Factorial ANOVA

• FactorialANOVA


  An extension of the one-way ANOVA to include two or more independent variables (IVs)
• ANOVA


  Allows us to analyze the effects of two or more IVs on a dependent variable (DV) in one analysis

  
• TWO-WAY ANOVA


  Allows us to find out if there is an interactive effect of two IVs on the DV
• In the previous chapters, we have used the linear model to test for differences between group means when those groups have belonged to a single predictor variable
• This time, we are going to extend the linear model to situations in which we have two categorical predictors (independent variables)
• Independent factorial design


  Several independent variables or predictors have been measured using different entities (between groups)
• Repeated-measures (related) factorial design

  Several independent variables or predictors have been measured but the same entities have been used in all conditions

  
• Mixed design


  Several independent variables or predictors have been measured: some have been measured with different entities, whereas others used the same entities
• ANOVA allows us to analyze the effects of two or more IVs on a DV in one analysis → Factorial ANOVA
• ANOVA can also be used to find out if there is an interactive effect of two variables on the DV
• ANOVA is not just restricted to two IVs, you could have 3, 4, or more
• One-way independent ANOVA


  One independent variable measured using different entities
• Two-way repeated measures ANOVA


  Two independent variables both measured using the same entities
• Two-way mixed ANOVA


  Two independent variables: one measured using different entities and the other measured using the same entities
• Three-way independent ANOVA


  Three independent variables all of which are measured using different entities

  
• Often in the literature, you will see ANOVA designs expressed as 2 X 3 ANOVA or 3X 4 ANOVA
• This simply tells you how many IVs were used and how many conditions in each
• For example, 2 X 3 ANOVA, there are two IVs, the first has 2 conditions, the 2nd has 3 conditions
• For example, 3 X 4 ANOVA, two IVs, one with 3 conditions, and 1 with 4 conditions
• For example, 4 X 3 x 2 ANOVA, there are three IVs, the first has 4 conditions, the second has 3 conditions, and the third has 2 conditions
• In general, models that compared means are named as A [number of IV]-way of [how these variables were measured] ANOVA
• Two-way repeated-measures ANOVA


  • Two independent variables both measured using the same entities
• In the study by Reidy and Richards (1997), there were two independent variables: anxiety group (anxious vs non-anxious) and word type (neutral vs negative)

  
• The researchers were interested in three effects: the overall difference between anxious and non-anxious participants in the number of words recalled, whether memory was best for negative words or neutral words, and whether there was a difference between anxious and non-anxious participants in the type of words best remembered

  
• Main effect


  The overall effect of each of the IVs on the DV
• Interaction effect


  The interaction between the two IVs on the DV
• ANOVA allows us to test all three of these effects in one analysis
• Interaction effect


  When there is an interaction, the IVs analyzed together uncover a different pattern of variation in the DV than when the IVs are analyzed separately

  
• In ANOVA, we analyze all the possible sources of variance in our studies

  
• In one-way ANOVA, if the between-groups variance is greater than the within-groups variance, we can conclude that the between-groups difference was not due to sampling error

  
• In factorial ANOVA, we partition the total variance into that which represents the two IVs separately, and that which is attributable to the interaction between these IVs

  
• We then compare these sources of variance within-conditions (or error) variance

  
• The model assumptions in factorial designs are normality, outliers, and homoscedasticity
• If the homoscedasticity assumption is violated, we can use the Welch procedure or bootstrap the post hoc tests to make them robust
• In the study by Reidy and Richards (1997), there were three possible sources of variance: the main effect of anxiety conditions, the main effect of word type conditions, and the interaction between the two factors

  
• The main effect of word type was significant, the main effect of anxiety group was not significant, and the interaction between anxiety group and word type was significant
• When there is a significant interaction, the main effects should be interpreted with caution and only if they are meaningful in the context of the research
• Contrasts can be used to break down main effects and determine where the differences between groups lie, especially if an IV has 3 or more levels

  
• Post hoc tests can also be used to determine where the differences between groups lie
• Simple effects analysis can be used to determine the difference between any two conditions of one IV in one of the conditions of another IV when there is a significant interaction