1. CME Analysis: a New Method for Unraveling Aliased Effects in Two-Level Fractional Factorial Experiments
C. F. Jeff Wu (Joint work with Heng Su)
School of Industrial and Systems Engineering
Georgia Institute of Technology
Introduction
Properties
Method of analysis
Examples
Concluding remarks

2. Introduction Ever since the founding work by D. J. Finney
It has been widely known and accepted that aliased effects in two- level regular designs cannot be de-aliased without adding more runs
The following was shown in Wu (2011, Fisher Lecture; 2015 JASA to appear):
By using a new framework based on the conditional main effect (CME) the aliasing problems can sometimes be resolved

3. What is a conditional main effect (cme)
Consider two factors A and B, each at two levels denoted by + and –
Main effect of A: 𝑀𝐸 𝐴 = 𝑦 𝐴+ − 𝑦 𝐴− .
𝑀𝐸 𝐴 = 𝑦 𝐴+|𝐵+ + 𝑦 𝐴+ 𝐵− − 𝑦 𝐴−|𝐵+ + 𝑦 𝐴− 𝐵− (1)
Two-factor interaction of A and B:
𝐼𝑁𝑇 𝐴,𝐵 = 𝑦 𝐴+|𝐵+ − 𝑦 𝐴− 𝐵− − 𝑦 𝐴+|𝐵− − 𝑦 𝐴− 𝐵 (2)
Conditional main effect of A given B at +:
𝑐𝑚𝑒 𝐴 𝐵+ = 𝑦 𝐴+ 𝐵+ − 𝑦 (𝐴−|𝐵+) (3)
Conditional main effect of A given B at -:
𝑐𝑚𝑒 𝐴 𝐵− = 𝑦 𝐴+ 𝐵− − 𝑦 (𝐴−|𝐵−) (4)
By adding (1) and (2), we have
𝑀𝐸 𝐴 +𝐼𝑁𝑇 𝐴,𝐵 =𝑐𝑚𝑒(𝐴|𝐵+).
By subtracting (2) from (1), we have
𝑀𝐸 𝐴 −𝐼𝑁𝑇 𝐴,𝐵 =𝑐𝑚𝑒(𝐴|𝐵−).

4. Similar property for the design
Use short hand notation (A|B+) and (A|B-) instead of cme(A|B+) and cme(A|B-)
From the above relations, it is seen that a cme is related to a main effect and an interaction.
Take (A|B+) for example, we call A its parent effect, AB its interaction effect, B its conditioning effect and + its conditioning level A + - A B AB + - AB + -
A|B+
A|B- + -
A|B- + -
A|B+ + -
Construction definition of cme:
𝐴 𝐵+ = 1 2 𝐴+𝐴𝐵
𝐴 𝐵− = 1 2 (𝐴−𝐴𝐵) /2

5. Orthogonal modeling
For a 2 𝑘−𝑝 design with k factors, the set of candidate effects consists of
4× 𝑘 2 cme’s
𝑘 main effects
𝑘 2 2fi’s
Without any restriction, it is hard to find a good model from such a large candidate set.
In this work, we restrict the model search to orthogonal models, i.e., effects in a candidate model are orthogonal to each other.

6. Orthogonality relations
In this work, we restrict the orthogonality to the following notion:
Two effects are orthogonal if their inner product is zero
Orthogonality relations between cme’s and traditional effects
cme’s are orthogonal to all the traditional effects
except for their parent effects and interaction effects.

7. Grouping of cme’s and properties
Consider a 2 𝐼𝑉 4−1 design with I=ABCD
We categorize cme’s into different groups based on their orthogonality relations. A B C D + -

8. Twins
cme’s having the same parent effect and interaction effect are twins
Twin cme’s are orthogonal to each other.
The 2d space of twin cme’s is the same as
the 2d space of their parent and interaction effect.
Only one of the twin cme’s can be chosen.
Recall that we have
𝑐𝑚𝑒 𝐴 𝐵+ =𝑀𝐸 𝐴 +𝐼𝑁𝑇 𝐴,𝐵
𝑐𝑚𝑒(𝐴|𝐵−)=𝑀𝐸 𝐴 −𝐼𝑁𝑇 𝐴,𝐵
We can replace a main effect, say A, and a 2fi involving
A, say AK, by one of the twin cme’s.
We refer to A as AK’s parental main effect

9. Rule 1
Substitute a pair of 2fi and its parental main effect with similar magnitude with one of the corresponding twin cme’s.
If the pair have the same sign
𝑐𝑚𝑒 𝐴 𝐵+ =𝑀𝐸 𝐴 +𝐼𝑁𝑇 𝐴,𝐵 will have larger magnitude than both A and AB
Replace A and AB with (A|B+)
If the pair have the opposite signs
𝑐𝑚𝑒 𝐴 𝐵− =𝑀𝐸 𝐴 −𝐼𝑁𝑇 𝐴,𝐵 will have larger magnitude than both A and AB
Replace A and AB with (A|B-)

10. Siblings and family
cme’s having the same parent effect but different interaction effects are siblings.
Siblings are NOT orthogonal
cme’s having the same
or fully aliased interaction effects
are said to belong to the same family
Non-twin cme’s in the family
are NOT orthogonal

11. 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

12. Other cme’s and Rule 3
cme’s having different parent effect and interaction effect
are orthogonal to each other.
Rule 3
cme’s with different parent effects and different interaction
effects can be included in the same model.

13. CME Analysis Recall that we have three rules
Based on the three rules, we propose the CME analysis
(i). Use the traditional analysis methods such as ANOVA or half-normal plot, to select significant effects, including aliased pairs of effects. Go to (ii).
(ii). Among all the significant effects, use Rule 1 to find a pair of fully aliased 2fi 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.
Rule 1:
Substitute a pair of 2fi and its parental main effect with similar magnitude with one of the corresponding twin cme’s.
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:
cme’s with different parent effects and different interaction effects can be included in the same model.

14. Examples We show three examples to illustrate the CME analysis
All examples have designs of Resolution IV
All examples have significant interactions from traditional analysis

15. Example 1 (Injection Molding)
2 𝐼𝑉 6−2 design with
𝐼=𝐴𝐵𝐶𝐸=𝐵𝐶𝐷𝐹=𝐴𝐷𝐸𝐹
Traditional analysis:
𝑦~𝐵+𝐴+𝐴𝐵
Step (ii)
A and AB are both significant
Consider (A|B+) or (A|B-)
The CME analysis
(A|B+)

16. Summary of Example 1 In the traditional analysis, we have:
𝑦~𝐵 2.39× 10 −9 +𝐴 5.38× 10 −5 +𝐴𝐵 % . 𝑅 2 =96.24%
In the CME analysis, we have:
𝑦~𝐵 6.06× 10 −10 + 𝐴 𝐵 × 10 − 𝑅 2 =96.14%
The second model is more parsimonious and better than the first in terms of p values for significant effects. Two models have comparable 𝑅 2 values.
The cme (A|B+) has a good engineering interpretation: at high screw speed, pressure has a significant effect on shrinkage but not at low speed.

17. Example 2 (Filtration) 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 either (A|D+) or (A|D-)
D and DB(=AC) are both significant
Consider either (D|B+) or (D|B-)
The CME analysis
(A|D+)
(D|B-)

18. Summary of Example 2 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 % +𝐴𝐶 % +𝐷 % +𝐶 % . 𝑅 2 =99.79%
𝑦~ 𝐴 𝐷 × 10 −5 + 𝐷 𝐵− × 10 −5 +𝐶 % . 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 cme’s (A|D+) and (D|B-) in the last two models have good engineering interpretations.

19. Example 3 (Aluminum) (E|B+) (F|A+) 2 𝐼𝑉 6−2 design with
𝐼=𝐴𝐵𝐶𝐸=𝐵𝐶𝐷𝐹=𝐴𝐷𝐸𝐹
Traditional analysis:
𝑦~𝐵+𝐸+𝐴𝐶+𝐹+𝐴𝐹
Step (ii)
E and EB(=AC) are both significant
Consider either (E|B+) or (E|B-)
F and AF are both significant
Consider either (F|A+) or (F|A-)
E and ED(=AF) are also a possible pair
However (E|D+) and (E|D-) are siblings of (E|B-)
The CME analysis
(E|B+)
(F|A+)

20. Summary of Example 3 In the traditional analysis, we have:
𝑦~𝐵+𝐸+𝐹+𝐴𝐶+𝐴𝐹. ( 𝑅 2 =96.45%)
In the CME analysis, we have:
𝑦~ 𝐸 𝐵+ +𝐵+𝐹+𝐴𝐹. ( 𝑅 2 =94.93%)
𝑦~ 𝐸 𝐵+ +𝐵+ 𝐹 𝐴+ . ( 𝑅 2 =92.22%)
By comparing the 𝑅 2 values and p values for significant terms, the third model is not the best. But models 2 and 3 have fewer terms.
The cme’s (E|B+) and (F|A+) in the last two models have good engineering interpretations, while the 2fi’s AC(=BD) and AF(=DE) are aliased.
The last two models can be presented to the engineers as alternative choices to the first model.

21. Concluding remarks and future research
In this work, we have
Proposed the CME analysis to de-alias aliased effects in traditional analysis.
Explored the orthogonality relations between cme’s and traditional effects.
Used three examples to illustrate the advantages of the CME strategy.
Future research:
Minimum aberration designs for cme analysis (Mukerjee-Wu-Chang, 2015).
Using indicator function theory (Sabbaghi 2015) to study aliasing relations.
Need to extend to 3 𝑘−𝑝 designs? Ans. No. (Wu-Hamada 2009 book can handle estimation of interactions for resolution IV 3-level designs.)
CME’s provide a class of new basis functions in variable selection, new work.
Potential impact of CME outside physical experiments, e.g., in medical and social experiments.