Repeated ANOVA

Outline When to use a repeated ANOVA How variability is partitioned Interpretation of the F-ratio How to compute & interpret one-way ANOVA Post Hoc Tests

Hypotheses Null hypothesis: Ho:  1 =  2 =  3 = …=  k Researcher Hypothesis: H1: At least one population mean is different from the others

F-ratio For Repeated Measures ANOVA: The repeated measures design removes individual differences from both because the same people are in all treatments.

Partitioning Variability First Stage: –Compute SS Total –SS Total partitioned into between- and within- treatments Second Stage: –Compute variability between subjects (SS Between Subjects)

Total variability (SS total ) Between- groups variability (SS between ) Within- groups variability (SS within ) Between- subjects variability (SS btwn Ss ) Error variability (Ss error/residual ) Used to calculate numerator of F Used to calculate denominator of F

Formulas for Repeated- Measures ANOVA

Critical Value & F notation Use the F distribution table DF of Numerator: df between (k-1) DF of Denominator: df error/residual (N-k)-(n-1) F notation F(df between, df error ) = F score, p < 0.05

Calculating ANOVA by hand General Notation: – –n = number of scores in the group/treatment condition – –T = sum of the scores in a group/treatment condition – –k = number of treatments; number of levels of the IV/factor – –N = total number of scores in the study (n x k)

Calculating ANOVA by hand General Notation: – –G = sum of all the scores in the total study (ΣT) – –Σx 2 = sum of the squared values of each of the scores in the total study One new notational symbol for person total/ participant total: – –S = sum of the scores across treatments for each person

Calculating ANOVA by hand Example: Effects of label information on perceived quality of wine –French –Italian –Canadian DV: perceived quality of wine (1 to 20, with higher scores indicating better taste)

The Data…. ParticipantFrenchItalianCanadian

Step 1: State Hypotheses Ho: H1:

Step 2: Compute df df total =N-1 df btwn =k-1 df within =N-K df error = (N-K) – (n-1)

Step 3: Determine F-critical  =.05 df error = df between = F critical =

Step 4: Calculate SS SS TOTAL =  2 – G 2 N

Step 4: Calculate SS SS BETWEEN =  T 2 – G 2 n N n N

Step 4: Calculate SS SS within =  SS inside each treatment

Step 4: Calculate SS

Step 5: Calculate MS MS BETWEEN = SS BETWEEN df BETWEEN df BETWEEN MS error = SS error df error df error

Step 7: Summary Table SourceSSdfMSF Between Within Subjects Error Total

Step 9: Statistical Decision F notation: F (df between, df error ) = F score, p < 0.05

Strength of relationship Proportion of variance accounted for (  2 )

Tukey’s HSD: Honestly Significant Difference q= studentized range statistic MS error = error term n=number of scores in each treatment We use MS error instead of MS within

Assumptions for One-Way ANOVA 1. The observations within each treatment must be independent. 2.The population distribution within each treatment must be normally distributed. 3.The variances of the population distributions for each treatment should be equivalent. 4.Homogeneity of Covariance