Building Models for Complex Design of Experiments

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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
 Because both whole plot and subplot are arranged as
CR blocks, both Fit Models produce the same results.
JMP DefaultCR Blocks

19
Runs 20 – 42

20
 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
 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
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
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
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
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
Split-Split and Strip Plots
Split-Split
Strip

27
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
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
Additional Designs

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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
Other Designs: Split Plots
 Split-Split-Split
 Strip with multiple treatments assigned to the strips.

Building Models for Complex Design of Experiments

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