orthogonal arrays of strength 3 with full estimation capacities eric d. schoen; man v.m. nguyen

Orthogonal arrays Orthogonal arrays ofof strength 3 strength 3 withwith full estimation full estimation

capacitiescapacitiesEric D. Schoen; Man V.M. Eric D. Schoen; Man V.M.

NguyenNguyen

FullEC OA with strength 3 2Nankai University, July 10 2006

OutlineOutline

Motivating exampleMotivating example StrengthStrength or efficiency?or efficiency? Selection of arraysSelection of arrays Details of best arraysDetails of best arrays


Example from wood Example from wood technologytechnology

Response: strength of wood-glue-Response: strength of wood-glue-wood bondwood bond

Factors: Factors: glue (4 levels) glue (4 levels) wood (3 levels)wood (3 levels) moisture content, pretreatment, moisture content, pretreatment,

pressure, storage temperature (2 levels)pressure, storage temperature (2 levels) About 50 runsAbout 50 runs


Full Estimation CapacityFull Estimation Capacity

Most of the factors will be active.Most of the factors will be active. Many active interactions.Many active interactions. Design should permit estimation of all Design should permit estimation of all

main effects (me) and all 2-factor main effects (me) and all 2-factor interactions (2fi).interactions (2fi).

Parameters in wood technology example: Parameters in wood technology example: 1 intercept1 intercept 9 me9 me 32 2fi32 2fi


Design optionsDesign options

Choose orthogonal array (OA) of Choose orthogonal array (OA) of maximum strength compatible with maximum strength compatible with run-size & factor specifications.run-size & factor specifications.

Choose array that maximizes Choose array that maximizes DD--efficiency for the efficiency for the pp parameters of parameters of the full model.the full model.


Orthogonal arraysOrthogonal arrays Example: OA(4, 2Example: OA(4, 233, 2)., 2). Symbols arranged in Symbols arranged in

NN rows and rows and nn columns.columns.

Rows: runs of a Rows: runs of a design.design.

Columns: factors of a Columns: factors of a design.design.

Strength Strength tt : each : each tt--tuple of symbols tuple of symbols occurs equally often. occurs equally often.

00 00 00

11 00 11

00 11 11

11 11 00


Properties for strength Properties for strength tt = 2, 3, 4= 2, 3, 4

tt = 2: orthogonality among main effects. = 2: orthogonality among main effects. t t = 3: additional orthogonality me = 3: additional orthogonality me 2fi. 2fi. t t = 4: additional orthogonality among 2fi. = 4: additional orthogonality among 2fi.

In example: run sizes multiple of 24, 48, In example: run sizes multiple of 24, 48, 96 for strength 96 for strength tt = 2, 3, 4. = 2, 3, 4.

Find 48-run array of strength 3 and Find 48-run array of strength 3 and FullEC.FullEC.


Optimization of Optimization of DD--efficiencyefficiency

XX11: matrix with intercept + me ( length : matrix with intercept + me ( length NN ) )

XX22: matrix with 2fi ( length : matrix with 2fi ( length NN ) )

X = [XX = [X11 X X2 2 ]]

DD = |X’X| = |X’X|1/1/pp//NN 0 0 DD 1 1 Maximum Maximum DD minimizes volume of minimizes volume of

confidence region.confidence region.


Efficiency of me Efficiency of me estimationestimation

Primary goal: efficient estimation of main Primary goal: efficient estimation of main effects to separate active from inactive me.effects to separate active from inactive me.

DD11 = |X = |X11’(I - X’(I - X22(X(X22’X’X22))-1-1XX22’)X’)X11||1/1/pp11//NN residual sum of squares and products after residual sum of squares and products after

projecting Xprojecting X11 on X on X22

Orthogonal arrays of strength 3 have Orthogonal arrays of strength 3 have DD11 = = 1.1.


Efficiency of 2fi Efficiency of 2fi estimationestimation

Primary goal: distinguishing among 2fi.Primary goal: distinguishing among 2fi.

DD22 = |X = |X22’(I - X’(I - X11(X(X11’X’X11))-1-1XX11’)X’)X22||1/1/pp22//NN

|X’X|= (|X’X|= (NN DD22))pp22.|X.|X11’X’X11| | If an array has higher |X’X| than has a If an array has higher |X’X| than has a

strength-3 array, it must have higher strength-3 array, it must have higher DD22-efficiency too.-efficiency too.


Selection of arraysSelection of arrays

Strength is bestStrength is best Generate all non-Generate all non-

isomorphic isomorphic OA(48; (48; 4 4 x x 3 3 xx 2 244; 3): 19 ; 3): 19 arrays.arrays.

Select FullEC Select FullEC arrays: 14 arrays.arrays: 14 arrays.

Select array with Select array with maximum maximum DD22: 1 : 1 array.array.

DD-efficiency is best-efficiency is best Generate 50 Generate 50

random arrays.random arrays. Improve Improve DD--

efficiency with efficiency with modified Fedorov modified Fedorov algorithm.algorithm.

Keep the best Keep the best array.array.


Details of best arraysDetails of best arrays

strengthstrength DD11 DD22

33 11 0.730.73

22 0.850.85 0.860.86


Some other 48-run casesSome other 48-run cases

6 6 xx 2 255

tt=3: 30 non-=3: 30 non-isomorphic arrays.isomorphic arrays.

No FulEC array.No FulEC array.

BestBest D D22-efficiency -efficiency 0.77.0.77.

Strength 0.Strength 0. DD11-efficiency 0.72.-efficiency 0.72.

3 3 xx 2 277

tt=3: 3056 ni arrays.=3: 3056 ni arrays. 209 FulEC array.209 FulEC array. Best Best DD2 2 given given tt=3: =3:

0.77.0.77.

BestBest D D22-efficiency -efficiency 0.80.0.80.

Strength 0.Strength 0. DD11-efficiency 0.71.-efficiency 0.71.


ConclusionConclusion

Separation of main effects & 2fi?Separation of main effects & 2fi? Use FullEC array of strength 3.Use FullEC array of strength 3. Choose array with highest Choose array with highest DD22--

efficiency.efficiency.

Activity of 2fi?Activity of 2fi? Use FullEC array with optimum Use FullEC array with optimum DD22--

efficiency.efficiency.


Contact informationContact information

[email protected]@tno.nl [email protected]@tue.nl

(see also abstract)(see also abstract)

orthogonal arrays of strength 3 with full estimation capacities eric d. schoen; man v.m. nguyen

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