Appendix 2

A2.1 Four-Factor Screening and Characterization Designs

Screening Main Effects in 8 Runs

This is an 8-run standard fraction (1/2 replicate) for four factors. The design allows you to estimate all main effects aliased only by three-factor or higher-order interactions.

Screening Design Layout

Std	A	B	C	D
1	–	–	–	–
2	+	–	–	+
3	–	+	–	+
4	+	+	–	–
5	–	–	+	+
6	+	–	+	–
7	–	+	+	–
8	+	+	+	+

Alias Structure

[Intercept] = Intercept

[A] = A

[B] = B

[C] = C

[D] = D

[AB] = AB + CD

[AC] = AC + BD

[AD] = AD + BC

Characterizing Interactions with 12 Runs

This is a 12-run irregular* fraction (3/4 replicate) for four factors. The design allows you to estimate all main effects and two-factor interactions (2fi) aliased only by three-factor or higher-order interactions. However, if effects are calculated hierarchically starting with main effects, these will be partially aliased with one or more interactions. In this case, be sure to review the probabilities in the ANOVA for the 2fi model. Exclude any main effects that are not significant.

Characterization Design Layout

Std	A	B	C	D
1	–	–	–	–
2	+	+	–	–
3	–	–	+	–
4	+	–	+	–
5	–	+	+	–
6	+	+	+	–
7	–	–	–	+
8	+	–	–	+
9	–	+	–	+
10	+	+	–	+
11	+	–	+	+
12	–	+	+	+

Alias Structure for Factorial Two-Factor Interaction Model

[Intercept] = Intercept − 0.5 * ABC − 0.5 * ABD

[A] = A − ACD

[B] = B − BCD

[C] = C − ABCD

[D] = D − ABCD

[AB] = AB − ABCD

[AC] = AC − BCD

[AD] = AD − BCD

[BC] = BC − ACD

[BD] = BD − ACD

[CD] = CD − ABD

[ABC] = ABC − ABD

Alias Structure for Factorial Main Effect Model

[Intercept] = Intercept − 0.333 * CD − 0.333 * ABC − 0.333 * ABD

[A] = A − 0.333 * BC − 0.333 * BD − 0.333 * ACD

[B] = B − 0.333 * AC − 0.333 * AD − 0.333 * BCD

[C] = C − 0.5 * AB

[D] = D − 0.5 * AB

A2.2 Five-Factor Screening and Characterization Designs

Screening Main Effects in 12 Runs

This is a minimum-run resolution IV design with 2 runs added in reserve to cover for any that cannot be performed or turn out to be statistically discrepant. The design allows you to estimate all main effects aliased only by three-factor or higher-order interactions.

Screening Design Layout

Std	A	B	C	D
1	+	+	+	–
2	–	+	–	+
3	+	–	–	–
4	+	–	+	+
5	+	+	–	+
6	–	–	+	–
7	–	–	–	+
8	+	–	+	–
9	–	+	+	+
10	–	+	–	–
11	–	–	+	–
12	+	+	–	+

Alias Structure

All main effects are aliased only with four-factor or higher order interactions, which aren’t worth noting. Be very wary of any interactions that look significant. Due to aliasing issues, these must be verified via a follow-up characterization design.

[Intercept] = Intercept + BD – CD + DE

[A] = A

[B] = B

[C] = C

[D] = D

[E] = E

[AB] = AB – BD – CE

[AC] = AC – BE – CD

[AD] = AD + BD – CD + DE

[AE] = AE + 2 * BD – BE – 2 * CD + CE + DE

[BC] = BC + 2 * BD – BE – 2 * CD + CE + 2 * DE

Characterizing Interactions with 16 Runs

This is a 16-run standard fraction (1/2 replicate) for five factors. The design allows you to estimate all main effects and two-factor interactions aliased only by three-factor or higher-order interactions.

Design Layout

Std	A	B	C	D	E
1	–	–	–	–	+
2	+	–	–	–	–
3	–	+	–	–	–
4	+	+	–	–	+
5	–	–	+	–	–
6	+	–	+	–	+
7	–	+	+	–	+
8	+	+	+	–	–
9	–	–	–	+	–
10	+	–	–	+	+
11	–	+	–	+	+
12	+	+	–	+	–
13	–	–	+	+	+
14	+	–	+	+	–
15	–	+	+	+	–
16	+	+	+	+	+

Alias Structure for Factorial Two-Factor Interaction (2FI) Model

[Intercept] = Intercept + ABCDE

[A] = A + ABCD

[B] = B + BCDE

[C] = C + ABDE

[D] = D + ABCE

[E] = E + ABCD

[AB] = AB + CDE

[AC] = AC + BDE

[AD] = AD + BCE

[AE] = AE + BCD

[BC] = BC + ADE

[BD] = BD + ACE

[BE] = BE + ACD

[CD] = CD + ABE

[CE] = CE + ABD

[DE] = DE + ABC

A2.3 Six-Factor Screening and Characterization Designs

Screening Main Effects in 14 Runs

This is a minimum-run resolution IV design (see boxed text in Section A2.2 above for details on these MR4 templates) with 2 runs added in reserve to cover for any that cannot be performed or turn out to be statistically discrepant.

Screening Design Layout

Std	A	B	C	D	E	F
1	–	+	–	+	+	–
2	–	–	–	+	–	–
3	–	+	+	+	–	+
4	–	–	+	–	+	–
5	–	–	–	–	–	+
6	+	–	+	+	+	+
7	–	+	–	–	+	+
8	+	–	+	–	–	+
9	+	+	+	–	+	+
10	+	–	–	–	+	–
11	+	+	–	+	–	+
12	+	+	+	+	+	–
13	–	+	–	–	–	–
14	+	–	+	+	–	–

Alias Structure

All main effects are aliased only with four-factor or higher order interactions, which are not worth noting. Be very wary of any interactions that look significant. Due to aliasing issues, these must be verified via a follow-up characterization design.

[Intercept] = Intercept + BF – DE

[A] = A

[B] = B

[C] = C

[D] = D

[E] = E

[F] = F

[AB] = AB + 0.5 * BC + 0.5 * BE + 0.5 * BF – 0.5 * CD + 0.5 * CE – 0.5 * CF – 0.5 * DE – 0.5 * DF

[AC] = AC + BE – BF + DE – DF

[AD] = AD – 0.5 * BC – 0.5 * BE – 0.5 * BF + 0.5 * CD + 0.5 * CE – 0.5 * CF + 0.5 * DE + 0.5 * DF

[AE] = AE + 0.5 * BC – 0.5 * BE – 0.5 * BF + 0.5 * CD + 0.5 * CE + 0.5 * CF + 0.5 * DE + 0.5 * DF

[AF] = AF – 0.5 * BC – 0.5 * BE – 0.5 * BF – 0.5 * CD + 0.5 * CE + 0.5 * CF + 0.5 * DE + 0.5 * DF

[BD] = BD – BF + DE – EF

Characterizing Interactions with 22 Runs

This is a minimum-run resolution V design (see boxed text in Section 2.2 above for details on these MR5 templates). The design allows you to estimate all main effects and two-factor interactions aliased only by three-factor or higher-order interactions.

Design Layout

Std	A	B	C	D	E	F
1	–	–	+	+	–	+
2	+	+	–	–	–	–
3	–	–	–	–	+	–
4	+	+	+	+	–	–
5	–	+	–	+	–	+
6	+	+	+	–	–	+
7	+	–	–	+	–	+
8	–	+	–	+	+	–
9	+	–	+	–	–	–
10	+	–	+	+	+	+
11	+	–	–	+	+	–
12	+	+	–	+	+	+
13	–	+	–	–	+	+
14	+	+	+	–	–	–
15	–	–	+	+	+	–
16	+	–	–	–	+	+
17	–	+	+	–	–	–
18	–	–	–	–	–	+
19	–	–	–	+	–	–
20	–	+	+	+	+	+
21	+	+	+	–	+	–
22	–	–	+	–	+	+

Alias Structure for Factorial Two-Factor Interaction (2FI) Model

Each main effect and two-factor interaction is partially aliased only with three-factor interactions. Only the first main effect ([A]) and two-factor interaction ([AB]) on the entire list is shown below. The other aliasing is very similar, so this provides an idea of what to expect.

[Intercept] = Intercept

[A] = A + 0.333 * BCD + 0.333 * BCE + 0.333 * BCF – 0.333 * BDE – 0.333 * BDF – 0.333 * BEF – 0.333 * CDE – 0.333 * CDF – 0.333 * CEF + 0.333 * DEF …

[AB] = AB – 0.333 * ACD – 0.333 * ACE – 0.333 * ACF + 0.333 * ADE + 0.333 * ADF + 0.333 * AEF – 0.333 * BCD – 0.333 * BCE – 0.333 * BCF + 0.333 * BDE + 0.333 * BDF + 0.333 * BEF – 0.333 * CDE – 0.333 * CDF – 0.333 * CEF + 0.333 * DEF …

A2.4 Seven-Factor Screening and Characterization Designs

Screening Main Effects in 16 Runs

This is a 16-run standard fraction (1/8 replicate) for seven factors. Main effects are clear of two-factor interactions. Two-factor interactions are completely aliased with each other.

Screening Design Layout

Std	A	B	C	D	E	F	G
1	–	–	–	–	–	–	–
2	+	–	–	–	+	–	+
3	–	+	–	–	+	+	–
4	+	+	–	–	–	+	+
5	–	–	+	–	+	+	+
6	+	–	+	–	–	+	–
7	–	+	+	–	–	–	+
8	+	+	+	–	+	–	–
9	–	–	–	+	–	+	+
10	+	–	–	+	+	+	–
11	–	+	–	+	+	–	+
12	+	+	–	+	–	–	–
13	–	–	+	+	+	–	–
14	+	–	+	+	–	–	+
15	–	+	+	+	–	+	–
16	+	+	+	+	+	+	+

Alias Structure

[Intercept] = Intercept

[A] = A + BCE + BFG + CDG + DEF

[B] = B + ACE + AFG + CDF + DEG

[C] = C + ABE + ADG + BDF + EFG

[D] = D + ACG + AEF + BCF + BEG

[E] = E + ABC + ADF + BDG + CFG

[F] = F + ABG + ADE + BCD + CEG

[G] = G + ABF + ACD + BDE + CEF

[AB] = AB + CE + FG

[AC] = AC + BE + DG

[AD] = AD + CG + EF

[AE] = AE + BC + DF

[AF] = AF + BG + DE

[AG] = AG + BF + CD

[BD] = BD + CF + EG

Characterizing Interactions with 30 Runs

This is a minimum-run resolution V design (see boxed text in section A2.2 above for details on these MR5 templates). The design allows you to estimate all main effects and two-factor interactions aliased only by three-factor or higher-order interactions.

Design Layout

Std	A	B	C	D	E	F	G
1	–	–	–	+	–	–	+
2	–	+	–	–	+	–	–
3	–	+	+	+	–	–	+
4	–	–	+	–	–	–	+
5	+	+	+	+	+	+	+
6	–	–	–	–	+	+	–
7	+	+	–	–	–	+	–
8	–	+	–	+	+	+	–
9	+	–	+	–	+	–	–
10	+	+	–	+	–	+	+
11	–	–	–	+	–	+	–
12	–	–	–	+	+	+	+
13	–	–	+	+	+	–	–
14	+	–	–	+	+	–	–
15	+	+	–	+	–	–	–
16	–	+	+	–	–	–	–
17	+	+	–	–	+	+	+
18	+	+	+	–	–	+	+
19	–	+	+	–	+	+	–
20	+	+	+	+	+	–	–
21	+	–	–	–	+	–	+
22	+	–	–	–	–	+	+
23	+	–	+	+	+	+	–
24	–	–	+	–	–	+	–
25	+	–	+	+	+	–	–
26	–	+	+	–	–	–	+
27	+	+	–	–	–	–	+
28	–	–	+	+	–	+	+
29	+	–	+	+	+	–	+
30	–	+	–	–	–	+	+

Alias Structure for Factorial Two-Factor Interaction (2FI) Model

Each main effect and two-factor interaction is partially aliased only with three-factor interactions. Only the first main effect ([A]) and two-factor interaction ([AB]) on the entire list is shown below. The other aliasing is very similar so this provides an idea of what to expect.

[Intercept] = Intercept – 0.0667 * ABC + 0.2 * ABD – 0.333 * ABE + 0.333 * ABF + 0.0667 * ABG + 0.333 * ACD + 0.333 * ACE + 0.2 * ACF – 0.0667 * ACG + 0.0667 * ADE – 0.0667 * ADF – 0.333 * ADG – 0.0667 * AEF + 0.2 * AEG + 0.333 * AFG + 0.133 * BCE + 0.667 * BCG + 0.133 * BEF + 0.133 * BFG + 0.133 * CDE + 0.133 * CDG + 0.667 * DEF + 0.133 * DFG

[A] = A – 0.467 * ABC – 0.6 * ABD – 0.333 * ABE + 0.333 * ABF + 0.467 * ABG + 0.333 * ACD + 0.333 * ACE – 0.6 * ACF – 0.467 * ACG + 0.467 * ADE – 0.467 * ADF – 0.333 * ADG – 0.467 * AEF – 0.6 * AEG + 0.333 * AFG – 0.0667 * BCE – 0.333 * BCG – 0.0667 * BEF – 0.0667 * BFG – 0.0667 * CDE – 0.0667 * CDG – 0.333 * DEF – 0.0667 * DFG …

[AB] = AB + 0.25 * BCD + 0.417 * BCE + 0.0833 * BCF – 0.417 * BCG + 0.25 * BDE– 0.25 * BDF – 0.25 * BDG – 0.417 * BEF + 0.0833 * BEG + 0.417 * BFG + 0.583 * CDE – 0.0833 * CDF – 0.583 * CDG – 0.417 * CEF – 0.417 * CEG + 0.417 * CFG – 0.583 * DEF – 0.0833 * DEG + 0.583 * DFG + 0.417 * EFG …

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