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

FactorsLevels

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

Level1

Level2

Level 1 Group1

Group2

Level2

Group3

Group4

Factor B

Two-way ANOVA2 x 2 ANOVA

2 x 2 repeated measurement ANOVA

Level1

Level2

Level 1 Subj.1….12

Subj.1….12

Level2

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 Group1

Group2

Explicit Group3

Group4

Fact

or B

2 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

Fact

or 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

DFSSDFSSF//

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

DFSSDFSSF//

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 ANOVADo 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 ANOVADo 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 ANOVADo 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 = *

112131415112............4454

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.1Subj. 2Subj. 3

….

Subj.1Subj. 2Subj. 3

….

Subj.1Subj. 2Subj. 3

….

Subj.1Subj. 2Subj. 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

DFSSDFSSF//

errortreattotal SSSSSS

errortreattotal DFDFDF

errorresidual

treattreat

DFSSDFSSF//

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 = *

112131415112............4454

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 = *

112131415112............4454

eeeeee

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

Level1

Level2

Level 1

Subj.1….12

Subj.1….12

Level2

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

Fact

or 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

Fact

or B

semantic perception

Imagery

Picture

Words

Sounds