1. 2p factorial experiments. Read the description of the study in Exercise 40 (Chapter 11, Section 11.1, “In a study of processes used to remove impurities …”). An ANOVA table for analyzing the dataset is given below. Follow the instructions below.
Source
DF
SS
MS
F
P-Value
A
1
0.168
0.168
0.08
0.788
B
1
1.940
1.940
0.87
0.366
C
1
8.161
8.161
3.64
0.074
D
1
13.082
13.082
5.84
0.028
A*B
1
3.419
3.419
1.53
0.235
A*C
1
0.263
0.263
0.12
0.736
A*D
1
0.911
0.911
0.41
0.533
B*C
1
0.738
0.738
0.33
0.574
B*D
1
0.781
0.781
0.35
0.563
C*D
1
6.771
6.771
3.02
0.101
A*B*C
1
0.020
0.020
0.01
0.926
A*B*D
1
0.775
0.775
0.35
0.565
A*C*D
1
0.622
0.622
0.28
0.606
B*C*D
1
1.758
1.758
0.78
0.389
A*B*C*D
1
0.004
0.004
0.00
0.967
Error
16
35.851
2.241
Total
31
75.264
a. Identify the effects (main and interactions) that are significant at the =0.10 level.
b. Compute the coefficient of determination R2 and give a brief interpretation. (See Formula 11K and pages L-71 to L-72) of lecture notes.)
c. Computed R2adj, the adjusted R2, (See Formula 11K and pages L-71 to L-72) of lecture notes.)
d. Assume that only the main and interaction effects of C and D are important (two-way interaction model). Complete the ANOVA table below in this case.
Source
DF
SS
MS
F
P-Val
C
1
8.161
8.161
?
0.036
D
?
?
?
?
0.010
C*D
1
6.771
6.771
?
0.055
Error
?
?
?
Total
31
75.264
Hint: You can derive the missing values from the previous ANOVA table.
e. Compute R2adj for the ANOVA table in part (d).
f. Assume that only the main effects of C and D (two-way additive model) are important. Below is the corresponding ANOVA table. Compute R2adj
Source DF SS MS P-val
C 1 8.1608 8.1608 4.38 0.045
D 1 13.0816 13.0816 7.02 0.013 Error 29 54.0214 1.8628
Total 31 75.2638
g. Based on the R2adj values, which of the 3 fitted models do you prefer? Give a short explanation for your choice.
2. A Fractional Factorial Design. In a study of p=5 factors with two levels each, only 8 different combinations of levels of A, B, C, D, and E will be run (i.e. q=2 and the experiment is a quarter fractional). Use A, B, D as the independent factors and with generators
C = ABD and E = AD
A. List out the 8 treatments will be run.
B. Which treatments, used in the experiment, have all of B and D at their low levels?
C. What effects will be aliased with the A main effect?
