Repeated-Measures ANOVA

1. Repeated-Measures ANOVA

2. Old and New • One-Way ANOVA looks at differences between different samples exposed to different manipulations (or different samples that may come from different groups) within a single factor. • That is, 1 factor, k levels  k separate groups are compared. • NEW: Often, we can answer the same research questions looking at just 1 sample that is exposed to different manipulations within a factor. • That is, 1 factor, k levels  1 sample is compared across k conditions.

3. Multiple measurements for each sample. • That is, a single sample is measured on the same dependent variable once for each condition (level) within a factor. • Investigate development over time • (Quazi-Independent: time 1, time 2, time 3) • Chart learning (manipulate different levels of practice) • (Independent: 1 hour, 4 hours, 7 hours) • Compare different priming effects in a LDT • (Independent: Forward, backward, no-prime, non-word) • Simply examine performance under different conditions with the same individuals. • (Independent: suspect, equal, detective, audio format)

4. Extending t-tests T-test ANOVA cousin • Comparing two independent samples? • Independent-samples t-test! • Comparing the dependent measures of a single sample under two different conditions? • Related- (or dependent- or paired-) sample t-test! • Comparing more than two independent samples within a single factor? • One-way ANOVA! • Comparing the dependent measures of a single sample under more than two different conditions? • Repeated-Measures ANOVA!

5. R-M ANOVA • Like the Related-samples t, repeated measures ANOVA is more powerful because we eliminate individual differences from the equation. • In a One-way ANOVA, the F-ratio is calculated using the variance from three sources: • F = Treatment(group) Effect + Individual differences + Experimenter error/Individual differences + Experimenter error. •  The denominator represents “random” error and we do not know how much was from ID and EE. • This is error we expect from chance.

6. Why R-M ANOVA is COOL… • With a Repeated-Measure ANOVA, we can measure and eliminate the variability due to individual differences!!!! • So, the F ratio is conceptually calculated using the variance from two sources: • F = Treatment(group) Effect + Experimenter error/ Experimenter error. • The denominator represents truly random error that we cannot directly measure…the leftovers. • This is error we expect just from chance. • What does this mean for our F-value?

7. R-M vs. One-Way: Power • All else being equal, Repeated-Measures ANOVAs will be more powerful because they have a smaller error term  bigger Fs. • Let me demonstrate, if you will. • Assume treatment variance = 10, experimental-error variance = 1, and individual difference variance = 1000. •  In a One-Way ANOVA, F conceptually = (10 + 1 + 1000)/(1 + 1000) = 1.01 •  In a Repeated-Measures ANOVA, F conceptually = (10 + 1)/(1) = 11

8. What is a R-M ANOVA doing? • Again, SStotal = SSbetween + Sswithin • Difference: We break the SSwithin into two parts: • SSwithin/error = SSsubjects/individual differences + SSerror •  To get SSerrorwe subtract SSsubjectsfrom SSwithin/error. • This time, we truly have SSerror, or random variability. • Other than breaking the SSwithin into two components and subtracting out SSsubjects, repeated measures ANOVA is similar to One-Way ANOVA.

9. Let’s learn through example. • A Priming Experiment! • 10 participants engage in an LDT, within which they are exposed to 4 different types of word pairs. RT to click Word or Non-Word recorded and averaged for each pair type. • Forward-Prime pairs (e.g., baby-stork) • Backward-Prime pairs (e.g., stork-baby) • Unrelated Pairs (e.g., glass-apple) • Non-Word Pairs (e.g., door-blug)

10. Hypotheses • For the overall RM ANOVA: • Ho: µf = µb = µu= µn • Ha: At least one treatment mean is different from another. • Specifically: • Ho: µf < µb • Ho: µf and µb < µu and µn • Ho: µu < µn

11. The Data

12. Formula for SStotal • Ian’s SSt = Remember, this is the sum of the squared deviations from the grand mean. • So, SStotal = 12.39– (19.72/40) • = 12.39 – 9.70225 • = 2.68775 • Importantly, SStotal = SSwithin/error + SSbetween

13. The Data

14. Within-Groups Sum of Squares: SSwithin/error • SSwithin/error = the sum of each SS with each group/condition. • Measures variability within each condition, then adds them together. • So, SSwithin/error = • (1.36– ([3.4]2/10)) + (.58– ([2.2]2/10)) + (3.82– ([6]2/10) + (6.63– ([8.1]2/10) • = (1.36– 1.156) + (.58– .484) + (3.82– 3.6) + (6.63– 6.561) • = .204+ .096+ .22+ .069= .589

15. The Data

16. Breaking up SSwithin/error • We must find SSSUBJECTS and subtract that from total within variance to get SSERROR •  SSSUBJECTS = • K is generic for the number of conditions, as usual. • SSSUBJECTS = (1.42/4 +2.52/4 +2.12/4 +2.32/4 +2.62/4 +1.92/4 +1.42/4 +1.82/4 +1.62/4 +2.12/4 +) – 19.72/40 = .49+1.5625+1.1025+1.3225+1.69+.9025+.49+.81+.64+1.1025) -9.70225 = 10.1125-9.70225 • =.41025

17. Now for SSerror • SSerror= SSwithin/error – SSsubjects • SSerror= .589 - .41025 = .17875 or .179 • Weeeeeeeeee! We have pure randomness!

18. The Data

19. SSbetween-group: • The (same) Formula: • So = SSbetween = • [((3.4)2/10) + ((2.2)2/10) + ((6)2/10) + ((8.1)2/10)] – 19.72/40 • = (1.156+ .484+ 3.6+ 6.561) – 9.70225 • = 11.801 – 9.70225 = 2.09875 or 2.099

20. Getting Variance from SS • Need? …DEGREES OF FREEDOM! • K = 4 • Ntotal = total number of scores = 40 (4x10) • DfTOTAL= Ntotal– 1 = 39 • DfBETWEEN/GROUP = k – 1 = 3 • Dfwithin/error= N – K = 40 – 4 = 36 • Dfsubjects= s-1 = 10-1 (where s is the # of subjects) = 9 • Dferror= (k-1)(s-1) = (3)(9) = 27 OR Dfwithin/error - Dfsubjects= 36 – 9 = 27

21. Mean Squared (deviations from the mean) • We want to find the average squared deviations from the mean for each type of variability. • To get an average, you divide by n in some form (or k which is n of groups) and do a little correction with “-1.” • That is, you use df. • MSbetween/group = = 2.099/3 = .7 • MSwithin/error = = .179/27 = .007

22. How do we interpret these MS • MS error is an estimate of population variance. • Or, variability due to ___________?

23. F? • F = = .7/.007 = 105.671 • (looks like it should be 100, but there were rounding issues due to very small numbers. • OK, what is Fcrit? Do we reject the Null??

24. Pros and Cons • Advantages • Each participant serves as own control. • Do not need as many participants as one-way. Why? More power, smaller error term. • Great for investigating trends and changes. • Disadvantages • Practice effects (learning) • Carry-over effects (bias) • Demand characteristics (more exposure, more time to think about meaning of the experiment). • Control • Counterbalancing • Time (greater spacing…but still have implicit memory). • Cover Stories

25. Sphericity • Levels of our IV are not independent • same participants are in each level (condition). • Our conditions are dependent, or related. • We want to make sure all conditions are equally related to one another, or equally dependent. • We look at the variance of the differences between every pair of conditions, and assume these variances are the same. • If these variances are equal, we have Sphericity

26. More Sphericity • Testing for Sphericity • Mauchly’s test • Significant, no sphericity, NS… Sphericity! • If no sphericity, we must engage in a correction of the F-ratio. Actually, we alter the degrees of freedom associated with the F-ratio. • Four types of correction (see book) • Estimate sphericity from 0 (no sphericity) to 1 (sphericity) • Greenhouse-Geiser (1/k-1) • Huynh-Feldt • MANOVA (assumes measures are independent) • non-parametric, rank-based Friedman test (one-factor only)

27. Symmetry

28. Effect Sizes • The issue is not entirely settled. Still some debate and uncertainty on how to best measure effect sizes given the different possible error terms. • ω2 = See book for equation.

29. Specific tests • Can use Tukey post-hoc for exploration • Can use planned comparisons if you have a priori predictions. • Sphericity not an issue 

30. Contrast Formula Same as one way, except error term is different

31. Contrasts • Some in SPSS: • Difference: Each level of a factor is compared to the mean of the previous level • Helmert: Each level of a factor is compared to the mean of the next level • Polynomial: orthogonal polynomial contrasts • Simple: Each level of a factor is compared to the last level • Specific: • GLM forward backward unrelatenonword • /WSFACTOR = prime 4 special (1 1 1 1 • -1 -1 1 1 • -1 1 0 0 • 0 0 -1 1)

32. 2+ Within Factors • Set up. • Have participants run on tread mill for 30min. • Within-subject factors: • Factor A • Measure Fatigue every 10min, 3 time points. • Factor B • Do this once after drinking water, and again (different day) after drinking new sports drink. • 3 (time) x 2 (drink) within-subject design.

33. Much is the same, much different… • We have 2 factors (A and B) and an interaction between A and B. • These are within-subjects factors • All participants go through all the levels of each factor. • Again, we will want to find SS for the factors and interaction, and eventually the respective MS as well. • Again, this will be very similar to a one-way ANOVA. • Like a 1-factor RM ANOVA, we will also compute SSsubject so we can find SSerror.

34. What is very different? • Again, we can parse up SS w/e into SSsubject and SSerror. • NEW: We will do this for each F we calculate. • For each F, we will calculate: • SS Effect ; SS Subject(within that effect) ; and SS Error • What ISSSErrorfor each effect? • (We will follow the logic of factorial ANOVAS)

35. What are the SSErrors now? • Lets start with main effects. • Looking at Factor A, we have • Variability due to Factor A: SSFactor A • Variability due to individual differences. • How do we measure that? • By looking at the variability due to a main effect of Participant (i.e., Subject): SSSubject (within Factor A) • Variability just due to error. • How do we calculate that!?!?!? • Think about the idea that we actually have 2 factors here, Factor A and Subject.

36. The Error is in the INTERACTIONS with “Subject.” • For FFactorA (Time) • SSAS is the overall variability looking at Factor A and Subjects in Factor A (collapsing across Drink). • To find SSerrorfor the FFactor A (Time) Calculate: • SSAS; SSFactorA; SSSubject(within Factor A) • SSerroris: SSAS - (SSFactorA+ SSSubject (within Factor A)) • That is, SSerrorfor FFactor A (Time)is SS A*subj!!!!!! • Which measures variability within factor A due to the different participants (i.e., error)

37. The Same for Factor B. • For FFactorB (Drink) • SSAS is the overall variability looking at Factor Band Subjects in Factor B(collapsing across Time). • To find SSerrorfor the FFactorB (Drink) Calculate: • SSBS; SSFactorB; SSSubject (within Factor B) • SSerroris: SSBS- (SSFactorB+ SSSubject (within Factor B)) • That is, SSerrorfor FFactorB (Drink)is SS B*subj!!!!!! • SS B*subj: Which measures variability within factor B due to the different participants (i.e., error)

38. Similar for AxB Interaction • For FAxB • SSBetween groupsis the overall variability due to Factor A, Factor B,and Subjects. • To find SSerrorfor the FAxBCalculate: • SSBetween; SSFactorA ; SSFactorB ; SSSubject • SSerroris: SSBetween- (SSFactorA + SSFactorB + SSSubject) • That is, SSerrorfor FAxBis SS A*B*subj!!!!!! • SS A*B*subj: measures variability within the AxB interaction due to the different participants (i.e., error)

39. We are finely chopping SSW/E SS W/E SSSub (A) SSSub (B) SSSub (AxB)

40. Getting to F • Factor A (Time) • SSA = 91.2; dfA = (kA – 1) = 3 – 1 = 2 • SSError(A*S) = 16.467; dfError(A*S) = (kA – 1)(s – 1) = (3-1)(10-1) = 18 • So, MSA = 91.2/ 2 = 45.6 • So, MSError(A*S)= 16.467/ 18 = .915 • FA = 45.6/.915 = 49.846

41. Snapshot of other Fs • Factor B (Drink) • dfB= (kB– 1) = 2 – 1 = 1 • dfError(B*S) = (kB– 1)(s – 1) = (2-1)(10-1) = 9 • AxB (Time x Drink) • dfAxB= (kA– 1)(kB – 1)= 2 x 1= 2 • dfError(A*B*S) = (kA – 1)(kB – 1)(s – 1) = 2 x 1 x 9 = 18 • Use SS’s to calculate the respective MSeffect and MSerror for the other main effect and the Interaction F-values.