Lecture 13

• Today: 4.3-4.6

• Next day:

• Assignment #4: Chapter 4 - 13 (a,b), 14, 15, 23, additional question on D-optimality

Optimal Design Approach (4.4.2)

• Algorithm:

Assignment Question

• Suppose in the cable shrinkage example, effects A, E and AC=BE are identified as signifincat

• To resolve the aliasing of the interaction effects, a follow-up experiment with 4 trials is to be performed

• What 4 trails should be performed?

• Use the D-optimality criterion and report the value of Dmax

Additional Features of a Fractional Factorial

• Main effect or two-factor interactions (2fi) is clear if it is not aliased with other main effects or 2fi’s

• Main effect or 2fi is strongly clear if it is not aliased with other main effects, 2fi’s or 3fi’s

Blocking Fractional Factorial Designs

• Can perform a 2k-p fractional factorial design in 2q blocks

• That is, k factors are investigated in 2k-p runs with 2q blocks

• The design is constructed by assigning p treatment factors and q blocking factors to interactions between (k-p) of the factors

Example

• An experimenter wishes to explore the impact of 6 factors (A-F) on the response of a system

• There exists enough resources to run 16 experiment trials in 4 blocks

• A 26-2 fraction factorial design in 22 blocks is required

Example

• Design:

– Fractional factorial: E=ABC; F=ABD

– Blocking: b1=ACD; b2=BCD

• Defining Contrast sub-group:

Example

A B C D AB AC AD BC BD CD ABC ABD ACD BCD ABCD-1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 1 -1-1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1-1 -1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1-1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 1 -1-1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1-1 1 -1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1-1 1 1 1 -1 -1 -1 1 1 1 -1 -1 -1 1 -1-1 1 1 -1 -1 -1 1 1 -1 -1 -1 1 1 -1 11 -1 -1 1 -1 -1 1 1 -1 -1 1 -1 -1 1 11 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -11 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -11 -1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 11 1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 -11 1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1

Comment

• Must be careful when choosing the interactions to assign the factors

– Fractional factorial: E=AB; F=ABD

– Blocking: b1=ACD; b2=BCD

• Defining Contrast sub-group:

Additional Features

• Main effect or two-factor interactions (2fi) is clear if it is not aliased with other main effects, 2fi’s or block effects

• Main effect or 2fi is strongly clear if it is not aliased with other main effects, 2fi’s, 3fi’s or block effects

• As before, block by factor interactions are negligible

• Analysis is same as before

• Appendix 4 has blocked fractional factoria designs ranked by number of clear effects

Fractional Factorial Split-Plot Designs

• It is frequently impractical to perform the fractional factorial design in a completely randomized manner

• Can run groups of treatments in blocks

• Sometimes the restrictions on randomization take place because some factors are hard to change or the process takes place in multiple stages

• Fractional factorial split-plot (FFSP) design may be a practical option

Performing FFSP Designs

• Randomization of FFSP designs different from fractional factorial designs

• Have hard to change factors (whole-plot or WP factors) and easy to change factors (sub-plot or SP factors)

• Experiment performed by:

– selecting WP level setting, at random.

– performing experimental trials by varying SP factors, while keeping the WP factors fixed.

Example

• Would like to explore the impact of 6 factors in 16 trials

• The experiment cannot be run in a completely random order because 3 of the factors (A,B,C) are very expensive to change

• Instead, several experiment trials are performed with A, B, and C fixed…varying the levels of the other factors

Design Matrix

A B C p q r-1 -1 +1 -1 -1 -1

+1 -1 +1-1 +1 +1+1 +1 -1

+1 -1 -1 -1 -1 +1+1 -1 -1-1 +1 -1+1 +1 +1

-1 +1 -1 -1 -1 -1+1 -1 +1-1 +1 +1+1 +1 -1

+1 +1 +1 -1 -1 +1+1 -1 -1-1 +1 -1+1 +1 +1

Impact of the Randomization Restrictions

• Two Sources of randomization Two sources of error– Between plot error: ew (WP error)

– Within plot error: (SP error)

• Model:

• The WP and SP error terms have mutually independent normal distributions with standard deviations σw and σs

SWeXy

s

The Design

• Situation:– Have k factors: k1 WP factors and k2 SP factors

– Wish to explore impact in 2k-p trials

– Have a 2 k1-p1 fractional factorial for the WP factors

– Require p=p1+p2 generators

– Called a 2(k1+ p

2)-(k

1+ p

2) FFSP design

s

Constructing the Design

• For a 2(k1+ p

2)-(k

1+ p

2) FFSP design, have generators for WP and SP

designs

• Rules:

– WP generators (e.g., I=ABC ) contain ONLY WP factors

– SP generators (e.g., I=Apqr ) must contain AT LEAST 2 SP factors

• Previous design: I=ABC=Apqr=BCpqr

Analysis of FFSP Designs

• Two Sources of randomization Two sources of error– Between plot error: σw (WP error).

– Within plot error: σs (SP error).

• WP Effects compared to: aσs2

+ bσs2

• SP effects compared to : bσs2

• df for SP df for WP.

• Get more power for SP effects!!!

WP Effect or SP Effect?

• Effects aliased with WP main effects or interactions involving only WP factors tested as a WP effect.

• E.g., pq=ABCD tested as a WP effect.

• Effects aliased only with SP main effects or interactions involving at least one SP factors tested as a SP effect .

• E.g., pq=ABr tested as a SP effect.

Ranking the Designs

• Use minimum aberration (MA) criterion