Table of Contents
Week 2 - factorial between-Ps ANOVA I: Omnibus tests ................................................... 3
One-way ANOVA ........................................................................................................... 3
Two-way ANOVA ........................................................................................................... 5
Week 3 - Factorial Between-Participants ANOVA II: Following Up Significant Effects and
Effect Sizes ....................................................................................................................... 8
Follow-up main effects & interactions ............................................................................. 8
Effect sizes .................................................................................................................. 11
Week 4 - Factorial Between-Ps ANOVA III: Higher-Order ANOVA .................................. 13
Manipulation check ...................................................................................................... 13
Higher-Order Factorial Designs.................................................................................... 13
Omnibus tests in 3-way factorial ANOVA ..................................................................... 16
Overview of follow-up tests in 3-way ANOVA ............................................................... 17
Flow chart .................................................................................................................... 20
Week 5 - Power Analysis and Blocking Designs .............................................................. 22
Statistical decisions: ..................................................................................................... 22
Power .......................................................................................................................... 22
Blocking design ............................................................................................................ 26
Blocking VS Experimental designs ............................................................................... 27
Week 6 - Correlation, Bivariate Regression, and Analysis of Covariance (ANCOVA) ...... 29
Experiments vs Correlational Designs.......................................................................... 29
Covariance and Correlation ......................................................................................... 30
Covariance................................................................................................................... 30
Correlation (Pearson’s r / bivariate correlation) ............................................................ 30
Bivariate regression ..................................................................................................... 31
ANCOVA – Analysis of Covariance .............................................................................. 33
ANCOVA vs Blocking ................................................................................................... 34
Assumptions of ANCOVA ............................................................................................ 35
Week 7 – Mid semester break (no lecture) ...................................................................... 36
,Week 8 – Standard Multiple Regression and Hierarchical Multiple Regression ............... 36
Standard multiple regression........................................................................................ 36
Partial Correlation (pr²)................................................................................................. 38
Semi-partial Correlation (sr²) ........................................................................................ 38
Zero-order, Partial, and Semi-partial Correlations ........................................................ 39
Hierarchical multiple regression: .................................................................................. 41
Week 9: Moderated Multiple Regression ......................................................................... 44
Multicollinearity and Singularity .................................................................................... 45
Moderated Multiple Regression.................................................................................... 46
Week 10 – ANZAC Day (no lecture) ................................................................................ 49
Week 11 – Moderation, Mediation and Indirect Effects .................................................... 49
Mediation ..................................................................................................................... 49
Bootstrapping mediation .............................................................................................. 50
Suppression model ...................................................................................................... 50
Mediation VS Moderation ............................................................................................. 51
Week 12 – Within-Participants ANOVA ........................................................................... 53
Introduction to within-participants designs .................................................................... 53
One-Way Within-Participants ANOVA .......................................................................... 54
Two-way within-participants ANOVA ............................................................................ 55
Mixed-model Approach ................................................................................................ 56
Sphericity: Problem and Solutions ............................................................................... 57
Epsilon () adjustments................................................................................................ 58
Multivariate analysis of variance (MANOVA) ................................................................ 58
Pros and cons of within-participants designs:............................................................... 59
, Week 2 - factorial between-Ps ANOVA I: Omnibus tests
Important concepts:
✓ Three omnibus tests (2 main effects and interactions)
✓ Sources of variances in one-way & two-way ANOVA
✓ how F-ratio is calculated
✓ structural model of one-way & two-way ANOVA
Analysis of Variance (ANOVA)
- it is the partitioning of variables
- it compares 2/more conditions to test an association between variables
- it is an Omnibus test/technique
- Test statistic = F-ratio
Variance:
The dispersion or spread of scores around a point of central tendency (the mean)
—> how spread out are the scores when compared to the mean?
Error variance: Treatment variance:
- Due to random/unmeasured - Systematic differences due to the
influences manipulation of IV
- X be explained - can be explained
- Within-group variance - Between-group variance
One-way ANOVA
Source of variance:
Total variation
Between-group variation Within-group variation
Systematic variance due to membership in Error variance due to random chance /
different groups/treatment X unmeasured influences
n ∑(̅̅̅̅ ̅. )²
𝑋𝑗 + 𝑋 ∑(𝑋𝑖𝑗 + 𝑋̅𝑗 )²
, Hypothesis testing for one-way ANOVA
Hypotheses for 2 means
- Statistical Hypotheses
- 𝐻0 : 𝜇1 = 𝜇2
- 𝐻1 : 𝜇𝑗 ≠ 𝜇.
- Conceptual Hypotheses
- Null hypothesis: no differences between treatment means
- Alternative hypothesis: difference between treatment means
Hypotheses for 3+ means
- Statistical Hypotheses
- 𝐻0 : 𝜇1 = 𝜇2 = 𝜇3 = ⋯ 𝜇𝑗 𝜇𝑗 = mean of
- 𝐻1 : 𝜇𝑗 ≠ 𝜇. group j
- Conceptual Hypotheses 𝜇. = grand mean
- Null hypothesis: no differences between treatment means
- Alternative hypothesis: difference between treatment means
How to Calculate F-ratio?
Sum of Squares (SS):
- A measure of variability
Treatment sum of squares: 𝑆𝑆𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 or 𝑆𝑆𝑚𝑜𝑑𝑒𝑙 or 𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛
- between-groups variability
- How much each group mean varies from the grand mean
Error sum of squares: 𝑆𝑆𝑒𝑟𝑟𝑜𝑟 or 𝑆𝑆𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 or 𝑆𝑆𝑤𝑖𝑡ℎ𝑖𝑛
- within-groups variability
- How much do individual scores in each group vary from that group’s mean
Mean squares (MS):
- An index/measure of variability among sample statistics
- We calculate mean squares to estimate:
- Between-groups variance: 𝑀𝑆𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡
- Within-groups variance: 𝑀𝑆𝑒𝑟𝑟𝑜𝑟
- is roughly equivalent to 𝑠 2 (sample variance) in a t-test
- is a good estimate of 𝜎𝑒2 (population variance)
Week 2 - factorial between-Ps ANOVA I: Omnibus tests ................................................... 3
One-way ANOVA ........................................................................................................... 3
Two-way ANOVA ........................................................................................................... 5
Week 3 - Factorial Between-Participants ANOVA II: Following Up Significant Effects and
Effect Sizes ....................................................................................................................... 8
Follow-up main effects & interactions ............................................................................. 8
Effect sizes .................................................................................................................. 11
Week 4 - Factorial Between-Ps ANOVA III: Higher-Order ANOVA .................................. 13
Manipulation check ...................................................................................................... 13
Higher-Order Factorial Designs.................................................................................... 13
Omnibus tests in 3-way factorial ANOVA ..................................................................... 16
Overview of follow-up tests in 3-way ANOVA ............................................................... 17
Flow chart .................................................................................................................... 20
Week 5 - Power Analysis and Blocking Designs .............................................................. 22
Statistical decisions: ..................................................................................................... 22
Power .......................................................................................................................... 22
Blocking design ............................................................................................................ 26
Blocking VS Experimental designs ............................................................................... 27
Week 6 - Correlation, Bivariate Regression, and Analysis of Covariance (ANCOVA) ...... 29
Experiments vs Correlational Designs.......................................................................... 29
Covariance and Correlation ......................................................................................... 30
Covariance................................................................................................................... 30
Correlation (Pearson’s r / bivariate correlation) ............................................................ 30
Bivariate regression ..................................................................................................... 31
ANCOVA – Analysis of Covariance .............................................................................. 33
ANCOVA vs Blocking ................................................................................................... 34
Assumptions of ANCOVA ............................................................................................ 35
Week 7 – Mid semester break (no lecture) ...................................................................... 36
,Week 8 – Standard Multiple Regression and Hierarchical Multiple Regression ............... 36
Standard multiple regression........................................................................................ 36
Partial Correlation (pr²)................................................................................................. 38
Semi-partial Correlation (sr²) ........................................................................................ 38
Zero-order, Partial, and Semi-partial Correlations ........................................................ 39
Hierarchical multiple regression: .................................................................................. 41
Week 9: Moderated Multiple Regression ......................................................................... 44
Multicollinearity and Singularity .................................................................................... 45
Moderated Multiple Regression.................................................................................... 46
Week 10 – ANZAC Day (no lecture) ................................................................................ 49
Week 11 – Moderation, Mediation and Indirect Effects .................................................... 49
Mediation ..................................................................................................................... 49
Bootstrapping mediation .............................................................................................. 50
Suppression model ...................................................................................................... 50
Mediation VS Moderation ............................................................................................. 51
Week 12 – Within-Participants ANOVA ........................................................................... 53
Introduction to within-participants designs .................................................................... 53
One-Way Within-Participants ANOVA .......................................................................... 54
Two-way within-participants ANOVA ............................................................................ 55
Mixed-model Approach ................................................................................................ 56
Sphericity: Problem and Solutions ............................................................................... 57
Epsilon () adjustments................................................................................................ 58
Multivariate analysis of variance (MANOVA) ................................................................ 58
Pros and cons of within-participants designs:............................................................... 59
, Week 2 - factorial between-Ps ANOVA I: Omnibus tests
Important concepts:
✓ Three omnibus tests (2 main effects and interactions)
✓ Sources of variances in one-way & two-way ANOVA
✓ how F-ratio is calculated
✓ structural model of one-way & two-way ANOVA
Analysis of Variance (ANOVA)
- it is the partitioning of variables
- it compares 2/more conditions to test an association between variables
- it is an Omnibus test/technique
- Test statistic = F-ratio
Variance:
The dispersion or spread of scores around a point of central tendency (the mean)
—> how spread out are the scores when compared to the mean?
Error variance: Treatment variance:
- Due to random/unmeasured - Systematic differences due to the
influences manipulation of IV
- X be explained - can be explained
- Within-group variance - Between-group variance
One-way ANOVA
Source of variance:
Total variation
Between-group variation Within-group variation
Systematic variance due to membership in Error variance due to random chance /
different groups/treatment X unmeasured influences
n ∑(̅̅̅̅ ̅. )²
𝑋𝑗 + 𝑋 ∑(𝑋𝑖𝑗 + 𝑋̅𝑗 )²
, Hypothesis testing for one-way ANOVA
Hypotheses for 2 means
- Statistical Hypotheses
- 𝐻0 : 𝜇1 = 𝜇2
- 𝐻1 : 𝜇𝑗 ≠ 𝜇.
- Conceptual Hypotheses
- Null hypothesis: no differences between treatment means
- Alternative hypothesis: difference between treatment means
Hypotheses for 3+ means
- Statistical Hypotheses
- 𝐻0 : 𝜇1 = 𝜇2 = 𝜇3 = ⋯ 𝜇𝑗 𝜇𝑗 = mean of
- 𝐻1 : 𝜇𝑗 ≠ 𝜇. group j
- Conceptual Hypotheses 𝜇. = grand mean
- Null hypothesis: no differences between treatment means
- Alternative hypothesis: difference between treatment means
How to Calculate F-ratio?
Sum of Squares (SS):
- A measure of variability
Treatment sum of squares: 𝑆𝑆𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡 or 𝑆𝑆𝑚𝑜𝑑𝑒𝑙 or 𝑆𝑆𝑏𝑒𝑡𝑤𝑒𝑒𝑛
- between-groups variability
- How much each group mean varies from the grand mean
Error sum of squares: 𝑆𝑆𝑒𝑟𝑟𝑜𝑟 or 𝑆𝑆𝑟𝑒𝑠𝑖𝑑𝑢𝑎𝑙 or 𝑆𝑆𝑤𝑖𝑡ℎ𝑖𝑛
- within-groups variability
- How much do individual scores in each group vary from that group’s mean
Mean squares (MS):
- An index/measure of variability among sample statistics
- We calculate mean squares to estimate:
- Between-groups variance: 𝑀𝑆𝑡𝑟𝑒𝑎𝑡𝑚𝑒𝑛𝑡
- Within-groups variance: 𝑀𝑆𝑒𝑟𝑟𝑜𝑟
- is roughly equivalent to 𝑠 2 (sample variance) in a t-test
- is a good estimate of 𝜎𝑒2 (population variance)
