1. Design of Experiments
Introduction Part II

2. One-way ANOVA Source SS df MS F p Treatment, t 0.5201 3 0.1734 8.53
0.0026
Error, e
0.2438 12
0.0203
Total
0.7639
f-statistic = / = 8.53;
P(F(3,12) > 8.53) =
Ho can be rejected for any  >
There is a significant difference in the wear resistance among the 4 fabric.
Which (type of) fabric is better? Don’t know!

3. Which Level (Fabric)?
Student-Newman-Keuls (SNK) Range test (Pairwise Comparison)
Tukey’s test
Duncan’s multiple range test

4. SNK Range Test ; 0.49>0.30 so B>A. ; 0.36>0.27 so B>D.
; 0.26>0.22 so B>C.
; 0.23<0.27 so C may be equal A.
; 0.10<0.22 so C may be equal D.
; 0.13<0.22 so D may be equal A.
A,D,C are similar but B differs from the rest.
2.19
2.32
2.42
2.68 A D C B

5. F-Test on two Variances
Sampled populations are normally distributed
 ~N(0,2)
Obs A
eij B C D 1
1.93
-0.26
2.55
-0.13
2.4
2.33
0.015 2
2.38
0.19
2.72
0.04
2.68
0.2625
0.085 3
2.2
0.01
2.75
0.07
2.31
2.28
-0.035 4
2.25
0.06
2.7
0.02
-0.065
Average
2.19
2.4175
2.315

6. Normal Quantile Plot

7. What if the Model is Inadequate?
Randomization Restriction
Data Transformation
y3, y2 shorten the tail of left skew distribution.
shorten the tail of a right skewed distribution.

8. Two-way ANOVA Vacuum Tube - Used in high end audio system. Vacuum Pump
Exhaust index ( in seconds):
60, 90, 150
Pump Heater Voltage (in volts):
127, 220
On the pressure inside a tube ( m-6 Hg)
2 observations for each treatment

9. Example Two-factor Yijk = m + Ei + Vj + EVij + ek(ij) Exhaust Index, i
Voltage, j
60 sec.
90 sec.
150 sec.
127 volts 48 58 28 33 7 15
220 volts 62 54 14 10 6 9

10. Two-way Anova Source df SS MS F Prob Exhaust 2 4608.17 2304.08 99.5
.001*
Voltage 1
96.33
4.2
.088
E-V Interact
283.16
141.58
6.1
.036*
Error 6
139
23.17
Which combination gives a better pressure (vacuum) performance?
Do an SNK range test!

11. SNK Range Test E3 V2 E3 V1 E2 V2 E2 V1 E1 V1 E1 V2 7.5 11 12 30.5 5358
Okay
Lower Pressure (Vacuum) is better, pick the Red section.
150s220V, 150s127V, 90s220V
Okay!
Don’t forget to assess the normality Assumption

12. Three-factorial experiment – Cutting Power Requirement
Tool Type (i)
Type 1
Type 2
Cut Type (k)
Bevel Angle (j)
Bevel Angle
15o
30o
Continuous
29.0
26.5
30.5
27.0
28.5
30.0
32.5
28.0
25.0
29.5
32.0
Interrupted
27.5
24.5
26.0

13. 3-way ANOVA Source DF SS MS F P Tool 1 2.820 1.27 0.272 Bevel 20.320
9.13
0.006
Tool*Bevel
0.195
0.09
0.770
Cut
31.008
13.93
0.001
Tool*Cut
0.008
0.00
0.953
Bevel Cut
0.945
0.42
0.521
Tool*Bevel*Cut
Error 24
53.437
2.227
Total 31

14. Summary Experiment factorial experiments
Problem – if factors have significant effect on response
Response
Factors
Level of factors
factorial experiments
Completely random design
ANOVA
F-test
Multiple Comparison Procedures
SNK Range Test
Normality Assumption
Normal Quantile Plot

15. For your Information
Hicks and Turner, Fundamental Concepts in the Design of Experiments, 5th Ed., Oxford 1999
Montgomery, Design and Analysis of Experiments, 5th Ed., John Wiley, 2001.
SPSS for Windows
Minitab 14,