Factorial Analysis of Variance - Rutgers Universitymmm431/quant_methods_F13/QM_Lecture15.pdf• We need two different experiments to determine the effects of these two factors on memory

Factorial Analysis of Variance

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Repeated-Measures ANOVA

Overview of the Factorial ANOVA

• In the context of ANOVA, an independent variable (or a quasi-

independent variable) is called a factor, and research studies

with multiple factors, in which every level of one factor is

paired with every level of the other factors, are called

factorial designs.

– Example: the Eysenck (1974) memory study, in which type-of-

processing was one factor and age was another factor.

Counting Rhyming Adjective Imagery Intentional

Younger

Older

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• A design with m factors (with m>1) is called an m-way factorial design – The Eysenck study described in the previous slide has two factors and

is therefore a two-way factorial design

• We can design factorial ANOVAs with an arbitrary number of factors. – For example, we could add gender as another factor in the Eysenck

memory study

• However, for simplicity, we will deal only with two-way factorial designs in this course. – We will also assume that each of our factorial samples (cells) contains

the same number of scores n

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Why might we want to use factorial designs?

• With a one-way ANOVA, we can examine the effect of

different levels of a single factor:

– E.g., How does age affect word recall?

– Or, How does type of processing affect word recall?

• We need two different experiments to determine the effects of

these two factors on memory if we use a one-way design.

• Moreover, using a factorial design allows us to detect

interactions between factors

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Example:

• We have developed a new drug for treating migraines, but

suspect that it affects women differently than men

– The scores represent the number of weekly migraines reported

following administration of the drug

Low

dose

High

dose

Total

Women

Men

Total

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• Notice that the study involves two dosage conditions and two

gender conditions, creating a two-by-two matrix with a total of

4 different treatment conditions.

• Each treatment condition is represented by a cell in the

matrix.

• For an independent-measures research study (which is the

only kind of factorial design that we will consider), a separate

sample would be used for each of the four conditions.

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• As with one-way ANOVAs the goal for the two-factor ANOVA

is to determine whether the mean differences that are

observed for the sample data are significant differences and

not simply the result of sampling error.

• For the example we are considering, the goal is to determine

whether different dosages of a drug and differences in gender

produce significant differences in the number of migraines

reported.

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Factorial ANOVA Main Effects

To evaluate the sample mean differences, a two-factor ANOVA conducts three separate and independent hypothesis tests. The three tests evaluate:

1. The Main Effect for Factor A: The mean differences between the

levels of factor A are obtained by computing the overall mean for each row in the matrix.

• In this example, the main effect of factor A would compare the overall mean number of migraines reported by women versus the overall mean number of migraines reported by men.

2. The Main Effect for Factor B: The mean differences between the levels of factor B are obtained by computing the overall mean for each column in the matrix.

• In this example, the ANOVA would compare the overall mean number of

migraines reported for the low and high dosage conditions.

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Example: Main Effects

Low

dose

High

dose

Total

Women

Men

Total

Main effect B

Main effect A

Main effect A: main effect of gender.

Women report more migraines than men

Main effect B: main effect of drug dose.

Administering a higher dose of the drug

leads to fewer reported migraines

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Interactions

• The A x B Interaction: Often two factors will "interact" so

that specific combinations of the two factors produce results

(mean differences) that are not explained by the overall

effects of either factor.

– For example, a particular drug may have different efficacies for men

vs. women. Different doses of the drug might produce very small

changes in men, but dramatic, or even opposite, effects in women.

This dependence on the effect of one factor (drug dosage) on another

(sex or gender) is called an interaction.

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Example: Interaction

Low

dose

High

dose

Total

Women

Men

Total

In this case, the drug seems to be much more effective for women than for men. We

would say that there is an interaction between the effect of gender and the effect of

drug dosage

Main effect B

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Example: No Interaction

Low dose High

dose

Total

Women

Men

Total

In this case, the main effect of the drug dosage is the same as in the previous case,

but there is no longer a difference between the effect of drug dosage on women

versus men. There is no interaction.

Main effect B

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Interactions

• This is the primary advantage of combining two factors

(independent variables) in a single research study: it allows

you to examine how the two factors interact with each other.

– That is, the results will not only show the overall main effects of each

factor, but also how unique combinations of the two variables may

produce unique results.

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More Examples

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Simple Effects

• To interpret significant interactions, researchers often conduct a fourth type of hypothesis test for simple effects

• Simple effects (or simple main effects) involve testing the effect of one factor at a particular value of the second factor – In our example, testing the effect of the drug for women only or for men only

are examples of simple effects, as are testing the effect of gender in low-dose only or high-dose only conditions

• Testing for simple effects essentially consists of running a separate one way ANOVA across all levels of one factor at a fixed level of the second factor – For example, in the Eysenck memory study, we could run a one-way

ANOVA to determine the effect of processing condition on word recall for young subjects only

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Example: Interaction

Low

dose

High

dose

Total

Women

Men

Total

Main effect B

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Structure of the Analysis for a Two-Way ANOVA

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The Two-Way ANOVA

• Each of the three hypothesis tests in a two-factor ANOVA will

have its own F-ratio and each F-ratio has the same basic

structure:

• Each MS value equals SS/df, and the individual SS and df

values are computed in a two-stage analysis.

• The first stage of the analysis is identical to the single-factor

(one-way) ANOVA and separates the total variability (SS and

df) into two basic components: between treatments and within

treatments.

between

within

MSF

MS

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The Two-Way ANOVA

• The second stage of the analysis separates the between-treatments variability into the three components that will form the numerators for the three F-ratios: 1. Variance due to factor A

2. Variance due to factor B

3. Variance due to the interaction.

• Each of the three variances (MS) measures the differences for a specific set of sample means. The main effect for factor A, for example, will measure the mean differences between rows of the data matrix.

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The Two-Way ANOVA: Possible Outcomes

1. All 3 hypothesis tests are not significant

2. Only main effect of Factor A is significant

3. Only main effect of Factor B is significant

4. Both main effects (for A and B) are significant

5. Only the interaction (A×B) is significant

6. Main effect of Factor A and A×B interaction are significant

7. Main effect of Factor B and A×B interaction are significant

8. All 3 hypothesis tests are significant

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The Two-Way ANOVA: Steps

1. State Hypotheses

2. Compute F-ratio statistic: for each main effect and their interaction

– For data in which I give you cell means and SS’s, you will have to

compute: • marginal means

• SStotal, SSbetween, SSwithin, SSfactor A, SSfactor B, & SSA×B

• dftotal, dfbetween, dfwithin, dffactor A, dffactor B, & dfA×B

3. Use F-ratio distribution table to find critical F-value(s) representing rejection region(s)

4. For each F-test, make a decision: does the F-statistic for your test fall into the rejection region?

, whateverwhatever error

error

MSF df df

MS

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Source df SS MS F p

Between groups

Gender

Dose

Dose x gender

Within (error)

Total

Set up a summary ANOVA table:

1. Compute degrees of freedom

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

cells

dose levels 1 1

gender levels

1 19

# 1 3

16

#

#

1

1 1

total

between

within total between

dose

dose gender between dose

gender

gender

df N

df

df df

df

df

df df df

d

df

f

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Source df SS MS F p

Between groups 3

Gender 1

Dose 1

Dose x gender 1

Within (error) 16

Total 19


2. Compute SSwithin (SSerror)

20 918 32 24 4

withinSS SS

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

01:830:200:01-05 Fall 2013


Source df SS MS F p

Between groups 3

Gender 1

Dose 1

Dose x gender 1

Within (error) 16 94

Total 19


3. Compute SSbetween (SScells)

2

2 2 2 25 20 13.25 10 13.25 12 13.25 11 13.25

5 45.563 10.563 1.56

313

3 5.063

5 62. .7575

cell TbetweenSS n M M

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

01:830:200:01-05 Fall 2013


Source df SS MS F p

Between groups 3 313.75

Gender 1

Dose 1

Dose x gender 1


Total 19 407.75


4. Compute SSgender (SSFactor A)

2

2 210 15 13.25 11.5 13.25

10 3.0625 3.0625

10 6. 61.25125

gender row gend r TeSS n M M

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

01:830:200:01-05 Fall 2013


Source df SS MS F p


Gender 1 61.25

Dose 1

Dose x gender 1


Total 19 407.75


5. Compute SSdose (SSFactor B)

2

2 210 16 13.25 10.5 13.25

151.25

10 7.5625 7.5625

10 15.125

dose col d ToseSS n M M

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

01:830:200:01-05 Fall 2013


Source df SS MS F p


Gender 1 61.25

Dose 1 151.25

Dose x gender 1


Total 19 407.75


6. Compute SSdose×gender (SSA×B)

313.75 151.25 61.25

313.75 2 101.1 . 52 5 2

gender between gender dod seoseSS SS SS SS

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

01:830:200:01-05 Fall 2013


Source df SS MS F p


Gender 1 61.25

Dose 1 151.25

Dose x gender 1 101.25


Total 19 407.75


7. Compute the MS values needed to compute the 3 required F ratios:

661.25

11.25

gender

gender

gender

SSMS

df

95.875

4

16

errorerror

error

SSMS

df 1

151.25

151.25dose

dos

e

e

dos

SSMS

df

101101.25

1.25

dose gender

dose gender

dose gender

SSMS

df

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

01:830:200:01-05 Fall 2013


Source df SS MS F p


Gender 1 61.25 61.25

Dose 1 151.25 151.25

Dose x gender 1 101.25 101.25

Within (error) 16 94 5.875

Total 19 407.75


8. Compute F ratios for each of the two main effects (gender and dose) and

their interaction:

, whateverwhatever ewhatever rror

error

dfMS

F dfMS

61.25

1,165.8

10.45

37

gender

e

ge

rro

nder

r

MSF

MS

151.25

1,165.875

25.74dose

err

e

or

dos

MSF

MS

101.25

1,1 165.

7.8

375

2genddose

dose

erro

er

g nd

r

e er

MSF

MS

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

01:830:200:01-05 Fall 2013


Source df SS MS F p


Gender 1 61.25 61.25 10.43

Dose 1 151.25 151.25 25.74

Dose x gender 1 101.25 101.25 17.23


Total 19 407.75


9. Finally, look up Fcrit for each of your obtained F values

In this case, we happen to be lucky that they all have the same degrees of

freedom (1,16), so we only have to look up one Fcrit

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

01:830:200:01-05 Fall 2013


1 2 3 4 5 6 7 8 9 10

1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88

2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40

3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79

4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96

5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74

6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06

7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64

8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35

9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14

10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98

11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85

12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75

13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67

14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60

15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54

16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49

17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45

18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41

19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38

20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35

22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30

24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25

26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22

28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19

30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16

40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08

50 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03

60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99

120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91

200 3.89 3.04 2.65 2.42 2.26 2.14 2.06 1.98 1.93 1.88

500 3.86 3.01 2.62 2.39 2.23 2.12 2.03 1.96 1.90 1.85

1000 3.85 3.00 2.61 2.38 2.22 2.11 2.02 1.95 1.89 1.84

dfnumerator

F table for α=0.05

reject H0

df e

rro

r

01:830:200:01-05 Fall 2013


1 2 3 4 5 6 7 8 9 10

1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54 241.88

2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38 19.40

3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81 8.79

4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00 5.96

5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77 4.74

6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10 4.06

7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68 3.64

8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39 3.35

9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18 3.14

10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02 2.98

11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90 2.85

12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80 2.75

13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71 2.67

14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65 2.60

15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59 2.54

16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54 2.49

17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49 2.45

18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46 2.41

19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42 2.38

20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39 2.35

22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34 2.30

24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30 2.25

26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27 2.22

28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24 2.19

30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21 2.16

40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12 2.08

50 4.03 3.18 2.79 2.56 2.40 2.29 2.20 2.13 2.07 2.03

60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04 1.99

120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96 1.91

200 3.89 3.04 2.65 2.42 2.26 2.14 2.06 1.98 1.93 1.88

500 3.86 3.01 2.62 2.39 2.23 2.12 2.03 1.96 1.90 1.85

1000 3.85 3.00 2.61 2.38 2.22 2.11 2.02 1.95 1.89 1.84

dfbetween

df e

rro

r

F table for α=0.05

reject H0

01:830:200:01-05 Fall 2013


Source df SS MS F p


Gender 1 61.25 61.25 10.43* <0.05

Dose 1 151.25 151.25 25.74* <0.05

Dose x gender 1 101.25 101.25 17.23* <0.05


Total 19 407.75


Both main effects are significant, as is their interaction. This suggests that:

1. The number of reported migraines differs between men and women • Women report more migraines than men

2. The number of reported migraines is affected by the drug • Fewer migraines are reported in the high dose condition

3. The effect of the drug differs across genders • Women are more sensitive to the drug than are men

Low

dose

High

dose

Overall

(gender)

Women M=20

SS=20

M=10

SS=18

Men M=12

SS=32

M=11

SS=24

Overall

(dose)

13.25, 20

5, 10, 10

T

row col

M N

n n n

Documents

Factorial Analysis of Variance - Rutgers Universitymmm431/quant_methods_F13/QM_Lecture15.pdf• We need two different experiments to determine the effects of these two factors on memory