SPSS meets SPM

All about Analysis of Variance

• Introduction and definition of terms

• One-way between-subject ANOVA: An example

• One-way repeated measurement ANOVA

• Two-way repeated measurement ANOVA:

• Pooled and partitioned errors

• How to specify appropriate contrasts to test main effects

and interactions

SPSS meets SPM

Two-sample t-test Paired-sample-t-test

ANOVA Repeated ANOVA

between-subject ANOVA within-subject ANOVA

F-test F-test

Factors

Levels

K1 x K2 ANOVA

Two Factors

with K1 levels of one factor and

K2 level of the second factor

Repeated MeasuresSingle Measures

Analysis of Variance

Level

1

Level

2

Level 1 Group

1

Group

2

Level

2

Group

3

Group

4

Factor B

Two-way ANOVA

2 x 2 ANOVA

2 x 2 repeated measurement ANOVA

Level

1

Level

2

Level 1 Subj.

1….12

Subj.

1….12

Level

2

Subj.

1….12

Subj.

1….12

Factor A

Factor B

Factor A

Mixed Design

Drug Placebo

Patient Subj.

1…12

Subj.

1…12

Control Subj.

13...24

Subj.

13...24

Factor AWithin-subject Factor

Factor BBetween-subject

Factor

ImagingDesigns

Fearful Neutral

Implicit Group

1

Group

2

Explicit Group

3

Group

4

Fac

tor

B2 x 2 ANOVA 2 x 2 repeated

measurement ANOVA

Fearful Neutral

Implicit Subj.

1….12

Subj.

1….12

Explicit Subj.

1….12

Subj.

1….12

Factor A

Fac

tor

B

Factor A

Main Effect A

Main E

ffect B

Interaction A X B

Implicit Explicit

Fearful

Neutral

Fearful Neutral Happy

Implicit

Explicit

3 x 2 ANOVA

Contrasts

ijjijX

An individual score is specified by

jj

Grand mean

Treatment effect

jijij X Residual error

One-way between-subject ANOVA

ijjijX FULL MODEL

General Principle of ANOVA

ijijX

Is the full model a significantly better model then the reduced model?

Data represent a random variation around the grand mean

REDUCED MODEL

Total Variation (SStotal)

Treatment effect (SStreat)

Error (SSerror)

errorerror

treattreat

DFSS

DFSSF

/

/

errortreattotal SSSSSS

errortreattotal DFDFDF

Partitions of Sums of Squares

____________________________________

4-different drug treatments (Factor A with p levels)____________________________________

1 2 3 4____________________________________

2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2

_____________________________________Sums(Ai) 10 15 35 20_____________________________________Means(Ai) 2 3 7 4

_____________________________________

One factor with p levels; i = 1…4M subjects with n subjects per level Number of total observations = 20

1.

2.

3.

4.

5.

m

mii

tot GxSS 2)(

pn

AG i

i

*

m

imii

error AxSS 2)(

i

itreat GAnSS 2)(*

1* pndftot

1pdftreat

)1(* npdferror

errorerror

treattreat

DFSS

DFSSF

/

/

420

20351510

G

..)41()42( 22

..)43(*5)42(*5 22

..)21()22( 22

One-way ANOVA between subjects1st levels betas from one voxel in amygdala

Multiple Regression

____________________________________Drug treatment (Factor A with p levels)____________________________________ 1 2 3 4____________________________________ 2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2

Dependent variable = 1st level betas extracted from the amygdala

One way ANOVA

Do the drug treatment affect differentlyaffect differently mean activation in the amygdala ?

Do the drug treatments relaterelate to the mean activation in the amygdala?

213313435068741055532

1st level betas

Drug treatments

11111222223333344444

y = a + bX

11111111111111111111

Multiple Regression

____________________________________Drug treatment (Factor A with p levels)____________________________________ 1 2 3 4____________________________________ 2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2

Dependent variable = 1st level betas extracted from the amygdala

One way ANOVA

Do the drug treatment affect differentlyaffect differently mean activation in the amygdala ?

Do the drug treatments relaterelate to the mean activation in the amygdala?

213313435068741055532

1st level betas

Drug treatments

00000111110000000000

11111111111111111111

00000000001111100000

00000000000000011111

11111000000000000000

y y xx11 xx22 xx33 xx44

Multiple Regression

____________________________________Drug treatment (Factor A with p levels)____________________________________ 1 2 3 4____________________________________ 2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2

Dependent variable = 1st level betas extracted from the amygdala

One way ANOVA

Do the drug treatment affect differentlyaffect differently mean activation in the amygdala ?

Do the drug treatments relaterelate to the mean activation in the amygdala?

213313435068741055532

1st level betas

Drug treatments

00000111110000000000

11111111111111111111

00000000001111100000

00000000000000011111

11111000000000000000

Y Y bb11xx11= bb22xx22 bb33xx33 bb44xx44 bb00

+ + + +

Multiple Regression

____________________________________Teaching Methods (Factor A with p levels)____________________________________ 1 2 3 4____________________________________ 2 3 6 5 1 4 8 5 3 3 7 5 3 5 4 3 1 0 10 2

Dependent variable = reading score

One way ANOVA

1 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 1

ijjijX

y = *

11

21

31

41

51

12

.

.

.

.

.

.

.

.

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.

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.

44

54

eeeeee

ee

+

ebXbXbXbXby 044332211

b1

b2

b3

b4

b0

jj bb

;0

Do the drug treatment affect differentlyaffect differently mean activation in the amygdala ?

Do the drug treatments relaterelate to the mean activation in the amygdala?

Two-sample t-test Paired-sample-t-test

ANOVA Repeated ANOVA

between-subject ANOVA within-subject ANOVA

F-test F-test

Repeated MeasuresSingle Measures

Repeated ANOVA

Assumptions

• Homogeneity of Variance

• Homogeneity of Correlations

• Normality

Assumptions

• Homogeneity of Variance

• Normality

• Independence of observations

Drug 1 Drug 2 Drug 3 Placebo

Subj.1

Subj. 2

Subj. 3

….

Subj.1

Subj. 2

Subj. 3

….

Subj.1

Subj. 2

Subj. 3

….

Subj.1

Subj. 2

Subj. 3

…..

Drug 1 Drug 2 Drug 3 Placebo

Group 1 Group2 Group3 Group4

One-way within-subject

ANOVA

ijjijX

An individual score is specified by

One-way between-subject

ANOVA

An individual score is specified by

ijjiijX

jj

Grand mean

Treatment effect

jijij X Residual error

i Grand mean

Subject effect

Residual error

j Treatment effect (within-subject effect)

ij

Total Variation (SStotal)

Treatment effect (SStreat)

Error (SSerror)

Total Variation (SStotal)

Between subj (SSbetween)Subject effects

Subj. x Treat & Error

Within subj. (SSwithin)

Treatment effect (SStreat)

Residual (SSres)

errorerror

treattreat

DFSS

DFSSF

/

/

errortreattotal SSSSSS

errortreattotal DFDFDF

errorresidual

treattreat

DFSS

DFSSF

/

/

residualtreatbetweentotal SSSSSSSS

residualtreatmentbetweentotal DFDFDFDF

Partitions of Sums of Squares

Within subjectsWithin subjectsBetween SubjectsBetween Subjects

ijjijX

1 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 1

y = *

11

21

31

41

51

12

.

.

.

.

.

.

.

.

.

.

.

.

44

54

eeeeee

ee

+

b1

b2

b3

b4

b0

ijjiijX jj

ebXbXbXbXby 044332211 ebXbXbXbXby 044332211

1 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 11 0 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 1 0 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 1 0 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 10 0 0 1 1

y = *

11

21

31

41

51

12

.

.

.

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.

44

54

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ee

+

b1

b2

b3

b4

b0

+

1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 11 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 11 0 0 0 00 1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

1 2 3 4

Drug 1 Drug 2 Drug3 Placebo

1

2

3

4

5

Within subjectsWithin subjectsBetween SubjectsBetween Subjects

ijjijX ijjiijX jj

ebXbXbXbXby 044332211 ebXbXbXbXby 044332211

00000111110000000000

11111111111111111111

00000000001111100000

00000000000000011111

11111000000000000000

00000111110000000000

00000000001111100000

00000000000000011111

11111000000000000000

10000000000000000000

01000000000000000000

00100000000000000000

00010000000000000000

00000000000000000000

11111111111111111111

1 2 3 4

Drug 1 Drug 2 Drug3 Placebo

Level

1

Level

2

Level 1

Subj.

1….12

Subj.

1….12

Level

2

Subj.

1….12

Subj.

1….12

Factor A

Factor B

ijkABjk

Bk

AjiijkX

2 x 2 Repeated Measurement ANOVA

ABijk

Bik

Aij

ABjk

Bk

AjiijkX

Pooled Error

Partitioned Error Interaction between effect and subject

Within-Subjects Two-Way ANOVAWithin-Subjects Two-Way ANOVA

1 2 3 4Fear-implicit neutral-implicit fear-explicit neutral-explicit

ijjiijX

ebXbXbXbXby 044332211

00000111110000000000

00000000001111100000

00000000000000011111

11111000000000000000

10000000000000000000

01000000000000000000

00100000000000000000

00010000000000000000

00000000000000000000

11111111111111111111

Repeated Measurement Repeated Measurement ANOVA in SPMANOVA in SPM

One way ANOVA = 1st level betas 2nd level + subjects effects

Two way ANOVA = 1st level differential effects betweenlevels of a factors for main effects

differences of differential effects for interactions

2nd level (T-test for 2x2 ANOVA F-test for 3x3 ANOVA)

Pooled errors

Partitioned errors

What contrast to take from 1st level?Two way ANOVA (2*2) with repeated measured

Fearful Neutral

Implicit

Explicit

Factor A

Fac

tor

B

Fear/

implicit

Fear/ explicit

Neutral/ implicit

Neutral/ explicit

What contrast to take from 1st level?Two way ANOVA (3*3) with repeated measured

Factor A

Fac

tor

B

semantic perception

Imagery

Picture

Words

Sounds