1. Diploma in Statistics Design and Analysis of Experiments Lecture 5.11 © 2010 Michael Stuart Lecture 5.1 Part 1 "Split Plot" experiments 1.Review of –randomised block designs –hierarchical / nested designs 2.Examples 3.Analysis of Whole Units 4.Analysis of Sub Units 5.Split plot analysis 6.Expected Mean Squares –Error terms for tests 7.Interactions

2. Diploma in Statistics Design and Analysis of Experiments Lecture 5.12 © 2010 Michael Stuart Minute Test: How Much

3. Diploma in Statistics Design and Analysis of Experiments Lecture 5.13 © 2010 Michael Stuart Minute Test: How Fast

4. Diploma in Statistics Design and Analysis of Experiments Lecture 5.14 © 2010 Michael Stuart Randomised Blocks, Again An experiment was conducted to assess the effects of applying four chemicals to soybean seeds with a view to improving germination rates. Each treatment was applied to 100 seeds planted in adjacent plots. As a check, another plot was planted with 100 seeds which received no treatment. The experiment was replicated in five blocks of five plots each, with each treatment being assigned to plots at random within each block. The number of failures in each plot was recorded.

5. Diploma in Statistics Design and Analysis of Experiments Lecture 5.15 © 2010 Michael Stuart Testing for Interaction with Blocks Analysis of Variance for Failures Source DF Seq SS Adj SS Adj MS F P Block 4 49.8400 49.8400 12.4600 2.30 0.103 Treatment 4 83.8400 83.8400 20.9600 3.87 0.022 Block*Treatment 16 86.5600 86.5600 5.4100 ** Error 0 * * * Total 24 220.2400 ** Denominator of F-test is zero. Without a valid reference term, it is not possible to have an F test for interaction. To check for interaction, –replicate each design point within each block, –replicates provide estimate of pure error

6. Diploma in Statistics Design and Analysis of Experiments Lecture 5.16 © 2010 Michael Stuart Assumption of No Interaction With no replication, use block by treatment interaction mean square as error mean square. With interaction present, this means –estimate of  is inflated, –power of the F test for treatment effects is reduced.

7. Diploma in Statistics Design and Analysis of Experiments Lecture 5.17 © 2010 Michael Stuart Blocking as Random Effect Source Expected Mean Square 1 Block (3) + 5.0000 (1) 2 Treatment (3) + Q[2] 3 Error (3)

8. Diploma in Statistics Design and Analysis of Experiments Lecture 5.18 © 2010 Michael Stuart  SS TT eBeB eSeS BB eTeT e = e B + e S + e T Hierarchy of components of variation Batch variation Sampling variation Testing variation y

9. Diploma in Statistics Design and Analysis of Experiments Lecture 5.19 © 2010 Michael Stuart Hierarchical Design for Estimating Components of Variation 60 measurements nested in 30 samples nested in 15 batches

10. Diploma in Statistics Design and Analysis of Experiments Lecture 5.110 © 2010 Michael Stuart Part 2Examples Example 1 3 varieties of wheat are planted in a homogeneous block of 3 plots, with varieties randomly assigned to plots; the experiment is replicated 4 times, with separate randomisations in each block, as follows:

11. Diploma in Statistics Design and Analysis of Experiments Lecture 5.111 © 2010 Michael Stuart A L B H A H B L A L B L A H B H Example 1 Following planting, it was decided to try two new fertilisers. Each plot was divided in four subplots and a 2 2 was implemented in each, with the possible combinations being assigned at random to subplots within each plot, as shown for one plot below.

12. Diploma in Statistics Design and Analysis of Experiments Lecture 5.112 © 2010 Michael Stuart Plot structure 48 subplots nested in 12 whole plots nested in 4 blocks Treatment structure 3 varieties randomly allocated to whole plots within blocks 2 2 = 4 fertiliser combinations randomly allocated to subplots within whole plots least variation in between variation most variation

13. Diploma in Statistics Design and Analysis of Experiments Lecture 5.113 © 2010 Michael Stuart Example 2 Electronic components are baked in an oven at a set temperature for a set time. Two factors thought to influence the life times of the components were the oven temperature and the bake time. Trial settings for these factors were chosen as follows: Oven Temperature (T), °F, 580, 600, 620, 640, Baking time (B), min,5, 10, 15. To save on costly runs, three components were baked together at each temperature, with one withdrawn at each of the set times. This plan was replicated 3 times. The results follow.

14. Diploma in Statistics Design and Analysis of Experiments Lecture 5.114 © 2010 Michael Stuart Example 2 Results of accelerated life time tests for electronic components

15. Diploma in Statistics Design and Analysis of Experiments Lecture 5.115 © 2010 Michael Stuart Example 2 What are the whole units? What are the whole unit treatments? What are the sub units? What are the sub unit treatments? What is the plot structure? What is the treatment structure?

16. Diploma in Statistics Design and Analysis of Experiments Lecture 5.116 © 2010 Michael Stuart Example 2 What are the whole units? What are the whole unit treatments? What are the sub units? What are the sub unit treatments? oven load of 3 components oven temperatures single components baking times

17. Diploma in Statistics Design and Analysis of Experiments Lecture 5.117 © 2010 Michael Stuart Unit and Treatment Structures

18. Diploma in Statistics Design and Analysis of Experiments Lecture 5.118 © 2010 Michael Stuart Unit structure 36 sub units nested in 12 whole units nested in 3 blocks Treatment structure 4 temperatures randomly allocated to whole units within blocks 3 baking times randomly allocated to sub units within whole units least variation in between variation most variation

19. Diploma in Statistics Design and Analysis of Experiments Lecture 5.119 © 2010 Michael Stuart Implications of unit and treatment structures Treatment effects assessed relative to variation between units to which they are applied.

20. Diploma in Statistics Design and Analysis of Experiments Lecture 5.120 © 2010 Michael Stuart Case study Paper manufactured in two stages: pulp prepared in large batches, long process, batches divided into small parts, each of which is put through a short cooking process. Experiment to investigate effects of three pulp preparation methods, four cooking temperature settings on tensile strength of the paper.

21. Diploma in Statistics Design and Analysis of Experiments Lecture 5.121 © 2010 Michael Stuart Case study Protocol: batch made using one method, randomly selected, each of four samples "cooked" at one of the four different temperatures, random order repeated for the other two methods, replicated on successive days, new random orderings Results

22. Diploma in Statistics Design and Analysis of Experiments Lecture 5.122 © 2010 Michael Stuart Randomised Blocks analysis for Methods

23. Diploma in Statistics Design and Analysis of Experiments Lecture 5.123 © 2010 Michael Stuart Randomised Blocks analysis for Methods Analysis of Variance for Y, no interaction term Source DF Seq SS Adj SS Adj MS F P B 2 19.389 19.389 9.694 4.28 0.102 M 2 32.097 32.097 16.049 7.08 0.049 Error 4 9.069 9.069 2.267 Total 8 60.556 S = 1.50578 Analysis of Variance for Y, with interaction term Source DF Seq SS Adj SS Adj MS F P B 2 19.3889 19.3889 9.6944 4.28 0.102 M 2 32.0972 32.0972 16.0486 7.08 0.049 B*M 4 9.0694 9.0694 2.2674 ** Error 0 * * * Total 8 60.5556

24. Diploma in Statistics Design and Analysis of Experiments Lecture 5.124 © 2010 Michael Stuart Diagnostics

25. Diploma in Statistics Design and Analysis of Experiments Lecture 5.125 © 2010 Michael Stuart Diagnostics

26. Diploma in Statistics Design and Analysis of Experiments Lecture 5.126 © 2010 Michael Stuart Randomised Blocks analysis for Temperature

27. Diploma in Statistics Design and Analysis of Experiments Lecture 5.127 © 2010 Michael Stuart Randomised Blocks analysis for Temperature Analysis of Variance for Y, no interaction term Source DF Seq SS Adj SS Adj MS F P B 2 25.879 25.879 12.940 11.30 0.009 T 3 144.636 144.636 48.212 42.10 0.000 Error 6 6.871 6.871 1.145 Total 11 177.386 S = 1.07013 Analysis of Variance for Y, with interaction term Source DF Seq SS Adj SS Adj MS F P B 2 25.879 25.879 12.940 11.30 0.009 T 3 144.636 144.636 48.212 42.10 0.000 B*T 6 6.871 6.871 1.145 ** Error 0 * * * Total 11 177.386

28. Diploma in Statistics Design and Analysis of Experiments Lecture 5.128 © 2010 Michael Stuart Diagnostic analysis

29. Diploma in Statistics Design and Analysis of Experiments Lecture 5.129 © 2010 Michael Stuart Diagnostic analysis

30. Diploma in Statistics Design and Analysis of Experiments Lecture 5.130 © 2010 Michael Stuart Response:Strength (Y) Factors:Day (Block), B Method, M Temperature, T Effects to include in model: B M B*M T B*T M*T Split plot analysis assessed at whole unit level assessed at sub unit level

31. Diploma in Statistics Design and Analysis of Experiments Lecture 5.131 © 2010 Michael Stuart Split plot analysis Analysis of Variance for Y Source DF Seq SS Adj SS Adj MS F P B 2 77.556 77.556 38.778 4.68 0.126 x M 2 128.389 128.389 64.194 7.08 0.049 B*M 4 36.278 36.278 9.069 2.14 0.138 T 3 434.083 434.083 144.694 42.01 0.000 B*T 6 20.667 20.667 3.444 0.81 0.580 M*T 6 75.167 75.167 12.528 2.96 0.052 Error 12 50.833 50.833 4.236 Total 35 822.972 x Not an exact F-test. S = 2.05818

32. Diploma in Statistics Design and Analysis of Experiments Lecture 5.132 © 2010 Michael Stuart Split plot analysis Exercise: Check calculation of F ratios for M and T and corresponding degrees of freedom; cross check with previous analyses.

33. Diploma in Statistics Design and Analysis of Experiments Lecture 5.133 © 2010 Michael Stuart 4Expected Mean Squares Source Expected Mean Square for Each Term 1 B (7) + 3.0000 (5) + 4.0000 (3) + 12.0000 (1) 2 M (7) + 4.0000 (3) + Q[2, 6] 3 B*M (7) + 4.0000 (3) 4 T (7) + 3.0000 (5) + Q[4, 6] 5 B*T (7) + 3.0000 (5) 6 M*T (7) + Q[6] 7 Error (7) Exercise: Translate into  2 notation.

34. Diploma in Statistics Design and Analysis of Experiments Lecture 5.134 © 2010 Michael Stuart Diagnostics

35. Diploma in Statistics Design and Analysis of Experiments Lecture 5.135 © 2010 Michael Stuart Diagnostics

36. Diploma in Statistics Design and Analysis of Experiments Lecture 5.136 © 2010 Michael Stuart Interaction effect?

37. Diploma in Statistics Design and Analysis of Experiments Lecture 5.137 © 2010 Michael Stuart Interaction effect?

38. Diploma in Statistics Design and Analysis of Experiments Lecture 5.138 © 2010 Michael Stuart Reasons for using split plots Adding another factor after the experiment started Some factors require better precision than others Changing one factor is –more difficult –more expensive –more time consuming than changing others

39. Diploma in Statistics Design and Analysis of Experiments Lecture 5.139 © 2010 Michael Stuart Reading DCM §4.1, §14.4

40. Diploma in Statistics Design and Analysis of Experiments Lecture 5.140 © 2010 Michael Stuart Lecture 5.1 Part 2 Further Developments Repeated measures Robust design Analysis of Covariance Non-normal error Strategies for experiments

41. Diploma in Statistics Design and Analysis of Experiments Lecture 5.141 © 2010 Michael Stuart Robust design Seek optimal settings of experimental factors that remain optimal, irrespective of uncontrolled environmental factors. Run the experimental design, the inner array, at a range of settings of the environmental variables, the outer array. Popularised by Taguchi.

42. Diploma in Statistics Design and Analysis of Experiments Lecture 5.142 © 2010 Michael Stuart Analysis of Covariance Objective:take account of variation in uncontrolled environmental variables. Solution:measure the environmental variables at each design point and incorporate in the analysis through regression methods (Analysis of Covariance) Effects:reduces "error" variation, makes factor effects more significant adjusts factor effect estimates to take account of extra variation source.

43. Diploma in Statistics Design and Analysis of Experiments Lecture 5.143 © 2010 Michael Stuart Analysis of Covariance; Illustration Breaking strength of monofilament fibre produced by three different machines, allowing for variation in fibre thickness.

44. Diploma in Statistics Design and Analysis of Experiments Lecture 5.144 © 2010 Michael Stuart Analysis of Covariance; Minitab

45. Diploma in Statistics Design and Analysis of Experiments Lecture 5.145 © 2010 Michael Stuart Analysis of Covariance; Minitab General Linear Model: Y versus Machine Source DF Seq SS Adj SS Adj MS F P X 1 305.13 178.01 178.01 69.97 0.000 Machine 2 13.28 13.28 6.64 2.61 0.118 Error 11 27.99 27.99 2.54 Total 14 346.40 S = 1.59505 One-way ANOVA: Y versus Machine Source DF SS MS F P Machine 2 140.4 70.2 4.09 0.044 Error 12 206.0 17.2 Total 14 346.4 S = 4.143

46. Diploma in Statistics Design and Analysis of Experiments Lecture 5.146 © 2010 Michael Stuart Analysis of Covariance; Minitab

47. Diploma in Statistics Design and Analysis of Experiments Lecture 5.147 © 2010 Michael Stuart Further Developments Non-Normal errors –transformations –generalised linear models, including Logistic

48. Diploma in Statistics Design and Analysis of Experiments Lecture 5.148 © 2010 Michael Stuart Changing spread with log

49. Diploma in Statistics Design and Analysis of Experiments Lecture 5.149 © 2010 Michael Stuart Changing spread with log

50. Diploma in Statistics Design and Analysis of Experiments Lecture 5.150 © 2010 Michael Stuart Changing spread with log

51. Diploma in Statistics Design and Analysis of Experiments Lecture 5.151 © 2010 Michael Stuart Changing spread with log

52. Diploma in Statistics Design and Analysis of Experiments Lecture 5.152 © 2010 Michael Stuart Changing spread with log

53. Diploma in Statistics Design and Analysis of Experiments Lecture 5.153 © 2010 Michael Stuart Changing spread with log

54. Diploma in Statistics Design and Analysis of Experiments Lecture 5.154 © 2010 Michael Stuart Changing spread with log

55. Diploma in Statistics Design and Analysis of Experiments Lecture 5.155 © 2010 Michael Stuart Changing spread with log

56. Diploma in Statistics Design and Analysis of Experiments Lecture 5.156 © 2010 Michael Stuart Changing spread with log

57. Diploma in Statistics Design and Analysis of Experiments Lecture 5.157 © 2010 Michael Stuart Why the log transform works High spread at high X transformed to low spread at high Y Low spread at low X transformed to high spread at low Y

58. Diploma in Statistics Design and Analysis of Experiments Lecture 5.158 © 2010 Michael Stuart Strategies for Experimenting –Consultation –Planning –Resources –Ethical issues –Implementation of design –Application of results

59. Diploma in Statistics Design and Analysis of Experiments Lecture 5.159 © 2010 Michael Stuart

60. Diploma in Statistics Design and Analysis of Experiments Lecture 5.160 © 2010 Michael Stuart Strategy for Experimentation Shewhart's PDCA Cycle

61. Diploma in Statistics Design and Analysis of Experiments Lecture 5.161 © 2010 Michael Stuart Strategy for Experimentation Shewhart's PDCA Cycle Plan:Plan a change to the process, predict its effect, plan to measure the effect Do:Implement the change as an experiment and measure the effect Check:Analyse the results to learn what effect the change had, if any Act:If successful, make the change permanent, proceed to plan the next improvement or if not, proceed to plan an alternative change

62. Diploma in Statistics Design and Analysis of Experiments Lecture 5.162 © 2010 Michael Stuart Strategy for Experimentation: new vs old manufacturing process Plan: Compare defect rates for old process and new (cheaper) process –predict reduction, or no increase, in number of defectives using new process Sample output over an eight week period, six days per week –select 50 components at random per day Count number of defectives per sample

63. Diploma in Statistics Design and Analysis of Experiments Lecture 5.163 © 2010 Michael Stuart Do: Implement plan Record daily numbers of defectives Assessing experimental process for manufacturing electronic components

64. Diploma in Statistics Design and Analysis of Experiments Lecture 5.164 © 2010 Michael Stuart Check: Analyse data test statistical significance of the change Assessing experimental process for manufacturing electronic components

65. Diploma in Statistics Design and Analysis of Experiments Lecture 5.165 © 2010 Michael Stuart Act: If no worse, make the change permanent, –proceed to plan the next improvement or if not, proceed to plan an alternative change Assessing experimental process for manufacturing electronic components

66. Diploma in Statistics Design and Analysis of Experiments Lecture 5.166 © 2010 Michael Stuart Resources e.g. sample size Need to know  Also, need to know €

67. Diploma in Statistics Design and Analysis of Experiments Lecture 5.167 © 2010 Michael Stuart Ethical issues –withholding medical treatment? –double-blind experiments, –inadequate budget puts patients at risk for non-informative results

68. Diploma in Statistics Design and Analysis of Experiments Lecture 5.168 © 2010 Michael Stuart Strategy When you see the credits roll at the end of a successful movie you realize there are many more things that must be attended to in addition to choosing a good script. Similarly in running a successful experiment there are many more things that must be attended to in addition to choosing a good experimental design. Ref:Robinson, G.K., Practical Strategies for Experimenting, Wiley, 2000.