Implicit in the definition of orthogonal arrays is a projection propertyþ

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Gwendolyn Hall
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1 Projection properties Implicit in the definition of orthogonal arrays is a projection propertyþ factor screening effect ( factor) sparsity A design is said to be of projectivity : if in every subset of : factors, a complete factorial, possibly with some combinations replicated, is produced. -1-

2 Projection properties of regular designs are straightforward and can be studied through the defining relations. A regular design of resolution V has projectivity V ", but cannot have projectivity V. -2-

3 L (the 12-run Plackett-Burman design) is of 12 projectivity 3 The projection of L 12 onto any three factors consists 3 of a complete 2 and a half replicate defined by M œ EFG or M œ EFG. -3-

4 Projections of OA ÐR, 2 R ", 2)'s onto 3 factors: Rœ : one full 2 factorial one 2. 3 Rœ"': two full 2 factorials or one full 2 two 2 designs or 3 1 four 2 designs. Rœ : two full 2 factorials one " or one full 2 three 2 designs -4-

5 3 Rœ24: three full 2 factorials or two full 2 two 2 designs or one full 2 four 2 designs or 3 1 six 2 designs. -5-

6 > " Theorem. If R is not a multiple of 2, then an 8 OA ÐR, 2, >Ñwith 8 > 2 has projectivity > 1. In particular, if R is not a multiple of 8, then an 8 OA ÐR, 2, 2 Ñ with 8 4 has projectivity 3. L 12 is not of projectivity 4, but its projection onto any four factors has the hidden projection property that all the main effects and 2-factor interactions of these four factors are estimable. -6-

7 Theorem. Suppose R is not a multiple of 8. Then 8 for any OAÐRß 2 ß 2 Ñ with 8 4, its projection onto any four factors has the property that all the main effects and #-factor interactions are estimable when the higher-order interactions are negligible. In order for a regular design to have the hidden projection property as described in this theorem, the resolution must be at least five. -7-

8 Theorem. Let \ be an OAÐR ß 2 ß 3 Ñ with 8 &. If R is not a multiple of 16, then \ has the property that, in the projection onto any five factors, all the main effects and #-factor interactions are estimable when the higher-order interactions are negligible. 8-8-

9 Weld-repaired castings experiment (Wu and Hamada) A 12-run design was used to study the effects of 7 factors on the fatigue life of weld-repaired castings. Factor A. initial structure as received " treat B. bead size small large C. pressure treat none HIP D. heat treat anneal solution treat/ age E. cooling rate slow rapid F. polish chemical mechanical G. final treat none peen -9-

10 Design: 7 columns of the 12-run Plackett-Burman design Response: logged lifetime of the casting -10-

11 logged Run A B C D E F G lifetime 1 'Þ!&) # %Þ($$ $ %Þ'#& % &Þ)** & (Þ!!! ' &Þ(&# ( &Þ')# ) 'Þ'!( * &Þ)") "! &Þ*"( "" &Þ)'$ "# %Þ)!* -11-

12 Half-normal plot of main effects shows that F is the only # significant main effect. The model with F alone has V œ!þ%&ß and the model with F and D (the next largest main effect) has # V œ!þ&*þ The main effect analysis does not succeed in explaining the variation in the data very well. A significant interaction FG is obtained by entertaining all the (two-factor) interactions with F. Adding FG to F doubles the V # to Adding D to the model (F,FG) only increases the V # to Based on F and FG, the model for predicted logged lifetime is sc œ &Þ(!Þ%&) G!Þ%&* FG. -12-

13 ( OAÐ")ß 2 3 ß #Ñ:!!!!!!!!!! " " " " " "!! # # # # # #! "!! " " # #! " " " # #!!! " # #!! " "! #! "! # " #! # " # "! #!! # #! # "! " "!! # # " "! "! "!! # # " "! # " "!! # " "! " #! # " " " " #! "! # " " #! " # "! " #! # " #! " " # "! #! " # " # # "! " #! -13-

14 Construction of asymmetrical (mixed-level) orthogonal arrays 1. Method of replacement 2. Method of grouping 3. Difference matrices -14-

15 8 b an OAÐRß = ß #Ñ Ê 8 Ÿ ÐR "ÑÎÐ= "Ñ If = is a prime or power of a prime and R œ = 5 for 8 some 5, then ba regular OA ÐRß= ß#Ñwith 8 œ ÐR "ÑÎÐ= "Ñ. -15-

16 4 OA(9, 3,2): B B B B B #B!!!! 0 " " # 0 # # " ! " #! " # " # " # " # -16-

17 13 OA(27, 3,2): B" B# B$ B" B# B" #B# B" B$ B" #B$!!!!!!!! "! " #!!! #! # "!!!! "!! " #! " " " # " # â! # " # " " #!! #!! # "! " # " # # "! # # # " # " ã -17-

18 OA (16, 2 15 ß#Ñ 15 columns: 1, 2, 3, 4, 12, 13, 14, 23, 24, 34, 123, 124, 134, 234, 1234 These columns can be grouped into 5 sets of the form Ö+ß,ß+, : 1, 2, 12; 3, 4, 34; 14, 123, 234; 13, 24, 1234; 23, 124,

19 Identify the three two-level factors in the same group with a four-level factor, e.g., Ð!ß!ß!ÑÄ!ß Ð!ß"ß"ÑÄ"ßÐ"ß!ß"ÑÄ#ß Ð"ß"ß!ÑÄ$. & Then we have an OAÐ"'ß % ß #Ñ -19-

20 Conversely, from an OA Ð"'ß % ß #Ñ, we can replace each four-level factor with three two-level factors: 0 Ä Ð!ß!ß!Ñß 1 Ä Ð!ß "ß "Ñß 2 Ä Ð"ß!ß "Ñß 3 Ä Ð"ß "ß!Ñ. Then we obtain an OAÐ16 ß 2, #Ñ. This shows that " 5 $7 7 for any!ÿ7ÿ&, ban OAÐ16ß 2 % ß#Ñ. "& "# OAÐ"'ß # ß #Ñß OAÐ"'ß # %ß #Ñß * # ' $ OAÐ"'ß # % ß #Ñß OAÐ"'ß # % ß #Ñß OAÐ"'ß # $ % % ß #Ñß OAÐ"'ß % & ß #Ñ 15 & -20-

21 In general, if 5 is even, then for any! Ÿ 7 Ÿ Ð2 "ÑÎ$, there is an OAÐ2 ß 2 5 " $7 7 % ß #Ñ. If 5 is odd, then for any! Ÿ 7 Ÿ Ð2 5 ÑÎ$, there $7 7 is an OAÐ2 ß 2 % ß#Ñ. Each of these designs can be constructed by either the method of grouping or replacement, but it's easier to use the method of grouping. Wu (1989) provided an algorithm for grouping the factors

22 Difference matrices Let K be an additive group of : elements. A -: < matrix with elements from K, denoted by H -:ß<à: is called a difference matrix if, among the differences of the corresponding elements of any two columns, each element of K occurs - times. -22-

23 !!!! " #! # " is a difference matrix H 333 ßà. Each Hadamard matrix of order R is a H RßRß# -23-

24 For two matrices E œò+ 34 Ó of order ; < and F of order 7 5, both with entries from K, define their Kronecker sum as + E F œ ÒF Ó where F 5 œ ÒF 5NÓ. 34, "Ÿ3Ÿ;ß"Ÿ4Ÿ< The Kronecker sum of an OA Ð :ß: ß#Ñand a # <5 difference matrix H is an OA Ð -.:, : ß#Ñ. -:ß<à:

25 Let E be a : " matrix consisting of all the elements of K, then the Kronecker sum of E and a # < difference matrix H is an OA Ð -:, : ß#Ñ. -:ß<à: For each prime number :, the : : matrix H:ß:à: œ c! + â Ð: "Ñ+ d, where + œ Ð!ß "ß âß : "Ñ X, is a difference matrix. -25-

26 # : This can be used to construct an OA Ð: ß: ß#Ñ. Adding the column Ð0ß "ß âß : "ß â ß!ß "ß âß : "Ñ X, we obtain a saturated regular # : " OA Ð: ß: ß#Ñ. # : " Taking the Kronecker sum of an OAÐ: ß: ß#Ñwith the difference matrix H :ß:à:, we obtain an $ :Ð: "Ñ OA Ð: ß: ß#Ñ. This can be expanded to a # $ : : " saturated OA Ð: ß: ß#Ñby adding a column. Iterating this construction, we obtain saturated 5 5 Ð: "ÑÎÐ: "Ñ OA Ð: ß : ß #Ñ for all

27 H 'ß'à$!!!!!!! " #! " #! # " "! #!! # " # "! #! # " "! " " # #! -27-

28 H can be used to construct an OA Ð")ß $ ß #Ñ. 'ß'à$ Adding the column (!ß "ß #ß $ß %ß &ß âß!ß "ß #ß $ß %ß &) X, we obtain an ' OAÐ")ß $ 6 ß #Ñ, which is saturated. Replacing the 6-level factor with a two-level factor and a threelevel factor, we obtain an OA Ð")ß # $ Ñ, often ( denoted as. P "8 ' -28-

29 A general construction result (Wang and Wu, JASA 1991) " 2 Denote an OA ÐRß= " â = 2 ß#Ñby 5" 5 P Ð= â= 2 Ñ. R " 2 5 If R is a multiple of =, then PR Ð=Ñ is the R " vector in which each of 0, 1, â and = " appears RÎ= times. Let be the vector of zeros.! = " = 5-29-

30 Let EœÒE" ßâßE2 Óand F œòf" ßâßF2Óbe two partitioned matrices such that for each 3, both E 3 and F3 have entries from an additive group K3. Suppose 5" 52 E is an PRÐ= " â= 2 Ñ where each E3 consists of the columns corresponding to the 53 = 3-level factors, and each F3 is a difference matrix HQß< à=. Then 3 3 5< " " 5< 2 2 ÒE " F" ß âß E 2 F2Ó is an PQR Ð= " â= 2 Ñ. > Furthermore, if G is an PQ Ð; " " â; > 7Ñ. Then ÒE " F" ß âß E 2 F2ß! R GÓ is an 5< P Ð= " " 5< > â= 2 2 ; " â; 7 Ñ. QR " 2 " 7 > -30-

31 12-run arrays: "" # % "# "# "# P Ð# Ñ, P Ð' # ÑßP Ð$ # Ñ # P"# Ð' # Ñ À Ô! " # $ % &! " # $ % &!!!!!! " " " " " " Õ!!! " " " " " "!!! Ø X -31-

32 % P"# Ð$ # Ñ À Ô!!!! " " " " # # # #!! " "!! " "!! " "! "! "! "! "! "! " Ö Ù!! " " " "!! "!! " Õ! " "! "! "!! "! " Ø X -32-

33 18-run arrays: ÒP Ð$Ñ H ß! P Ó $ 'ß'à$ $ ' By choosing PÐÑPÐ 6 3, 6 2 3ÑßPÐÑ for P6, we obtain P18Ð3 ÑßP18Ð 2 3 Ñ and 6 P Ð 6 3 Ñ, respectively

34 24-run arrays: ÒP# Ð#Ñ L"# ß!# P"# Ó, where L"# is the Hadamard matrix of order 12. "" # % By choosing P"# Ð# Ñ, P"# Ð' # ÑßP"# Ð$ # Ñ # 3 " 4 for P"#, we obtain P#% Ð# Ñß P#% Ð' # Ñ and 16 P#% Ð 3 # Ñ, respectively. -34-

35 Let B be a two-level column of any of these P "#. Then it is also a nonzero column of L "#. Both P# Ð#Ñ! "# and P# Ð#Ñ B are columns of P# Ð#Ñ L"#, and 02 B is a column of!# P"#. We have ÐP# Ð#Ñ! "# Ñ ÐP# Ð#Ñ BÑ œ!# BÞ We can replace these three two-level columns in the aforementioned arrays with a four-level column to obtain P#% Ð% # #! ÑßP#% Ð' % # "" Ñ and P#% Ð 3 13 % # Ñ, respectively. -35-

36 36-run arrays: ÒP Ð$Ñ H ß! P Ó $ "#ß"#à$ $ "# By choosing P"# Ð$Ñ, P"# Ð# Ñ, " # % P"# Ð"# Ñß P"# Ð' # Ñß P"# Ð$ %Ñß P"# Ð$ # Ñ "$ "# "" for P"#, we obtain P$' Ð$ Ñß P$' Ð$ # Ñ, "# 2 13 P$' Ð$ 12 Ñ, P$' Ð$ 6 # Ñ, P$' Ð 3 %Ñ and 13 % P Ð 3 2 Ñ, respectively. $' "" -36-

37 H "#ß"#à$:!!! " "!! "! # #!!!!! #! #! #!! "!! "!! # " #!! "!!! # #! "!! " "!!! " # #!! " " #! # #! " # " # " # # # # "!! "!! # #! # " " # #! " " # " # #!! #! #! # " # "!! # # " " "! # "!! " # " " # # "! # # " # # " "! "! "! #! " " " "! "! " # -37-

38 40-run designs: ÒP Ð#Ñ L ß! P Ó # #! # #! " "* " # By choosing P#! Ð#! Ñ, P#! Ð# Ñß P#! Ð"! # Ñß P#! Ð& # ) Ñ for P#!, we obtain P%! Ð#! " # #! Ñß $ 9 " P%! Ð# ÑßP%! Ð"! # Ñ and P%! Ð 5 # Ñ, respectively. Applying the method of grouping to $ 9 " P%! Ð# ÑßP%! Ð"! # Ñ and P%! Ð 5 # Ñßwe obtain P Ð% # $' ÑßP Ð"! " % # "* Ñ and P Ð 5 25 % # Ñ. %! %! %! -38-

39 " # " " P#! Ð"! # Ñ œ ÒP# Ð# Ñ Hß!# P"! Ð"! ÑÓ, where Hœ!!!!!!!!!! X!!!!! " " " " ". ) P#! Ð& # Ñ can be found in Wang and Wu (1991). -39-

40 Two : : Latin squares are said to be orthogonal if each ordered pair of the : symbols appears exactly once when the two Latin squares are superimposed.! " #! # " " #! "! # #! " # "! -40-

41 A set of 5 mutually orthogonal : : Latin squares is # 5 # " equivalent to an OA Ð: ß: ß#Ñ, a -fraction of a : 5 5 # : experimentþ " A : : Latin square can be used for a : -fraction of a : $ experiment. b5 mutually orthogonal : : Latin squares Ê 5Ÿ: ". This upper bound is achievable when : is a prime or power of a prime. -40-

Ð"Ñ + Ð"Ñ, Ð"Ñ +, +, + +, +,,

ÐÑ + ÐÑ, ÐÑ +, +, + +, +,, Handout #11 Confounding: Complete factorial experiments in incomplete blocks Blocking is one of the important principles in experimental design. In this handout we address the issue of designing complete