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Introduction to Factorial Design Study the effects of 2 or more factors All possible combinations of factor levels are investigated For example, if there are a levels of factor A and b levels l of ffactor B, each replicate contains all ab treatment combinations DOX 6E Montgomery 2
Some Basic Definitions Definition of a factor effect: The change in the mean response when the factor is changed from low to high Chapter 5 A y y 40 52 20 30 21 A A 2 2 B y y 30 52 20 40 B B 11 2 2 52 20 30 40 AB 1 2 2 3
The Case of Interaction: Chapter 5 A y y 50 12 20 40 A A 1 2 2 B y y 40 12 20 50 9 B B 2 2 12 20 40 50 AB 29 2 2 4
Regression Model & The Associated Response Surface y x x x x 0 1 1 2 2 12 1 2 The least squares fit is yˆ 355 35.5 10.5 105 x 5.5 55 x 0.5 05 x x 35.5 355 10.5 105 x 5.5 55 x 1 2 1 2 1 2 Chapter 5 5
The Effect of Interaction on the Response Surface Suppose that we add an interaction term to the model: yˆ 35.5 10.5x 5.5x 8x x 1 2 1 2 Interaction ti is actually a form of curvature Chapter 5 6
Advantages of Factorial More efficient than one-factor-at-a-time experiments Avoid misleading conclusions especially when interactions are present Allow the effects of a factor to be estimated at several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions DOX 6E Montgomery 7
Example 5.1 The Battery Life Experiment Text reference pg. 167 A = Material type; B = Temperature (A quantitative variable) 1. What effects do material type & temperature have on life? 2. Is there a choice of material that would give long life regardless of temperature (a robust product)? Chapter 5 8
The General Two-Factor Factorial Experiment a levels of factor A; b levels of factor B; n replicates This is a completely randomized design Chapter 5 9
Statistical (effects) model: i 1,2,..., a yijk i j ( ) ij ijk j 1,2,..., b k 1, 2,..., n Other models (means model, regression models) can be useful Chapter 5 10
Extension of the ANOVA to Factorials (Fixed Effects Case) pg. 168 a b n a b 2 2 2 ( yijk y... ) bn ( yi.. y... ) an ( y. j. y... ) i 1 j 1 k 1 i 1 j 1 a b a b n 2 2 (........) ij i j ( ijk ij. ) i 1 j 1 i 1 j 1 k 1 breakdown: n y y y y y y SS SS SS SS SS df T A B AB E abn 1 a 1 b 1 ( a 1)( b 1) ab( n 1) Chapter 5 11
ANOVA Table Fixed Effects Case Design-Expert will perform the computations ti Text gives details of manual computing (ugh!) see pp. 171 Chapter 5 12
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Table 5.4 (p. 172) Life Data (in hours) for the Battery Design Experiment. Circle numbers are the cell totals DOX 6E Montgomery 14
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Figure 5.9 (p. 171) Material type-temperature plot for Example 5-1. DOX 6E Montgomery 16
Figure 5.10 (p. 173) Design-Expert Output for Example 5-1. DOX 6E Montgomery 17
Design-Expert Output Example 5.1 Chapter 5 18
Interaction Plot DESIGN-EXPERT Plot Life X = B: Temperature Y = A: Material 188 Interaction Graph A: Material A1 A1 A2 A2 A3 A3 146 Life 104 2 62 2 2 20 15 70 125 B: Temperature Chapter 5 19
JMP output Example 5.1 Chapter 5 20
Residuals for the two-factor factorial model are e ijk y ijk y ijk For example, for the 1 st entry, y ijk = 130 while j = 539/4 = 134.75 and therefore e ijk = 130 134.75 = - 4.75 y ijk DOX 6E Montgomery 21
Residual Analysis Example 5.1 Normal plot looks ok, although the largest ve residual does stand out Plot shows a mild tendency for the variance of the residuals to increase as the battery life increases Chapter 5 22
Residual Analysis Example 5.1 Both plot show a mild inequality of variance, with the treatment combination of 15 0 F and material 1 possibly having larger variance than others. From Table 5-6, it can be seen that the 15 0 F - material 1 cell contains both extreme residuals (-60.75 and 45.25). Reexamination of the data does not reveal any obvious problem. Possible that this treatment combination produces slightly more erratic battery life. Problem is not severe enough to have a dramatic impact on the analysis and conclusion. Chapter 5 23
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Figure 5.15 (p. 182) Plot of ŷ j ŷ ijk versus ŷ ijk, battery life data. Any patterns in these quantities is suggestive of the presence of interaction. The figure shows a distinct pattern as the quantities ŷ j ŷ ijk move from positive to negative to positive to negative again. This structure is the result of interaction between material types and temperature. DOX 6E Montgomery 25
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Factorials with More Than Two Factors Basic procedure is similar to the two-factor case; all abc kn treatment combinations are run in random order ANOVA identity is also similar: SS SS SS SS SS T A B AB AC SS SS SS ABC AB K E Complete three-factor example in text, Example 5-3 DOX 6E Montgomery 29
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Figure 5.16 (p. 189) Main effects and interaction plots for Example 5-3. (a) Percentage of carbonation (A). (b) Pressure (B). (c) Line speed (C). (d) Carbonation-pressure interaction. DOX 6E Montgomery 33
Figure 5.17 (p. 190) Average fill height ht deviation at high h speed and low pressure for different carbonation levels. DOX 6E Montgomery 34
Quantitative and Qualitative Factors The basic ANOVA procedure treats every factor as if it were qualitative Sometimes an experiment will involve both quantitative and qualitative ti factors, such as in Example 5.1 This can be accounted for in the analysis to produce regression ess models for the quantitative tat factors at each level (or combination of levels) of the qualitative factors These response curves and/or response surfaces are often a considerable aid in practical interpretation i of the results Chapter 5 35
Quantitative and Qualitative Factors A = Material type Candidate model terms from Design- B = Linear effect of Temperature Expert: B 2 = Quadratic effect of Intercept Temperature A B AB = Material type Temp Linear B 2 AB 2 = Material type - Temp Quad AB B B 3 = Cubic effect of 3 AB Temperature (Aliased) 2 Chapter 5 36
Quantitative and Qualitative Factors Chapter 5 37
Regression Model Summary of Results Chapter 5 38
Regression Model Summary of Results Chapter 5 39
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