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Assignment  Factorial Design at Two Levels and Response Surface Method
Question 1: Factorial design at two levels
An experiment was conducted on an operating chemical process in which four factors were studied in a 2^{4} factorial design. Shown in the table below are the factor levels, the design, the random order in which the runs were made, and the response (impurity) at each of the 16 reaction conditions:
Factor Levels




+

1

Concentration of Catalyst (%)

5

7

2

Concentration of NaOH(%)

40

45

3

Agitation speed (rpm)

10

20

4

Temperature (F)

150

180

Run Order

Coded Factor Levels of the Experiment Design

Impurity

1

2

3

4

2









0.38

6

+







0.4

12



+





0.27

4

+

+





0.3

1





+



0.58

7

+



+



0.56

14



+

+



0.3

3

+

+

+



0.32

8







+

0.59

10

+





+

0.62

15



+



+

0.53

11

+

+



+

0.5

16





+

+

0.79

9

+



+

+

0.75

5



+

+

+

0.53

13

+

+

+

+

0.54

(a) Make the table of contrast (a table of plus and minus signs from which all the main effects and interaction effects be calculated).
(b) Calculate all the main and interaction effects. Show the steps of your calculation and only run software to check your answer.
(c) Assuming the threefactor and higher order interactions to be noise, compute an estimate of the error variance of the effects. Show the steps of your calculation and only run software to check your answer.
(d) Based on your results from Question 1(b) and (c), which of the estimated effects are likely to be the real effects rather than noise? Why? How would you interpret each of the real effect(s)? Show your interpretation graphically.
(e) Assume that the present conditions of operation are 1=5%, 2=40%, 3=10 rpm, 4=180F. Make recommendations of better operation conditions to reduce the impurity.
(f) Assume your conclusions from Question 1(d) remain valid for operation conditions beyond the range enclosed by the two specified levels of the four factors. If you are able to conduct a few more experiment runs, describe how you would set and/or vary the four factors in the additional experiments so that, with as few experiments as possible,
i. You may confirm/ further investigate each of the real effects found in Question 1(d), and
ii. You could most likely find even better operation conditions than your answer to Question 1(e).
Question 2: Estimate the error variance of the effects
A chemist used a 2^{4} factorial design to study the effects of temperature, PH, concentration, and agitation rate on yield (measured in grams), without replicated runs. If we know the standard deviation of an individual observation is 6 grams, what is the variance of the temperature effect? Hint: use the basic properties of variances of random variables.
Question 3: Application of response surface method
The data table below came from a tire radial runout study. Lower runout values are desirable.
The factors under study were:
 x_{1} = Postinflation time (minutes),
 x_{2} = Postinflation pressure (psi),
 x_{3} = Cure temperature (^{0}F),
 x_{4} = Rim size (inches), and
The response variable was:
 y = Radial runout (mils).

x_{1}

x_{2}

x_{3}

x_{4}

y

1

9.6

40

305

4.5

0.51

2

32.5

40

305

4.5

0.28

3

9.6

70

305

4.5

0.65

4

32.5

70

305

4.5

0.51

5

9.6

40

335

4.5

0.24

6

32.5

40

335

4.5

0.38

7

9.6

70

335

4.5

0.45

8

32.5

70

335

4.5

0.49

9

9.6

40

305

7.5

0.3

10

32.5

40

305

7.5

0.35

11

9.6

70

305

7.5

0.45

12

32.5

70

305

7.5

0.82

13

9.6

40

335

7.5

0.24

14

32.5

40

335

7.5

0.54

15

9.6

70

335

7.5

0.35

16

32.5

70

335

7.5

0.51

17

60

55

320

6

0.53

18

5

55

320

6

0.56

19

17.5

85

320

6

0.67

20

17.5

25

320

6

0.45

21

17.5

25

350

6

0.41

22

17.5

25

290

6

0.23

23

17.5

25

320

9

0.41

24

17.5

25

320

3

0.47

25

17.5

25

320

6

0.32

(a) Suppose you want to fit a secondorder polynomial model to the data. Write the equations for least square regression in vector/matrix form. Define all the variables in your equation and specify the dimensions of each. Write down the matrix of independent variable values, X.
(b) Make what you think is an appropriate analysis of the data to obtain your best model for radial runout, y:
i. Describe your approach and steps of analysis
ii. Report your model expression, the overall goodnessoffit (e.g. ANOVA, esp. Rsquare and Adjusted Rsquare), estimated parameters, and their significance results (tratio or pvalue).
Hint: use stepwise least square regression. The terms included in the final model should be either a significant variable or part of a significant variable.
(c) Based on your final model, make a twodimensional contour plot of yˆ versus x_{1} and x_{3 }over the ranges 5 ≤ x_{1} ≤ 60 and 290 ≤ x_{3} ≤350, for each of the following three sets of conditions:
i. x_{2} = 40, x_{4} = 3
ii. x_{2 }= 55, x_{4 }= 9
iii. x_{2 }= 55, x_{4} = 4.5
Condition iii is the current condition used in the plant for production.
(d) Comment on this conclusion: "It was surprising to observe that either very wide (9inch) or very narrow (3inch) rims could be used to reach low radial runout levels". Is it true? Use you model to explain why it is true/not true.
(e) Based on your model, how you would vary the factors studied above to achieve low radial runout? Besides the model results, discuss the practical/operational considerations in determining how the factors should be varied.
Question 4: More than one response variables
The coded factor settings and results from an experiment are shown below. The objective is to look for the settings to achieve a high yield and low filtration time.
Trial

x_{1}

x_{2}

Yield (gs)

Field Color

Crystal Growth

Filtration Time

1





21.1

Blue

None

150

2



0

23.7

Blue

None

10

3



+

20.7

Red

None

8

4

0



21.1

Slightly Red

None

35

5

0

0

24.1

Blue

Very slight

8

6

0

+

22.2

Unobserved

Slight

7

7

+



18.4

Slightly Red

Slight

18

8

+

0

23.4

Red

Much

8

9

+

+

21.9

Very Red

Much

10


Variable Level

Variable



0

+

x_{1} = Condensation temperature (C)

90

100

110

x_{2} = Amount of B (cm^{3})

24.4

29.3

34.2

(a) Analyze the data to obtain your best model:
i. Describe your approach and steps of analysis.
ii. Report your model expression, the overall goodnessoffit (e.g. ANOVA, esp. Rsquare and Adjusted Rsquare), estimated parameters, and their significance results (tratio or pvalue).
Hint: fit a secondorder polynomial model by stepwise least square regression with each of the two responses. Take the logarithm of filtration time to fit a better model.
(b) Based on your model, draw contour diagrams for yield and filtration time.
(c) Find the settings of x_{1} and x_{2} within the range enclosed by the experiment design levels that give
i. The highest predicted yield, and
ii. The lowest predicted filtration time.
Hint: optimize each response separately. If you use an analytical method, you should check whether the true maximum/minimum condition is satisfied.
(d) Specify the optimal set of conditions for x_{1} and x_{2} that will simultaneously give high yield and low filtration time. At this set of conditions what field color and how much crystal growth would you most likely expect?
Hint: Give a best approximation of the optimal condition, either graphically or using a more formal analytical approach.
Analyzing experiments of twolevel full factorial design using least square regression in JMP:
1. Set up the coded factor levels from DOE>Custom Design or DOE>Full Factorial Design and make the design table.
2. Enter the response values from the experiments the design table.
3. Run the script "Model" from the generated design table or click Analyze>Fit model.
4. Check and configure the model specification in the popup window and click "Run".
5. Note that an effect is half the value of the corresponding parameter estimated in least square regression.
6. In the model results window, red triangle next to "Prediction Profiler" > Interaction Profiler will make the interaction profiles appear.
Attachment: Assignment File.rar