Chapter 5 Introduction to Factorial Designs

5. Basic Definitions and Principles Stud the effects of two or more factors. Factorial designs Crossed: factors are arranged in a factorial design Main effect: the change in response produced b a change in the level of the factor 2

Definition of a factor effect: The change in the mean response when the factor is changed from low to high 40+ 52 20+ 30 A + 2 A A 2 2 30+ 52 20+ 40 B + B B 2 2 52+ 20 30+ 40 AB 2 2 3

50+ 2 20+ 40 A + A A 2 2 40+ 2 20+ 50 B + 9 B B 2 2 2+ 20 40+ 50 AB 29 2 2 4

Regression Model & The Associated Response Surface β + β x + β x 0 2 2 + β x x + ε 2 2 The least squares fit is ˆ 35.5+ 0.5x + 5.5x + 0.5x x 2 2 35.5+ 0.5x + 5.5x 2 5

The Effect of Interaction on the Response Surface Suppose that we add an interaction term to the model: ˆ 35.5+ 0.5x + 5.5x + 8x x 2 2 Interaction is actuall a form of curvature 6

When an interaction is large, the corresponding main effects have little practical meaning. A significant interaction will often mask the significance of main effects. 7

5.2 The Advantage of Factorials One-factor-at-a-time desgin Compute the main effects of factors A: A + B - - A - B - B: A - B - - A - B + Total number of experiments: 6 Interaction effects A + B -, A - B + > A - B - > A + B + is better??? 8

5.3 The Two-Factor Factorial Design 5.3. An Example a levels for factor A, b levels for factor B and n replicates Design a batter: the plate materials (3 levels) v.s. temperatures (3 levels), and n 4: 3 2 factorial design Two questions: What effects do material tpe and temperature have on the life of the batter? Is there a choice of material that would give uniforml long life regardless of temperature? 9

The data for the Batter Design: 0

Completel randomized design: a levels of factor A, b levels of factor B, n replicates

Statistical (effects) model: i,2,..., a ijk µ + τi + β j+ ( τβ ) ij + εijk j, 2,..., b k,2,..., n µ is an overall mean, τ i is the effect of the ith level of the row factor A, β j is the effect of the jth column of column factor B and (τ β) ij is the interaction between τ i and β j. Testing hpotheses: H 0 : τ L τ a 0 v.s. H : at least oneτ i 0 H 0 : β L β b 0 v.s. H : at least oneβ j 0 H 0 : ( τβ ) ij 0 i, j v.s. H : at least one ( τβ) ij 2 0

5.3.2 Statistical Analsis of the Fixed Effects Model i... j. ij.... b j k i k n k a a n b n ijk n ijk ijk i j k ijk i... j. ij.... i.. bn ij. n. j. an... abn 3

a b n a b 2 2 2 ( ijk... ) bn ( i..... ) + an (. j.... ) i j k i j a b a b n 2 2 ( ij. i... j....) ( ijk ij. ) i j i j k + n + + SS SS + SS + SS + SS T A B AB E df breakdown: abn a + b + ( a )( b ) + ab( n ) 4

5 Mean squares 2 2 2 2 2 2 2 ) ) ( ( ) ( ) )( ( ) ( ) ) )( ( ( ) ( )) /( ( ) ( )) /( ( ) ( σ τβ σ β σ τ σ + + + n ab SS E MS E b a n b a SS E MS E b an b SS E MS E a bn a SS E MS E E E a i b j ij AB AB b j j B B a i i A A

The ANOVA table: 6

Example 5. Response: Life ANOVA for Selected Factorial Model Analsis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 5946.22 8 7427.03.00 < 0.000 A 0683.72 2 534.86 7.9 0.0020 B 398.72 2 9559.36 28.97 < 0.000 AB 963.78 4 2403.44 3.56 0.086 Pure E 8230.75 27 675.2 C Total 77646.97 35 Std. Dev. 25.98 R-Squared 0.7652 Mean 05.53 Adj R-Squared 0.6956 C.V. 24.62 Pred R-Squared 0.5826 PRESS 3240.22 Adeq Precision 8.78 8

DESIGN-EXPERT Plot Life X B: T em perature Y A: M ateri al 88 Interactio n Graph A: Material A A A2 A2 A3 A3 46 Life 04 2 62 2 2 20 5 70 25 B: Tem perature 9

Multiple Comparisons: Use the methods in Chapter 3. Since the interaction is significant, fix the factor B at a specific level and appl Turke s test to the means of factor A at this level. See Page 74 Compare all ab cells means to determine which one differ significantl 20

5.3.3 Model Adequac Checking Residual analsis: e ijk ˆ ijk ijk ijk ij DE SIG N-EX P ERT Plo t Life Normal plot of residuals DE SIGN-EXPER T Plo t L ife Res iduals vs. P redicted 45.25 99 N orm al % probabilit 95 90 80 70 50 30 20 0 5 Residuals 8.75-7.75-34.25-60.75-60.75-34.25-7.75 8.75 45.25 49.50 76.06 02.62 29.9 55.75 Res idual Pre dicte d 2

DESIG N-E XPERT Pl ot L ife Residuals vs. Run 45.25 8.75 Re sid uals - 7.75-34.25-60.75 6 6 2 26 3 36 R un N um ber 22

DE SIG N-EX P ERT Plo t Life Residuals vs. Material DES IGN-E XPE RT P l ot L i fe Residuals vs. Temp erature 45.25 45.25 8.75 8.75 R es iduals -7.75 R esid ua ls -7.75-34.25-34.25-60.75-60.75 2 3 2 3 Material Tem pera tu re 23

5.3.4 Estimating the Model Parameters The model is µ + τ + β + ( τβ) + ε ijk The normal equations: µ : abnµ + bn Constraints: i j ij a i i τ : bnµ + bnτ + β : anµ + n ( τβ ) i a : nµ + nτ a i τ i τ i + i i n + + b j nβ b an β j j j anβ + j j b + + β n n j b + j a i n( τβ) ij ij n a ( τβ) ij ijk i j ij ( τβ) ij b ( τβ) i j ij ( τβ) ( τβ) 0 τ 0, 0, i β j ij j a i b j ij 24

Estimations: ˆ µ ˆ τ i ˆ β j i j ( τβ) ij ij i j + The fitted value: ˆ ijk ( τβ) ij ˆ µ + ˆ τ i + ˆ β j + ij Choice of sample size: Use OC curves to choose the proper sample size. 25

Consider a two-factor model without interaction: Table 5.8 The fitted values: ˆijk i + j 26

One observation per cell: The error variance is not estimable because the two-factor interaction and the error can not be separated. Assume no interaction. (Table 5.9) Tuke (949): assume (τβ) ij rτ i β j (Page 83) Example 5.2 27

5.4 The General Factorial Design More than two factors: a levels of factor A, b levels of factor B, c levels of factor C,, and n replicates. Total abc n observations. For a fixed effects model, test statistics for each main effect and interaction ma be constructed b dividing the corresponding mean square for effect or interaction b the mean square error. 28

Degree of freedom: Main effect: # of levels Interaction: the product of the # of degrees of freedom associated with the individual components of the interaction. The three factor analsis of variance model: µ + τ + β + γ + ( τβ) ijkl + ( ) ik + ( ) jk + ( ) The ANOVA table (see Table 5.2) Computing formulas for the sums of squares (see Page 86) Example 5.3 i τγ j k βγ ij τβγ ijk + ε ijkl 29

Example 5.3: Three factors: the percent carbonation (A), the operating pressure (B); the line speed (C) 3

5.5 Fitting Response Curves and Surfaces An equation relates the response () to the factor (x). Useful for interpolation. Linear regression methods Example 5.4 Stud how temperatures affects the batter life Hierarch principle 33

Involve both quantitative and qualitative factors This can be accounted for in the analsis to produce regression models for the quantitative factors at each level (or combination of levels) of the qualitative factors A Material tpe B Linear effect of Temperature B 2 Quadratic effect of Temperature AB Material tpe Temp Linear AB 2 Material tpe - Temp Quad B 3 Cubic effect of Temperature (Aliased) 34

5.6 Blocking in a Factorial Design A nuisance factor: blocking A single replicate of a complete factorial experiment is run within each block. Model: ijk µ + τ + β + ( τβ ) + δ + ε No interaction between blocks and treatments ANOVA table (Table 5.20) i j ij k ijk 38

Example 5.6: Two factors: ground clutter and filter tpe Nuisance factor: operator 40

Two randomization restrictions: Latin square design An example in Page 200. Model: ijkl µ + α + τ + β + ( τβ) + δ + ε Tables 5.23 and 5.24 i j k jk l ijkl 4