Chapter 10 2 way

Two-way Between-Subjects ANOVA

Two-way ANOVA Two-way ANOVA is one type of Factorial

ANOVA. Factorial ANOVAs are designs with two or

more between-subjects independent variables If there are within-subjects IVs, then they are

often called Mixed ANOVAs

Two-way ANOVA Grouping factors (IVs)

Example IV: experience with three levels - rookie, novice, and veteran IV: pitcher type with two levels - starter and relief DV: physical stamina

3 X 2 factorial design Computes a separate F ratio for each independent variable

(called main effects) and the interaction between the variables F for experience F for pitcher type F for experience*pitch interaction

Example: 3 X 2 Factorial Design

Group 1 Group 2

Group 3 Group 4

Starter Relief

Group 5 Group 6

rookie

novice

veteran

ExperienceLevel

Pitcher Type

Practice

A 3 X 3 design How many independent variables?

1. 1

2. 2

3. 3

4. 4

Practice A 3 X 3 design

How many levels of the first independent variable listed?

1. 1

2. 2

3. 3

4. 4

Practice A 3 X 3 design

How many conditions?

1. 2

2. 3

3. 6

4. 9

Practice A researcher tests male and female doctors

for manual dexterity . She tests dexterity for different tools: scalpel, scissors, scope, and probe.

How many independent variables?1. 12. 23. 34. 4

Practice A researcher tests male and female doctors for

manual dexterity . She tests dexterity for different tools: scalpel, scissors, scope, and probe. How many levels of the second independent variable

(tools)?

1. 1

2. 2

3. 3

4. 4

Practice A researcher tests male and female doctors

for manual dexterity . She tests dexterity for different tools: scalpel, scissors, scope, and probe.

How many conditions?1. 22. 43. 64. 8

Practice You are studying factors that are associated with problem-

solving skills. You place subjects into groups according to the following: student involvement with extracurricular activities (involved vs. not involved), student university type (public vs. private), and student self-reported procrastination (high, medium, and low). You then test the students’ problem-solving ability with a test.

What is the design?1. 1 X 2 X 2 X 32. 2 X 2 X 33. 3 X 2 X 64. 4 X 3



How many independent variables?1. 12. 23. 34. 4



How many cells?1. 32. 83. 124. 24

Two-way ANOVA Main Effects

Think of them as one-way ANOVAs for each independent variable.

If you have 2 IVs, then you have two possible main effects

Example A main effect for experience would look at the three

levels ignoring (collapsed across) pitcher type A main effect for pitcher type looks at starters vs.

relief pitchers regardless of (collapsed across) experience

Group 1 Group 2

Group 3 Group 4

Starter Relief

Group 5 Group 6

rookie

novice

veteran

ExperienceLevel

Pitcher Type

Marginal MeanFor Rookie

Marginal MeanFor Novice

Marginal MeanFor Veteran

Marginal MeanFor Starter

Marginal MeanFor Relief

Two-way ANOVA Interaction

For a two-way ANOVA there is one possible interaction Interactions occur if the effects of one IV are different

under different levels of the other IV Example

Something about being an expert makes you behave differently if you are a starter as opposed to being a relief pitcher.

As the number of factors increases, the number of possible interaction increases



How many main effects are possible?1. 12. 23. 34. 4



How many interactions are possible?1. 12. 23. 34. 4

Two-way ANOVA Identifying main effects and interactions

First test for significance (will discuss how that is done later)

Then use either table or graph to see the relationship that exists between variables

For instructional purposes we will assume the tests for significance have been done and that the main effects and interactions identified are significant

Remember if not significant than no exploration of that particular main effect or interaction.

Group 1 Group 2

Group 3 Group 4

Starter Relief

rookie

veteranExperienceLevel

Pitcher Type

Marginal MeanFor Rookie

Marginal MeanFor Veteran

Marginal MeanFor Starter

Marginal MeanFor Relief

No Main Effects or Interaction

20 20

Starter Relief

rookie

veteran

20

20 20 20

20 20

0

5

10

15

20

25

rookie veteran

starter

relief

Main Effect for Experience

10 10

Starter Relief

rookie

veteran

10

20 20 20

15 15

0

5

10

15

20

25

rookie veteran

starter

relief

Main Effect for Pitcher Type

20 10

Starter Relief

rookie

veteran

15

20 10 15

20 10

0

5

10

15

20

25

rookie veteran

starter

relief

Two Main Effects

20 15

Starter Relief

rookie

veteran

17.5

15 10 12.5

17.5 12.5

0

5

10

15

20

25

rookie veteran

starter

relief

Interaction with No Main Effects

20 10

Starter Relief

rookie

veteran

15

10 20 15

15 15

0

5

10

15

20

25

rookie veteran

starter

relief

Interaction and Main Effect for Experience

20 15

Starter Relief

rookie

veteran

17.5

10 15 12.5

15 15

0

5

10

15

20

25

rookie veteran

starter

relief

Interaction and Main Effect for Pitcher Type

20 10

Starter Relief

rookie

veteran

15

15 15 15

17.5 12.5

0

5

10

15

20

25

rookie veteran

starter

relief

Interaction and Two Main Effects

9 11

Starter Relief

rookie

veteran

10

28 12 20

16 14

0

5

10

15

20

25

30

rookie veteran

starter

relief

Assumptions of Between Factor ANOVAs DV data are interval or ratio level Data are normally distributed Variances are equivalent

(homogeneity of variance) Independence of observations Same statistical ratio

ANOVA = Treatment VarianceError Variance

Stating Hypotheses Two levels of hypotheses

Main effects Hypothesis for each IV Hypothesis for Main Effect A (also sometimes called

Main Effect Row) Ho: μ1 = μ2 … Ha: not all of the μi are equal.

Hypothesis for Main Effect B (also sometimes called Main Effect Column) Ho: μ1 = μ2 … Ha: not all of the μi are equal.

Stating Hypotheses Hypothesis for Interaction (sometimes written

A*B or Row*Column) Hypothesis for each combination of IVs

Ho: There is no interaction between factors A and B. All differences are explained by main effects.

Ha: There is an interaction. The mean difference between treatments are not what would be predicted from main effects only

Partitioning

One-Way Subjects SSbetween

SSwithin

Two-Way Subjects SSbetween (divided up)

SSrow

SScolumn

SSrow*column

SSwithin

Computation

N

GX

22

)()()(222222

N

G

n

T

N

G

n

T

N

G

n

T

column

column

row

row

SStotal

formula df

N - 1

SSrow

SScolumn

SSrow*column

N

G

n

T

row

row22

SSwithin SStotal – SSrow – SScolumn – SS row*column

a – 1Where a is number of rows

b – 1Where b is number of columns

(a – 1)(b – 1)

(a)(b)(n-1)

N

G

n

T

column

column22

Example

Starter Relief

Rookie 4

3

3

5

1

0

2

1

Veteran 3

2

1

2

2

3

1

1

Calculate the means for each of these cells

Rookie/Starter 3.75

Rookie/Relief 1

Veteran/Starter 2

Veteran/Relief 1.75

What are the Total df?

Source SS df MS F

Main Effect for Row 9.00

Main Effect for Column 1.00

Interaction 6.25

Error

(within groups)

9.50

Total 25.75

What are the Total df?

Source SS df MS F



Interaction 6.25

Error

(within groups)

9.50

Total 25.75 15

What are the df row?

Source SS df MS F



Interaction 6.25

Error

(within groups)

9.50

Total 25.75 15

What are the df row?

Source SS df MS F

Main Effect for Row 9.00 1


Interaction 6.25

Error

(within groups)

9.50

Total 25.75 15

What are the df column?

Source SS df MS F



Interaction 6.25

Error

(within groups)

9.50

Total 25.75 15

What are the df column?

Source SS df MS F


Main Effect for Column 1.00 1

Interaction 6.25

Error

(within groups)

9.50

Total 25.75 15

What are the df interaction?

Source SS df MS F



Interaction 6.25

Error

(within groups)

9.50

Total 25.75 15

What are the df interaction?

Source SS df MS F



Interaction 6.25 1

Error

(within groups)

9.50

Total 25.75 15

What are the df error?

Source SS df MS F



Interaction 6.25 1

Error

(within groups)

9.50

Total 25.75 15

What are the df error?

Source SS df MS F



Interaction 6.25 1

Error

(within groups)

9.50 12

Total 25.75 15

What is the MS for row?

Source SS df MS F



Interaction 6.25 1

Error

(within groups)

9.50 12

Total 25.75 15

What is the MS for row?

Source SS df MS F

Main Effect for Row 9.00 1 9.00


Interaction 6.25 1

Error

(within groups)

9.50 12

Total 25.75 15

What is the MS for column?

Source SS df MS F



Interaction 6.25 1

Error

(within groups)

9.50 12

Total 25.75 15

What is the MS for column?

Source SS df MS F


Main Effect for Column 1.00 1 1.00

Interaction 6.25 1

Error

(within groups)

9.50 12

Total 25.75 15

What is the MS interaction?

Source SS df MS F



Interaction 6.25 1

Error

(within groups)

9.50 12

Total 25.75 15

What is the MS interaction?

Source SS df MS F



Interaction 6.25 1 6.25

Error

(within groups)

9.50 12

Total 25.75 15

What is the MS error?

Source SS df MS F




Error

(within groups)

9.50 12

Total 25.75 15

What is the MS error?

Source SS df MS F




Error

(within groups)

9.50 12 0.79

Total 25.75 15

What is the main effect for row?

Source SS df MS F




Error

(within groups)

9.50 12 0.79

Total 25.75 15

What is the main effect for row?

Source SS df MS F

Main Effect for Row 9.00 1 9.00 11.37



Error

(within groups)

9.50 12 0.79

Total 25.75 15

What is the main effect for column?

Source SS df MS F




Error

(within groups)

9.50 12 0.79

Total 25.75 15

What is the main effect for column?

Source SS df MS F


Main Effect for Column 1.00 1 1.00 1.26


Error

(within groups)

9.50 12 0.79

Total 25.75 15

What is the interaction?

Source SS df MS F




Error

(within groups)

9.50 12 0.79

Total 25.75 15

What is the interaction?

Source SS df MS F



Interaction 6.25 1 6.25 7.90

Error

(within groups)

9.50 12 0.79

Total 25.75 15

Example Critical Value

For each test use the df associated with it. Is not necessarily the same for all three tests

although in our example it is. 4.747

Evaluate each effect separately

Is Main Effect row sig? Yes No

Is Main Effect column sig? Yes No

Is Interaction sig? Yes No

Now what? If you have no significant interaction, then

you can talk about what main effects are significant in the same way that you evaluated one-way ANOVAs.

If the interaction is significant you must be careful interpreting main effects. The main effect could be present simply because of the interaction. So concentrate on the interaction interpretation.

Now what? If it is a 2X2 ANOVA and the interaction is

significant then graph the means and interpret. Our example

0

0.5

1

1.5

2

2.5

3

3.5

4

starter relief

rookie

veteran

Now what? If interaction is significant

Plot interaction Interpret interaction

Level of one IV is influenced by level of a second IV

If Main Effects are significant Conduct Post Hoc tests

Report results Effect sizes

• Use omega squared to report effect size

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Chapter 10 2 way