Question

Using a 2 × 2 matrix structure, with factor A defining the rows and factor B defining the columns, create a set of means that produce each of the following patterns: a. A main effect for factors A and B, but no interaction b. A main effect for factor A and an interaction, but no main effect for factor B c. A main effect for both factors and an interaction. Assume that a difference of at least 5 points is significant.

Answer 1

Examples were provided to illustrate a 2 × 2 matrix set of means showing a main effect for factors A and B without interaction, a main effect for factor A with interaction but no main effect for factor B, and a main effect for both factors with an interaction.

Step-by-step explanation:

To answer the student's question about creating a set of means in a 2 × 2 matrix with factor A defining the rows and factor B defining the columns to produce specified patterns of effects in an ANOVA framework, we need to consider the main effects and interactions. Here are examples that meet the criteria for each case:

• Main effect for A and B, but no interaction:
  
  A11015
  A22025
  
• Main effect for factor A and interaction, but no main effect for factor B:
  
  A11015
  A22520
  
• Main effect for both factors and an interaction:
  
  A11020
  A22535
  

In these matrices, the numbers represent the group means. A main effect of a factor is seen when the means across the levels of that factor vary significantly, and an interaction is present when the difference between the means across one factor changes depending on the level of the other factor. Here, a significant difference is considered to be at least 5 points. These patterns can then be statistically tested using ANOVA to assess the effects and interactions.