W9: Two-Way Between-Subjects ANOVA

Question 1

What is an alternative name for groups and what is an alternative name for categories within one group

Answer 1

Group: Factor

Categories: Levels of a factor

Question 2

What is another name for two-way between-subject designs

Answer 2

Factorial designs

Question 3

What is the omnibus hypothesis for a 2-way factorial design with 2 IVs and 1 DV

Answer 3

3

• 1 for Main Effect A
• 1 for Main Effect B
• 1 for Interaction Effect
  
  • Note, it does not tell us where the difference lies if there are more than 1 degrees of freedom.

Question 4

What does the focused research questions for a 2-way factorial design allows us to explain?

Answer 4

If the design is balanced and if the corresponding contrasts are orthogonal,

It allows us to explain all possible difference that is contained in the 2-way table

Question 5

In two-way factorial designs, what sources are the differences driven by?

Answer 5

1. Direct effects of Factor A
2. Direct effects of Factor B
3. Combined effects of both Factor A and B, over and above that of their indiviudla direct effects

Question 6

In a two-way ANOVA, define interaction

Answer 6

AxB:

• Due to the combined effects of both Factor A and Factor B, over and above that of individual direct effects
• Effect of Factor A is in part dependent upon the Effects of levels of Factor B

In other words, effect left over after row and column marginal effects have been accounted for

Question 7

What is the grand mean

Answer 7

• Average of all cells
• Mean that ignores Factor A and Factor B

Question 8

What is the formula for marginal effects

Answer 8

Marginal Effects = Marginal Cell Mean - Grand Mean

• x = X - m
  
  • x is deviation score
  • X is observed score
  • m is grand mean
• All the deviation scores added up will sum to 0 (Remeber lecture 2)

Question 9

What do all the interaction effects add up to?

Answer 9

Zero

Question 10

What is the formula for interaction effects. Interpret it

Answer 10

Interaction = Cell Mean - Grand Mean - (Main effect A + Main Effect B)

Therefore, interaction effect is after all main effects have been accounted for.

Likewise, all the interaction scores added up will sum to 0 (Remeber lecture 2)

Question 11

In the two-way effects, what kind of decomposition is it

Answer 11

It is an additive decomposition, where

1. Grand Mean
2. Main Effect A
3. Main Effect B
4. Interaction A x B

=

Cell Mean

Question 12

In a visual plot, how do we know that a two-way factorial design has an interaction

Answer 12

Think of “It Depends”

Interaction Effect

.If the lines are converging or cross over when

• Y-Axis: Interaction effect

Not Necessary Interaction effect

If the Y-Axis is mean level of DV, we won’t be able to observe the inteaction effect (even if it is not parallel). We MUST remove the marginal effects first

Question 13

What is an interaction? And hence, what do we require to observe an interaction?

Answer 13

Interaction

• Effect that occurs over and above the marginal effects of the two factors.
  
  • Requires marginal effects to be removed before it can be observed.

Question 14

Can we use linear regression in a two-way between-subjects anova? Hence, what should we use?

Answer 14

Yes

• Dummy variables provides a full interpretation
• But it may not be the one that correspond to our focused research questions
  
  • Main effects and interaction linear contrasts provide a way to achieve a desired focused approach

Question 15

Define Main Effect Contrasts

Answer 15

Main effect contrasts

• Compares the cell means of the two-way table to investigate contrasts of each factor in the two design

Question 16

Define Interaction Effect Contrasts

Answer 16

Interaction Contrasts

• Compares the cell means of the two-way table using the cross-product of contrast weights from the linear comparisons for the main effect linear contrasts
  
  • A:(1, -1, 0)
  • B (1, -1)
    
    • AxB (1,-1,0; -1,1,0)

Question 17

What is the scaling of planned contrasts for main effects and interaction effects?

Answer 17

• Planned contrasts for main effects have order-0 scaling
• Interaction contrasts have order-1 scaling

Question 18

What is order-0 scaling

Answer 18

Scaling for a difference between two means.

Question 19

What is order-1 scaling

Answer 19

Scaling for the difference two sets of differences between
means (i.e. difference for differences)

(which is technically what an interaction represents)

Question 20

If a two-way between-subjects factorial design has 3 levels in Factor A and 2 levels in Factor B, how many different types of means can be identified in the two-way cross-classified table?

Answer 20

12

In number of cell means:

• Factor A and B: 6
• Marginal Means A: 3
• Marginal Means B: 2
• Grand Mean: 1
  
  • = 12

Question 21

What do we need to know before figuring out the interaction effect?

Answer 21

We need to know the grand mean, marginal mean and cell mean to figure out the interaction effect.

• Cell mean = Grand mean + Marginal Effect A + Marginal Effect B + Interaction AxB
• Therefore, Interaction AxB = Cell mean - Grand mean - Marginal Effect A - Marginal Effect B