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Other experimental designs

Other experimental designs. Randomized Block design Repeated Measures designs. The Randomized Block Design. Suppose a researcher is interested in how several treatments affect a continuous response variable (Y).

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Other experimental designs

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  1. Other experimental designs Randomized Block design Repeated Measures designs

  2. The Randomized Block Design

  3. 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.

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

  5. 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.

  6. Experimental Designs The objective of Experimental design is to reduce the magnitude of random error resulting in more powerful tests to detect experimental effects

  7. The Completely Randomizes Design Treats 1 2 3 … t Experimental units randomly assigned to treatments

  8. Randomized Block Design Blocks All treats appear once in each block

  9. 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) .

  10. 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).

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

  12. Randomized Block Design

  13. 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.

  14. 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.

  15. Diagram: Blocks (Bolt Samples)

  16. Table Blocks (Bolt Samples) Chemical 1 2 3 A 10.1 12.2 11.9 B 11.4 12.9 12.7 C 9.9 12.3 11.4 D 12.1 13.4 12.9

  17. The Model for a randomized Block Experiment i = 1,2,…, t j = 1,2,…, b yij = the observation in the jth block receiving the ith treatment m = overall mean ti = the effect of the ith treatment bj = the effect of the jth Block eij = random error

  18. The Anova Table for a randomized Block Experiment

  19. 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.

  20. The Anova Table for Diet Experiment

  21. The Anova Table forTextile Experiment

  22. 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

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

  24. Using SPSS to analyze a randomized Block Design • Treat the experiment as a two-factor experiment • Blocks • Treatments • Omit the interaction from the analysis. It will be treated as the Error term.

  25. The data in an SPSS file Variables are in columns

  26. Select General Linear Model->Univariate

  27. Select the dependent variable, the Block factor, the Treatment factor. Select Model.

  28. Select a Custom model.

  29. Put in the model only the main effects.

  30. Obtain the ANOVA table If I want to break apart the Diet SS into components representing Source of Protein (2 df), Level of Protein (1 df), and Source Level interaction (2 df) - follow the subsequent steps

  31. Replace the Diet factor by the Source and level factors (The two factors that define diet)

  32. Specify the model. There is no interaction between Blocks and the diet factors (Source and Level)

  33. Obtain the ANOVA table

  34. The ANOVA table for the Completely Randomized Design The ANOVA table for the Randomized Block Design

  35. Comments The error term, , for the Completely Randomized Design models variability in the reponse, y, between experimental units The error term, , for the Completely Block Design models variability in the reponse, y, between experimental units in the same block (hopefully the is considerably smaller than . The ability to detect treatment differences depends on the magnitude of the random error term

  36. Example – Weight gain, diet, source of protein, level of protein (Completely randomized design)

  37. 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

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

  39. Using SPSS to analyze a randomized Block Design • Treat the experiment as a two-factor experiment • Blocks • Treatments • Omit the interaction from the analysis. It will be treated as the Error term.

  40. The data in an SPSS file Variables are in columns

  41. Select General Linear Model->Univariate

  42. Select the dependent variable, the Block factor, the Treatment factor. Select Model.

  43. Select a Custom model.

  44. Put in the model only the main effects.

  45. Obtain the ANOVA table If I want to break apart the Diet SS into components representing Source of Protein (2 df), Level of Protein (1 df), and Source Level interaction (2 df) - follow the subsequent steps

  46. Replace the Diet factor by the Source and level factors (The two factors that define diet)

  47. Specify the model. There is no interaction between Blocks and the diet factors (Source and Level)

  48. Obtain the ANOVA table

  49. Repeated Measures Designs

  50. 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)

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