1 the general factorial design two-factor factorial design general case: factor a – a levels...

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1 The General Factorial Design Two-factor factorial design General case: Factor A a levels Factor B b levels n replicate (n2) 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

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Page 1: 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:

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The General Factorial Design

• Two-factor factorial design

• General case: Factor A – a levels

Factor B – b levels

n replicate (n2)

• 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

Page 2: 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:

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• Effects model

• Partitioning sum of squares

Special case: a three-factor analysis of variance model

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Page 3: 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:

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• Objective: to achieve uniform fill heights• Response variable: average deviation from the

target fill height• Variables:

o percent carbonation (A, 10, 12, 14%)o operating pressure (B, 25, 30 psi)o line speed (C, 200, 250 bpm)

• Two replicates. 3222=24 runs in random order

Example 5-3: Soft Drink Bottling Problem

Page 4: 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:

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Page 5: 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:

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• 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

Page 6: 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:

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Page 7: 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:

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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

Page 8: 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:

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Quantitative and Qualitative Factors

Candidate model terms from Design- Expert: Intercept

A B A2

AB A3

A2B

Battery Life Example

A = Linear effect of Temperature

B = Material type

A2 = Quadratic effect of Temperature

AB = Material type–TempLinear

A2B = Material type–TempQuad

A3 = Cubic effect of Temperature (Aliased)

Page 9: 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:

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Quantitative and Qualitative Factors

Response: Life

ANOVA for Response Surface Reduced Cubic ModelAnalysis of variance table [Partial sum of squares]

Sum of Mean FSource Squares DF Square Value Prob > FModel 59416.22 8 7427.03 11.00 < 0.0001A 39042.67 1 39042.67 57.82 < 0.0001B 10683.72 2 5341.86 7.91 0.0020A2 76.06 1 76.06 0.11 0.7398AB 2315.08 2 1157.54 1.71 0.1991A2B 7298.69 2 3649.35 5.40 0.0106Pure E 18230.75 27 675.21C Total 77646.97 35

Std. Dev. 25.98 R-Squared 0.7652Mean 105.53 Adj R-Squared 0.6956C.V. 24.62 Pred R-Squared 0.5826

PRESS 32410.22 Adeq Precision 8.178

Page 10: 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:

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Regression Model Summary of Results

The levels of temperature are A = -1, 0, +1 (15, 70, 125o)

B[1] and B[2] are coded indicator variables for materials

Material Type: 1 2 3B[1]: 1 0 -1B[2]: 0 1 -1

Page 11: 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:

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Regression Model Summary of Results

Final Equation in Terms of Actual Factors:

Material B1 Life =+169.38017-2.50145 * Temperature+0.012851 * Temperature2

Material B2 Life =+159.62397-0.17335 * Temperature-5.66116E-003 * Temperature2

Material B3 Life =+132.76240+0.90289 * Temperature-0.010248 * Temperature2

Page 12: 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:

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Regression Model Summary of ResultsDESIGN-EXPERT Plot

Life

X = B: TemperatureY = A: Material

A1 A1A2 A2A3 A3

A: MaterialInteraction Graph

Life

B: Temperature

15.00 42.50 70.00 97.50 125.00

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Page 13: 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:

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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

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Page 14: 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:

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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

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Page 15: 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:

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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