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 3 Factor 2 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°

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 Default CR 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

8 7

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

B2 B1 B2 B1

A1

A1

A2

A2

1

2

3

4

1

2

3 4 WP2

WP1

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

B2 B1 B2 B1

A1

A1

A2

A2

Rep - 1

26


Split-Split and Strip Plots

Split-Split

Strip

27


Example – Split-Strip Plot

Fertilizer

S3 S2 S1

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

30


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 Testing Radar Detection

31


Other Designs: Split Plots

Split-Split-Split

Strip with multiple treatments assigned to the strips.

building models for complex design of experiments

Technology

strain b factor

sas institute

factor changes

affected factor

restricted factor

growth media factor

plot oven run

splitsplit plots crossover