SPSS meets SPM All about Analysis of Variance

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

Analysis of Variance Single Measures Repeated Measures Two-sample t-test Paired-sample-t-test ANOVA between-subject ANOVA F-test Repeated ANOVA within-subject ANOVA F-test Factors Levels K 1 x K 2 ANOVA Two Factors with K 1 levels of one factor and K 2 level of the second factor

2 x 2 repeated measurement ANOVA Two-way ANOVA 2 x 2 ANOVA Factor B Level 1 Level 2 Group 1 Group 2 Group 3 Group 4 Level 1 Level 2 Level 1 Subj. 1…. 12 Level 2 Subj. 1…. 12 Factor B Mixed Design Factor A Within-subject Factor Drug Factor B Between-subject Factor Placebo Patient Subj. 1… 12 Control Subj. 13. . . 24 Imaging Designs

2 x 2 repeated measurement ANOVA 2 x 2 ANOVA Factor A Explicit Group 1 Group 2 Group 3 Group 4 Factor B Implicit Factor A Neutral Main Effect B Factor B Fearful Neutral Implicit Subj. 1…. 12 Explicit Subj. 1…. 12 Main Effect A Interaction A X B 3 x 2 ANOVA Fearful Neutral Fearful Happy Implicit Neutral Explicit Implicit Contrasts Explicit

One-way between-subject ANOVA An individual score is specified by Grand mean Treatment effect Residual error

General Principle of ANOVA FULL MODEL REDUCED MODEL Data represent a random variation around the grand mean Is the full model a significantly better model then the reduced model?

Partitions of Sums of Squares Total Variation (SStotal) Treatment effect (SStreat) Error (SSerror)

One-way ANOVA between subjects 1 st levels betas from one voxel in amygdala 1. __________________ 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 2. 2. 3. 3. 4. 4. ___________________ One factor with p levels; i = 1… 4 M subjects with n subjects per level Number of total observations = 20 5. 5.

One way ANOVA Do the drug treatment affect differently mean activation in the amygdala ? __________________ 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 = 1 st level betas extracted from the amygdala Multiple Regression Do the drug treatments relate to the mean activation in the amygdala? 1 st level betas Drug treatments 2 1 3 3 1 3 4 3 5 0 6 8 7 4 10 5 5 5 3 2 1 1 1 2 2 2 3 3 3 4 4 4 y = a. X 1 1 1 1 1 + b

One way ANOVA Multiple Regression Do the drug treatment affect differently mean activation in the amygdala ? __________________ 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 = 1 st level betas extracted from the amygdala Do the drug treatments relate to the mean activation in the amygdala? 1 st level betas 2 1 3 3 1 3 4 3 5 0 6 8 7 4 10 5 5 5 3 2 y Drug treatments 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 x 2 x 3 x 4 1 1 1 1 1

One way ANOVA Multiple Regression Do the drug treatment affect differently mean activation in the amygdala ? __________________ 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 = 1 st level betas extracted from the amygdala Do the drug treatments relate to the mean activation in the amygdala? 1 st level betas 2 1 3 3 1 3 4 3 5 0 6 8 7 4 10 5 5 5 3 2 Y Drug treatments 1 1 1 0 0 0 0 = 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 b 1 x 1+b 2 x 2+b 3 x 3+b 4 x 4 + b 0

Multiple Regression One way ANOVA Do the drug treatment affect differently mean activation in the amygdala ? Do the drug treatments relate to the mean activation in the amygdala? __________________ 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 y= 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 b 1 * b 2 b 3 b 4 b 0 + 11 21 31 41 51 12. . . 44 54

Repeated ANOVA Single Measures Two-sample t-test ANOVA between-subject ANOVA F-test Drug 1 Drug 2 Drug 3 Placebo Group 1 Group 2 Group 3 Group 4 Repeated Measures Paired-sample-t-test Repeated ANOVA within-subject ANOVA F-test Drug 1 Drug 2 Drug 3 Placebo Subj. 1 Subj. 2 Subj. 3 …. . Assumptions • Homogeneity of Variance • Normality • Homogeneity of Correlations • Independence of observations • Normality

One-way between-subject One-way within-subject ANOVA An individual score is specified by Grand mean Subject effect Treatment effect Residual error Treatment effect (within-subject effect) Residual error

Partitions of Sums of Squares Total Variation (SStotal) Treatment effect (SStreat) Error (SSerror) Total Variation (SStotal) Within subj. (SSwithin) Between subj (SSbetween) Subject effects Treatment effect (SStreat) Residual (SSres) Subj. x Treat & Error

Between Subjects y= 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 Within subjects 1 2 3 4 Drug 1 Drug 2 Drug 3 Placebo 1 1 1 1 1 b 1 b 2 * b 3 b 4 b 0 + 11 21 31 41 51 12. . . 44 54 y= 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 b 2 * b 3 b 4 b 0 + 11 21 31 41 51 12. . . 44 54 + 10000 01000 00100 00010 00001 p 1 p 2 p 3 p 4 p 5

Between Subjects 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 Within subjects 1 2 3 4 Drug 1 Drug 2 Drug 3 Placebo 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 x 2 Repeated Measurement ANOVA Factor A Level 1 Level 2 Level 1 Subj. 1…. 12 Level 2 Subj. 1…. 12 Factor B Pooled Error Partitioned Error Interaction between effect and subject

Within-Subjects Two-Way ANOVA 1 2 3 4 Fear-implicit neutral-implicit fear-explicit neutral-explicit 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Repeated Measurement ANOVA in SPM Pooled errors One way ANOVA = 1 st level betas 2 nd level + subjects effects Partitioned errors Two way ANOVA = 1 st level differential effects between levels of a factors for main effects differences of differential effects for interactions 2 nd level (T-test for 2 x 2 ANOVA F-test for 3 x 3 ANOVA)

What contrast to take from 1 st level? Two way ANOVA (2*2) with repeated measured Factor A Factor B Fearful Neutral Implicit Explicit Fear/ implicit Fear/ explicit Neutral/ implicit Neutral/ explicit

What contrast to take from 1 st level? Two way ANOVA (3*3) with repeated measured Factor A semantic Factor B Picture Words Sounds perception Imagery