1. The Randomized Block Design

2. Suppose a researcher is interested in how several treatments affect a continuous response variable (Y). The treatments may be the levels of a single factor or they may be the combinations of levels of several factors. Suppose we have available to us a total of N = nt experimental units to which we are going to apply the different treatments.

3. The Completely Randomized (CR) design randomly divides the experimental units into t groups of size n and randomly assigns a treatment to each group.

4. The Randomized Block Design divides the group of experimental units into n homogeneous groups of size t. These homogeneous groups are called blocks. The treatments are then randomly assigned to the experimental units in each block - one treatment to a unit in each block.

5. Example 1: Suppose we are 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  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).

6. Suppose we have available to us a total of N = 60 experimental rats to which we are going to apply the different diets based on the t = 6 treatment combinations. Prior to the experimentation the rats were divided into n = 10 homogeneous groups of size 6. The grouping was based on factors that had previously been ignored (Example - Initial weight size, appetite size etc.) Within each of the 10 blocks a rat is randomly assigned a treatment combination (diet).

7. The weight gain after a fixed period is measured for each of the test animals and is tabulated on the next slide:

8. Randomized Block Design

9. Example 2: The following experiment is interested in comparing the effect four different chemicals (A, B, C and D) in producing water resistance (y) in textiles. A strip of material, randomly selected from each bolt, is cut into four pieces (samples) the pieces are randomly assigned to receive one of the four chemical treatments.

10. This process is replicated three times producing a Randomized Block (RB) design. Moisture resistance (y) were measured for each of the samples. (Low readings indicate low moisture penetration). The data is given in the diagram and table on the next slide.

11. Diagram: Blocks (Bolt Samples)

12. Table Blocks (Bolt Samples) Chemical123 A10.112.211.9 B11.412.912.7 C9.912.311.4 D12.113.412.9

13. The Model for a randomized Block Experiment i = 1,2,…, tj = 1,2,…, b y ij = the observation in the j th block receiving the i th treatment  = overall mean  i = the effect of the i th treatment  j = the effect of the j th Block  ij = random error

14. The Anova Table for a randomized Block Experiment SourceS.S.d.f.M.S.Fp-value TreatSS T t-1MS T MS T /MS E BlockSS B n-1MS B MS B /MS E ErrorSS E (t-1)(b-1)MS E

15. A randomized block experiment is assumed to be a two-factor experiment. The factors are blocks and treatments. The is one observation per cell. It is assumed that there is no interaction between blocks and treatments. The degrees of freedom for the interaction is used to estimate error.

16. The Anova Table for Diet Experiment

17. The Anova Table forTextile Experiment

18. If the treatments are defined in terms of two or more factors, the treatment Sum of Squares can be split (partitioned) into: –Main Effects –Interactions

19. The Anova Table for Diet Experiment terms for the main effects and interactions between Level of Protein and Source of Protein

20. Repeated Measures Designs

21. 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)

22. 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.

23. 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

24. 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

25. 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

26. 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.

27. 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.

28. 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)

29. 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

30. 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

31. 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