Statistics

Problem 1:

Write down a 16-run design for 10 factors.

For a 10 factor each at two levels, the 16-run design will be as provided below;

RunOrder	A	B	C	D	E	F	G	H	J	K
1	-1	1	1	1	-1	1	-1	-1	-1	-1
2	1	-1	1	1	-1	-1	1	-1	-1	-1
3	1	1	-1	1	-1	-1	-1	1	-1	1
4	-1	-1	-1	1	-1	1	1	1	-1	1
5	-1	-1	1	-1	1	1	1	-1	-1	1
6	-1	-1	1	1	1	-1	-1	1	1	1
7	1	-1	1	-1	-1	1	-1	1	1	-1
8	-1	-1	-1	-1	-1	-1	-1	-1	1	1
9	-1	1	1	-1	-1	-1	1	1	1	-1
10	1	1	-1	-1	-1	1	1	-1	1	1
11	1	-1	-1	-1	1	-1	1	1	-1	-1
12	1	1	1	1	1	1	1	1	1	1
13	1	-1	-1	1	1	1	-1	-1	1	-1
14	-1	1	-1	1	1	-1	1	-1	1	-1
15	-1	1	-1	-1	1	1	-1	1	-1	-1
16	1	1	1	-1	1	-1	-1	-1	-1	1

a) Give the notation of the design.

210-6 design.

b) How might you decide which 10 columns to use for a balanced design?

In deciding which 10columns to use for a balanced design, an experimenter has to ensure that for each of the 10 columns there are eight 1 (high level) and eight -1 (low level) since the experiment involves 16 runs. According to Moen, Nolan and Provost (2013), this selection would provide a balance to the design (p. 164). In turn, any four factors of the ten factors form a full factorial design and can be incorporated if the experimenters believe that there are six factors whose effect is negligible on the response (low-level current knowledge) and sequentially does not interact with the other factors.

Problem 2

Define a definitive screening design

a) What is a DSD?

This is a resolution IV design in which the main effects are not aliased with 2-way interactions proposed by Jones and Nachtsheim and constitute a recognized class of screening designs for three level factors. In a DSD, all factors are numeric and subsequently tested at three levels with a characteristic feature of a self-foldover (Errore, Jones, Li & Nachtsheim, 2017). This consecutively forms the basic reason why DSDs are dissimilar from other standard designs. The term “screening” inclusion in the label of the design is an implication that the design is considered in determining factors with significant linear effects. As such, DSD have dependable properties for determining the significant linear effects irrespective of active second-order effects which subsequently is appropriate for the requirements of a screening design.

b) How many levels are usually in a DSD?

According to Jones and Nachtsheim (2011), Definitive Screening Design (DSD) usually have three levels as this allows for assessment of curvature in the factor response interaction (p. 2).

c) Is a DSD better for a small number of factors (less than 5)?

NO; Definitive Screening Designs as the name “screening” suggest are important in screening out factors at the early stages of experimentation when there are large number of factors and helps identify a normally reduced number of extremely significant factors. Jones and Nachtsheim (2011) as such proposes the use of more than 5 factors in DSD.

d) What is the notation for a DSD of 10 factors?

General notation for a DSD is 2m+1 where m represents the number of factors and the result of the notation is the number of runs for the experiment, the notation for 10 factors in a DSD will be 2*10+1.

Problem 3:

a) Provide a run order chart

b) Any trends in the chart?

The data does not show any upward or downward trend or sudden shift from the Minitab output above; however the chart shows a significant amount of cyclic movements the chart does not indicate any cause for variation in Clustering, Mixtures, Trends and Oscillation as they have p-values higher than α=0.05 level of significance.

c) Dotplot

The data is not symmetrically distributed as the Dotplot indicates the data is skewed to the left as a result there are more observations of higher values.

d) Effects

Term Effect Coef SE Coef T-Value P-Value VIF

Constant 68.81 4.94 13.93 0.000

A -10.38 -5.19 4.94 -1.05 0.314 1.00

B 22.38 11.19 4.94 2.26 0.043 1.00

A*B -2.13 -1.06 4.94 -0.22 0.833 1.00

e) Normal plot of effects

Effect as a result of factor B is significant.

f) Prediction equation

y = 68.81 - 5.19 A + 11.19 B - 1.06 A*B

g) ANOVA table

Below is an output for Minitab ANOVA table;

Analysis of Variance

Source DF Adj SS Adj MS F-Value P-Value

Model 3 2451.19 817.06 2.09 0.155

Linear 2 2433.13 1216.56 3.12 0.081

A 1 430.56 430.56 1.10 0.314

B 1 2002.56 2002.56 5.13 0.043

2-Way Interactions 1 18.06 18.06 0.05 0.833

A*B 1 18.06 18.06 0.05 0.833

Error 12 4685.25 390.44

Total 15 7136.44

h) Any significant factors or interactions?

From the Minitab output above, Factor B is significant with a p-value 0.043 which is less than α=0.05 level of significance. Factor A and the 2-way interaction A*B are not significant having p-values of 0.314 and 0.833 respectively and are greater than α=0.05 level of significance. The normal probability plot of effects also iterates that the effect as a result of factor B is significant.

i) Recommendations

Factor B has significant influence on the response variable as such more research should be conducted to determine the causes for the importance of factor B on the response variable y. Other factor combination can be considered to examine their influence on the response variable.

References

Errore, A., Jones, B., Li, W., and Nachtsheim, C. (July 2017). Using definitive screening designs to identify active first- and second-order factor effects. Journal of Quality Technology, 49(3), 244-264.

Jones, B., & Nachtsheim, C. J. (2011). A class of three-level designs for definitive screening in the presence of second-order effects. Journal of Quality Technology, 43(1), 1.

Moen, R. D., Nolan, T. W., & Provost, L. P. (2013). Reducing the size of experiments. In Quality improvement through planned experimentation (3rd ed., pp. 160-190). New York: McGraw-Hill Education.

standard

orderrun orderABy

115-1-190

2101-135

314-1185

471180

511-1-140

661-178

718-1190

851180

99-1-130

1031-153

1119-1187

1281185

1313-1-187

1411-148

1517-1183

1641150

16151413121110987654321

90

80

70

60

50

40

30

Number of runs about median:10

Expected number of runs:8.9

Longest run about median:3

Approx P-Value for Clustering:0.723

Approx P-Value for Mixtures:0.277

Number of runs up or down:11

Expected number of runs:10.3

Longest run up or down:2

Approx P-Value for Trends:0.663

Approx P-Value for Oscillation:0.337

Observation

y

Run Chart of y

8880726456484032

y

Dotplot of y

3210-1-2-3

99

95

90

80

70

60

50

40

30

20

10

5

1

AA

BB

FactorName

Standardized Effect

P

e

r

c

e

n

t

Not Significant

Significant

Effect Type

B

Normal Plot of the Standardized Effects

(response is y, α = 0.05)