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 Two-sample t-test Paired-sample-t-test Single Measures Repeated Measures 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

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

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

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 1st levels betas from one voxel in amygdala 2. 3. 4. 5. ____________________________________ 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…4 M subjects with n subjects per level Number of total observations = 20

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

Do the drug treatments relate to the mean activation in the amygdala? One way ANOVA Multiple Regression Do the drug treatment affect differently mean activation in the amygdala ? Do the drug treatments relate to the 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 = 1st level betas extracted from the amygdala 1st level betas Drug treatments 2 1 3 4 5 6 8 7 10 1 y x1 x2 x3 x4

Do the drug treatments relate to the mean activation in the amygdala? One way ANOVA Multiple Regression Do the drug treatment affect differently mean activation in the amygdala ? Do the drug treatments relate to the 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 = 1st level betas extracted from the amygdala 1st level betas Drug treatments 2 1 3 4 5 6 8 7 10 1 Y = b1x1 + b2x2 + b3x3 + b4x4 + b0

Do the drug treatments relate to the mean activation in the amygdala? One way ANOVA Multiple Regression 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 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 11 21 31 41 51 12 . 44 54 b1 b2 b3 b4 b0 y = * +

Repeated ANOVA Two-sample t-test Paired-sample-t-test Repeated Measures Single Measures Two-sample t-test Paired-sample-t-test ANOVA Repeated ANOVA between-subject ANOVA within-subject ANOVA F-test F-test Drug 1 Drug 2 Drug 3 Placebo Subj.1 Subj. 2 Subj. 3 …. ….. Drug 1 Drug 2 Drug 3 Placebo Group 1 Group2 Group3 Group4 Assumptions Homogeneity of Variance Normality Independence of observations Assumptions Homogeneity of Variance Homogeneity of Correlations Normality

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

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

* * y = y = + + + Between Subjects Within subjects p1 p2 p3 p4 p5 b1 Drug 1 Drug 2 Drug3 Placebo p1 p2 p3 p4 p5 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 y = * 11 21 31 41 51 12 . 44 54 + b1 b2 b3 b4 b0 1 0 0 0 1 1 0 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 0 1 1 11 21 31 41 51 12 . 44 54 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 b1 b2 b3 b4 b0 y = * + +

Between Subjects Within subjects 1 1 1 2 3 4 Drug 1 Drug 2 Drug3 Placebo 1 1

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

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

Repeated Measurement ANOVA in SPM Pooled errors One way ANOVA = 1st level betas 2nd level + subjects effects Partitioned errors Two way ANOVA = 1st level differential effects between levels of a factors for main effects differences of differential effects for interactions 2nd level (T-test for 2x2 ANOVA F-test for 3x3 ANOVA)

What contrast to take from 1st level? Two way ANOVA (2*2) with repeated measured Factor A Fearful Neutral Implicit Explicit Factor 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 semantic perception Imagery Picture Words Sounds Factor B