Copyright © 2010 SAS Institute Inc. All rights reserved.
Building Models for
Complex DOEs
Donald McCormack, JMP
2
Copyright © 2010, SAS Institute Inc. All rights reserved.
Intro
 Basic Designs
 Adding nuisance variables – Latin Squa...
3
Copyright © 2010, SAS Institute Inc. All rights reserved.
Basic Designs
 Typical DOE − Completely Randomized Design (CR...
4
Copyright © 2010, SAS Institute Inc. All rights reserved.
Basic Designs
 Typical DOE −
Completely Randomized Block Desi...
5
Copyright © 2010, SAS Institute Inc. All rights reserved.
Latin Squares
 Two blocking variables, rows and columns, used...
6
Copyright © 2010, SAS Institute Inc. All rights reserved.
Latin Squares - Examples
 Emissions
 Box, Hunter, & Hunter p...
7
Copyright © 2010, SAS Institute Inc. All rights reserved.
Latin Squares - Summary
 Treat nuisance (blocking) variables ...
8
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots
 Am I free to let any factors change at any run?
...
9
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots
 Two columns are needed
 One for the block (nois...
10
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots – Set Up: Example
 Heat treatment in oven.
 Thr...
11
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots – Set Up: Example Scenario 1
 Only one temperatu...
12
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots – Set Up: Example Scenario 2
 Include Oven Run.
...
13
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots – Summary
 The hard to change/batch factor needs...
14
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split-Split and Strip Plots
 Randomization restriction on tw...
15
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots: Two Hard to Change Factors
Change Simultaneously...
16
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots: Two Hard to Change Factors
Change Simultaneously...
17
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots: Split-Split Plot
 Two additional sources of err...
18
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots: Split-Split Plot
 Because both whole plot and s...
19
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots: Split-Split Plot
Runs 20 – 42
20
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots: Split-Split Plot
 How to ID the blocks – Whole ...
21
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split Plots: Split-Split Plot
 How to ID the blocks – Subplo...
22
Copyright © 2010, SAS Institute Inc. All rights reserved.
Strip Plots: Example
 Two step semiconductor process: ion im...
23
Copyright © 2010, SAS Institute Inc. All rights reserved.
Strip Plots: Example
1, 1, 1
Implant
1, 1, -1
1, -1, 1
-1, 1,...
24
Copyright © 2010, SAS Institute Inc. All rights reserved.
Strip Plots
 How to ID the blocks – CR blocks
A1
A2
A1
A2
B1...
25
Copyright © 2010, SAS Institute Inc. All rights reserved.
Strip Plots
 How to ID the blocks – RCB blocks
 Count each ...
26
Copyright © 2010, SAS Institute Inc. All rights reserved.
Split-Split and Strip Plots
Split-Split
Strip
27
Copyright © 2010, SAS Institute Inc. All rights reserved.
Example – Split-Strip Plot
F
e
r
t
i
l
i
z
e
r
S3S2S1
Soil Ty...
28
Copyright © 2010, SAS Institute Inc. All rights reserved.
Crossover Designs
 Only one random effect – Subject[Sequence...
29
Copyright © 2010, SAS Institute Inc. All rights reserved.
Additional Designs
30
Copyright © 2010, SAS Institute Inc. All rights reserved.
Other Designs: Latin Squares
 Two factor full factorial in L...
31
Copyright © 2010, SAS Institute Inc. All rights reserved.
Other Designs: Split Plots
 Split-Split-Split
 Strip with m...
Copyright © 2010 SAS Institute Inc. All rights reserved.
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Building Models for Complex Design of Experiments

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This talk was presented live at JMP Discovery Summit 2012 in Cary, North Carolina, USA. More information about design of experiments is available at http://www.jmp.com/applications/doe/

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Building Models for Complex Design of Experiments

  1. 1. Copyright © 2010 SAS Institute Inc. All rights reserved. Building Models for Complex DOEs Donald McCormack, JMP
  2. 2. 2 Copyright © 2010, SAS Institute Inc. All rights reserved. Intro  Basic Designs  Adding nuisance variables – Latin Squares  When blocks matter – Split Plots  Three random effects – Strip and Split-Split Plots  Crossover Designs  Other designs – Split Plot and Latin Square variations.
  3. 3. 3 Copyright © 2010, SAS Institute Inc. All rights reserved. Basic Designs  Typical DOE − Completely Randomized Design (CRD) Temp: 25° Temp: 30° pH: 6.0 pH: 7.0 Strain A Strain B Factor 3Factor 2Factor 1 A, 6.0, 30° B, 7.0, 25° A, 6.0, 25° B, 6.0, 30° B, 7.0, 30° A, 7.0, 30° A, 7.0, 25° B, 6.0, 25°
  4. 4. 4 Copyright © 2010, SAS Institute Inc. All rights reserved. Basic Designs  Typical DOE − Completely Randomized Block Design (CRBD) Temp: 25° Temp: 30° Factor 3 pH: 6.0 pH: 7.0 Factor 2 Strain A Strain B Factor 1 A, 6.0, 30° B, 7.0, 25° A, 6.0, 25° B, 6.0, 30° B, 7.0, 30° A, 7.0, 30° A, 7.0, 25° B, 6.0, 25° CRD1 Growth Media 1 B, 6.0, 25° A, 7.0, 25° A, 6.0, 30° A, 7.0, 30° B, 7.0, 30° B, 7.0, 25° A, 6.0, 25° B, 6.0, 30° CRD2 Growth Media 2 Growth Media Factor 4
  5. 5. 5 Copyright © 2010, SAS Institute Inc. All rights reserved. Latin Squares  Two blocking variables, rows and columns, used for nuisance variables.  Two restrictions on randomization – there must be unique combinations of treatments across rows and down columns.  Number of levels must be identical for row, column, and treatment variables.  Assumption: No two way or higher interaction between row, column, and treatment factors.  More than two nuisance variables? Graeco-Latin and Hyper-Graeco Latin designs.  JMPer Cable Spring 2002
  6. 6. 6 Copyright © 2010, SAS Institute Inc. All rights reserved. Latin Squares - Examples  Emissions  Box, Hunter, & Hunter p. 157  Fuel additive is the treatment.  Drivers and cars are blocking variables, 4 of each.  Emissions 2  Example 1 with two replicated LS  Same Drivers and Cars? 1 2 3 4 1 A B D C 2 D C A B 3 B D C A 4 C A B D Emissions Example Car Driver
  7. 7. 7 Copyright © 2010, SAS Institute Inc. All rights reserved. Latin Squares - Summary  Treat nuisance (blocking) variables as random effects  Unbound the variance components  No nesting or crossing unless there is replication  If there are different sets of nuisance variables across replication, nest the nuisance variable in the replication variable. For example, if the cars in Rep 1 were different than the cars in Rep two, next Car in Rep (Car [Rep]).
  8. 8. 8 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots  Am I free to let any factors change at any run?  Yes – CRD  No, I have to restrict where, when, or how often one or more factors is changed. » Test for statistical differences in at least one restricted factor? » No – RCBD, Latin Square » Yes – Split Plot  What’s the difference?  RCBD, Latin Square – I’m estimating (nuisance) variability so it can be removed from experimental variability.  Split Plot – I’m estimating both the signal and noise variability of the affected factor and comparing the former to the later as my statistical test.
  9. 9. 9 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots  Two columns are needed  One for the block (noise variability)  One for the factor (signal)  Two ways block column can be arranged:  CR – Each time a factor level changes the block ID changes.  RCB – Blocks correspond to groups of unrepeated factor levels.  The nature of the factor often dictates whether you’ll have CR or RCB blocks. Customer Designer uses CR.  You’ll need at least the number of factor levels plus one CR blocks or two RCBD blocks with the same level appearing at least once in both blocks. More is better.  Block arrangement affects how the model is built.
  10. 10. 10 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots – Set Up: Example  Heat treatment in oven.  Three factors: Temperature, Time, and Power.  Oven can fit four units.  Scenario 1 – Only one temp per oven run.  Scenario 2 – Two temperature zones in an oven with two items per zone.
  11. 11. 11 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots – Set Up: Example Scenario 1  Only one temperature per whole plot (Oven Run). Set Temp to Nominal and nest Oven Run in Temp.  JMP default –Leave Temp continuous and ignore the nesting (keep Oven Run random). You’ll get the same results.  In both cases, use REML and unbounded variance components. Oven Run as CR Block JMP Default Both give the same results
  12. 12. 12 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots – Set Up: Example Scenario 2  Include Oven Run.  Cross Temp with Oven Zone.  Make both Random.  Oven Run*Temp&Random is used as the noise estimate to test for differences in Temp. It removes the run to run variability between ovens.
  13. 13. 13 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots – Summary  The hard to change/batch factor needs two columns, one for the factor and one for the block  CR blocks  Each time the factor changes so does the block ID  Nest the block variable in the hard to change/batch factor. Make it a random effect.  You can also use the JMP default and ignore the nesting.  RCB blocks  Group sets of the factor changes into blocks such that no level is repeated in a given block.  Cross the hard to change factor with the block factor and make it random.
  14. 14. 14 Copyright © 2010, SAS Institute Inc. All rights reserved. Split-Split and Strip Plots  Randomization restriction on two factors A1B1 A2B1 A1B2 A2B2 B1 B2 A1 B1 B2 A2 Split Split-Split Strip A1 A2 A1 A2 B1 B2 B1 B2
  15. 15. 15 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots: Two Hard to Change Factors Change Simultaneously  Just like a split plot: one additional source of error.  CR Block – ID changes if either factor changes.  RCB Block – Grouping based on unique combinations of both factors. CR Blocks RCB Blocks JMP Default
  16. 16. 16 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots: Two Hard to Change Factors Change Simultaneously  How to ID the blocks A1B1 A2B1 A1B2 A2B2 A1B1 A2B1 A1B2 A2B2 1 2 2 5 4 3 6 7 8 1 CR Blocks RCB Blocks
  17. 17. 17 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots: Split-Split Plot  Two additional sources of error: whole plot and subplot  Subplot is more frequently changing, but still restricted, block inside of whole plots. Whole plots are very hard to change and subplot are hard to change.  Example: High throughput reactor (see Castillo, Quality Engineering 2010) Reactor Module Temperature Pressure Catalyst Type Concentration Reactor Block Purge Type
  18. 18. 18 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots: Split-Split Plot  Because both whole plot and subplot are arranged as CR blocks, both Fit Models produce the same results. JMP DefaultCR Blocks
  19. 19. 19 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots: Split-Split Plot Runs 20 – 42
  20. 20. 20 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots: Split-Split Plot  How to ID the blocks – Whole Plots B1 B2 A1 B1 B2 A2 B1 B2 A1 B1 B2 A2 2 3 4 1 2 1 CR Blocks RCB Blocks
  21. 21. 21 Copyright © 2010, SAS Institute Inc. All rights reserved. Split Plots: Split-Split Plot  How to ID the blocks – Subplots B1 B2 A1 B1 B2 A2 B1 B2 A1 B1 B2 A2 2 3 4 1 1 RCB Blocks 87 2 3 6 4 5 CR Blocks
  22. 22. 22 Copyright © 2010, SAS Institute Inc. All rights reserved. Strip Plots: Example  Two step semiconductor process: ion implant followed by a thermal anneal.  Implant: Three factors – O+ Dose, Energy, Implant Temp  Anneal: Three factors - O+ Conc, Anneal Temp, Time  Both are batch processes.  The treatment combinations for each step come from a full factorial (32) plus center point. Nine unique combinations possible.  Nine wafers are processed at each step.  For each implant run (i.e., for a unique implant treatment combination) randomly assign each wafer to a unique anneal treatment combination.  Replicate the experiment for 162 wafers total.
  23. 23. 23 Copyright © 2010, SAS Institute Inc. All rights reserved. Strip Plots: Example 1, 1, 1 Implant 1, 1, -1 1, -1, 1 -1, 1, 1 -1, -1, -1 1, 1, 1 Anneal 1, 1, -1 1, -1, 1 -1, 1, 1 -1, -1, -1 9 wafers each step 1 wafer from each implant step randomly assigned to anneal step X 2
  24. 24. 24 Copyright © 2010, SAS Institute Inc. All rights reserved. Strip Plots  How to ID the blocks – CR blocks A1 A2 A1 A2 B1 B2 B1 B2 1 2 3 4 1 2 3 4 WP1 WP2 B2B1B2B1 A1 A1 A2 A2 1 2 3 4 1 2 3 4 WP2 W P 1
  25. 25. 25 Copyright © 2010, SAS Institute Inc. All rights reserved. Strip Plots  How to ID the blocks – RCB blocks  Count each set of treatment combinations A1 A2 A1 A2 B1 B2 B1 B2 Rep - 1 B2B1B2B1 A1 A1 A2 A2 Rep - 1
  26. 26. 26 Copyright © 2010, SAS Institute Inc. All rights reserved. Split-Split and Strip Plots Split-Split Strip
  27. 27. 27 Copyright © 2010, SAS Institute Inc. All rights reserved. Example – Split-Strip Plot F e r t i l i z e r S3S2S1 Soil Type Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 Ca0 Ca1 F0 F1 F2 F3
  28. 28. 28 Copyright © 2010, SAS Institute Inc. All rights reserved. Crossover Designs  Only one random effect – Subject[Sequence]  Biggest challenge is setting up the dataset to estimate the carryover effect.  Example - Three periods, two treatments  JMPer Cable Fall 2006
  29. 29. 29 Copyright © 2010, SAS Institute Inc. All rights reserved. Additional Designs
  30. 30. 30 Copyright © 2010, SAS Institute Inc. All rights reserved. Other Designs: Latin Squares  Two factor full factorial in LS: Radar Detection  Montgomery DOE 7th Ed, table 5.23  Hyper-Graeco-Latin Square: Wear testing  Box, Hunter, & Hunter p. 163 Wear TestingRadar Detection
  31. 31. 31 Copyright © 2010, SAS Institute Inc. All rights reserved. Other Designs: Split Plots  Split-Split-Split  Strip with multiple treatments assigned to the strips.
  32. 32. Copyright © 2010 SAS Institute Inc. All rights reserved.
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