factorial anova statistics for the social sciences psychology 340 spring 2010

Factorial ANOVA

Statistics for the Social SciencesPsychology 340

Spring 2010

PSY 340Statistics for the

Social SciencesOutline

• Basics of factorial ANOVA– Interpretations

• Main effects

• Interactions

– Computations

– Assumptions, effect sizes, and power

– Other Factorial Designs• More than two factors

• Within factorial ANOVAs

• Mixed factorial ANOVAs


Social Sciences

Statistical analysis follows design

• The factorial (between groups) ANOVA:

– More than two groups

– Independent groups

– More than one Independent variable


Social Sciences Factorial experiments

• Two or more factors– Factors - independent variables– Levels - the levels of your independent variables

• 2 x 3 design means two independent variables, one with 2 levels and one with 3 levels

• “condition” or “groups” is calculated by multiplying the levels, so a 2x3 design has 6 different conditions

B1 B2 B3

A1

A2


Social Sciences Factorial experiments

• Two or more factors (cont.)– Main effects - the effects of your independent variables

ignoring (collapsed across) the other independent variables

– Interaction effects - how your independent variables affect each other

• Example: 2x2 design, factors A and B

• Interaction:– At A1, B1 is bigger than B2

– At A2, B1 and B2 don’t differ


Social Sciences Results

• So there are lots of different potential outcomes:• A = main effect of factor A• B = main effect of factor B• AB = interaction of A and B

• With 2 factors there are 8 basic possible patterns of results:

5) A & B6) A & AB7) B & AB8) A & B & AB

1) No effects at all2) A only3) B only4) AB only


Social Sciences 2 x 2 factorial design

Condition

mean

A1B1

Condition

mean

A2B1

Condition

mean

A1B2

Condition

mean

A2B2

A1 A2

B2

B1

Marginal means

B1 mean

B2 mean

A1 mean A2 mean

Main effect of B

Main effect of A

Interaction of ABWhat’s the effect of A at B1?What’s the effect of A at B2?


Social Sciences

Main effect of AMain effect of BInteraction of A x B

A

B

A1 A2

B1

B2

Main Effect of A

Main Effect of B

60

45

45

30

6030

6030

A

A1 A2

Dependent

Vari

able

B1

B2

√X

X

Examples of outcomes


Social Sciences


A

B

A1 A2

B1

B2

Main Effect of A

Main Effect of B

45

60

30

45

3030

6060

A

A1 A2

Dependent

Vari

able

B1

B2

√X

X



Social Sciences


A

B

A1 A2

B1

B2

Main Effect of A

Main Effect of B

45

45

45

45

6030

3060

A

A1 A2

Dependent

Vari

able

B1

B2

√X

X



Social Sciences


A

B

A1 A2

B1

B2

Main Effect of A

Main Effect of B

45

45

30

30

3030

6030

A

A1 A2

Dependent

Vari

able

B1

B2

√

√

√



Social Sciences Factorial Designs

• Benefits of factorial ANOVA (over doing separate 1-way ANOVA experiments)– Interaction effects

– One should always consider the interaction effects before trying to interpret the main effects

– Adding factors decreases the variability– Because you’re controlling more of the variables that

influence the dependent variable– This increases the statistical Power of the statistical tests


Social Sciences Basic Logic of the Two-Way ANOVA

• Same basic math as we used before, but now there are additional ways to partition the variance

• The three F ratios– Main effect of Factor A (rows)

– Main effect of Factor B (columns)

– Interaction effect of Factors A and B


Social Sciences Partitioning the variance

Total variance

Stage 1

Between groups variance Within groups variance

Stage 2Factor A variance Factor B variance Interaction variance


Social Sciences Figuring a Two-Way ANOVA

• Sums of squares

SSWithin = SSgroups∑

SSA = nAgroups(X∑

Agroups

−GM )2

SSAB =SSBetween − SSA +SSB( )

SSBetweenGroups = n(XAllGroups∑ −GM )2

SSTotal = (X∑ −GM )2

SSB = nBgroups(X∑

Bgroups

−GM )2



• Degrees of freedom

dfA =NA −1

dfB =NB −1

dfAB =NConditions −dfA −dfB −1

dfWithin = dfeach group∑Number of levels of A

Number of levels of A

Number of levels of B

Number of levels of B



• Means squares (estimated variances)

MSA =SSA

dfA

MSB =SSB

dfB

MSAB =SSAB

dfAB

MSWithin =SSWithin

dfeach group



• F-ratios

FA =MSA

MSWithin

FB =MSB

MSWithin

FAB =MSAB

MSWithin


Social Sciences ExampleFactor B: Arousal Level

LowB1

MediumB2

HighB3

FactorA:

Task

Difficulty

A1

Easy

3

1

1

6

4

2

5

9

7

7

7

9

11

10

8

A2

Difficult

3

0

0

2

0

3

8

3

3

3

0

0

0

5

0

XA1B1 =3

SSA1B1 =18

nA1B1 =5

XA2 B1 =1 XA2 B2 =4

XA1B2 =6 XA1B3 =9

XA2 B3 =1

SSA1B2 =28 SSA1B3 =10

SSA2 B1 =8 SSA2 B2 =20 SSA2 B3 =20

nA1B2 =5 nA1B3 =5

nA2 B1 =5 nA2 B2 =5 nA2 B3 =5



LowB1

MediumB2

HighB3

FactorA:

Task

Difficulty

A1

Easy

3

1

1

6

4

2

5

9

7

7

7

9

11

10

8

A2

Difficult

3

0

0

2

0

3

8

3

3

3

0

0

0

5

0

XA1B1 =3

SSA1B1 =18

nA1B1 =5

XA2 B1 =1 XA2 B2 =4

XA1B2 =6 XA1B3 =9

XA2 B3 =1

SSA1B2 =28 SSA1B3 =10

SSA2 B1 =8 SSA2 B2 =20 SSA2 B3 =20

nA1B2 =5 nA1B3 =5

nA2 B1 =5 nA2 B2 =5 nA2 B3 =5

GM =4N =30

SSTotal =344

SSBetweenGroups = n(XAllGroups∑ −GM )2

=5(3 − 4)2 + 5(6 − 4)2 + 5(9 − 4)2 +5(1−4)2 + 5(4 −4)2 + 5(1−4)2

=240

SSWithin = SSgroups∑=18 + 28 +10 + 8 + 20 + 20

=104



LowB1

MediumB2

HighB3

FactorA:

Task

Difficulty

A1

Easy

3

1

1

6

4

2

5

9

7

7

7

9

11

10

8

A2

Difficult

3

0

0

2

0

3

8

3

3

3

0

0

0

5

0

XA1B1 =3

SSA1B1 =18

nA1B1 =5

XA2 B1 =1 XA2 B2 =4

XA1B2 =6 XA1B3 =9

XA2 B3 =1

SSA1B2 =28 SSA1B3 =10

SSA2 B1 =8 SSA2 B2 =20 SSA2 B3 =20

nA1B2 =5 nA1B3 =5

nA2 B1 =5 nA2 B2 =5 nA2 B3 =5

GM =4N =30

SSTotal =344

SSA = nAgroups(X∑

Agroups

−GM )2

=15 6 − 4( )2

+ 15 2 − 4( )2

=120

SSB = nBgroups(X∑

Bgroups

−GM )2

=10 2 − 4( )2

+10 5 − 4( )2

+10 5 − 4( )2

=60

=XB1∑

nB1

=20

10= 2.0 =

50

10= 5.0 =

50

10= 5.0

=XA1∑

nA1

=90

15= 6.0

=XA2∑

nA1

=30

15= 2.0



LowB1

MediumB2

HighB3

FactorA:

Task

Difficulty

A1

Easy

3

1

1

6

4

2

5

9

7

7

7

9

11

10

8

A2

Difficult

3

0

0

2

0

3

8

3

3

3

0

0

0

5

0

XA1B1 =3

SSA1B1 =18

nA1B1 =5

XA2 B1 =1 XA2 B2 =4

XA1B2 =6 XA1B3 =9

XA2 B3 =1

SSA1B2 =28 SSA1B3 =10

SSA2 B1 =8 SSA2 B2 =20 SSA2 B3 =20

nA1B2 =5 nA1B3 =5

nA2 B1 =5 nA2 B2 =5 nA2 B3 =5

GM =4N =30

SSTotal =344

SSBetweenGroups =240

SSA =120

SSB =60

SSAB =SSBetween − SSA +SSB( )=240 − 120 + 60( )

=60


Social Sciences Example: ANOVA table

Source SS df MS FBetween

A

B

AB

120

60

60

1

2

2

120

30

30

27.7

6.9

6.9

Within

Total

104

344

24 4.33

√

√

√


Social Sciences Factorial ANOVA in SPSS

• What we covered today is a completely between groups Factorial ANOVA– Enter your observations in one column, use separate

columns to code the levels of each factor

– Analyze -> General Linear Model -> Univariate

– Enter your dependent variable (your observations)

– Enter each of your factors (IVs)

• Output– Ignore the corrected model, intercept, & total (for now)

– F for each main effect and interaction


Social Sciences Assumptions in Two-Way ANOVA

• Populations follow a normal curve• Populations have equal variances• Assumptions apply to the populations that go with

each cell


Social Sciences

Note: if you downloaded the lecture Tues. there were two errors

Effect Size in Factorial ANOVA (completely between groups)

η2 = RB2 =

SSB

SSTotal − SSA − SSAB

=SSB

SSB + SSwithin

η2 = RA2 =

SSA

SSTotal − SSB − SSAB

=SSA

SSA + SSwithin

η2 = RAB2 =

SSAB

SSTotal − SSB − SSA

=SSAB

SSAB + SSwithin


Social Sciences

Approximate Sample Size Needed in Each Cell for 80% Power (.05 significance level)


Social SciencesOther ANOVA designs

• Basics of repeated measures factorial ANOVA– Using SPSS

• Basics of mixed factorial ANOVA– Using SPSS

• Similar to the between groups factorial ANOVA– Main effects and interactions– Multiple sources for the error terms (different

denominators for each main effect)


Social Sciences Example

• Suppose that you are interested in how sleep deprivation impacts performance. You test 5 people on two tasks (motor and math) over the course of time without sleep (24 hrs, 36 hrs, and 48 hrs). Dependent variable is number of errors in the tasks.– Both factors are manipulated as within subject

variables– Need to conduct a within groups factorial

ANOVA



Factor B: Hours awake24B1

36B2

48B3

Factor A:

Task

A1

Motor

0

1

0

4

0

0

3

1

5

1

6

5

5

9

5

A2

Math

1

1

0

3

1

1

2

1

2

3

4

6

6

4

4


Social Sciences Within factorial ANOVA in SPSS

• Each condition goes in a separate column– It is to your benefit to systematically order those columns to reflect

the factor structure

– Make your column labels informative

• Analyze -> General Linear Model -> Repeated measures– Enter your factor 1 & number of levels, then factor 2 & levels, etc.

(remember the order of the columns)

– Tell SPSS which columns correspond to which condition

• As was the case before, lots of output– Focus on the within-subject effects

– Note: each F has a different error term



Source SS df MS F p A

Error (A)

1.20

13.13

1

4

1.20

3.28

0.37 0.58

B

Error (B)

AB

Error (AB)

104.60

6.10

2.60

8.10

2

8

2

8

52.30

0.76

1.30

1.01

69.00 < 0.01

1.29 0.33



• It has been suggested that pupil size increases during emotional arousal. A researcher presents people with different types of stimuli (designed to elicit different emotions). The researcher examines whether similar effects are demonstrated by men and women.– Type of stimuli was manipulated within subjects

– Sex is a between subjects variable

– Need to conduct a mixed factorial ANOVA



Factor B: StimulusNeutral

B1

PleasantB2

AversiveB3

FactorA:

Sex

A1

Men

4

3

2

3

3

8

6

5

3

8

3

3

2

6

1

A2

Women

3

2

4

1

3

6

4

6

7

5

2

1

6

3

2


Social Sciences Mixed factorial ANOVA in SPSS

• Each within condition goes in a separate column– It is to your benefit to systematically order those columns to reflect the factor

structure– Make your column labels informative

• Each between groups factor has a column that specifies group membership

• Analyze -> General Linear Model -> Repeated measures– Enter your within groups factors: factor 1 & number of levels, then factor 2

& levels, etc. (remember the order of the columns)– Tell SPSS which columns correspond to which condition– Enter your between groups column that specifies group membership

• As was the case before, lots of output– Need to look at the within-subject effects and the between groups effects



Source SS df MS F pBetween

Sex (A)

Error (A)

0.83

20.00

1

8

0.83

2.50

0.33 0.58

Within

Stimulus (B)

Sex * Stimulus

Error (B)

58.10

0.07

39.20

2

2

16

29.00

0.03

2.45

11.85 0.001

0.01 0.97


Social Sciences Partitioning the variance

dftotal = N - 1

Total

SX2 -G2

NSStotal =

1Stage

2Stage

( / )Factor A is Between w r levels ( / )Factor B is Within w q levels

Between subjects var

SSbtwnsub =

dfbtwnsub = - 1nr

SpA

2

r-

G2

N

Sub within groups var

SSwthngrps = SSbtwnsubs - SS btwn A

dferror = ( - 1)r n

Factor A variability

SS btwn A =

dfA = - 1r

STA

2

nr-

G2

N

Factor B variability

SS btwn B =

dfB = - 1q

STA

2

nq- G2

N

Within subjects var

dfwthnsub = ( - 1)nr q

SSwthnsub = SS total- SSbtwnsub

Interaction AxB variability

SS A x B = SScells -SS btwn A - SSbtwnB

dfAxB = ( - 1)( - 1)r q

SScells = STij

2

n- G2

N

Factor B x Within Subs var

dfBxWthnsub = ( - 1)( - 1)r n q

SSBxSubswingrps = SS /w insubs -SS btwn B - SSAxB

• Stage 1 partition is same as usual

• Stage 2 combines the other partitioning that we’ve done:

– The between subjects var is broken into 2 parts

– The within subjects is broken into different parts.

– Note: the interaction, because it involves a within groups variable, comes out in the partitioning of the within groups par

factorial anova statistics for the social sciences psychology 340 spring 2010

Documents

main effect of b main

main effect of b interaction

social sciences main

b a1 a2 b1 b2 main effect

main effect of factor

effect sizes

factorial anova statistics

social sciences results