Analysis of variance John W. Worley AudioGroup, WCL Department of Electrical and Computer Engineering University of Patras, Greece

Slide 2 of 25 Experiment Hypothesis Null Hyp. (H 0 )Exp. Hyp. (H 1 ) I.V.D.V. Data NominalOrdinalIntervalRatio Analysis Modify Hyp. Type I Type II

Slide 3 of 25 Data Analysis  Descriptive Mean. Standard error the mean (SEM).  Inferential statistics t-test. –Related-means t-test. –Independent-means t-test.  Univariate Analysis of Variance (ANOVA).  Multivariate ANOVA (MANOVA).

Slide 4 of 25 Variance µ1µ1 µ2µ2 Small Variance

Slide 5 of 25 Variance µ1µ1 µ2µ2 Large Variance

Slide 6 of 25 µ1µ1 µ2µ2 P < 0.05 Small Variance

Slide 7 of 25 µ1µ1 µ2µ2 P > 0.05 Large Variance

Slide 8 of 25 Errors in statistical decisions  Type I error: Rejecting Null Hyp. when it’s true.  Type II error: Retaining Null Hyp. when its false.

Slide 9 of 25 Factorial design: Definitions  Factor is a categorical predictor variable e.g. A treatment.  Level is amount of the factor. e.g. the amount of treatment.  One-way ANOVA One factor, multiple levels.  Two-way ANOVA Two factors, different levels within the factors.

Slide 10 of 25 ANOVA: Assumptions  Normal distributed data. Histogram Kolmogorov-Smirnov test Shapiro-Wilk test  Interval or ratio data.  Independence.  Homogeneity of variance. Levens test

Slide 11 of 25 Mnemonic aid YESNO Levels of Factor Mnemonic n1n1 n2n2 n Subjects One-way, Between-Subjects Design: - One between groups factor (with 2 levels).

Slide 12 of 25 Memory recall with practise Levels of Factor Time Day-1Day-2Day-3 One-way, Within-subjects Design: - One within groups factor (with 3 levels) n1n1 n2n2 n Subjects

Slide 13 of 25 Mnemonic aid YES NO Levels of Factors Mnemonic Levels of Factors Time Day-1Day-2Day-3 n1n1 n2n2 n Subjects Mixed Design: - One between groups factor (with 2 levels). - One within groups factor (with 3 levels) Day-1Day-2Day-3

Slide 14 of 25 Variable Interaction  Story recall is improved by mnemonic aid and practice, with no interaction.  An interaction, practice has a greater effect upon recall with a mnemonic aid.

Slide 15 of 25 Post-hoc tests  Least-significant difference (LSD) pairwise comparison. No Type I error control.  Studentized Newman-Keuls (SNK). Liberal, no Type I error control.  Bonferroni Method. Controls Type I error Good when comparisons small.  Tukey Test Controls Type I error Good when comparisons large.

Slide 16 of 25 MANOVA Memory recall with practise Levels of Factor Time Day-1Day-2Day-3 n1n1 n2n2 n Subjects Recall Comprehension Confidence

Slide 17 of 25 Multivariate analysis of variance (MANOVA)  2-stage test  For more than one DV.  Avoids Type I error, with multiple ANOVA’s.  Detects differences among a combination of variables

Slide 18 of 25 MANOVA: Pre-requisites.  What DV’s: Correlations*. Discriminate function variates.  Assumptions:  Independence.  Random sampling.  Interval data  Multivariate normality  Equality of covariance F Leven’s test F Box’s test *Cole et al (1994) Psych. Bulletin, 115(3),

Slide 19 of 25 MANOVA: Output  Pillai-Bartlett Trace (V).  Wilk’s Lamda (Λ).  Hotelling’s T 2.  Roy’s Largest Root.

Slide 20 of 25 Distributions platykurtic leptokurtic

Slide 21 of 25 MANOVA: Output  Pillai-Bartlett Trace (V).  Wilk’s Lamda (Λ).  Hotelling’s T 2.  Roy’s Largest Root.

Slide 22 of 25 MANOVA: Follow-up analysis  If MANOVA sig. F ANOVA Type I error –Bonferroni correction F Discriminant analysis

Slide 23 of 25 Practical Considerations  Controls  Standardisation. F Instructions  Practise/fatigue effects F Counterbalance conditions. F Randomise trial order.  Within- or Between-subjects design?  For MANOVA:  Choose DV’s theoretically  Do Follow-up analysis.  Experimentwise errors

Slide 24 of 25 Conclusions  Identify your DV type.  Within- or Between-subjects design.  Control for everything.  Report descriptive and inferential statistics.  ANOVA is highly useful.  MANOVA is logical progression. F Follow-up analysis. F Bonferroni correction.

AudioGroup, WCL Department of Electrical and Computer Engineering University of Patras, Greece