1. Design of Engineering Experiments Part 4 – Introduction to Factorials
Text reference, Chapter 5
General principles of factorial experiments
The two-factor factorial with fixed effects
The ANOVA for factorials
Extensions to more than two factors
Quantitative and qualitative factors – response curves and surfaces

2. Basic Principles and Advantages
A factorial design – all possible combinations of the levels of the factors are investigated in each complete trial or replication
Most efficient for two or more factor experiments
Both main effects and interaction effects can be dealt with
The effects of a factor can be estimated at several levels of the other factors – wider range of validity of conclusions
Factor A (a levels), factor B (b levels), ab treatment combinations.

3. Some Basic Definitions
Definition of a factor effect (main effect): The change in the mean response when the factor is changed from low to high
The procedure will be different when having more than two levels.

4. Some Basic Definitions
Interaction occurs when differences in response between the levels of one factor is not the same at all levels of the other factors
At the low level of B (B-):
A = 40 – 20 = 20
At the high level of B (B+):
A = 52 – 30 = 22
The magnitude of the interaction effect is the average difference in the two A effects:

5. The Case of Interaction:

6. Regression Model & The Associated Response Surface
Assume the factors are quantitative. Another way of illustrating the concept of interaction.

7. 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

8. Some Basic Definitions
The knowledge of the interaction is more useful than knowledge of the main effect
A significant interaction will often mask the significance of main effects.
The main effect of one factor needs to be examined with the levels of other factors fixed when there is a significant interaction
The factorial design is more efficient than one-factor-at-a-time design, and the efficiency increases with the number of factors

9. Example 5-1 The Battery Life Experiment Text reference pg. 175
A = Material type; B = Temperature (A quantitative variable)
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)?

10. The General Two-Factor Factorial Experiment
a levels of factor A; b levels of factor B; n replicates
This is a completely randomized design

11. Statistical (effects) model:
Other models (means model, regression models) can be useful

12. Hypotheses testing Equality of row treatment effects
Ho: t1 = t2 =    = ta = 0
H1: at least one ti  0
Equality of column treatment effects
Ho: b1 = b2 =    = bb = 0
H1: at least one bj  0
Equality of row and column treatment interactions
Ho: (tb)ij = 0 for all i, j
H1: at least one (tb)ij  0

13. Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 167

14. ANOVA Table – Fixed Effects Case
Design-Expert will perform the computations
Text gives details of manual computing

15. Design-Expert Output – Example 5-1
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 < A B < AB Pure E C Total
Std. Dev R-Squared Mean Adj R-Squared C.V Pred R-Squared
PRESS Adeq Precision 8.178

16. Residual Analysis – Example 5-1

17. Residual Analysis – Example 5-1

18. Residual Analysis – Example 5-1

19. Interaction Plot

20. Choice of Sample Size
OC curves can be used to determine sample size (n)
An effective way is to specify a min. diff. between any two treatment means. E.g. if the difference in any two row means is D, the min. value of F2 is
If the difference in any two columns is specified,
If the difference in any two interactions is specified,

21. One Observation per Cell
If there are two factors and only one observation per cell, the effect model is
Reminder: when n  2,
The two-factor interaction effect (tb)ij and the experimental error cannot be separated in obvious manner. Therefore, there are no tests on main effects unless the interaction effect is zero. And the model will be
The main effects may be tested by comparing MSA and MSB to MSResidual