2. within & mixed design ANOVA

within subjects

AKA. repeated measures

• type of ANOVA where the same people take part in all levels of an independent event

within groups ANOVA

• risk of carryover effects

• different assumptions made (need assumption of sphericity)

• sources of systematic and unsystematic variability are different to between groups ANOVA

• degrees of freedom differ

sphericity

• an assumption about variances

• difference between each pair of treatment levels should have equal variance

variance_A-B = Variance_A-C = Variance_B-C

• only important if a variable has 3 or more levels

• violated sphericity → increased risk for type I error

• within subjects equivalent of Levenes test

violating sphericity

• use mauchley’s test to assess equal variances

• if assumption is violated p<0.05 we make corrections to degrees of freedom

• id data was perfectly spherical our green-house geyser & Huhyn-Feidt estimates would be 1 (for this data)

• if >0.05 we have met assumption

adjusting degrees of freedom

• sphericity is corrected by multiplying original df by estimated of sphericity shown in previous table

• knock on effects for MS & F-values

• generally ok to use green-house geiser

sources of variability

total variability SST

• within SSW

  • effects of experiment SSM

  • unexplained variability/ error SSR

• between SSB

SST

• same as between groups ANOVA

• subtract score from grans mean

SSW

• manipulated IV within the person

• SSW = SS_person1 + SS_person2 etc…

• calculate individual sum of squares & add them up

SSM

same as between groups ANOVA

1. subtract group mean from grand eman & square

2. multiply by number of participants in group

3. add together

SSR

SSW- SSM

calculating degrees of freedom

main effect df (SSM) → a-1 (no levels of factor A-1)

• e.g. 3 times of day = 3-1 =2 2

error term df (SSR) → (a-1)(S-1) (main effect df x subject df)

df SSW → no. participants -1

main effect

MSM

MS - SSM / main effect df (SSM)

error

MSR

MS = SSR / error df (SSR)

F- ratio

main effect MSM / error MSR

two way within subjects ANOVA

Within (SSW)

• effects of experiment SSM

  • effects of factor A SSA

  • effects of factor B SSB

  • interaction SSAxB

• unexplained variability/ error SSR

partitioning variability requires different error terms for each main effect and interaction

Factor A

MS (main effect): SSA / dfA

MS (error): SSAxS / dfAxS

F-ratio: MSA / MSAxS

interaction

MS (main effect): SSAxB / dfAxB

MS (error): SSAxBxS / dfAxBxS

F-ratio: MSAxB / MSAxBxS

mixed design ANOVA

contains both between groups & within subjects IVs

partitioning variability in mixed design ANOVA

error term used in repeated measures & interaction effects are different

MSBxS / A = SSBxS/A / dfBxS/A

where dfBxS/A = a(b-1)(s-1)

A → between groups factor

B → within groups factor

mixed design assumptions

1. equality of error variance in between groups factors → Levenes test

2. equality of variances across different levels of within-subjects factors → Mauchley

3. equality of covariances between within-subjects factors at each between groups factor level → Box’s M test (very sensitive to deviations from normality, disregard use if group size equal)

mixed design interaction

tests of within-subjects effects

simple main effects interaction

• same as factorial ANOVA

• multivariate tests table report Pilai’s trace

• simple main effects - pairwise comparison