1. Repeated Measures Designs

2. In a Repeated Measures Design We have experimental units that may be grouped according to one or several factors (the grouping factors) Then on each experimental unit we have not a single measurement but a group of measurements (the repeated measures) The repeated measures may be taken at combinations of levels of one or several factors (The repeated measures factors)

3. Example In the following study the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery. The enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for n = 15 cardiac surgical patients.

4. The data is given in the table below. Table: The enzyme levels -immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery

5. The subjects are not grouped (single group). There is one repeated measures factor - Time – with levels –Day 0, –Day 1, –Day 2, –Day 7 This design is the same as a randomized block design with –Blocks = subjects

6. The Anova Table for Enzyme Experiment The Subject Source of variability is modelling the variability between subjects The ERROR Source of variability is modelling the variability within subjects

7. Example : (Repeated Measures Design - Grouping Factor) In the following study, similar to example 3, the experimenter was interested in how the level of a certain enzyme changed in cardiac patients after open heart surgery. In addition the experimenter was interested in how two drug treatments (A and B) would also effect the level of the enzyme.

8. The 24 patients were randomly divided into three groups of n= 8 patients. The first group of patients were left untreated as a control group while the second and third group were given drug treatments A and B respectively. Again the enzyme was measured immediately after surgery (Day 0), one day (Day 1), two days (Day 2) and one week (Day 7) after surgery for each of the cardiac surgical patients in the study.

9. Table: The enzyme levels - immediately after surgery (Day 0), one day (Day 1),two days (Day 2) and one week (Day 7) after surgery for three treatment groups (control, Drug A, Drug B)

10. The subjects are grouped by treatment –control, –Drug A, –Drug B There is one repeated measures factor - Time – with levels –Day 0, –Day 1, –Day 2, –Day 7

11. The Anova Table There are two sources of Error in a repeated measures design: The between subject error – Error 1 and the within subject error – Error 2

12. Tables of means DrugDay 0Day 1Day 2Day 7Overall Control118.6377.8860.5055.7578.19 A103.2568.2552.0051.5068.75 B103.3869.3854.1351.5069.59 Overall108.4271.8355.5452.9272.18

14. Example : Repeated Measures Design - Two Grouping Factors In the following example, the researcher was interested in how the levels of Anxiety (high and low) and Tension (none and high) affected error rates in performing a specified task. In addition the researcher was interested in how the error rates also changed over time. Four groups of three subjects diagnosed in the four Anxiety-Tension categories were asked to perform the task at four different times patients in the study.

15. The number of errors committed at each instance is tabulated below.

16. The Anova Table

17. Latin Square Designs

18. Selected Latin Squares 3 x 34 x 4 A B CA B C DA B C DA B C DA B C D B C AB A D CB C D AB D A CB A D C C A BC D B AC D A BC A D BC D A B D C A BD A B CD C B AD C B A 5 x 56 x 6 A B C D EA B C D E F B A E C DB F D C A E C D A E BC D E F B A D E B A CD A F E C B E C D B AF E B A D C

19. Definition A Latin square is a square array of objects (letters A, B, C, …) such that each object appears once and only once in each row and each column. Example - 4 x 4 Latin Square. A B C D B C D A C D A B D A B C

20. In a Latin square You have three factors: Treatments (t) (letters A, B, C, …) Rows (t) Columns (t) The number of treatments = the number of rows = the number of colums = t. The row-column treatments are represented by cells in a t x t array. The treatments are assigned to row-column combinations using a Latin-square arrangement

21. Example A courier company is interested in deciding between five brands (D,P,F,C and R) of car for its next purchase of fleet cars. The brands are all comparable in purchase price. The company wants to carry out a study that will enable them to compare the brands with respect to operating costs. For this purpose they select five drivers (Rows). In addition the study will be carried out over a five week period (Columns = weeks).

22. Each week a driver is assigned to a car using randomization and a Latin Square Design. The average cost per mile is recorded at the end of each week and is tabulated below:

23. The Model for a Latin Experiment i = 1,2,…, tj = 1,2,…, t y ij(k) = the observation in i th row and the j th column receiving the k th treatment  = overall mean  k = the effect of the i th treatment  i = the effect of the i th row  ij(k) = random error k = 1,2,…, t  j = the effect of the j th column No interaction between rows, columns and treatments

24. A Latin Square experiment is assumed to be a three-factor experiment. The factors are rows, columns and treatments. It is assumed that there is no interaction between rows, columns and treatments. The degrees of freedom for the interactions is used to estimate error.

25. The Anova Table for a Latin Square Experiment SourceS.S.d.f.M.S.F p-value TreatSS Tr t-1MS Tr MS Tr /MS E RowsSS Row t-1MS Row MS Row /MS E ColsSS Col t-1MS Col MS Col /MS E ErrorSS E (t-1)(t-2)MS E TotalSS T t 2 - 1

26. The Anova Table for Example SourceS.S.d.f.M.S.F p-value Week 51.17887412.7947216.060.0001 Driver 69.44663417.3616621.790.0000 Car 70.90402417.7260122.240.0000 Error 9.56315120.79693 Total 201.0926724

27. Example In this Experiment the we are again interested in how weight gain (Y) in rats is affected by Source of protein (Beef, Cereal, and Pork) and by Level of Protein (High or Low). There are a total of t = 3 X 2 = 6 treatment combinations of the two factors. Beef -High Protein Cereal-High Protein Pork-High Protein Beef -Low Protein Cereal-Low Protein and Pork-Low Protein

28. In this example we will consider using a Latin Square design Six Initial Weight categories are identified for the test animals in addition to Six Appetite categories. A test animal is then selected from each of the 6 X 6 = 36 combinations of Initial Weight and Appetite categories. A Latin square is then used to assign the 6 diets to the 36 test animals in the study.

29. In the latin square the letter A represents the high protein-cereal diet B represents the high protein-pork diet C represents the low protein-beef Diet D represents the low protein-cereal diet E represents the low protein-pork diet and F represents the high protein-beef diet.

30. The weight gain after a fixed period is measured for each of the test animals and is tabulated below:

31. The Anova Table for Example SourceS.S.d.f.M.S.F p-value Inwt1767.08365353.41673111.10.0000 App2195.43315439.08662138.030.0000 Diet4183.91325836.78263263.060.0000 Error63.61999203.181 Total 8210.049935

32. Diet SS partioned into main effects for Source and Level of Protein SourceS.S.d.f.M.S.F p-value Inwt1767.08365353.41673111.10.0000 App2195.43315439.08662138.030.0000 Source631.221732315.6108799.220.0000 Level2611.20971 820.880.0000 SL941.481722470.74086147.990.0000 Error63.61999203.181 Total 8210.049935

33. Graeco-Latin Square Designs Mutually orthogonal Squares

34. Definition A Greaco-Latin square consists of two latin squares (one using the letters A, B, C, … the other using greek letters , , , …) such that when the two latin square are supper imposed on each other the letters of one square appear once and only once with the letters of the other square. The two Latin squares are called mutually orthogonal. Example: a 7 x 7 Greaco-Latin Square A  B  C  D  E  F  G  B  C  D  E  F  G  A  C  D  E  F  G  A  B  D  E  F  G  A  B  C  E  F  G  A  B  C  D  F  G  A  B  C  D  E  G  A  B  C  D  E  F 

35. Note: At most (t –1) t x t Latin squares L 1, L 2, …, L t-1 such that any pair are mutually orthogonal. It is possible that there exists a set of six 7 x 7 mutually orthogonal Latin squares L 1, L 2, L 3, L 4, L 5, L 6.

36. The Greaco-Latin Square Design - An Example A researcher is interested in determining the effect of two factors the percentage of Lysine in the diet and percentage of Protein in the diet have on Milk Production in cows. Previous similar experiments suggest that interaction between the two factors is negligible.

37. For this reason it is decided to use a Greaco-Latin square design to experimentally determine the two effects of the two factors (Lysine and Protein). Seven levels of each factor is selected 0.0(A), 0.1(B), 0.2(C), 0.3(D), 0.4(E), 0.5(F), and 0.6(G)% for Lysine and 2(  ), 4(  ), 6(  ), 8(  ), 10(  ), 12(  ) and 14(  )% for Protein ). Seven animals (cows) are selected at random for the experiment which is to be carried out over seven three-month periods.

38. A Greaco-Latin Square is the used to assign the 7 X 7 combinations of levels of the two factors (Lysine and Protein) to a period and a cow. The data is tabulated on below:

39. The Model for a Greaco-Latin Experiment i = 1,2,…, t j = 1,2,…, t y ij(kl) = the observation in i th row and the j th column receiving the k th Latin treatment and the l th Greek treatment k = 1,2,…, t l = 1,2,…, t

40.  = overall mean  k = the effect of the k th Latin treatment  i = the effect of the i th row  ij(k) = random error  j = the effect of the j th column No interaction between rows, columns, Latin treatments and Greek treatments l = the effect of the l th Greek treatment

41. A Greaco-Latin Square experiment is assumed to be a four-factor experiment. The factors are rows, columns, Latin treatments and Greek treatments. It is assumed that there is no interaction between rows, columns, Latin treatments and Greek treatments. The degrees of freedom for the interactions is used to estimate error.

42. The Anova Table for a Greaco-Latin Square Experiment SourceS.S.d.f.M.S.F p-value LatinSS La t-1MS La MS La /MS E GreekSS Gr t-1MS Gr MS Gr /MS E RowsSS Row t-1MS Row MS Row /MS E ColsSS Col t-1MS Col MS Col /MS E ErrorSS E (t-1)(t-3)MS E TotalSS T t 2 - 1

43. The Anova Table for Example SourceS.S.d.f.M.S.F p-value Protein 160242.82 6 26707.1361 41.23 0.0000 Lysine 30718.24 6 5119.70748 7.9 0.0001 Cow 2124.24 6 354.04082 0.55 0.7676 Period 5831.96 6 971.9932 1.5 0.2204 Error 15544.41 24 647.68367 Total 214461.67 48