2. Split-Plots Designs Used when one (or more ) factors require more experimental material for evaluation Different size experimental units Whole plots Split Plots

3. Split Plots Common in Process Experiments When different factors are from different process steps When randomizing the levels of one factor is difficult or time consuming

4. Process Experiments Factor in Earlier Step become Whole Plot Factor Factors in Later Steps can be varied within and become subplot factors Process Step 1 Process Step 2 Process Step 3

5. Example: Cookie Baking Experiment Orange Cookies Chocolate Cookies Problem: Although made from the same recipe, except the syrup, the chocolate cookies stay thick and round after baking, but the orange cookies spread thin and flat.

6. Research Question: want to study factors that may affect the final diameter of the orange cookies. Factors to study: 1) Amount of shortening in cookie dough 2) Bake Temperature

7. Experimental Procedure : 1. Mix the ingredients in the dough 2. Add the orange syrup Note: one dough batch makes enough for several trays of cookies

8. 3. Set the oven temperature and allow the temperature to stabilize 4. Place the cookies in the oven, and set the timer. 5. Remove tray of cookies, measure diameter of each cookie and calculate the average per tray

9. Experimental Plan 2 2 Factorial Replicated Factor Run Amount of Shortening Bake Temperature Batch 1 Low Low 1 2 High Low 2 3 Low High 3 4 High High 4 5 Low Low 5 6 High Low 6 7 Low High 7 8 High High 8 This plan requires 8 batches of Cookies

10. Note: one dough batch makes enough for several trays of cookies so we could test all four combinations with two batches Batch 1 Batch 2 Factor Run Amount of Shortening Bake Temperature 1 Low Low 2 Low High 3 High Low 4 High High What is the problem with this plan?

11. Experimental Plan 2 2 Factorial Its difficult to completely Randomize the order, because once I make a batch I must use it up before I can change the level of Amount of Shortening. However, I can vary the Bake Temperature from tray to tray within the batch. Any difference due to Amount of Shortening could be confused with differences in other ingredients (eggs, flour, etc. ) that change between batch 1 and batch 2 Batch 1 Batch 2 Factor Run Amount of Shortening Bake Temperature 1 Low Low 2 Low High 3 High Low 4 High High

13. Solution: make replicate batches with the same amount of shortening Factor Run Amount of Shortening Batch 1 Low 1 2 Low 2 3 High 3 4 High 4 Now I can randomize which batches get high or low amount of shortening, and the experimental unit for amount shortening is batch. Fixed Effect of Amount of Shortening Random (nested) effect of Batch This is the error term for testing Amount of Shortening

14. I can still vary the Bake Temperature from Tray to Tray within a Batch. I can randomize the order of Bake Temperatures within each Batch What is the Experimental Unit for Bake Temperature? What is the model for the combined Plan? Factor Amount of Bake Run Batch Shortening Tray Temperature 1 1 Low 1 Low 2 1 Low 2 High 3 2 High 1 Low 4 2 High 2 High 5 3 Low 1 Low 6 3 Low 2 High 7 4 High 1 Low 8 4 High 2 High

15. Run Amount of Shortening Batch 1 Low 1 2 Low 2 3 High 3 4 High 4 Bake Temperature Low High y 111 y 112 y 121 y 122 y 211 y 212 y 221 y 222 Presenting Split Plot as a Cross Product Design

16. Fixed Amount of Shortening Effect Random (nested) effect of Batch Fixed Bake Temperature Effect Interaction of Amount and Temperature This is called a Split Plot Model with Completely Randomized Design in the Whole Plots

17. Expanded Example in the Book: Factors to study: 1) Amount of shortening in cookie dough 2) Bake Temperature 3) Tray Temperature

20. BakeT Low High Low High Low High Batch Shortening TrayT RoomT Hot RoomT Hot RoomT Hot 1 80% 1.18 1.77 1.33 2.09 1.33 1.74 2 100% 2.11 2.28 2.08 2.45 2.46 2.37 3 100% 2.01 2.14 2.07 2.26 2.19 2.34 4 80% 0.94 1.34 1.23 1.72 1.28 1.72 Subplot Design 3×2 Whole Plot Design

21. Must be a power of nlev

27. Analysis with proc mixed

33. Hard to Vary Factors Some factors are hard or time consuming to vary Complete randomization is impractical

34. Example, John Ward (2006) Objective - study effects of factors such as: lure weight, line weight, and casting method upon distance to cast a lure with a fishing pole.

35. FactorDescription1st/Low Level (-1)2nd/High Level (1) ALure Weight1/16 oz. - Small Lure1/4 oz. - Big Lure B Hangs Off Tip of the Rod3-6 inches18-24 inches CLine Weight6lb10lb Factors

36. A= Lure wt. - = light + = heavy Easy to change

37. B = Hangs of tip of Rod - = 3 to 6 inches + = 18 to 24 inches Easy to change

38. C = line weight - = 6 lb + = 10 lb Hard to change

39. Experimental Plan 2 3 Randomize within Randomize within RunABC 1Light 3 to 610lb 2Heavy3 to 610lb 3Light18 to 2410lb 4Heavy 18 to 2410lb 5Light 3 to 6 6lb 6Heavy3 to 6 6lb 7Light 18 to 24 6lb 8Heavy18 to 24 6lb Problem: Line wt confounded with anything that changes during experiments such as wind

40. More variability over longer blocks of time, thus whole plots likely to vary more than subplots

41. Solution Run C Block 110lbJohnny 2 6lbJohnny 3 10lbMatt 4 6lbMatt Randomize within This is an RCB Design

42. RunABCBlock 1Light 3 to 610lbJohnny 2Heavy3 to 610lbJohnny 3Light 18 to 2410lbJohnny 4Heavy 18 to 2410lbJohnny 5Light3 to 66lbJohnny 6Heavy 3 to 66lbJohnny 7Light 18 to 246lbJohnny 8Heavy18 to 246lbJohnny 9Light 3 to 66lbMatt 10Heavy3 to 66lbMatt 11Light 18 to 246lbMatt 12Heavy 18 to 246lbMatt 13Light3 to 610lbMatt 14Heavy 3 to 610lbMatt 15Light 18 to 2410lbMatt 16Heavy18 to 2410lbMatt Randomize within Randomize within Randomize within Randomize within Solution What is experimental unit for A and B? What is Experimental unit for C?What is the model ?

43. Line weight crossed (fixed) j = 1,2 Block by Treatment Interaction (random) Hangs off End (fixed) l = 1, 2 Lure weight (fixed) k = 1, 2 Block Effect (random) i = 1, 2 This is called a Split Plot Model with Randomized Block Design in the Whole Plots

44. The Whole and Split-Plot Factors Have Different Experimental Units and Error Terms for F-tests Whole PlotSplit-Plot

45. Source of Variation Degrees of Freedom Mean Square Expected Mean Square Aa-1MSA Error(w)a(r-1)MSE(w) Tb-1MST AT(a-1)(b-1)MS(AT) Error(s)a(r-1)(b-1)MSE(s) ANOVA Table for Split Plot Design with Completely Randomized Design in Whole Plots

46. Source of Variation Degrees of Freedom Mean Square Expected Mean Square Blocksr-1MS Blocks Aa-1MSA Error(w)(a-1)(r-1)MSE(1) Cc-1MSB AC(a-1)(c-1)MS(AB) Error(s)a(r-1)(c-1)MSE(2) ANOVA Table for Split Plot Design with Randomized Complete Block Design in Whole Plots

47. CompletelyRandomized Complete Block Whole Plot Sourced.f.Sourced.f. Blocksr-1 Aa-1A Error(w)a(r-1)Error(w)(r-1)(a-1) Sub Plots Bb-1B AB(a-1)(b-1)AB(a-1)(b-1) Error(s)a(r-1)(b-1)Error(s)a(a-1)(b-1) Summary

52. Grind Collagen Dissolve to make Gel Batch Extrude Gel to make Casing Tube

57. Analysis of the Data in SAS

63. No Replicates Determine Significant Effects Graphically -Normal Plot -Half Normal Plot -Lenth Plot -Bayes Plot

64. Levels Factors - + A: Pressure Low High B: Power Low High C: Gas Flow Low High D: Type of Gas Oxygen SiCl 4 E: Paper Type A B

65. Because the plasma was created in a vacuum it takes up to a half hour of pumping to get the reactor down to the appropriate vacuum level each time the chamber is opened to insert new paper samples. Therefore rather than completely randomizing the 2 5 factorial …

66. Whole Plot Factors: A, B, C, D, AB, AC, AD, BC, BD, CD, ABC, ABD, ACD, BCD, ABCD Split Plot Factors: E, AE, BE, CE, DE, ABE, ACE, ADE, BCE, BDE, CDE, ABCE, ABDE, ACDE, BCDE, ABCDE

68. Chooses whole plot factors: A, B, C, D, AB, AC, AD, BC, BD, CD, ABC, ABD, ACD, BCD, ABCD

72. 4×4=16 Total subplots 3 factors whole plot and 3 subplot factors would require 8×8=64 subplots

73. Resolution III

74. Resolution III, but less aberration

85. Creating the design in proc factex