2. Unsorted Treatments Random Numbers 1 0.533 1 0.683 2 0.702 2 0.379 3 0.411 3 0.962 3 0.139 Sorted Sorted Experimental Treatments Random Units Numbers 3 0.139 1 2 0.379 2 3 0.411 3 1 0.533 4 1 0.683 5 2 0.702 6 3 0.962 7 Randomization

3. Bread Rise Experiment 1. Mix The Dough 2. Divide the dough into 12 small loaves of the same size. 3. Randomly assign 4 loaves to rise 35 minutes, 4 to rise 40 minutes, etc. 4. After allowing each loaf to rise the specified time, measure the height of the loaf.

11. Model for CRD Design Cell Means Model

12. Alternate Model Effects Model

13. Notation Sample means Grand Mean

14. Least Squares Estimates Cell Means Choose estimates to minimize

15. Matrix Notation for Alternate Model LS Estimators are solution to Problem is singular

16. SAS proc glm Non-singular

17. is a generalized inverse for Biased Estimates

18. Bread Rise Experiment

22. = ^

23. :

24. Matrix Notation for Estimable Functions is an unbiased estimator for when the rows of L are linear combination of the rows of for example

25. is a linear

30. ssE must represent varation in experimental units not subsamples, repeated measures or duplicates Teaching Example (illustration of problems) Classes randomized to different teaching methods experimental unit=class No replicate classes no way to compute ssE Teaching method confounded with difference in classes Use of student to student variability (i.e. subsamples) to calculate ssE Could be totally misleading

31. ● Independence of error terms ε ij ● Equality of variance across levels of treatment factor ● Normal distribution of ε ij

32. Check equal variance assumption 1. plot data vs treatment factor level 2. plot residuals vs predicted values or cell means

34. Check normality with normal plot of residuals

35. ods graphics on; proc glm data=bread plots=diagnostics; class time; model height=time/solution; run; ods graphics off;

40. λ = 1 -1.294869

43. Solutions►Analysis►Design of Experiments Two-Level Factorial Response Surface MixtureMixed-Level Factorial Optimal Design Split-Plot Design General Factorial

44. Define Variables►Add> ►Add qualitative factorial variable

45. Customize…►Replicate Runs Edit Responses… Design►Randomize Design …

46. Fit …

47. Model ► Fit Details…Model►Check Assumptions►Perform Residual AnalysisModel►Check Transformation►Box-Cox Plot

51. Teaching Experiment Objective: Compare student satisfaction between 3 different teaching methods Experimental unit: class Two replicate classes for each teaching method. Response: rating given by each student, summarized over class as multinomial vector of counts

55. power, 1-β practical significance

56. power Size of a practical difference

67. 3. H 0 : μ 3 = μ 4 Does a mix of artificial fertilizers enhance yield? Is there a difference in plowed and broadcast? Does Timing of Application change Yield?

71. ^^ Option 1Option 2

72. Option 1

73. Option 2

77. Review Important Concepts Experimental Unit Randomization Replication Practical Difference Determining the number of replicates by calculating the power