Lecture 11: One Way ANOVA Repeated Measures

1. Lecture 11:One Way ANOVARepeated Measures Laura McAvinue School of Psychology Trinity College Dublin

2. Analysis of Variance One way ANOVA Factorial ANOVA More than One Independent Variable One Independent Variable Between subjects Repeated measures / Within subjects Two way Three way Four way Different participants Same participants

3. One Way Repeated Measures ANOVA • A statistical technique for testing for differences between the means of several groups • Groups are related in some way • Usually same participants in each group • Similar to the paired samples T-Test • But no restriction on the number of groups

4. A few examples… • What are the independent & dependent variables in each of the following studies? • Longitudinal study of annual growth of children over the first five years of life • The effects of three levels of methylphenidate on reaction time performance of the same participants • The measurement of depression before therapy, after therapy and at a nine month follow up • Same group of people is tested under different conditions or at different times

5. Repeated Measures ANOVA Data points in each group are related Between Subjects ANOVA Data points in each group are unrelated

6. Population draw one sample Did the manipulation alter the sample to such an extent that the sample represents another population during at least one condition? Manipulate the sample Cond1 Cond 2 Cond 3 Number of trials required to pass test µ1 µ2 µ3 measure effect of IV on a DV

7. Null Hypothesis Significance Testing • Step 1: Halt – At least one condition mean is significantly different from the others • Step 2: Ho – All of the condition means are equal • Step 3: Collect your data • Step 4: Run the ANOVA • Step 5: Obtain the F statistic and associated p value • Step 6: Decide whether to reject or fail to reject Ho on the basis of the p value

8. Total Variance, Between Subjects ANOVA Between Groups Within Groups Variance between the means Variance within each group Captures the effect of the manipulation / treatment Variance due to random error / chance The bigger the effect of your manipulation, the bigger the Between Groups Variance

9. F Ratio Compares the variance due to the treatment / manipulation to the variance due to random error / chance MS between groups MS within groups Between Groups Variance Within Groups Variance Treatment Effect + Differences due to chance / error Differences due to chance / error If the treatment / manipulation has an effect, then… MS between groups > MS within groups F > 1

10. Total Variance, Between Subjects ANOVA Between Groups Within Groups Variance due to random error / chance Variance due to individual differences between subjects Variance due to random error

11. Repeated Measures Design • Powerful • Error Variance • Variance between participants • Unexplained Random Error • Because we test participants repeatedly with this design, we can quantify this variance between participants • We can remove this variance from the error term • Making the error term smaller, • Making the F ratio bigger, • Making it easier to obtain a statistically significant F

12. Four Kinds of Variance • Total Variance • SStotal • ∑ (xij - Grand Mean )2 • Variance due to the manipulation • SStreatment • n∑ (Group meanj - Grand Mean )2

13. Four Kinds of Variance • Variance due to individual differences between participants • SSparticipants • No. of conditions ∑ (Participant mean - Grand Mean )2 • Variance due to random error • SSrandom error = SStotal - SStreatment - SSparticipants MS Treatment MS Random error F

14. Condition / Treatment

15. Condition / Treatment SStotal : ∑ (xij - Grand Mean )2 ∑(2 – 18.67) 2 + (10 – 18.67)2 +…+ (34 – 18.67) 2

16. Condition / Treatment SStreatment : n ∑ (Treatment Mean - Grand Mean )2 4 ∑(16 – 18.67) 2 + (19 – 18.67)2 + (21 – 18.67) 2

17. Condition / Treatment SSparticipants : no. of conditions ∑ (Participant Mean - Grand Mean )2 3 ∑(4.33 – 18.67) 2 + (11.67 – 18.67)2 +…+ (31.67 – 18.67) 2

18. Condition / Treatment SSrandom error : SStotal - SStreatment - SSparticipants

19. ANOVA table N = Total no. of observations, n = no. of people in each condition, K = no. of conditions

20. Post hoc testing • Significant F value • At least one condition mean is significantly different from the others • But which one? • Posthoc tests • Bonferroni • Tukey • Sidak

21. A few assumptions… Data in each group should be… • Interval scale • Normally distributed • Histograms, box plots • Homogeneity of variance • Variance within each condition should be roughly equal

22. Extra Assumption • Sphericity • Variance of the differences between conditions is the same • Variance t1-t2 Variance t1-t3 Variance t2-t3 • Correlation between pairs of groups is the same • corr t1-t2 corr t1-t3 corrt2-t3

23. Testing Sphericity • Mauchly’s Test • If p > .05, assume equality of variances • If p < .05, then the data fails to meet the assumption of sphericity • Need to use one of the correction factors • E.g. Greenhouse-Geisser • A more conservative test, using different dfs

24. Advantages & Disadvantages of Repeated Measures Designs • Major Advantage • Individual differences between participants can be removed from the analysis • More power • Disadvantage • Risk of carry-over effects from one condition to the next • Practice effects • Reduce by counterbalancing