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Repeated Measures ANOVA Two-Factor ANOVA

Repeated Measures ANOVA Two-Factor ANOVA. Introduction to Statistics Chapter 14 Apr 16-21, 2009 Classes #25-26. Recalling between vs. within subject distinction…. Difference between?

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Repeated Measures ANOVA Two-Factor ANOVA

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  1. Repeated Measures ANOVATwo-Factor ANOVA Introduction to Statistics Chapter 14 Apr 16-21, 2009 Classes #25-26

  2. Recalling between vs. within subject distinction… • Difference between? • When we have a within subjects design (e.g., repeated measures, matched subjects), can’t use one-way ANOVA from last chapter • That ANOVA assumes each observation is independent • Within-subjects design = non-independent observations (e.g., same person answering questions over time  each response from same person will be related)

  3. Repeated Measures ANOVA Reaction Time Study - could be modified into RM design by having one group of subjects perform under all conditions Main effects – the effect of 1 IV on 1 DV Interactions – the effect of multiple IV’s on 1 DV

  4. Repeated Measures ANOVA “Within-subjects” design - subjects are measured on DV more than once as in training studies or learning studies Advantages increased power - due to decreased variance can use smaller sample sizes allows for study over time

  5. Repeated Measures ANOVA Disadvantages carryover effects - early treatments affect later ones. practice effects - subjects’ experience with test can influence their score on the DV. fatigue sensitization - subjects’ awareness of treatment is heightened due to repeated exposure to test.

  6. The strength of a within subjects design • Key to statistics = see where variability is coming from • Less noise have to deal with, the better – easier for the picture of the data to come through • Same person answers questions multiple times  variability that’s unique to that person is constant across each participant  less noise in the data

  7. Comparing repeated ANOVA with one-way ANOVA • Still involves F ratio • Still same interpretation – greater than 1 = more variance due to condition than to error • But variance due to participant is removed from the model •  F = variance between conditions, divided by amount of variance would expect by chance (excluding variance due to individual differences – error variance, or residual variance)

  8. In other words • F captures how much signal there is (how much difference across conditions) compared to noise (how much variability there is that can’t be explained by individual differences) • = MS between conditions / MS error

  9. Step 1:Set Hypothesis • STEP 1: State the Hypothesis • H0: m1 = m2 = m3 = m4 • HA : m1≠ m2≠m3≠ m4

  10. Step 2: Determining F-critical • See p. 533 • a = .01 (boldface) or .05 (lightface) • Numerator = df between treatments • Denominator = df error

  11. Step 3: STAGE 1 • SSbetween treatments = (T2/n) – (G2/N) • SSwithin treatments = SS inside each treatment = (SS1+SS2+SS3+...+SSk) • SStotal = X2 – (G2/N) or SSbetween + SSwithin n = # of scores in a tx condition N = total # of scores in whole study T = sum of scores for each tx condition G = sum of all scores in the study (Grand Total)

  12. Step 3: STAGE 1 • Calculate dfbetween treatments = k – 1 • Calculate dfwithin treatments = dfinside each treatment • Calculate dftotal = N-1 • dfbetween + dfwithin = dftotal k = number of factor levels n = number of scores in a treatment condition N = total number of scores in whole study (N = nk) T = sum of scores for each treatment condition G = sum of all scores in the study (Grand Total)

  13. Step 3: STAGE 2 • SSbetween subjects = p2 - G2 k N • SSerror = SSwithin treatments - SSbetween treatments • dfbetween subjects = n – 1 • dferror = dfwithin treatments - dfbetween subjects

  14. Step 3: STAGE 3F-ratio calculation • For numerator: • MSbetween treatments = SS between treatments df between treatments

  15. STEP 3: STAGE 3F-ratio calculation • For denominator: • MSerror = SS error df error

  16. STEP 3: STAGE 3F-ratio calculation • Fcalculated = MS between treatments MS error

  17. Step 4: Summary table and decision Source SS df MS F Between treatments Within treatments Between Subjects Error Total

  18. The similarity in logic continues… • Effect size: • measured by eta squared, still captures proportion of variability that’s explained by condition n2= SS between treatments between treatments + SS error

  19. The similarities continue… • To report in APA style: • F (df numerator, df denominator) = value, p information • df numerator = number conditions – 1 • df denominator = error df = df total – df numerator

  20. Effect size n2 = SS between treatments = SS between treatments SS between treat + SS within treat SS total

  21. Still need post hoc tests • Still have options such as Tukey and Scheffé

  22. Two-Factor ANOVA • In Chapter 13, we looked at ANOVAs with several levels of one IV • Here, we are looking at ANOVAs with several levels of more two IVs • We will now be looking at three types of mean differences within this analysis

  23. Main Effects • The mean differences that might among the levels of each factor (IV) • The mean differences of the rows (IV1) • The mean differences of the columns (IV2)

  24. STEP 1: State the Hypotheses for the Main Effects • Null Hyp:H0: µA1 = µA2 • Alternative Hyp: HA: µA1≠ µA2 • Null Hyp:H0: µB1 = µB2 • Alternative Hyp: HA: µB1≠ µB2

  25. Interactions • This occurs when the mean differences between individual treatment conditions (cells) are different than what would be predicted from the overall main effects

  26. STEP 2: State the Hypotheses for the Interactions • H0: There is no interaction between factors A and B. • HA: There is an interaction between factors A and B.

  27. STEP 3: Draw a matrix • This will allow us to determine how many groups we have and where different participants fall within these groups • For an example see page 401

  28. STEP 4: Determine df • dfbetween treatments = number of cells – 1 • dfwithin = dfeach treatment • dftotal = N-1

  29. STEP 5: Stage 1: Analyses • Total Variability: • SStotal = X2 – (G2/N) • Between Treatments Variability • SSbetween treatments = (T2/n) – (G2/N) • Within Treatments Variability • SSwithin treatments = SSeach treatment

  30. STEP 5: Stage 2: Analyses • Factor A: • SSfactor A = (T2ROW/nROW) – (G2/N) • dffactor A = number of rows – 1 • Factor B: • SSfactor B = (T2COL/nCOL) – (G2/N) • dffactor B = number of columns – 1 • A X B Interaction: • SSAXB = SSbetween treatments – SSfactor A – SSfactor B

  31. STEP 6: Calculations for MS and F • MSwithin treatments = SSwithin treat/dfwithin treat • MSA = SSA/dfA • MSB = SSB/dfB • MSAXB = SSAXB/dfAXB

  32. STEP 6: Calculations for MS and F • FA = MSA/MSwithin treatments • FB = MSB/MSwithin treatments • FAXB = MSAXB/MSwithin treatments

  33. Step 7: Summary table and decision Source SS df MS F Between treatments Factor A Factor B A X B Interaction Within treatments Total

  34. Effect Size • For Factor A: • n2 = SSA__________________ SS total – SSB - SSAXB • For Factor B: • n2 = SSB__________________ SS total – SSA - SSAXB • For AXB: • n2 = SSB__________________ SS total – SSA - SSB

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