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Chapter 5 Introduction to Factorial Designs 1 5.1 Basic Definitions and Principles • Study the effects of two or more factors. • Factorial designs • Crossed: factors are arranged in a factorial design • Main effect: the change in response produced by 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 3 4 Regression Model & The Associated Response Surface 5 The Effect of Interaction on the Response Surface Suppose that we add an interaction term to the model: Interaction is actually 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 5.3.1 An Example • a levels for factor A, b levels for factor B and n replicates • Design a battery: the plate materials (3 levels) v.s. temperatures (3 levels), and n = 4: 32 factorial design • Two questions: – What effects do material type and temperature have on the life of the battery? – Is there a choice of material that would give uniformly long life regardless of temperature? 9 • The data for the Battery Design: 10 • Completely randomized design: a levels of factor A, b levels of factor B, n replicates 11 • Statistical (effects) model: m is an overall mean, ti is the effect of the ith level of the row factor A, bj is the effect of the jth column of column factor B and (t b)ij is the interaction between ti and bj . • Testing hypotheses: 12 • 5.3.2 Statistical Analysis of the Fixed Effects Model 13 14 • Mean squares 15 • The ANOVA table: 16 17 Example 5.1 Response: Life ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of Mean F Source Squares DF Square Value Prob > F Model 59416.22 8 7427.03 11.00 < 0.0001 A 10683.72 2 5341.86 7.91 0.0020 B 39118.72 2 19559.36 28.97 < 0.0001 AB 9613.78 4 2403.44 3.56 0.0186 Pure E 18230.75 27 675.21 C Total 77646.97 35 Std. Dev. 25.98 R-Squared 0.7652 Mean 105.53 Adj R-Squared 0.6956 C.V. 24.62 Pred R-Squared 0.5826 PRESS 32410.22 Adeq Precision 8.178 18 19 • Multiple Comparisons: – Use the methods in Chapter 3. – Since the interaction is significant, fix the factor B at a specific level and apply Turkey’s test to the means of factor A at this level. – See Page 174 – Compare all ab cells means to determine which one differ significantly 20 5.3.3 Model Adequacy Checking • Residual analysis: 21 22 23 5.3.4 Estimating the Model Parameters • The model is • The normal equations: • Constraints: 24 • Estimations: • The fitted value: • 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: 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) – Tukey (1949): assume (tb)ij = rtibj (Page 183) – 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 may be constructed by dividing the corresponding mean square for effect or interaction by the mean square error. 28 • Degree of freedom: – Main effect: # of levels – 1 – Interaction: the product of the # of degrees of freedom associated with the individual components of the interaction. • The three factor analysis of variance model: – – The ANOVA table (see Table 5.12) – Computing formulas for the sums of squares (see Page 186) – Example 5.3 29 30 • Example 5.3: Three factors: the percent carbonation (A), the operating pressure (B); the line speed (C) 31 32 5.5 Fitting Response Curves and Surfaces • An equation relates the response (y) to the factor (x). • Useful for interpolation. • Linear regression methods • Example 5.4 – Study how temperatures affects the battery life – Hierarchy principle 33 – Involve both quantitative and qualitative factors – This can be accounted for in the analysis to produce regression models for the quantitative factors at each level (or combination of levels) of the qualitative factors A = Material type B = Linear effect of Temperature B2 = Quadratic effect of Temperature AB = Material type – TempLinear AB2 = Material type - TempQuad B3 = Cubic effect of Temperature (Aliased) 34 35 36 37 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: – No interaction between blocks and treatments • ANOVA table (Table 5.20) 38 39 • Example 5.6: – Two factors: ground clutter and filter type – Nuisance factor: operator 40 • Two randomization restrictions: Latin square design • An example in Page 200. • Model: • Tables 5.23 and 5.24 41

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posted: | 4/1/2014 |

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