Chapter 5 Introduction to Factorial Designs

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

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

Regression Model & The Associated Response Surface

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

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

5.3 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 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?

The data for the Battery Design:

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

Statistical (effects) model: Testing hypotheses:

5.3.2 Statistical Analysis of the Fixed Effects Model

Mean squares

The ANOVA table: See Page 180 Example 5.1

Response:. Life. ANOVA for Selected Factorial Model 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

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 Pages 182, 183 Compare all ab cells means to determine which one differ significantly

5.3.3 Model Adequacy Checking Residual analysis:

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

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

Consider a two-factor model without interaction: Table 5.8 The fitted values: Figure 5.15 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 192) Example 5.2

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.

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 196) Example 5.3

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 Example 5.5

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 5.18) Example 5.6

Two randomization restrictions: Latin square design An example in Page 209 Model: Table 5.22