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

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• Effects model

• Partitioning sum of squares

Special case: a three-factor analysis of variance model

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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

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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

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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

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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)

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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

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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

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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

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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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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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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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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