Multivariate Statistics Final Exam

when to use ANOVA

differences between 3 or more groups

• DV; Interval/ratio

• IV; nominal (categorical)

between-subjects design

Participants are divided into different groups and each group receives a different treatment

determining causality in between-subjects design

• random sampling

• random assignment to conditions

ANOVA Hypotheses

H₀: u₁ = u₂ = u₃

H1: not all u_i are equal

reason for post-hoc testing

ANOVA tells you whether at least one mean differs, but not which groups

ANOVA as analysis of variance

compares the between-group variance with the within-group variance

between group variance

variance explained by the model

within-group variance

unexplained variance inside each group

ANOVA F-test equation

= (observed variance)/(expected variance)

= (between-groups variance)/(within-groups variance)

ANOVA F-test interpretation

• Large F —> group means differ more than expected by chance

• F = 1 —> Between-group variance = within group variance (no effect)

ANOVA Assumptions

1.) random sample

2.) Independent observation

3.) DV at least interval

4.) Normality (DV normally distributed in each group)

5.) Homogeneity or variance

Homogeneity of variances

formal rules

• largest sample <4 x smallest sample

• largest variance <10 x smallest variance

What to do when homogeneity of variances is violated

use Welch or Brown-Forsythe correction

Steps in ANOVA analysis

1.) Check assumptions

2.) Test the significance of the factor (F-test)

3.) Determine effect size

4.) conduct post-hoc tests (if >2 groups)

5.) Report results

eta squared calculation

n² = (SS_m)/(SStotal)

ANOVA effect size rules of thumb

• .01 = small

• .09 = medium

• .25 = large

what does ANOVA effect size measure?

(…) % of variance in DV explained by group differences

reason to use corrections in post-hoc

if significance level =.05 per test, the overall error rate increases

multiple testing problem calculation

significance_ew = 1 - (1-significance)^c

c = number of comparisons

Bonferroni correction

= significance / c

Turkey’s HSD

• better when there are many groups

• Controls family-wise error rate

The difference between Frequentist and Bayesian ANOVA

Bayesian uses the Bayes Factors instead of p-values

Factorial ANOVA

ANOVA with more than one factor

hypotheses in factorial ANOVA

1.) Main effect of Factor A

2.) Main effect of Factor B

3.) Interaction effect (A x B)

factorial ANOVA interaction effect

The effect of one factor depends on the level of the other factor

• lines not parallel

• effect of A changes depending on B

Steps in factorial ANOVA

1.) check assumptions

2.) test main effects and interaction

3.) if significant

• profile plot

4.) If not significant;

• rerun model without interaction

5.) calculate effect sizes

6.) conduct post-hoc if needed

7.) if interaction is significant test simple main effects

8.) report results

partial eta squared calculation

partial n² = (SSeffect)/(SSeffect + SSresidual)

significant interaction in factorial ANOVA

you cannot interpret main effects alone, look at simple main effects

simple main effects

test effects of one factor within each level of the other factor

ANCOVA

if the DV also depends on a continuous variable, it;

• compares group means

• while controlling for a continuous covariate

why a covariate in ANCOVA

increases statistical power and corrects for group differences

ANCOVA model

yi = b0 + biz + b2x

• z = group dummy (factor)

• x = covariate

ANCOVA unadjusted means

compares raw group means, but may differ on covariate

ANCOVA adjusted means

groups corrected for covariate —> “what would the group means be if both groups had the same average value on the covariate”

ANCOVA assumptions (in addition to ANOVA assumptions)

homogeneity of regression slopes

homogeneity of regression slopes

regression lines must be parallel (no interaction between factor and covariate)

testing for homogeneity of regression slopes

test if interaction term is significant

Steps in Frequentist ANCOVA

1.) check homogeneity of regression slopes

• interaction must NOT be significant

2.) Check homogeneity of variances

3.) test for main effects:

• factor

• covariate

4.) conduct post-hoc

5.) interpret regression coefficient of covariate

Repeated measures ANOVA

within-subjects design, same participants measured multiple times where observations are dependent

Assumptions of repeated measures ANOVA

1.) random sample

2.) normality of DV at each time point

3.) Sphericity

sphericity

The variance of all pairwise difference scores are equal

why sphericity is needed in repeated measures ANOVA

because it assumes equal covariance between time points a violation inflates type I error

Test for sphericity

Mauchly’s test, is significant sphericity is violated

when sphericity is violated

use a correction

epsilon <.75

Greenhouse-Geisser correction

epsilon > .75

Huyn-Feldt correction

what Greenhouse Geisser does

adjusts degrees of freedom of the p-value

mixed design ANOVA

between-subject + within-subject factor