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Reply to R.A. Bailey's response
Computational Statistics & Data Analysis 3 (1985) 123-124 North-Holland
123
Reply to R.A. Bailey's response R.F. CROMP and N.V. F I N D L E R A rtificml Intelligence Laboratory, Computer Science Department, Arizona State University, Tempe, A Z 85287, USA Received March 1985
The suitability of the experimental design depends upon the definition of the function F. I f p; is replaced by Pi +- b then F will not have been defined to yield an even distribution over the range [0, b - 1]. However, if the values of Pi are chosen so as to be positioned around the center of the range (b, 2b), with pi and b being relatively prime, then F will distribute the experiments more evenly. By choosing pi from the interval (b, 2b), F > a and thus the whole of T is not reproduced. F is then able to yield experimental designs for any fraction a/b, a
For instance: If b = 23 and n = 4, then Pl = 32, P2 = 33, P3 = 35, P4 = 36. If b = 25 and n = 4 then Pl = 34, P2 = 36, P3 = 38, P4 = 39. If b = 7 and n = 3 then Pl = 8, P2 = 10, P3 = 11. We stress that this procedure is heuristic in nature and as such there are examples for which it will perform inadequately. However, in general we have found that the method produces the desired results. The striping effect displayed by the above technique can be eliminated by evaluating F and then permuting along the various dimensions in the computed design. The attached table is the result of applying this method to each of Dr. Bailey's examples, with the striping effect left intact.
0167-9473/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
R.F. Cromp, N. V. Findler / Reply to R.A. Bailey
124 Table
A : a = 2, b = 5, Pl = 6, Pa = 8. E x p e c t e d : 25.6. A c t u a l IMI = 26. B: a = 3, b = 8, P l = 1 1 , P2 = 1 3 . E x p e c t e d : 24.0. A c t u a l IMI = 24. C: a = 3, b = 7, P l = 9, P2 = 1 1 . E x p e c t e d : 27.4. A c t u a l I M l - - 27. X1
X2
1 1 2 3
B AC A
4
B
5
C
6
C
7 8
AB A
2 B C AC AB C B
3 ABC AC B
4 B AC AB
C
5
6
7
8
AC
C B
B
B AC AC
A AC B
C
B
AC AB
ABC
AB
C
A
A C
BC A
BC
BC A A
BC C
C ABC A B
123
Reply to R.A. Bailey's response R.F. CROMP and N.V. F I N D L E R A rtificml Intelligence Laboratory, Computer Science Department, Arizona State University, Tempe, A Z 85287, USA Received March 1985
The suitability of the experimental design depends upon the definition of the function F. I f p; is replaced by Pi +- b then F will not have been defined to yield an even distribution over the range [0, b - 1]. However, if the values of Pi are chosen so as to be positioned around the center of the range (b, 2b), with pi and b being relatively prime, then F will distribute the experiments more evenly. By choosing pi from the interval (b, 2b), F > a and thus the whole of T is not reproduced. F is then able to yield experimental designs for any fraction a/b, a
For instance: If b = 23 and n = 4, then Pl = 32, P2 = 33, P3 = 35, P4 = 36. If b = 25 and n = 4 then Pl = 34, P2 = 36, P3 = 38, P4 = 39. If b = 7 and n = 3 then Pl = 8, P2 = 10, P3 = 11. We stress that this procedure is heuristic in nature and as such there are examples for which it will perform inadequately. However, in general we have found that the method produces the desired results. The striping effect displayed by the above technique can be eliminated by evaluating F and then permuting along the various dimensions in the computed design. The attached table is the result of applying this method to each of Dr. Bailey's examples, with the striping effect left intact.
0167-9473/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
R.F. Cromp, N. V. Findler / Reply to R.A. Bailey
124 Table
A : a = 2, b = 5, Pl = 6, Pa = 8. E x p e c t e d : 25.6. A c t u a l IMI = 26. B: a = 3, b = 8, P l = 1 1 , P2 = 1 3 . E x p e c t e d : 24.0. A c t u a l IMI = 24. C: a = 3, b = 7, P l = 9, P2 = 1 1 . E x p e c t e d : 27.4. A c t u a l I M l - - 27. X1
X2
1 1 2 3
B AC A
4
B
5
C
6
C
7 8
AB A
2 B C AC AB C B
3 ABC AC B
4 B AC AB
C
5
6
7
8
AC
C B
B
B AC AC
A AC B
C
B
AC AB
ABC
AB
C
A
A C
BC A
BC
BC A A
BC C
C ABC A B