1 The General Factorial Design Two-factor factorial design General case: Factor A – a levels Factor B – b levels … n replicate (n  2) Total observations: abc…n Test hypotheses about the main effects and interactions may be formed Numbers of degrees of freedom for (1) total sum of squares (2) main effects (3) interactions (4) error sum of squares Mean squares F tests: upper-tail, one-tail

2 Effects model Partitioning sum of squares Special case: a three-factor analysis of variance model

3 Objective: to achieve uniform fill heights Response variable: average deviation from the target fill height Variables: opercent carbonation (A, 10, 12, 14%) ooperating pressure (B, 25, 30 psi) oline speed (C, 200, 250 bpm) Two replicates. 3  2  2  2=24 runs in random order Example 5-3: Soft Drink Bottling Problem

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5 Significant effects of percentage of carbonation, operating pressure, and line speed Some interaction between carbonation and pressure Residual analysis Positive main effects Low level of operating pressure, high level of line speed are preferred for production rate

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7 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 factors, such as in Example 5-1 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 These response curves and/or response surfaces are often a considerable aid in practical interpretation of the results

8 Quantitative and Qualitative Factors Candidate model terms from Design- Expert: Intercept A B A 2 AB A 3 A 2 B Battery Life Example A = Linear effect of Temperature B = Material type A 2 = Quadratic effect of Temperature AB = Material type–Temp Linear A 2 B = Material type–Temp Quad A 3 = Cubic effect of Temperature (Aliased)

9 Quantitative and Qualitative Factors Response:Life ANOVA for Response Surface Reduced Cubic Model Analysis of variance table [Partial sum of squares] Sum ofMeanF SourceSquaresDFSquareValueProb > F Model < A < B A AB A 2 B Pure E C Total Std. Dev.25.98R-Squared Mean105.53Adj R-Squared C.V.24.62Pred R-Squared PRESS Adeq Precision8.178

10 Regression Model Summary of Results The levels of temperature are A = -1, 0, +1 (15, 70, 125 o ) B[1] and B[2] are coded indicator variables for materials Material Type: 12 3 B[1]: 10-1 B[2]: 01-1

11 Regression Model Summary of Results Final Equation in Terms of Actual Factors: MaterialB1 Life = * Temperature * Temperature 2 MaterialB2 Life = * Temperature E-003 * Temperature 2 MaterialB3 Life = * Temperature * Temperature 2

12 Regression Model Summary of Results

13 Blocking in A Factorial Design So far, completely randomized in the factorial designs Very often, it is not feasible or practical, and it may require that the experiment be run in blocks Consider a factorial experiment with two factors (A and B) and n replicates, the linear effects model is

14 Blocking in A Factorial Design Assume different batches of raw materials have to be used, and each contains enough materials for ab observations, then each replicate must use a separate batch of material The material is a randomization restriction or a blocking factor. The effects model for the new design is Within a block the order in which the treatment combinations are run is completely randomized

15 Blocking in A Factorial Design The model assumes that the interaction between blocks and treatments is completely negligible If such interactions exist, they cannot be separated from the error component ANOVA is outlined in Table 5-18 In the case of two randomization restrictions, if the number of treatment combinations equals the number of restriction levels, then the factorial design may be run in a Latin square