Effect Size in the Two-Factor Mixed Design

Below are the equations for the partial  measure for the two-factor design. The overall formula for the partial  squared measure is a measure of the effect divided by the measure of the effect plus error variance. The formulas below calculate effect size based on the F statistic for each of the three effects in the two-factor model. Note that (a)(b)(n) in the numerator is the number of levels of factor a times the number of levels of factor a times the number of subjects at each level of the groups formed by crossing factor a and factor b. See material in Kirk (1995 p. 518-519). Note the partial  measure ignores other effects in the ANOVA model.

Measures of Population Effect Size



Example Effect Size Calculations

Rehearsal Experiment. The Rehearsal Study conducted on the first day of class had two levels of rehearsal (primary and secondary) and two levels of type of word (concrete and not concrete).The interaction F =2.82 was not statistically significant.


ATLAS Study. Here there were three waves of measurement (baseline, immediate followup and one-year followup) and two groups (ATLAS program or control). The interaction F=24.46 for nutrition behavior was statistically significant. I used the average n for the calculations below. The harmonic mean would be more accurate, however.

Power for the Two-Factor ANOVA Interaction Effect

The power calculations for a Two-factor ANOVA are based on the means for the levels of each factor. Here are the retrospective power calculations for Interaction of Drive and Drug for the monkey study described in Keppel. Below are the means in each of 6 groups that are used in computing .  is in the formula for the  and f measures. Remember that MS _S/AB = 18.33 for the experiment.

           a₁       a₂      a₃      mean
b₁         3      10     14      9       mean for b₁
b₂       11     12     10     11     mean for b₂
mean    7      11    12     10     grand mean

Computation of Interaction Deviations

-3 = 3 -7 -9 + 10
0 =10 -11 -9 + 10
3 =14 -12 -9 + 10
3 =11 -7 -11 + 10
0 =12 -11 -11 + 10
-3 =10 -12 -11 + 10
=(-3)²+(0)²+(3)²+(3)²+(0)²+(-3)²=36

Pearson-Hartley Tables
Looking up  =1.62 in the Pearson-Hartley tables on page 510 for numerator degrees of freedom equal to 2, and denominator degrees of freedom equal to 18, we find power equal to .62.

Cohen's Tables

The f effect size measure is obtained using the following formula:

We also need to calculate the correct sample size for the interaction given the two main effects in the model. For A X B Interaction the new n is equal to 7.

Rounding the f = .572 to .6, and sample size of 7, we find that the power is about .62 reading from the table with numerator degrees of freedom equal to 2 (page 313).