Supplementary material for “A study of two types of split-plot designs” by Tsai (2016)

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Supplementary material for “A study of two types of split-plot designs” by Tsai (2016)

This supplementary file describes the use of the R functions for the study of split-plot designs with few whole-plot factors and blocked split-plot designs discussed in Tsai (2016). Two func- tions find.SSP() and find.BSP() are available for generating optimal designs with respect to our criterion for a given design configuration. Four functions SSP(), SSP.wd(), BSP(), BSP.wd() are given for computing the m

i

-values, wordlength pattern, and the criterion val- ues of Section 3 for a given design. All the functions are included in a program file, and we can source the entire script by typing the following source() command in the R console.

source("http://math.ntnu.edu.tw/~pwtsai/doe/sspbsp_prg.R")

• find.SSP(nfat, nUnit) — this is used to generate optimal split-plot designs with few whole-plot factors for nfat = c(n1, n2) and nUnit = c(w, s), where n1, n2, w, s are the numbers of whole-plot factors, subplot factors, whole-plots, and subplots per whole-plot, respectively.

• SSP.wd(nfat, nUnit, wd1, wd2): — this is used to compute the m

_i

-values, wordlength pattern, and the criterion values in (b) and (c) of Section 3 for a split-plot design with few whole-plot factors, where wd1 is the subplot words and wd2 is the splitting words.

For example, SSP.wd(c(3,4), c(16,2), c("ABPR", "ABCQS"), c("PQ")) is used to evaulate a 32-run design with n1=3, n2=4, w=16, s=2, and the design has R = ABP and S = ABCQ as subplot words and P Q as the splitting word.

• SSP(nfat, nUnit, vec1, vec2) — this function is similar to that of SSP.wd() and is useful for the cases when, instead of subplot and splitting words, subplot columns (vec1) and splitting columns (vec1) are given, as in Tables 3 and 4 of Tsai (2016).

• find.BSP(nfat, nUnit) — this is used to generate optimal blocked split-plot designs for nfat = c(n1, n2) and nUnit = c(b, w, s), where n1, n2, w, s are defined as before and b is the number of blocks.

• BSP.wd(nfat, nUnit, wd1, wd2) — this is used to compute the m

_i

-values, word-length pattern, and the criterion values in (a), (b) and (c) of Section 3 for a given blocked split-plot design, where wd1 are the whole-plot and subplot defining words, and wd2 is the blocking and splitting words. We note that in all the cases discussed in McLeod and Brewster (2004), and therefore in Tsai (2016), splitting words (if they are needed) are a subset of blocking words and we do not need to separate these two words in this function.

• BSP(nfat, nUnit, vec1, vec2) — this function is similar to that of BSP.wd() with

vec1 as the whole-plot and subplot columns and vec2 the blocking and splitting

columns.

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Split-plot designs with few whole-plot factors

(1). Consider the design configuration with (n

₁

n

₂

; w s) = (3 4; 16 2). We use the following function to generate the split-plot design with few whole-plot factors with respect to our criterion.

find.SSP(c(3,4),c(16,2))

defining words: ABCPR ABCQS ( 15 23 ) splitting words: PQ ( 24 )

mi in whole-plot stratum: 2 2 2 1 1 1 0 0 0 0 0 0 mi in subplot stratum: 1 1 1 1 1 1 1 1 1 1 1 1 (b) and (c) : ( 21 27 ); ( 12 12 )

word length pattern: 0 1 2 0 0 user system elapsed

0.051 0.000 0.051

The best design we found has ABCP R and ABCQS as subplot words (columns 15 and 23) and P Q (column 24) as the splitting word. The m

^W_i

’s and m

^S_i

’s are given. The sums of m

_i

and m

²_i

in conditions (b) and (c) are (21, 27) and (12, 12 ), respectively.

The word length pattern is A

₃

= 0, A

₄

= 1, and A

₅

= 2. We note that this design is denoted d

₂

in Tsai (2016).

(2). For the case (n

₁

n

₂

; w s) = (3 4; 16 2), consider a design which has ABP R and ABCQS as the independent words and P Q as the splitting word. This design is denoted d

₁

in Tsai (2016). To evaluate this design, we use the following function.

SSP.wd(c(3,4), c(16,2), c("ABPR", "ABCQS"), c("PQ"))

mi in whole-plot stratum: 2 1 1 1 1 1 1 1 0 0 0 0 mi in subplot stratum: 2 2 1 1 1 1 1 1 1 1 0 0 (b) and (c) : ( 21 27 ); ( 12 16 )

word length pattern: 0 1 2 0 0

Comparing (1) and (2), we note that d

₁

and d

₂

are equally good with respect to the usual minimum aberration criterion but d

₂

is a better design with respect to our criterion.

(3). Function SSP() is used to evaluate a design when, instead of subplot and splitting words, subplot and splitting columns are given.

For example, for the case (n

₁

n

₂

; w s) = (1 13; 8 4), the design in Table 2 of BSS has 3,5,9,14,15,17,22,26,28 as subplot columns and 6,24 as splitting columns. To evaluate this design, we use the following function.

SSP(c(1, 13), c(8, 4), c(3,5,9,14,15,17,22,26,28), c(6, 24))

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mi in whole-plot stratum: 6 5 5 5 5 5

mi in subplot stratum: 6 6 5 5 5 5 5 5 1 1 1 (b) and (c) : ( 76 386 ); ( 45 225 )

word length pattern: 5 55 45 96 106 87 82 16 17 1 1 0

For the case (1 13; 8 4), we consider the design given in Table 3 of Tsai (2016).

> SSP(c(1,13), c(8,4), c(3,7,13,14,19,22,25,26,28), c(10,20)) mi in whole-plot stratum: 6 6 6 1 1 1

mi in subplot stratum: 5 5 5 5 5 5 5 5 5 5 5 (b) and (c) : ( 76 386 ); ( 55 275 )

word length pattern: 5 55 45 96 106 87 82 16 17 1 1 0

We note that these designs are equally good with respect to the usual minimum aber- ration criterion but our design has more two-factor interactions in the subplot stratum.

Clearly, our design is a better design, as discussed on Page 10 of Tsai (2016).

(4). Three cases where there are more than one admissible design.

> find.SSP(c(1,8),c(8,4))

defining words: APQT PQRU PQSV AQRSW ( 7 14 22 29 ) splitting words: PR QS ( 10 20 ) (b) and (c) : ( 36 72 ); ( 27 57 ) #BSS

defining words: APQT PQRU AQRV APRSW ( 7 14 13 27 ) splitting words: PR QS ( 10 20 ) (b) and (c) : ( 36 78 ); ( 28 64 ) #New

defining words: APQT PQRU APSV QRSW ( 7 14 19 28 ) splitting words: PR QS ( 10 20 ) (b) and (c) : ( 36 90 ); ( 29 73 ) #New

user system elapsed 10.058 0.052 10.110

> find.SSP(c(1,9),c(8,4))

defining words: APQT PQRU PQSV AQRSW APRSX (7 14 22 29 27) splitting words:PR QS (10 20) (b) and (c) : ( 45 105 ); ( 33 81 ) #BSS

defining words: APQT PQRU APSV QRSW ARSX (7 14 19 28 25) splitting words:PR QS (10 20) (b) and (c) : ( 45 135 ); ( 36 108 ) #New

> find.SSP(c(2,8),c(8,4))

defining words: ABPS ABQT APQRU BPQRV ABRW ( 7 11 29 30 19 ) splitting words: PR ( 20 ) (b) and (c) : ( 45 105 ); ( 32 64 ) #BSS

defining words: APQS ABQT PQRU BQRV ABPQRW ( 13 11 28 26 31 ) splitting words: PR ( 20 ) (b) and (c) : ( 45 135 ); ( 36 108 ) #New

defining words: ABPS ABQT PQRU AQRV BQRW ( 7 11 28 25 26 ) splitting words: PR ( 20 ) (b) and (c) : ( 45 141 ); ( 37 117 ) #New

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(5). Eight cases where better 32-run split-plot designs are listed in Table 3 of Tasi (2016).

Clearly, for each class, our design is better when compared with BSS’s designs.

(n1, n2; w s) = ( 1 5 ; 8 4 )

subplot columns: 15 splitting column: 10 22 #New (b) and (c) : ( 15 15 ); ( 13 13 )

word length pattern: 0 0 1 0

subplot columns: 31 ; splitting column: 10 18 #BSS (b) and (c) : ( 15 15 ); ( 12 12 )

=====

(n1, n2; w s) = ( 1 6 ; 8 4 )

subplot columns: 14 27 splitting column: 10 22 #New (b) and (c) : ( 21 27 ); ( 18 22 )

subplot columns: 15 30 splitting column: 10 18 #BSS (b) and (c) : ( 21 27 ); ( 17 23 )

=====

(n1, n2; w s) = ( 1 13 ; 8 4 ) ( discussed in (3)) (b) and (c) : ( 76 386 ); ( 55 275 ) #New (b) and (c) : ( 76 386 ); ( 45 225 ) #BSS

=====

(n1, n2; w s) = ( 1 14 ; 8 4 )

subplot columns: 3 5 9 14 15 17 22 23 26 28 splitting column: 10 20 #New (b) and (c) : ( 87 519 ); ( 51 303 )

word length pattern: 6 77 62 168 188 203 188 56 62 7 6 0 0

subplot columns: 3 5 9 14 15 17 22 23 26 28 splitting column: 6 24 #BSS (b) and (c) : ( 87 519 ); ( 50 290 )

word length pattern: 6 77 62 168 188 203 188 56 62 7 6 0 0

=====

(n1, n2; w s) = ( 2 4 ; 8 4 )

subplot columns: 15 splitting column: 20 #New (b) and (c) : ( 15 15 ); ( 13 13 )

subplot columns: 31 splitting column: 20 #BSS (b) and (c) : ( 15 15 ); ( 12 12 )

=====

(n1, n2; w s) = ( 2 5 ; 8 4 )

subplot columns: 15 28 splitting column: 20 #New (b) and (c) : ( 21 27 ); ( 18 22 )

subplot columns: 25 30 splitting column: 20 #BSS (b) and (c) : ( 21 27 ); ( 18 24 )

=====

(n1, n2; w s) = ( 2 6 ; 8 4 )

subplot columns: 15 27 28 splitting column: 20 #New (b) and (c) : ( 28 46 ); ( 24 36 )

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subplot columns: 15 21 27 splitting column: 29 #BSS (b) and (c) : ( 28 46 ); ( 24 40 )

word length pattern: 0 3 4 0 0 0

=====

(n1, n2; w s) = ( 3 4 ; 16 2 )

subplot columns: 15 23 splitting column: 24 #New (b) and (c) : ( 21 27 ); ( 12 12 )

subplot columns: 11 23 splitting column: 24 #BSS (b) and (c) : ( 21 27 ); ( 12 16 )

Blocked split-plot designs

(1). Consider the design configuration (n

1

n

2

; b w s) = (3 4; 2 4 4). We use the following function to generate optimal blocked split-plot design with respect to our criterion.

find.BSP(c(3,4),c(2,4,4))

defining words: ABPQR ACPQS ( 27 29 ) splitting/blocking words: ABC ( 7 ) mi in block stratum: 0

mi in whole-plot stratum: 2 1 1

mi in subplot stratum: 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 (a), (b) and (c) : ( 21 27 ); ( 21 27 ); ( 17 21 )

The best design we found has ABP QR and ACP QS as defining words (columns 27 and 29) and ABC (column 7) as the blocking word. The sums of m

_i

and m

²_i

in conditions (a), (b) and (c) are (21, 27) (21, 27) and (17, 21 ), respectively. The word length pattern for this design is A

₃

= 0, A

₄

= 1, and A

₅

= 2. This design is denoted d

₄

in Tsai (2016).

(2). For the case (n

₁

n

₂

; b w s) = (3 4; 2 4 4), consider another design with ABP R and ABCQS as the defining words and ABC as the blocking word. This design is denoted d

₃

in Tsai (2016). We evaluate this design by running the following function.

BSP.wd(c(3,4), c(2,4,4), c("ABPR", "ABCQS"), c("ABC"))

mi in block stratum: 1

mi in whole-plot stratum: 2 1 1

MB word length pattern: 0 1 1 2 2 0 0 0 0 1

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Comparing (1) and (2), we note that d

3

and d

4

are equally good with respect to the usual minimum aberration criterion, but clearly d

4

is a better design as discussed in Section 3 of Tsai (2016).

(3). Four cases where more than one design are listed in MB’s Table 1, only the first design is admissible and an additional admissible design up to equivalence is found.

> find.BSP(c(2,4), c(4,4,2))

defining words: ABPQRS ( 31 ) splitting/blocking words: PQ APR ( 12 21 ) #New (a), (b) and (c) : ( 15 15 ); ( 14 14 ); ( 8 8 )

defining words: PQRS ( 28 ) splitting/blocking words: APQ BPR ( 13 22 ) #MB1 (a), (b) and (c) : ( 15 21 ); ( 15 21 ); ( 8 8 )

> find.BSP(c(3,3), c(4,4,2))

defining words: ABCPR ( 15 ) splitting/blocking words: AC APQ ( 5 25 ) #New (a), (b) and (c) : ( 15 15 ); ( 14 14 ); ( 9 9 )

defining words: ABPR ( 11 ) splitting/blocking words: ABC APQ ( 7 25 ) #MB1 (a), (b) and (c) : ( 15 21 ); ( 15 21 ); ( 9 13 )

> find.BSP(c(3,4), c(4,4,2))

defining words: ABCPR ABCQS ( 15 23 ) splitting/blocking words: AC APQ ( 5 25 ) #New (a), (b) and (c) : ( 21 27 ); ( 20 26 ); ( 12 12 )

defining words: ABPR ABQS ( 11 19 ) splitting/blocking words: ABC APQ ( 7 25 ) #MB1 (a), (b) and (c) : ( 21 39 ); ( 21 39 ); ( 12 20 )

> find.BSP(c(5,3), c(4,4,2)) number of admissible design(s): 2

defining words: ABCE ABPQ BCDPR ( 7 19 30 ) splitting/blocking words: AC ABD ( 5 11 ) #New (a), (b) and (c) : ( 28 46 ); ( 25 41 ); ( 15 23 )

defining words: ABCE ABPQ BDPR ( 7 19 26 ) splitting/blocking words: AC ABD ( 5 11 ) #MB1 (a), (b) and (c) : ( 28 58 ); ( 26 54 ); ( 15 31 )

(4). Seven cases where only one design is listed in MB’s Table 1 and an additional admissible design up to equivalence is found.

> find.BSP(c(1,5), c(4,2,4))

defining words: APQRT ( 15 ) splitting/blocking words: PR PQS ( 10 22 ) #New (a), (b) and (c) : ( 15 15 ); ( 14 14 ); ( 13 13 )

defining words: PQRT ( 14 ) splitting/blocking words: APR PQS ( 11 22 ) #MB (a), (b) and (c) : ( 15 21 ); ( 15 21 ); ( 13 17 )

(7)

> find.BSP(c(1,6), c(4,2,4))

defining words: APQRT APRSU ( 15 27 ) splitting/blocking words: PR PQS ( 10 22 ) #New (a), (b) and (c) : ( 21 27 ); ( 20 26 ); ( 18 22 )

defining words: PQRT PRSU ( 14 26 ) splitting/blocking words: APR PQS ( 11 22 ) #MB (a), (b) and (c) : ( 21 39 ); ( 21 39 ); ( 18 30 )

> find.BSP(c(3,4), c(4,2,4))

defining words: ABPR ACPQS ( 11 29 ) splitting/blocking words: AB AC ( 3 5 ) #New (a), (b) and (c) : ( 21 27 ); ( 17 21 ); ( 17 21 )

defining words: APQR BCPQS ( 25 30 ) splitting/blocking words: AB AC ( 3 5 ) #MB (a), (b) and (c) : ( 21 27 ); ( 18 24 ); ( 17 23 )

> find.BSP(c(3,5), c(4,2,4))

defining words: ABPR ABQS ACPQT ( 11 19 29 ) splitting/blocking words: AB AC ( 3 5 ) (a), (b) and (c) : ( 28 46 ); ( 23 35 ); ( 23 35 ) #New

defining words: APQR BCPQS ABCPT ( 25 30 15 ) splitting/blocking words: AB AC ( 3 5 ) (a), (b) and (c) : ( 28 46 ); ( 25 43 ); ( 23 39 ) #MB

> find.BSP(c(6,2), c(4,4,2))

defining words: ABCE ACDF ABDPQ ( 7 13 27 ) splitting/blocking words: AC ABD ( 5 11 ) (a), (b) and (c) : ( 28 46 ); ( 24 36 ); ( 12 12 ) #New

defining words: ABCE ACDF ABPQ ( 7 13 19 ) splitting/blocking words: AC ABD ( 5 11 ) (a), (b) and (c) : ( 28 58 ); ( 25 49 ); ( 12 20 ) #MB

> find.BSP(c(6,3), c(2,8,2))

defining words: ABCE ACDF ACPQ ABDPR ( 7 13 21 27 ) splitting/blocking words: ABD ( 11 ) (a), (b) and (c) : ( 36 72 ); ( 35 71 ); ( 18 30 ) #New

word length pattern: 0 6 8 0 0 1 0

defining words: ABCE ACDF ABPQ BDPR ( 7 13 19 26 ) splitting/blocking words: ABD ( 11 ) (a), (b) and (c) : ( 36 90 ); ( 36 90 ); ( 18 42 ) #MB

> find.BSP(c(7,2), c(2,8,2))

defining words: ABCE ACDF BCDG ABDPQ ( 7 13 14 27 ) splitting/blocking words: ABD ( 11 ) (a), (b) and (c) : ( 36 78 ); ( 35 77 ); ( 14 14 ) #New

defining words: ABCE ACDF BCDG ABPQ ( 7 13 14 19 ) splitting/blocking words: ABD ( 11 ) (a), (b) and (c) : ( 36 96 ); ( 36 96 ); ( 14 26 ) #MB

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(5). Function BSP() is used to evaluate a design when, instead of defining and split- ting/blocking words, defining and splitting/blocking columns are given.

(a) Consider the case (n

₁

n

₂

; b w s) = (1 7; 2 2 8). The design on Page 15 of Tsai (2016) has 15,26,29 as the defining columns and 22 as the splitting/blocking column. We use the following function to evaluate the design.

BSP(c(1,7),c(2,2,8), c(15,26,29), c(22))

mi in block stratum: 0 mi in whole-plot stratum: 0

mi in subplot stratum: 3 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 0 (a), (b) and (c) : ( 28 46 ); ( 28 46 ); ( 28 46 )

MB word length pattern: 0 0 3 3 4 4 0 0 0 0 0 1

For (n

₁

n

₂

; b w s) = (1 7; 2 2 8), the design in MB’s Table 1 has 15,23,30 as the defining columns and 27 as the splitting/blocking column. We have the following output.

> BSP(c(1,7),c(2,2,8), c(15,23,30), c(27)) #MB mi in block stratum: 0

mi in whole-plot stratum: 1

mi in subplot stratum: 3 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 0 0 (a), (b) and (c) : ( 28 46 ); ( 28 46 ); ( 27 45 )

MB word length pattern: 0 0 3 3 4 4 0 0 0 0 0 1

(b) Consider the case (n

₁

n

₂

; b w s) = (1 8; 2 2 8)

> BSP(c(1,8), c(2,2,8), c(15,21,25,30), c(19)) #New mi in block stratum: 0

MB word length pattern: 0 0 6 4 8 8 0 0 0 0 1 4 0 0

> BSP(c(1,8), c(2,2,8), c(15,23,27,29), c(30)) #MB’s design mi in block stratum: 0

MB word length pattern: 0 0 6 4 8 8 0 0 0 0 1 4 0 0

In each case, the new design is a better design, as discussed in Tsai (2016).

References

Tsai, P. W. (2016), “A study of two types of split-plot designs”, Journal of Quality Technology 48,