Advanced ANOVA/Factorial ANOVA

DesignEdit
Factorial ANOVA involves testing of differences between group means based on two or more categorical independent variables (IVs), with a single, continuous dependent variable (DV). In other words, a factorial ANOVA could involve:
 Two or more betweensubjects categorical/ordinal IVs
 One interval or ratio DV
The results of interest are:
 Main effect for IV1
 Main effect for IV2
 Interaction between IV1 and IV2
If significant effects are found and more than two levels of an IV are involved, then provide followup tests, which could be either:
 A priori (planned) contrasts
 Posthoc contrasts
(Note: Use appropriate control of the familywise Type I error rate)
Effect sizes should also be reported. Etasquared provides an estimate of the percentage of variance in the DV explained by each main effect and interaction effect. Cohen's p provides an estimate of the size of differences between two groups in standard deviation groups.
ExampleEdit
"What is the effect of Gender (2) and Degree Type (3) on Overall Student Satisfaction?" This could be described as a 2 (Gender) by 3 (Degree Type) factorial ANOVA.
General stepsEdit
 Establish hypothesis/hypotheses
 Make sure you have separate hypotheses for:
 Main effect for each IV
 Interactions between IVs
 Planned contrasts (if warranted)
 Make sure you have separate hypotheses for:
 Examine assumptions:
 IVs (categorical; betweensubjects) and DV (at least interval)
 The data in each cell is normally distributed
 Homogeneity of variance (the variance in each cell is similar)
 Cells are independent
 Examine descriptive statistics, particularly the four moments (M, SD, Skewness, Kurtosis) overall, and also for each cell
 Examine graphs
 Conduct inferential test (ANOVA) and interpret significance of F scores
 Conduct followup tests (planned contrasts or posthoc tests) if F is significant
 Interpret interactions
 Calculate and interpret effect sizes
 Etasquare (omnibus  equivalent to R^{2})
 Standardised mean effect size (difference b/w two means) e.g., Cohen's d
Example SPSS outputsEdit
 Factorial ANOVA (example)  Are there differences in University Student Satisfaction levels between Gender and Age?
 Factorial ANOVA (example)  Are there differences in University Student Satisfaction levels between Gender and Age?  Are there differences in Locus of Control between Gender and Age?
DescriptivesEdit
 A table of descriptive statistics (M, SD, Skewness, and Kurtosis) for each cell and for each marginal total, and grand total should be presented when reporting results.
 For a 2way ANOVA, the descriptives table it is recommended to provide a breakdown of one IV in the columns and the other IV in the rows, such as illustrated in the following tables.
Age  
Gender  Younger  Older  Total 
Males  
Females  
Total 
Note that each of the three columns on the right should be further split into five columns to allow reporting of M, SD, Skewness, and Kurtosis, n. So, the expanded descriptive tables layout could look like this: (note that rows and columns have been swapped around from the table above  this is somewhat arbitrary):
Gender
 
Age 
Males

Females

Overall
 
M

SD

Sk

Kurt

M

SD

Sk

Kurt

M

SD

Sk

Kurt
 
Young  
Middle 
 
Older  
Overall 
Here's an example of such an APA style table:
Understanding interactionsEdit
 One of the keys to understanding Factorial ANOVA is being able to intepret interactions.
 A recommended experiential exercise for learning about interactions is to fabricate a dataset which can be used to demonstrate factorial ANOVAs in which there are:
 No effects
 Main effect A, no main effect B, no interaction
 Main effect A, no main effect B, interaction
 No main effect A, main effect B, no interaction
 No main effect A, main effect B, interaction
 Main effect A, main effect B, no interaction
 Main effect A, main effect B, interaction
 Interaction, no main effects
Francis exercisesEdit
Effect sizesEdit
WriteupEdit
A writeup checklist for a factorial ANOVA might include:
 What is the goal/purpose of the analysis?
 What is the design  i.e., what are the IVs and the DV?
 What are the relevant descriptive statistics for the cells and the marginal descriptives? (M, D, skewness, kurtosis, n)
 To what extent does the data meet the assumptions for ANOVA (e.g., independent observations (often assumed), normality, and homogeneity of variance)?
 What are the main and interaction effect statistics, including the direction, statistical significance, and size of effects (consider the merits of reporting etasquared and/or a standardised mean difference statistic such as Cohen's d; see effect sizes)?
 Provide a figure  often a line graph is used.
 Are followup tests necessary and/or warranted? e.g., if there are planned contrasts and/or a significant effect with three or more levels, then appropriate followup tests should be conducted using a method for controlling the familywise Type I error rate as appropriate.
 What are the conclusions?
See alsoEdit
External linksEdit
 Factorial ANOVA (ucspace)
 Factorial ANOVA Notes (Handout; 2007)