Recall: Conditional Main Effect

1. 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. • Conditional main effect of A given B at : • CME. • Switching the roles of A and B, CME and CME can be similarly defined. (1) (2) 1

2. Rule 1 of CME Analysis • By adding 𝑀𝐸(𝐴) and 𝐼𝑁𝑇(𝐴,𝐵) (Check (1)+(2)), • 𝑀𝐸(𝐴) + 𝐼𝑁𝑇(𝐴,𝐵) = CME(𝐴|𝐵). (3) • 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 signand 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

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

4. Siblings and Family • CMEs having the same parent effect and interaction effects are called twin effects, e.g.,CME and CME. • CMEs having the same parent effect but different interaction effects are called siblings effects, e.g.,CME and CME • The group of CMEs having the same or aliased interaction effects belongs to the same family, e.q., CME and CMEin Table 1.

5. 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: and in the Table.)

6. Rules 2 and 3 of CME Analysis • Rule 2: • 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.

7. 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.

8. Illustration with Filtration Experiment • Four factors: • Temperature (A) • Pressure (B) • Concentration of formaldehyde (C) • Stirring rate (D) • design with , aliasing relations • like , etc.

9. Illustration with Filtration Experiment • design with • Traditional analysis: • Step (ii) • A and AD are both significant • Consider A|D sinceAandDhavesamesign • Dand DB are both significant • ConsiderD|B since D and B have opposite sign The CME analysis (A|D) (D|B)

10. Summary of Filtration Experiment • In the traditional analysis, we have: • In the CME analysis, we have: • The third model is the most parsimonious and best in terms of p values for significant effects. All three models have c values. • The CMEs (A|D ) and (D|B ) in the lasttwo models have good engineering interpretations.