The following data set reflects a between-subjects Factorial design with two factors (with two levels for each factor). The data are presented in the format suitable for entry into statistical software.
| FactorA | FactorB | Outcome | |
|---|---|---|---|
| 1 | 1.00 | 1.00 | .00 |
| 2 | 1.00 | 1.00 | .00 |
| 3 | 1.00 | 1.00 | 3.00 |
| 4 | 1.00 | 1.00 | 5.00 |
| 5 | 2.00 | 1.00 | 4.00 |
| 6 | 2.00 | 1.00 | 7.00 |
| 7 | 2.00 | 1.00 | 4.00 |
| 8 | 2.00 | 1.00 | 9.00 |
| 9 | 1.00 | 2.00 | 9.00 |
| 10 | 1.00 | 2.00 | 6.00 |
| 11 | 1.00 | 2.00 | 4.00 |
| 12 | 1.00 | 2.00 | 9.00 |
| 13 | 2.00 | 2.00 | 3.00 |
| 14 | 2.00 | 2.00 | 6.00 |
| 15 | 2.00 | 2.00 | 8.00 |
| 16 | 2.00 | 2.00 | 3.00 |
The following tables represent typical output from statistical software. Options, labels, and layout vary from program to program.
The table of descriptive statistics can be used to determine the inferential statistics.
| FactorA | FactorB | Mean | Std. Deviation | N |
|---|---|---|---|---|
| Level 1 | Level 1 | 2.000 | 2.449 | 4 |
| Level 2 | 7.000 | 2.449 | 4 | |
| Level 2 | Level 1 | 6.000 | 2.449 | 4 |
| Level 2 | 5.000 | 2.449 | 4 |
The table of inferential statistics shows the key elements to be calculated.
| Source | SS | df | MS | F | p | Partial Eta2 |
|---|---|---|---|---|---|---|
| FactorA | 4.000 | 1 | 4.000 | 0.667 | 0.430 | 0.053 |
| FactorB | 16.000 | 1 | 16.000 | 2.667 | 0.128 | 0.182 |
| Interaction | 36.000 | 1 | 36.000 | 6.000 | 0.031 | 0.333 |
| Within | 72.000 | 12 | 6.000 | |||
| Total | 128.000 | 15 |
Error (Within Groups) Statistics: Within-groups error statistics are a function of the within group variabilities.
\[SS_1 = ( SD_1^2 ) ( df_1 ) = ( 2.44949^2 ) ( 3 ) = 18.000\] \[SS_2 = ( SD_2^2 ) ( df_2 ) = ( 2.44949^2 ) ( 3 ) = 18.000\] \[SS_3 = ( SD_3^2 ) ( df_3 ) = ( 2.44949^2 ) ( 3 ) = 18.000\] \[SS_4 = ( SD_4^2 ) ( df_4 ) = ( 2.44949^2 ) ( 3 ) = 18.000\] \[SS_{ERROR} = SS_1 + SS_2 + SS_3 + SS_4 = 18.000 + 18.000 + 18.000 + 18.000 = 72.000\] \[df_{ERROR} = df_1 + df_2 + df_3 +df_4 = 3 + 3 + 3 + 3 = 12\] \[MS_{ERROR} = \frac{SS_{ERROR}}{df_{ERROR}} = \frac{72.000}{12} = 6.000\]
Grand (or Total) Mean: A grand mean can be determined by taking the weighted average of all of the group means.
\[M_{TOTAL} = \frac{\sum n_{GROUP} (M_{GROUP})}{N} = \frac{ 4 (2.000) + 4 (7.000) + 4 (6.000) + 4 (5.000) }{( 4 + 4 + 4 + 4 )} = 5.000\]
Marginal Means: A level (marginal) mean can be determined by taking the weighted average of the appropriate group means.
For Factor A:
\[M_{A1} = \frac{\sum n_{GROUP} (M_{GROUP})}{N_{LEVEL}} = \frac{ 4 (2.000) + 4 (7.000) }{( 4 + 4 )} = 4.500\] \[M_{A2} = \frac{\sum n_{GROUP} (M_{GROUP})}{N_{LEVEL}} = \frac{ 4 (6.000) + 4 (5.000) }{( 4 + 4 )} = 5.500\]
For Factor B:
\[M_{B1} = \frac{\sum n_{GROUP} (M_{GROUP})}{N_{LEVEL}} = \frac{ 4 (2.000) + 4 (6.000) }{( 4 + 4 )} = 4.000\] \[M_{B2} = \frac{\sum n_{GROUP} (M_{GROUP})}{N_{LEVEL}} = \frac{ 4 (7.000) + 4 (5.000) }{( 4 + 4 )} = 4.000\]
Effect (Between Groups) Statistics: The Model statistics represent the overall differences among the groups. The Factor A and Factor B statistics are a function of the level (marginal) means and sample sizes. The interaction statistics reflect the between-groups variability not accounted for by the factors individually.
For the Model:
\[SS_{MODEL} = \sum n_{GROUP} (M_{GROUP} - M_{TOTAL})^2 = 4 ( 2.000 - 5.000 )^2 + 4 ( 7.000 - 5.000 )^2 + 4 ( 6.000 - 5.000 )^2 + 4 ( 5.000 - 5.000 )^2 = 56.000\] \[df_{MODEL} = \text{# groups} − 1 = 4 − 1 = 3\]
For Factor A:
\[SS_{FACTORA} = \sum n_{LEVEL} (M_{LEVEL} - M_{TOTAL})^2 = 8 ( 4.500 - 5.000 )^2 + 8 ( 5.500 - 5.000 )^2 = 4.000\] \[df_{FACTORA} = \text{# levels} − 1 = 2 − 1 = 1\] \[MS_{FACTORA} = \frac{SS_{FACTORA}}{df_{FACTORA}} = \frac{4.000}{1} = 4.000\]
For Factor B:
\[SS_{FACTORB} = \sum n_{LEVEL} (M_{LEVEL} - M_{TOTAL})^2 = 8 ( 4.000 - 5.000 )^2 + 8 ( 6.000 - 5.000 )^2 = 16.000\] \[df_{FACTORB} = \text{# levels} − 1 = 2 − 1 = 1\] \[MS_{FACTORB} = \frac{SS_{FACTORB}}{df_{FACTORB}} = \frac{16.000}{1} = 16.000\]
For the Interaction:
\[SS_{INTER} = SS_{MODEL} - SS_{FACTORA} - SS_{FACTORB} = 56.000 - 4.000 - 16.000 = 36.000\] \[df_{INTER} = df_{MODEL} - df_{FACTORA} - df_{FACTORB} = 3 - 1 - 1 = 1\] \[MS_{INTER} = \frac{SS_{INTER}}{df_{INTER}} = \frac{36.000}{1} = 36.000\]
Statistical Significance: The F statistic is the ratio of the between-and within-group variance estimates.
For the Factor A Main Effect:
\[F = \frac{MS_{FACTORA}}{MS_{ERROR}} = \frac{4.000}{6.000} = 0.667\]With dfFACTORA = 1 and dfERROR = 12, FCRITICAL = 4.747
Because FFACTORA < FCRITICAL, p > .05
This would not be considered a statistically significant finding.
For the Factor B Main Effect:
\[F = \frac{MS_{FACTORB}}{MS_{ERROR}} = \frac{16.000}{6.000} = 2.667\]With dfFACTORB = 1 and dfERROR = 12, FCRITICAL = 4.747
Because FFACTORB < FCRITICAL, p > .05
This would not be considered a statistically significant finding.
For the Interaction:
\[F = \frac{MS_{INTER}}{MS_{ERROR}} = \frac{36.000}{6.000} = 6.000\]With dfINTER = 1 and dfERROR = 12, FCRITICAL = 4.747
Because FINTER > FCRITICAL, p < .05
This would be considered a statistically significant finding.
Effect Size: The partial eta-squared statistic is a ratio of the between-subjects effect and the remaining variability (Sum of Squares) estimates after within-subjects error has been partialled out.
For the Factor A Main Effect:
\[\text{Partial} \; \eta^2 = \frac{SS_{FACTORA}}{( SS_{FACTORA} + SS_{ERROR} )} = \frac{4.000}{( 4.000 + 72.000 )} = 0.053\]Thus, 5.3% of the variability among the scores is accounted for by Factor A.
For the Factor B Main Effect:
\[\text{Partial} \; \eta^2 = \frac{SS_{FACTORB}}{( SS_{FACTORB} + SS_{ERROR} )} = \frac{16.000}{( 16.000 + 72.000 )} = 0.182\]Thus, 18.2% of the variability among the scores is accounted for by Factor B.
For the Interaction:
\[\text{Partial} \; \eta^2 = \frac{SS_{INTER}}{( SS_{INTER} + SS_{ERROR} )} = \frac{36.000}{( 36.000 + 72.000 )} = 0.333\]Thus, 33.3% of the variability among the scores is accounted for by interaction of Factor A and Factor B.
Confidence Intervals: For Factorial ANOVA, calculate the confidence intervals around (centered on) each mean separately (not shown here).
The Factorial ANOVA provides statistics for the main effects and interactions in a factorial design. Each effect would be summarized in a style analogous to a One Way ANOVA. The first example focuses on statistical significance testing, whereas the second version includes and emphasizes interpretation of the effect size.
A 2 (Factor A) x 2 (Factor B) ANOVA showed that neither Factor A, F(1,12) = 0.67, p = .430, nor Factor B, F(1,12) = 2.67, p = .128, had a statistically significant impact on the Outcome. However, the interaction was statistically significant, F(1,12) = 6.00, p = .031.
Analyses revealed that neither Factor A, partial eta2 = .05, F(1,12) = 0.67, p = .430, nor Factor B, partial eta2 = .18, F(1,12) = 2.67, p = .128, had an appreciable impact on the Outcome. However, the interaction had a large impact on the Outcome, partial eta2 = .33, F(1,12) = 6.00, p = .031.
Typically, the means, standard deviations, and confidence intervals would be presented in a table or figure associated with this text.