Understanding Rowwise Complementary Designs in Statistics

1. Rowwise ComplementaryDesigns Shao-Wei Cheng Institute of Statistical Science Academia Sinica Taiwan (based on a joint work with a Ph.D student Chien-Yu Peng)

2. Outline • Introduction • Columnwise and Rowwise Complementary Design • Indicator Function • Some Properties • Isomorphism • Generalized Wordlength Pattern and Resolution • Minimum Aberration • Orthogonality • Moment Aberration • Uniformity • Generalization • Summary

3. (Columnwise) Complementary Design • Tang and Wu (1996), Chen and Hedayat (1996) number of factors Hadamard matrix • useful when the number of factors in A is large

4. Rowwise Complementary Design (RCD) number of factors run size full factorial design with r replicates • r= the maximum number of appearances of design points in design RCD

5. Indicator Function • Fontana et al. (2000), Ye (2003), Cheng and Ye (2004), Cheng, Li, and Ye (2004), Tang (2001, J-characteristic) • Definition Let D be a full factorial design of s factors and A be a subset of D. The indicator function of A is a function FA(x) defined on D such that where rx is the number of appearances of the point x in design A.

6. Let P be the collection of all subset of {1,2, …,s}, then the indicator function can be uniquely represented by where . • cor( xI , xJ) = bIJ/b, where IJ=(IJ)\(IJ) (Note: xi2 = 1)

7. b0 = fraction aliasing (correlation) word Example • A1: 26-2 (regular) FFD, 3=12 and 6=145 • FA1= (1/4)(1+x1x2x3)(1+x1x4x5x6) • = (1/4)(1 + 1*x1x2x3 + 1*x1x4x5x6 + 1*x2x3x4x5x6) I=123=1456=23456 • A2: non-regular OA • FA2=(1/4)[1 + (1/2)*x2x4x6 + (-1/2)*x2x5x6 + (1/2)*x3x4x6 + (1/2)*x3x5x6 • + (1/2)*x1x2x4x6 + (1/2)*x1x2x5x6 + (-1/2)*x1x3x4x6 • + (1/2)*x1x3x5x6 + (1)*x1x2x3x4x5] wordlength = 4

8. Indicator Function and RCD general case r = max{r1, …, rk} r = 1 0 0 0 . . . 0 1 1 1 . . . 1 r r r . . . r 1 1 1 . . . 1 r1 r2 r3 . . . rk r - r1 r - r2 r - r3 . . . r - rk full factorial design 1 . 1 1 . 1 r . r r . r 0 . 0 0 . 0 • Let be the indicator function of , and be the RCD of , then the indicator function of is

9. Example • Let A be a 24-2 design • the indicator function of its RCD is

10. Property 1: Isomorphism • Suppose . If and are isomorphic, then and are also isomorphic.

11. Property 2: Generalized WLP and Resolution • Generalized Wordlength Pattern (GWLP, Deng and Tang, 1999) where ai=  (bI/b)2 • Generalized resolution (Tang and Deng, 1999) #I= i

12. Theorem Let and be the indicator functions of A and its RCD, then (i) GWLPs of A and its RCD are and , respectively. (ii) the design with larger run size has large generalized resolution and the difference between their generalized resolutions is .

13. Example • Let A be a 24-2 design with four factors. GWLP: (0, 1, 2, 0) • its RCD GWLP: (0, 1, 2, 0)/9

14. A is an orthogonal array of strength tif and only if its RCD is an orthogonal array of strength t. Property 3: Orthogonality

15. Example • Cheng (1995) shows that • 4-factor, 12-run designs • there is only one OA with strength 2 • the OA has 11 distinct runs (one run repeats twice) • Q: why no OA with 12 distinct runs? • 12 distinct runs  the indicator function of its RCD is 1-FA  its RCD has 4 runs, 4 factors • the OA  the indicator function of its RCD is 2-FA  its RCD has 20 runs, 4 factors

16. Property 4: Minimum aberration • The minimum aberration criterion is to sequentially minimize GWLP within a class of designs, called search class. • Denote the search class of A by and the search class of its RCD by . • example: if , then .

17. Theorem Design A is the MA design in the search class if and only if its RCD is the MA design in the search class S.

19. Property 5: Moment aberration • Xu (2003), based on coding theory • Let be the rowwise complementary design of , then its u-th power moment is where and are Stirling numbers of second kind. In particular, and .

20. Property 6: Uniformity • Fang and his colleagues • Symmetric L2-discrepancy (SL2) • Centered L2-discrepancy (CL2) • Wrap-around L2-discrepancy (WL2)

21. Theorem Let A be an ns two-level factorial design, then the discrepancies of its RCD can be expressed in terms of the discrepancies and GWLP of A as follows

22. Generalization Type I designs Hadamard matrix full factorial design number of factors

23. Type II designs number of factors full factorial design Hadamard matrix

24. Blocked factorial designs • blocked indicator function (Cheng, Li, and Ye, 2004) • If is a blocked indicator function which has q block factors (i.e., blocks and block size = runs), then the can be regarded as a blocked indicator function with blocks and block size = runs.

25. Summary • propose RCD, useful when run size is large • indicator function is a useful framework for studying RCD • various relations between a design and its RCD are explored • methods that combine the techniques of CCD and RCD can generate more designs