Abstract
When the levels of some factors in an experiment are difficult to be changed or controlled, fractional factorial split-plot (FFSP) designs are commonly used in which the factors are classified as the whole plot (WP) and sub-plot (SP) factors. Mixed-level designs are used in practice when the levels of the factors are not equal to each other. This paper considers the mixed-level FFSP designs with the WP factors being more important than the SP factors. It proposes the combined minimum aberration criterion of type WP (WP-MA\(^{c}\)) for mixed-level FFSP designs. Some optimal mixed-level FFSP designs are constructed under the WP-MA\(^{c}\) criterion.
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References
Ai MY, Zhang RC (2005) Characterization of minimum aberration mixed factorials in terms of consulting designs. Stat Pap 46(2):157–171
Bingham D, Sitter RR (1999a) Minimum-aberration two-level fractional factorial split-plot designs. Technometrics 41(1):62–70
Bingham D, Sitter RR (1999b) Some theoretical results for fractional factorial split-plot designs. Annals Stat 27(4):1240–1255
Bingham D, Sitter RR (2001) Design issues in fractional factorial split-plot experiments. J Quality Technol 33(1):2–15
Fries A, Hunter WG (1980) Minimum aberration \(2^{k-p}\) designs. Technometrics 22:601–608
Han XX, Chen JB, Liu MQ, Zhao SL (2020a) Asymmetrical split-plot designs with clear effects. Metrika 83:779–798
Han XX, Liu MQ, Yang JF, Zhao SL (2020b) Mixed 2- and \(2^r\)-level fractional factorial split-plot designs with clear effects. J Stat Plan Inf 204:206–216
Li PF, Liu MQ, Zhang RC (2007) \(2^{m}4^{1}\) designs with minimum aberration or weak minimum aberration. Stat Pap 48(2):235–248
Montgommery DC (2013) Design and analysis of experiments, 8th edn. Wiley, New York
Mukerjee R, Wu CFJ (2001) Minimum aberration designs for mixed factorials in terms of complementary sets. Stat Sin 11(1):225–240
Mukerjee R, Wu CFJ (2006) A modern theory of factorial design. Springer, New York
Potcner KJ, Kowalski SM (2004) How to analyze a split-plot experiment. Quality Prog 37(12):67–74
Wang CC, Zhao QQ, Zhao SL (2019) Optimal fractional factorial split-plot designs when the whole plot factors are important. J Stat Plan Inference 199:1–13
Wu CFJ, Hamada M (2009) Experiments: planning, analysis, and parameter design optimization, 2nd edn. Wiley, New York
Wu CFJ, Zhang RC (1993) Minimum aberration designs with two-level and four-level factors. Biometrika 80(1):203–209
Yang JF, Liu MQ, Zhang RC (2009) Some results on fractional factorial split-plot designs with multi-level factors. Commun Stat Theory Methods 38(20):3623–3633
Zhang RC, Shao Q (2001) Minimum aberration \((s^{2})s^{n-k}\) designs. Stat Sin 11:213–223
Zhao QQ, Zhao SL (2020) Constructing minimum aberration split-plot designs via complementary sets when the whole plot factors are important. J Stat Plan Inference 209:123–143
Zhao SL, Chen XF (2012a) Mixed-level fractional factorial split-plot designs containing clear effects. Metrika 75(7):953–962
Zhao SL, Chen XF (2012b) Mixed two- and four-level fractional factorial split-plot designs with clear effects. J Stat Plan Inference 142(7):1789–1793
Acknowledgements
The authors would like to thank the associate editor and two reviewers for their constructive comments and suggestions. This work was supported by National Natural Science Foundation of China (Grant Nos. 11771250, 11801308), Natural Science Foundation of Shandong Province (Grant No. ZR2018BA013),
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Appendices
Appendix A: Proofs of lemmas and theorems
Proof of Lemma 1
We prove that a design with some two-level factor or component of the four-level factor not involved in its defining contrast subgroup cannot be a WP-MA\(^{c}\) design.
(1) Suppose that \(D_1\) is a \(2^{(n_{1}+n_{2})-(k_{1}+k_{2})}4^{1}_{w}\) design and the two-level factor \(b_{1}\) is not involved in its defining contrast subgroup. Denote the independent defining words of design \(D_{1}\) by
where \(a_{1}z_{1},a_{2}z_{2},\ldots ,z_{k_{1}}\) are WP defining words, \(w_1,w_2,\ldots ,w_{k_{2}}\) are SP defining words. The letters \(a_{1},a_{2}\) and \(b_{1}\) are not included in \(z_{1},z_{2},\ldots ,z_{k_{1}},w_1,w_2,\ldots , w_{k_{2}}\). Then the defining contrast subgroup of design \(D_{1}\) is
where \(G_{1}\) is a subgroup generated by \(z_{3},\ldots ,z_{k_{1}},w_1,w_2,\ldots ,w_{k_{2}}\) and \(\gamma G_{1}=\{\gamma g: g\in G_{1}\}\) with \(\gamma =a_{1}z_{1}, a_{2}z_{2}, a_{3}z_{1}z_{2}\). Consider the \(2^{(n_{1}+n_{2})-(k_{1}+k_{2})}4^{1}_{w}\) design \(D'_{1}\) determined by the independent defining words
The defining contrast subgroup of \(D'_{1}\) is
Note that \(D_{1}\) and \(D'_{1}\) have the same defining words in \(G_{1}\cup (a_{1}z_{1}G_{1})\), and each defining word \(w\in (a_{2}z_{2}G_{1})\cup (a_{3}z_{1}z_{2}G_{1})\) has one less letter than the corresponding defining word \(b_1w\in (a_{2}b_{1}z_{2}G_{1})\cup (a_{3}b_{1}z_{1}z_{2}G_{1})\). Then \(D'_{1}\) has less combined aberration of type WP than \(D_{1}\).
(2) The second case can be proved similarly. Now suppose that \(D_2\) is a \(2^{(n_{1}+n_{2})-(k_{1}+k_{2})}4^{1}_{w}\) design and the component \(a_3\) of the four-level factor is not involved in its defining contrast subgroup. Note that any two \(a_{i}\) that appear in a defining word can be replaced by the third \(a_{i}\) because of \(I=a_{1}a_{2}a_{3}\). Thus, at most one of \(a_1\) and \(a_2\) is involved in \(G(D_{2})\). Without loss of generality, denote the independent defining words of design \(D_{2}\) by
where \(z_{1},z_{2},\ldots ,z_{k_{1}},w_1,w_2,\ldots ,w_{k_{2}}\) do not contain \(a_{1}, a_{2}, a_{3}\). The defining contrast subgroup of design \(D_{2}\) is
where the meaning of the symbols are the same as in (6). Now, consider the \(2^{(n_{1}+n_{2})-(k_{1}+k_{2})}4^{1}_{w}\) design \(D'_{2}\) determined by
The defining contrast subgroup of \(D'_{2}\) is
Note that \(D_{2}\) and \(D'_{2}\) have the same defining words in \(G_{1}\cup (a_{1}z_{1}G_{1})\), and each defining word \(a_1w\in a_{1}z_{1}z_{2}G_{1}\) has the same length as the corresponding defining word \(a_3w\in a_{3}z_{1}z_{2}G_{1}\), and each defining word \(w\in z_{2}G_{1}\) has one less letter than the corresponding defining word \(a_2w\in a_{2}z_{2}G_{1}\). Then \(D'_{2}\) has less combined aberration of type WP than \(D_{2}\). This completes the proof. \(\square \)
Proof of Lemma 2
(1) This is obvious from the fact that there are \(2^{k}-1\) defining words in the defining contrast subgroup of D.
(2) Suppose that D is a WP-MA\(^{c}\) \(2^{(n_{1}+n_{2})-(k_{1}+k_{2})}4^{1}_{w}\) design determined by C and the independent defining words
where \(z_{1},z_{2},\ldots ,z_{k_{1}},w_{1},w_{2},\ldots ,w_{k_{2}}\) contain all the two-level factors. The defining contrast subgroup of D is
where \(G_{1}\) is the same as in (6). Removing the column \(a_{3}\) in C we get a \(2^{(n_{1}+n_{2}+2)-(k_{1}+k_{2})}\) design \({\widetilde{D}}\). Then the defining contrast subgroup of \({\widetilde{D}}\) is
From Lemma 2(2) of Wang et al. (2019), we can get that the total number of letters in all the defining words of \({\widetilde{D}}\) is \((n_{1}+n_{2}+2)\times 2^{k_{1}+k_{2}-1}=(n+2)\times 2^{k-1}.\) By comparing G and \({\widetilde{G}}\), the designs D and \({\widetilde{D}}\) have the same defining words in \(G_{1}\cup (a_{1}z_{1}G_{1})\cup (a_{2}z_{2}G_{1})\), and each defining word \(a_3z_{1}z_{2}w\in a_{3}z_{1}z_{2}G_{1}\) has one less letter than the corresponding defining word \(a_1a_2z_{1}z_{2}w\in a_{1}a_2z_{1}z_{2}G_{1}\). Therefore, the total number of all letters in G is
(3) Removing the columns \(a_{1},a_{2},a_{3}\) in C, we get a \(2^{(n_{1}+n_{2})-(k_{1}+k_{2})}\) design \(D_{0}\) with defining contrast subgroup
From Lemma 2(3) of Wang et al. (2019), either all the defining words of \(G_{0}\) have even lengths or \(2^{k-1}\) of them have odd lengths. Since \(G_{1}\) is a subgroup generated by \(z_{3},\ldots ,z_{k_{1}},w_1, w_2,\ldots ,w_{k_{2}}\) and it can be regarded as the defining contrast subgroup of a two-level design, either all the defining words in \(G_{1}\) have even lengths or \(2^{k-3}\) of them have odd lengths.
By comparing \(G_{0}\) and G, the designs \(D_0\) and D have the same defining words in \(G_{1}\), and each defining word \(w\in (z_1G_{1})\cup (z_2G_{1})\cup (z_1z_2G_{1})\) has one less letter than the corresponding defining word \(a_iw\in (a_1z_1G_{1})\cup (a_2z_2G_{1})\cup (a_3z_1z_2G_{1})\), \(i=1,2,3\).
Let \(M=(z_{1}G_{1})\cup (z_{2}G_{1})\cup (z_{1}z_{2}G_{1})\). Note that there are \(2^{k-2}\) defining words in \(z_{1}G_{1}\), \(z_{2}G_{1}\) and \(z_{1}z_{2}G_{1}\), respectively. There are three cases for the wordlength of defining words in \(G_{0}\) and \(G_{1}\).
(i) If the lengths of defining words in \(G_{0}\) and \(G_{1}\) are all even, then the lengths of defining words in M are all even and there are \(3\times 2^{k-2}\) defining words with odd-length in G.
(ii) If \(2^{k-1}\) defining words in \(G_{0}\) have odd-length and all the defining words in \(G_{1}\) have even-length, then there are \(2^{k-1}\) odd-length and \(2^{k-2}\) even-length defining words in M. Thus there are \(2^{k-2}\) defining words with odd-length in G.
(iii) If \(2^{k-1}\) defining words have odd-length in \(G_{0}\) and \(2^{k-3}\) ones have odd-length in \(G_{1}\), then there are \(3\times 2^{k-3}\) defining words with odd-length and \(3\times 2^{k-3}\) defining words with even-length in M. So, there are \(3\times 2^{k-3}+2^{k-3}=2^{k-1}\) defining words with odd-length in G. This completes the proof. \(\square \)
Proof of Theorem 1
Note that for a \(2^{(n_{1}+n_{2})-(0+k_{2})}4^{1}_{w}\) design D, it has only the SP wordlength pattern \(W_{2}\). The result follows immediately if it is regarded as a \(2^{(n_{1}+n_{2})-k_{2}}4^{1}_{w}\) design. \(\square \)
Proof of Theorem 2
Without SP defining words in design D, the WP part of design D corresponds to a \(2^{n_{1}-k_{1}}4^{1}\) design. Then, the \(2^{(n_{1}+n_{2})-(k_{1}+0)}4^{1}_{w}\) design is a WP-MA\(^{c}\) design if and only if the \(2^{n_{1}-k_{1}}4^{1}\) design is an MA\(^c\) design. \(\square \)
Proof of Theorem 3
It is obviously that the WP part of D constitutes an MA\(^c\) \(2^{n_{1}-1}4^{1}\) design. Hence \(W^{c}_{1}(D)\) is sequentially minimized. We consider \(W^{c}_{2}(D)\) in what follows. When \(n_1=2m+1\), the SP wordlength pattern is \(W^{c}_{2}=(0,...,0,1,1,0,...,0)\), where \(B^{s}_{m+1+n_{2}}=1\), \(B^{s}_{m+2+n_{2}}=1\). Hence we consider the \(2^{(n_{1}+n_{2})-(1+1)}4^{1}_{w}\) designs satisfying \(B^{s}_{i}=0, i\le m+n_{2}\). From Lemma 2, \(\sum _{i=1}^{n+1} B_{i}^{s}=2\) and \(\sum _{i=1}^{n+1} iB_{i}^{s}=2n+3-(n_{1}+1)=2m+2n_{2}+3\). Then,
Subtracting \((m+1+n_{2})\) times the first equation from the second equation yields \(B^{s}_{m+2+n_{2}}=1\), then \(B^{s}_{m+1+n_{2}}=1\), \(B^{s}_{m+2+n_{2}}=1\) is the unique solution. Thus D is a WP-MA\(^{c}\) design.
When \(n_{1}=2m\), the SP wordlength pattern is \(W^{c}_{2}=(0,...,0,2,0,...,0)\), where \(B^{s}_{m+1+n_{2}}=2\). Hence we consider only those \(2^{(n_{1}+n_{2})-(1+1)}4^{1}_{w}\) designs satisfying \(B_{i}^{s}=0, i\le m+n_{2}\). From Lemma 2, \(\sum _{i=1}^{n+1} B_{i}^{s}=2\) and \(\sum _{i=1}^{n+1} iB_{i}^{s}=2n+3-(n_{1}+1)=2m+2n_{2}+2\). Then,
The unique solution of the two equations is \(B^{s}_{m+1+n_{2}}=2\). Thus D is a WP-MA\(^{c}\) design. \(\square \)
Proof of Theorem 4
By Section 4 in Zhang and Shao (2001), the WP part of D constitutes an MA\(^c\) \(2^{n_{1}-2}4^{1}\) design. Hence \(W^{c}_{1}(D)\) is sequentially minimized. We consider \(W^{c}_{2}(D)\) in what follows. The proof of WP-MA\(^{c}\) property of D when \(n_{1}=3\) and \(n_{1}=4\) are first presented, and then the WP-MA\(^{c}\) property of D when \(r=0\), \(r=1\) and \(r=2\) are proved.
(1) When \(n_{1}=3\), by (1), \(W_{1}^{c}=(0,0,3,0,\ldots ,0)\), where \(B^{w}_{3}=3\); \(W_{2}^{c}=(0,\ldots ,0,3,1,0,\ldots ,0)\), where \(B^{s}_{n_{2}+2}=3\) and \(B^{s}_{n_{2}+3}=1\). We consider only the designs satisfying \(B^{s}_{i}=0,i\le n_{2}+1\). From Lemma 2,
The unique solution of the two equations is \(B^{s}_{n_{2}+2}=3\), \(B^{s}_{n_{2}+3}=1\). Thus D is a WP-MA\(^{c}\) design.
(2) When \(n_{1}=4\), by (2), \(W_{1}^{c}=(0,0,1,2,0,\ldots ,0)\), where \(B^{w}_{3}=1\), \(B^{w}_{4}=2\); \(W_{2}^{c}=(0,\ldots ,0,1,3,0,\ldots ,0)\), where \(B^{s}_{n_{2}+2}=1\) and \(B^{s}_{n_{2}+3}=3\). We consider only the designs satisfying \(B^{s}_{i}=0,i\le n_{2}+1\). From Lemma 2,
The unique solution of the two equations is \(B^{s}_{n_{2}+2}=1\), \(B^{s}_{n_{2}+3}=3\). Thus D is a WP-MA\(^{c}\) design.
(3) When \(m>1\) and \(r=0\), by (3), we get
The WP wordlength pattern is \(W^{c}_{1}=(0,\ldots ,0,2,1,0,\ldots ,0)\), where \(B^{w}_{2m+2}=2\), \(B^{w}_{2m+3}=1\).
When \(l=0\), the SP wordlength pattern is \(W^{c}_{2}=(0,\ldots ,0,1,3,0,\ldots ,0)\), where \(B^{s}_{3t+1+n_{2}}=1\) and \(B^{s}_{3t+2+n_{2}}=3\). Hence we consider the \(2^{(n_{1}+n_{2})-(2+1)}4^{1}_{w}\) designs satisfying \(B^{s}_{i}=0, i\le 3t+n_{2}\). By Lemma 2, there are four SP defining words and the sum of the wordlength of them should be \((4n+6)-(6m+7)=6m+7+4n_{2}\). Then,
Subtracting \((3t+1+n_{2})\) times (7) from (8) yields
which forces \(B^{s}_{3t+5+n_{2}}=B^{s}_{3t+6+n_{2}}=\cdots =0\). From (7) and (9), we can get three solutions, which are \(B^{s}_{3t+1+n_{2}}=3\), \(B^{s}_{3t+4+n_{2}}=1\); \(B^{s}_{3t+1+n_{2}}=2\), \(B^{s}_{3t+2+n_{2}}=1\), \(B^{s}_{3t+3+n_{2}}=1\) and \(B^{s}_{3t+1+n_{2}}=1\), \(B^{s}_{3t+2+n_{2}}=3\). It shows that there does not exist any design having less combined aberration of type WP than D. Thus D is a WP-MA\(^{c}\) design.
When \(l=1\), the sub-plot wordlength pattern is \(W^{c}_{2}=(0,\ldots ,0,3,1,0,\ldots ,0)\), where \(B^{s}_{3t+3+n_{2}}=3\) and \(B^{s}_{3t+4+n_{2}}=1\). Hence we consider the \(2^{(n_{1}+n_{2})-(2+1)}4^{1}_{w}\) designs satisfying \(B^{s}_{i}=0, i\le 3t+2+n_{2}\). From Lemma 2,
Subtracting \((3t+3+n_{2})\) times (10) from (11) yields
which forces \(B^{s}_{3t+5+n_{2}}=B^{s}_{3t+6+n_{2}}=\cdots =0\). From (10) and (12), we can get the unique solution \(B^{s}_{3t+3+n_{2}}=3\), \(B^{s}_{3t+4+n_{2}}=1\), thus proving the WP-MA\(^{c}\) property of D.
When \(m=1\), \(r=0\), \(n_{1}=5\) and \(l=1\), from (3) we get
Then
It can be seen that the wordlength patterns of this design are
where \(B^{w}_{4}=2\), \(B^{w}_{5}=1\), \(B^{s}_{n_{2}+3}=3\), and \(B^{s}_{n_{2}+4}=1\). We consider only the designs satisfying \(B^{s}_{i}=0,i\le n_{2}+2\). From Lemma 2,
The unique solution of the two equations is \(B_{n_{2}+3}=3\), \(B_{n_{2}+4}=1\). Thus D is a WP-MA\(^{c}\) design.
When \(m>1\) and \(r=1,2\), the proofs are similar to that of \(m>1\) and \(r=0\) and hence are omitted to save space. This completes the proof of Theorem 4. \(\square \)
Appendix B: Tables of the WP-MA\(^c\) \(2^{(n_1+n_2)-(k_1+k_2)}4^{1}_{w}\) designs with 8, 16, 32 and 64 runs
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Yan, Z., Zhao, S. Mixed two- and four-level split-plot designs with combined minimum aberration. Metrika 85, 537–555 (2022). https://doi.org/10.1007/s00184-021-00838-x
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DOI: https://doi.org/10.1007/s00184-021-00838-x