ANOVA Designs Involving Repeated Measures 46 511 OneWay
- Slides: 48
ANOVA Designs Involving Repeated Measures 46 -511: One-Way Repeated Measures and Groups by Trials 1
Learning Objectives n Be able to identify the advantages and disadvantages of repeated measures designs n Understand how variance is partitioned in RM designs n Comprehend variations possible with such designs n Be familiar with post-hoc & planned comparisons using RM Designs n Including Trend Analysis 2
The Design n Repeated Measures Designs, a. k. a. n Dependent Measures Designs n Within-Subjects Designs n Mixed Randomized-Repeated Designs, a. k. a. n Groups by Trials n Split Plot Factorial n What are they n Relative Advantages n Relative Disadvantages 3
Some Examples n Completely Within Designs n 1 -Way n N-Way n Mixed Randomized-Repeated Designs n 1 -Between by 1 -Within n N-Between by N-Within n Characteristics of repeated measures designs n Nature of the repeated measures n Duration between measures 4
1 -Way Repeated Measures ANOVA: Sources of Variation n Between Subjects n Between Treatments n Within Subjects n Within Treatments 5
One-Way Example Person Drug 1 Drug 2 Drug 3 Drug 4 Pi Mean 1 2 3 4 5 30 14 24 38 26 28 18 20 34 28 16 10 18 20 14 34 22 30 44 30 108 64 92 136 98 27. 0 16. 0 23. 0 34. 0 24. 5 Tj 132 128 78 160 G= 4 9 8 Mean 26. 4 25. 6 15. 6 32. 0 24. 9 GM=24. 9 Five subjects, all are tested for reaction time after taking each of the four drugs, over a period of four days. 6
One-Way Example Person Drug 1 Drug 2 Drug 3 Drug 4 Pi Mean 1 2 3 4 5 30 14 24 38 26 28 18 20 34 28 16 10 18 20 14 34 22 30 44 30 108 64 92 136 98 27. 0 16. 0 23. 0 34. 0 24. 5 132 128 78 160 G = 498 Tj . . . Between Treatments Within Subjects Between Subjects Effects Where does Within Treatment variation come from? 7
Two Structural Models The rosy additive model: Xij = μ + πi + τj + εij The model that assumes people x treatment interaction: Xij = μ + πi + τj + πτij + εij 8
Partitioning Sums of Squares: or, here we go again Between People Between Treatments 9
Sums of Squares Within People Within Treatments 10
The error term n Two ways to get it: n SSRES = SSW. PEOPLE – SSTREAT n SSRES = SSW. TREATMENT – SSB. PEOPLE n How the error term differs from Between Subjects Design n What the error term represents/contains 11
Source Table Source SS df MS SSB. PEOPLE 680. 80 4 170. 20 SSW. PEOPLE 811. 00 15 54. 07 SSTREAT 698. 20 3 232. 73 SSRES 112. 80 12 9. 40 1, 491. 80 19 78. 52 SSTOT F p 24. 759 0. 000020 12
Missing Data in Within Subjects Designs n Due to such things as n Equipment failure n Experimenter or subject error n Loss of questionnaires n Usual missing data solutions ignore design • Y*ij = predicted (missing) score • s = number of subjects • S’i = sum of known values for the case • a = number of levels of A • A’j = sum of known values of A • T’ = sum of all known values 13
Example n Say subject #3 didn’t return to take drug 4 n #3’s sum is now 62 n Sum for A 4 is 130 (160 -30) n Sum of known scores for entire table = 49830=468 Error term must be reduced by number of imputed values (imputed values are not independent) 14
Assumptions n Observations within each treatment cell are independent. n Population treatment within each treatment must be normally distributed. n Variances for the population treatments should be equivalent. n Sphericity – that the variance of the difference scores for each pair of conditions is the same in the population. n Alternatives if assumptions do not hold. 15
Mean Comparison Procedures n Tukey n Same as in 1 -way between, substitute MS error (residual) for MS within & df error for df within n Scheffe’ n Same as in 1 -way between, substitute MS error for MS within & df error for df within n Bonferroni procedure n Šidák procedure 16
Contrasts n Unfortunately, contrasts are not quite a logical extension of contrasts from between subjects designs n Affected by mild violations of sphericity n Must determine variability specific to each contrast. n Two methods, t and F. n Let’s test the following contrast: n C 1: . 5*Drug 1+. 5*Drug 2 – Drug 3=0. 17
Method One Single Sample t-test: where and 18
Method 1 (Cont’d) (single sample t-test) Effect of Different Drugs on Reaction Time Person Drug 1 Drug 2 Drug 3 C 1 (Using t-test) 1 30 28 16 13 2 14 18 10 6 3 24 20 18 4 4 38 34 20 16 5 26 28 14 13 Mean 26. 4 25. 6 15. 6 ΣCi 52. 00 Mean Ci 10. 40 ΣCi 2 646. 00 19
Calculations for t: df = n-1; t(. 05, 4) = 2. 78 20
Method 2: Using the F statistic Sum of squares for contrast: Sum of squares for error term: 21
Method 2 (Cont’d) df = 1, (n – 1) Recalling that t 2 = F; 4. 5352 = 20. 566 22
Effect sizes Partial ω2: Partial η 2 / R 2: Cohen’s d: similar, have to decide on proper standard deviation 23
Power Use partial effect size Use G*Power or power charts. Assume ρ=. 50 unless you know different. 24
Trend Analysis Example Mean Subject # Time 1 Time 2 Time 3 Time 4 Time 5 Time 6 1 43. 00 31. 00 10. 00 9. 00 4. 00 2 40. 00 30. 00 9. 00 6. 00 4. 00 5. 00 3 49. 00 33. 00 12. 00 5. 00 6. 00 3. 00 4 39. 00 26. 00 11. 00 8. 00 5 41. 00 28. 00 6. 00 5. 00 6 44. 00 34. 00 9. 00 7. 00 6. 00 42. 67 30. 33 9. 83 6. 83 5. 17 4. 67 Experiment on forgetting: six participants master a list of 50 words, then are asked to recall them the next day (time 1), one week later (time 2), and so on. 25
SPSS Output: Test of Sphericity & Summary Table Epsilon(a) Within Subjects Effect Mauchly's W factor 1 Approx. Chi. Square . 004 Error(factor 1) Greenhous e-Geisser Sig. 14 Type III Sum of Squares Source factor 1 16. 948 df . 394 df . 373 Mean Square Huynh. Feldt. 587 F Lower-bound. 200 Sig. Sphericity Assumed 7694. 250 5 1538. 850 364. 369 . 000 Greenhouse-Geisser 7694. 250 1. 867 4121. 540 364. 369 . 000 Huynh-Feldt 7694. 250 2. 937 2620. 068 364. 369 . 000 Lower-bound 7694. 250 1. 000 7694. 250 364. 369 . 000 Sphericity Assumed 105. 583 25 4. 223 Greenhouse-Geisser 105. 583 9. 334 11. 311 Huynh-Feldt 105. 583 14. 683 7. 191 Lower-bound 105. 583 5. 000 21. 117 26
SPSS Output Polynomial Contrasts Source factor 1 Type III Sum of Squares factor 1 Linear 6179. 336 1 6179. 336 532. 505 . 000 Quadratic 1292. 161 1 1292. 161 502. 740 . 000 . 112 1 . 112 . 125 . 738 Order 4 143. 006 1 143. 006 39. 040 . 002 Order 5 79. 636 1 79. 636 33. 393 . 002 Linear 58. 021 5 11. 604 Quadratic 12. 851 5 2. 570 4. 471 5 . 894 Order 4 18. 315 5 3. 663 Order 5 11. 924 5 2. 385 Cubic Error(factor 1) Cubic df Mean Square F Sig. 27
Graph of linear & quadratic trend 28
Groups by Trials Design: What it is? n Combines between and within designs n Yields effects for… n n n Trials Groups Trials by Groups interaction n Can be used to answer questions such as… n n n Do test pattern scores (e. g. , pre-post) differ by experimental vs. control group… Do women and men differ significantly in their ability to detect smell in varying conditions. Do patients receiving drug A have a different course of improvement than those receiving drug B? 29
Assumptions n The usual assumptions for between subjects designs n The usual assumptions for repeated measures designs n Homogeneity of variance/covariance matrices by group 30
Partitioning Variance! n Between Groups Variance – n Subjects within Groups Variance – n Between Trials Variance – n Group by Trials Variance – n Subjects within Groups within Trials Variance (residual) – 31
Between Group Variance Definitional Formula Computational Formula 32
Subjects within Groups Definitional Formula Computational Formula 33
Between Trials Variance Definitional Formula Computational Formula 34
Group by Trials Variance Definitional Formula Computational Formula 35
Subjects within Groups within Trials (residual) Definitional Formula Computational Formula 36
Numerical Example Subject # B 1: Month 1 B 2: Month 2 B 3: Month 3 1 2 3 4 5 1 1 3 5 2 3 4 3 5 4 6 8 6 7 5 A 2: Mystery 6 7 8 9 10 3 4 5 4 4 1 4 3 2 5 0 2 2 0 3 A 3: Romance 11 12 13 14 15 4 2 3 6 3 2 3 0 1 3 1 2 A 1: Scifi. 37
Between Subjects Effects 38
Within Subjects & Interaction 39
Interaction Effect 40
Two-Within Subject Factors n Brief Example n Effects n Main effects for Factor A n Main effects for Factor B n Interaction effect for A x B 41
Numerical Example Number of Books Read each Month by Genre B 1: Science Fiction B 2: Mystery A 1: Month 1 A 2: Month 2 A 3: Month 3 s 1 1 3 6 5 4 1 s 2 1 4 8 8 8 4 s 3 3 3 6 4 5 3 s 4 5 5 7 3 2 0 s 5 2 4 5 5 6 3 A 3(month) x 2(genre) way within subjects ANOVA, where n=5 42
Summary Table 43
And, the Interaction Plot 44
Two-way Repeated Measures ANOVA: Main effects Main effect for A B 1: Science Fiction B 2: Mystery A 1: Month 1 A 2: Month 2 A 3: Month 3 s 1 1 3 6 5 4 1 s 2 1 4 8 8 8 4 s 3 3 3 6 4 5 3 s 4 5 5 7 3 2 0 s 5 2 4 5 5 6 3 A 1: Month 1 A 2: Month 2 A 3: Month 3 s 1 1+5=6 3+4=7 6+1=7 s 2 1+8=9 4+8=12 8+4=12 s 3 3+4=7 3+5=8 6+3=9 s 4 5+3=8 5+2=7 7+0=7 s 5 2+5=7 4+6=10 5+3=8 45
Why different error terms? n Recall SSRES=SSW. PEOPLE-SSTREAT A 1: Month 1 A 2: Month 2 A 3: Month 3 SSW. PEOPLE s 1 6 7 7 0. 667 s 2 9 12 12 6. 000 s 3 7 8 9 2. 000 s 4 8 7 7 0. 667 s 5 7 10 8 4. 667 =14. 000/2=7 SSRES=7 -2. 867=4. 133 46
For B Main effect… B 1: Sci. Fi B 2: Mystery SSW. PEOPLE s 1 10 10 0. 0 s 2 13 20 24. 5 s 3 12 12 0. 0 s 4 17 5 72. 0 s 5 11 14 4. 5 =101/3=33. 667 SSRES=33. 667 -0. 133=33. 534 47
Finally, for Ax. B… B 1: Science Fiction B 2: Mystery A 1: Month 1 A 2: Month 2 A 3: Month 3 SSW. PEOPLE s 1 1 3 6 5 4 1 21. 33 s 2 1 4 8 8 8 4 43. 50 s 3 3 3 6 4 5 3 8. 00 s 4 5 5 7 3 2 0 31. 33 s 5 2 4 5 5 6 3 10. 83 115. 00 SSRES=SSW. PEOPLE – SSA – SSB – SSAS – SSBS SSRES=115. 0 – 2. 867 – 0. 133 – 64. 467 – 4. 133 – 33. 533 = 9. 867 48
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