1 Chapter 5 Introduction to Factorial Designs

2 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

3 Definition of a factor effect: The change in the mean response when the factor is changed from low to high

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5 Regression Model & The Associated Response Surface

6 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

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

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

9 5.3 The Two-Factor Factorial Design 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: 3 2 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?

10 The data for the Battery Design:

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

12 Statistical (effects) model:  is an overall mean,  i is the effect of the ith level of the row factor A,  j is the effect of the jth column of column factor B and (   ) ij is the interaction between  i and  j. Testing hypotheses:

Statistical Analysis of the Fixed Effects Model

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15 Mean squares

16 The ANOVA table:

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18 Response:Life ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of squares] Sum of MeanF SourceSquares DFSquareValueProb > F Model < A B < AB Pure E C Total Std. Dev.25.98R-Squared Mean105.53Adj R-Squared C.V.24.62Pred R-Squared PRESS Adeq Precision8.178 Example 5.1

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20 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

Model Adequacy Checking Residual analysis:

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Estimating the Model Parameters The model is The normal equations: Constraints:

25 Estimations: The fitted value: Choice of sample size: Use OC curves to choose the proper sample size.

26 Consider a two-factor model without interaction: –Table 5.8 –The fitted values:

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 (  ) ij = r  i  j (Page 183) –Example

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.

29 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

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Example 5.3: Three factors: the percent carbonation (A), the operating pressure (B); the line speed (C) 31

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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

–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 34 A = Material type B = Linear effect of Temperature B 2 = Quadratic effect of Temperature AB = Material type – Temp Linear AB 2 = Material type - Temp Quad B 3 = Cubic effect of Temperature (Aliased)

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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)

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

41 Two randomization restrictions: Latin square design An example in Page 200. Model: Tables 5.23 and 5.24