / 0.777 / 0.471
Location / 1 / 106.408 / 106.488 / 0.122 / 0.730
A x B / 2 / 776.154 / 388.077 / 0.445 / 0.646
Residuals / 24 / 20948.923 / 872.872

Calculating Effectsize from SS in GPower

3. Finally, you have to enter the total sample size (N) and the appropriate degrees of freedom (df) for you test.

Numerator df = df for the model term for which you are testing power.

Groups = the product of the levels for each main effect, so for a two-way ANOVA shown below where Factor A has 3 levels and Factor B has 2 levels, groups = 3 x 2 = 6.

Note: After you Calculate the Power ensure that you have set your df correctly by looking at the df for the Critical F(df1, df2). The df1 should be equal to df for the model term, while df2 should equal the Error (Residuals) df.

SPLUS Output:

Type III Sums of Square