Recall: Conditional Main Effect Consider two factors A and B, each at two levels denoted by + and – : Interaction effect: Conditional main effect of A given B at + : CME 𝐴 𝐵+ = 𝑦 𝐴+ 𝐵+ − 𝑦 (𝐴−|𝐵+). Conditional main effect of A given B at −: CME 𝐴 𝐵− = 𝑦 𝐴+ 𝐵− − 𝑦 (𝐴−|𝐵−). Switching the roles of A and B, CME 𝐵 𝐴+ and CME 𝐵 𝐴− can be similarly defined. Main effect : 𝑀𝐸 𝐴 = 𝑦 𝐴+ − 𝑦 𝐴− = 1 2 𝑦 𝐴+|𝐵+ + 𝑦 𝐴+ 𝐵− − 1 2 𝑦 𝐴−|𝐵+ + 𝑦 𝐴− 𝐵− . (1) 𝐼𝑁𝑇(𝐴,𝐵)= 1 2 𝑦 𝐴+|𝐵+ − 𝑦 𝐴+ 𝐵− − 1 2 𝑦 𝐴−|𝐵+ − 𝑦 𝐴− 𝐵− . (2) 1

Rule 1 of CME Analysis By adding 𝑀𝐸(𝐴) and 𝐼𝑁𝑇(𝐴,𝐵) (Check (1)+(2)), By subtracting 𝑀𝐸(𝐴) and 𝐼𝑁𝑇(𝐴,𝐵), 𝑀𝐸(𝐴) − 𝐼𝑁𝑇(𝐴,𝐵) = CME (𝐴|𝐵 −). (4) If 𝑀𝐸(𝐴) and 𝐼𝑁𝑇(𝐴,𝐵) have the same sign and are comparable in magnitude, we can replace 𝑀𝐸(𝐴) and 𝐼𝑁𝑇(𝐴,𝐵) by CME(𝐴|𝐵 +). Similarly, when 𝑀𝐸(𝐴) and 𝐼𝑁𝑇(𝐴,𝐵) have the opposite sign, they can be replaced by CME(𝐴|𝐵 −). Rule 1: Substitute a pair of interaction effect and its parental main effect that have similar magnitudes with one of the corresponding two CMEs. Note: It achieves model parsimony (why?) 2

A 𝟐 𝐈𝐕 𝟒−𝟏 design and CMEs For CME(𝐴|𝐵+), we call 𝑀𝐸(𝐴) its parent effect and 𝐼𝑁𝑇(𝐴,𝐵) its interaction effect. Use 𝐴 𝐵+ , etc. as its shorthand notation. Table 1: CMEs and Factorial Effects from the 2 𝐼𝑉 4−1 Design with I=𝐴𝐵𝐶𝐷 3

Siblings and Family CMEs having the same parent effect and interaction effects are called twin effects, e.g., CME 𝐴 𝐵+ and CME 𝐴 𝐵− . CMEs having the same parent effect but different interaction effects are called siblings effects, e.g., CME 𝐴 𝐵+ and CME 𝐴 𝐶+ . The group of CMEs having the same or aliased interaction effects belongs to the same family, e.q., CME 𝐴 𝐵+ and CME 𝐶 𝐷− in Table 1.

More Relationships Summary of the relationships between various CMEs CMEs are orthogonal to all the traditional effects except for their parent effects and interaction effects. Sibling CMEs are not orthogonal to each other. CMEs in the same family are not orthogonal. CMEs with different parent effects and different interaction effects are orthogonal. (Example: 𝐴 𝐵+ and 𝐵 𝐷+ in the Table.)

Rules 2 and 3 of CME Analysis Only one CME among its siblings can be included in the model. Only one CME from a family can be included in the model. Rule 3: CMEs with different parent effects and different interaction effects can be included in the same model. Justification: In order to avoid generating too many incompatible models, only orthogonal effects are included in the model search.

CME Analysis Use the traditional analysis methods such as ANOVA or half- normal plot, to select significant effects, including aliased pairs of effects. Go to ii. Among all the significant effects, use Rule 1 to find a pair of interaction effect and its parental main effect, and substitute them with an appropriate CME. Use Rules 2 and 3 to guide the search and substitution of other such pairs until they are exhausted. In step i, a formal method like Lenth’s method (Section 4.9) can be used instead of the half-normal plots.

Illustration with Filtration Experiment Four factors: Temperature (A) Pressure (B) Concentration of formaldehyde (C) Stirring rate (D) design with I=𝐴𝐵𝐶𝐷, aliasing relations like 𝐴𝐵=𝐶𝐷, 𝐴𝐶=𝐵𝐷, etc.

Illustration with Filtration Experiment 2 𝐼𝑉 4−1 design with 𝐼=𝐴𝐵𝐶𝐷 Traditional analysis: 𝑦~𝐴+𝐴𝐷+𝐴𝐶+𝐷+𝐶 Step (ii) A and AD are both significant Consider A|D+ since A and D have same sign D and DB are both significant Consider D|B− since D and B have opposite sign The CME analysis (A|D +) (D|B −)

Summary of Filtration Experiment In the traditional analysis, we have: 𝑦~𝐴 0.45% +𝐴𝐷 0.45% +𝐴𝐶 0.47% +𝐷 0.59% +𝐶 0.82% . 𝑅 2 =99.79% In the CME analysis, we have: 𝑦~ 𝐴 𝐷+ 0.013% +𝐴𝐶 0.039% +𝐷 0.055% +𝐶 0.089% . 𝑅 2 =99.79% 𝑅 2 =99.79% 𝑦~ 𝐴 𝐷+ 1.96× 10 −5 + 𝐷 𝐵− 2.72× 10 −5 +𝐶 0.026% . 𝑅 2 =99.66% 𝑅 2 =99.66% The third model is the most parsimonious and best in terms of p values for significant effects. All three models have comparable 𝑅 2 values. The CMEs (A|D +) and (D|B −) in the last two models have good engineering interpretations.