1. Chapter 5: Introduction to Factorial Designs
Design & Analysis of Experiments 8E 2012 Montgomery
STT 511-STT411: Design of Experiments and Analysis of Variance Dr. Cuixian Chen
Chapter 5
Chapter 5: Introduction to Factorial Designs

2. Chapter 5
Design & Analysis of Experiments 8E 2012 Montgomery

3. STT215 Review: Experiment design with 2 factors
What are the effects of repeated exposure to an advertising message (digital camera)? The answer may depend on the length of the ad and on how often it is repeated. Outline the design of this experiment with the following information.
Subjects: 150 Undergraduate students.
Two Factors: length of the commercial (30 seconds and 90 seconds – 2 levels) and repeat times (1, 3, or 5 times – 3 levels)
Response variables: their recall of the ad, their attitude toward the camera, and their intention to purchase it. (see page 187 for the diagram.)

4. Design of Engineering Experiments – 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
Factorial design: in each complete trial or replicate of experiment, consider all possible combinations of the levels of factors.
Eg: if there are a levels of factor A and b levels of factor B, each replicate contains all a*b treatment combinations.
When factors are arranged in a factorial design, they are often said to be crossed.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

5. Some Basic Definitions
Parallel
Definition of a factor effect: The change in the mean response when the factor is changed from low to high
The main effect of a factor is the difference b/w average at low level and the average at high level.
Main effect of Factor A
Main effect of Factor B
Lack of Interaction effect b/w Factor A & Factor B
Chapter 5
Design & Analysis of Experiments 8E 2012 Montgomery

6. The Case of Interaction:
Not
Parallel
Main effect of Factor A
Main effect of Factor B
Interaction effect b/w Factor A & Factor B
Chapter 5
Design & Analysis of Experiments 8E 2012 Montgomery

7. Regression Model & The Associated Response Surface
Assume both factors are quantitative (eg: temperature, pressure, time etc.)
Factor A
Another way to illustrate interaction:
Factor B
Interaction effect b/w Factor A & Factor B
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5
Interaction contribution to this experiment was negligible.

8. 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
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5
Now suppose the interaction contribution to this experiment was not negligible.

9. The connection between ANOVA and Linear model
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

10. Review: ANOVA for one factor CRD
The reference distribution for F0 is the F(a-1, a(n-1)) distribution
Reject the null hypothesis (equal treatment means) if
Chapter 3
Design & Analysis of Experiments 8E 2012 Montgomery

11. Example 5.1 The Battery Life Experiment Text reference pg. 187
For a factorial design with two factors, an engineer is designing a battery in a device subjected to some extreme variations in temperature. The only design parameter that he can select at this point is plate material for the battery, with three possible choices.
When device is manufactured and is shipped to field, the engineer has no control over temperature extremes, which may affect effective battery life.
The engineer decides to test all three plate materials at three temperature levels—15, 70, and 125°F.
For two factors at three levels, this design is sometimes called a 3^2 factorial design.
Four batteries are tested at each combination of plate material and temperature, and all 36 tests are run in random order.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 4

12. Example 5.1 The Battery Life Experiment Text reference pg. 187Q:
# of levels of Material type?
# of levels of temperature?
# of replication in each combination?
Factor A = Material type; Factor B = Temperature (A quantitative variable)
Qestions:
What effects do material type & temperature have on battery life?
2. Is there a choice of material that would give long life regardless of temperature (a robust product)?
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

13. The General Two-Factor Factorial Experiment
a levels of factor A; b levels of factor B;
n replicates
This is a completely randomized design (CRD) for two factors.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

14. Statistical (effects) model:
Other models (means model, regression models) can be useful.
is overall mean effect, is effect of ith level of row factor A, is effect of jth level of column factor B, is the effect of interaction between and , and
is a random error component.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

15. Two-factor Factorial Design
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 4

16. Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 189
Let’s add and subtract four terms:
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

17. Extension of the ANOVA to Factorials (Fixed Effects Case) – pg. 189
Let’s add and subtract four terms:
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

18. ANOVA Table for Two-factor Factorial Design – Fixed Effects Case
The reference distribution for F0 is the F((numerator df), (denominator df)) distribution
Reject the null hypothesis (equal treatment means) if p-value<0.05,
Where p-value=1-pf(F0, (numerator df), (denominator df))
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

19. Example 5.1: The Battery Design Experiment
Table 5.4 presents the effective life (in hours) observed in the battery design example described in Section The row and column totals are shown in the margins of the table, and the circled numbers are the cell totals.
Total of each call
Q: ANOVA for Two-factor Factorial Design with fixed effects: We have
SSTotal = , SSMaterial = , SSTemperature = , SSIneraction = , 1) first find SSerror;
2) then find MSMaterial, MSTemperature, MSInteraction, MSerror, and
3) Find THREE F0’s and p-value, respectively for Materials, Temperature and Interaction.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 4

20. The Battery Design Experiment: ANOVA for two-factor CRD
For two factors at three levels, this design is sometimes called a 3^2 factorial design.
Q: How to input data for ANOVA for two-factor CRD?
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 3

21. The Battery Design Experiment: ANOVA for two-factor CRD
################ The Battery Design example: how to input your data ####################
group=c(rep(15, 4), rep(70, 4), rep(125, 4));
temp=rep(group, 3); print(temp);
material=c(rep(1, 12), rep(2, 12), rep(3, 12)); print(material);
y=c(130, 155, 74, 180, 34, 40, 80, 75, 20, 70, 82, 58, 150, 188, 159, 126, 136, 122, 106, 115, 25, 70, 58, 45, 138, 110, 168, 160, 174, 120, 150, 139, 96, 104, 82, 60);
#anova(lm(y~factor(temp)*factor(material)))
anova(lm(y~factor(temp)+factor(material)+factor(temp)*factor(material)))
Interaction b/w factors
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 3

22. The Battery Design Experiment: ANOVA for two-factor CRD
A significant interaction between material types and temperature.
Furthermore, the main effects of material type and temperature are also significant.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 4

23. The Battery Design Experiment: ANOVA for two-factor CRD
To assist in interpreting the results of this experiment, it is helpful to construct a graph of the average responses at each treatment combination.
The significant interaction is indicated by lack of parallelism of lines. In general, longer life is attained at low temperature, regardless of material type.
Changing from low to intermediate temperature, battery life with material type 3 may actually increase, whereas it decreases for types 1 and 2.
From intermediate to high temperature, battery life decreases for material types 2 and 3 and is essentially unchanged for type 1. Material type 3 seems to give the best results if we want less loss of effective life as the temperature changes.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 4

24. Design-Expert Output – Example 5.1
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

25. JMP output – Example 5.1
That is, about 77% of variability in battery life is explained by plate material, temperature, and material type–temperature interaction.
Chapter 5
Design & Analysis of Experiments 8E 2012 Montgomery

26. Residual Analysis: ANOVA for two-factor CRD
################ The Battery Design example ####################
Battery<-read.table("\\\\bearsrv\\classrooms\\Math\\wangy\\stt4511\\Battery-Life.TXT",header = TRUE);
model5.1<-lm(Battery$Life~as.factor(Battery$Temp)+as.factor(Battery$Material)+as.factor(Battery$Temp)*as.factor(Battery$Material))
anova(model5.1);
## Residual Analysis ##
## Model5.1=lm(y~as.factor(temp)+as.factor(material)+as.factor(temp)*as.factor(material));
res.model5.1<-resid(model5.1);
par(mfrow=c(3,2))
qqnorm(res.model5.1); #checking normality for residuals#
qqline(res.model5.1);
plot(c(1:36),res.model5.1); #time sequence plot for checking independence#
plot(Battery$Temp,res.model5.1); #checking equal variance for Temperature#
plot(Battery$Material,res.model5.1); #checking equal variance for Material#
plot(model5.1$fitted,res.model5.1); #Model validation/fitness#
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 3

27. Residual Analysis – Example 5.1
The normal probability plot of these residuals (Figure 5.11) does not reveal anything particularly troublesome, although the largest negative residual (60.75 at 15°F for material type 1) does stand out somewhat from the others.
There is some mild tendency for the variance of the
residuals to increase as the battery life increases.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

28. Residual Analysis – Example 5.1
Figures 5.13 and 5.14 plot the residuals versus material types and temperature, respectively.
Both plots indicate mild inequality of variance, with the treatment combination of 15°F and material type 1 possibly having larger variance than the others.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 5

29. Example 5.5: ANOVA for two-factor CRD
Total of each call
Q: Use R to do the ANOVA to analyze this two-factor CRD.
Design & Analysis of Experiments 8E 2012 Montgomery
Chapter 4

30. Chapter 5
Design & Analysis of Experiments 8E 2012 Montgomery

31. Example 5.5: ANOVA for two-factor CRD
################ The Battery Design example: how to input your data ####################
group=c(rep(125, 2), rep(150, 2), rep(175, 2));
Speed=rep(group, 3); print(Speed);
Angle=c(rep(15, 6), rep(20, 6), rep(25, 6)); print(Angle);
y=c(-2, -1, -3, 0, 2, 3, 0, 2, 1, 3, 4, 6, -1, 0, 5, 6, 0, -1);
cbind(Speed, Angle, y)
anova(lm(y~as.factor(Speed)+as.factor(Angle)+as.factor(Speed)*as.factor(Angle)))
#anova(lm(y~Speed+Angle+Speed*Angle+I(Speed^2)+I(Angle^2)))
Chapter 5
Design & Analysis of Experiments 8E 2012 Montgomery

32. An engineer suspects that the surface finish of a metal part is influenced by the feed rate and the depth of cut. She selects three feed rates and four depths of cut. She then conducts a factorial experiment and obtain the result:
Fill in the blanks
How many observations in total?
How many replications for each interaction of the Depth and the Feed?
What are the THREE null and alternative hypotheses?
May we reject all THREE null hypotheses?
What would be the output if we use RCBD?
With the ORIGINAL data, if the SS for AB is 55.7 instead of May we use RCBD instead?
Chapter 5
Design & Analysis of Experiments 8E 2012 Montgomery

33. State all the nulls and alternatives Do the two factor interacts?
Fill in the blanks.
State all the nulls and alternatives
Do the two factor interacts?
Is the average of the 1st phosphor different from 250?
Is the avg of the 1st phosphor at least 10 less than the average of the 2nd phosphor
Chapter 5
Design & Analysis of Experiments 8E 2012 Montgomery