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Annotated Output | Factorial ANOVA

Computer Output

The descriptive statistics can be used to determine the inferential statistics.

> Results <- aov(Outcome ~ FactorA * FactorB)
> model.tables(Results, "means")
Tables of means
Grand mean
  
5 

 FactorA 
FactorA
A1 A2 
 4  6 

 FactorB 
FactorB
 B1  B2 
4.5 5.5 

 FactorA:FactorB 
       FactorB
FactorA B1 B2
     A1 2  6 
     A2 7  5 
> lapply(
+   split(Outcome, interaction(FactorA, FactorB)),
+   function(x) c(n = length(x), mean = mean(x), sd = sd(x))
+ )
$A1.B1
      n    mean      sd 
4.00000 2.00000 2.44949 

$A2.B1
      n    mean      sd 
4.00000 7.00000 2.44949 

$A1.B2
      n    mean      sd 
4.00000 6.00000 2.44949 

$A2.B2
      n    mean      sd 
4.00000 5.00000 2.44949

The descriptive statistics are calculated separately for each group or condition. They are not equivalent to getting the statistics for the variable as a whole.

The grand mean and the marginal means for each factor are also presented.

The table of inferential statistics shows the key elements to be calculated.

> summary(Results)
                Df Sum Sq Mean Sq F value Pr(>F)  
FactorA          1     16      16   2.667 0.1284  
FactorB          1      4       4   0.667 0.4301  
FactorA:FactorB  1     36      36   6.000 0.0306 *
Residuals       12     72       6                 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

For the Factor A Main Effect, F = 0.667, p = .430. This would not be considered a statistically significant finding.

For the Factor B Main Effect, F = 2.667, p = .128. This would not be considered a statistically significant finding.

For the Interaction, F = 6.000, p = .031. This would be considered a statistically significant finding.

Effect size measures (partial eta-squared) are not in the output, but can be calculated from the available information.

For the Factor A Main Effect, partial eta 2 = 0.053. Thus, 5.3% of the variability among the scores is accounted for by Factor A.

For the Factor B Main Effect, partial eta 2 = 0.182. Thus, 18.2% of the variability among the scores is accounted for by Factor B.

For the Interaction, partial eta 2 = 0.333. Thus, 33.3% of the variability among the scores is accounted for by interaction of Factor A and Factor B.

For Factorial ANOVA, the confidence intervals around (centered on) each mean are appropropriate but not provided in the output.

APA Style

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.

A 2 (Factor A) x 2 (Factor B) ANOVA indicated a small and non-statistically significant main effect for Factor A, F(1,12) = 0.67, p = .430, partial η2 = .05, and for Factor B, F(1,12) = 2.67, p = .128, partial η2 = .18. In contrast, the interaction was large in magnitude and statistically significant, F(1,12) = 6.00, p = .031, partial η2 = .33.

Typically, the means, standard deviations, and confidence intervals would be presented in a table or figure associated with this text.