Design of Experiments Introduction Part II

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 = 0.1734/0.0203 = 8.53; P(F(3,12) > 8.53) = 0.0026 Ho can be rejected for any  > 0.0026. There is a significant difference in the wear resistance among the 4 fabric. Which (type of) fabric is better? Don’t know!

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

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

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 -0.0175 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 -0.1075 2.28 -0.035 4 2.25 0.06 2.7 0.02 -0.1375 -0.065 Average 2.19 2.4175 2.315

Normal Quantile Plot

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.

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

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

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!

SNK Range Test E3 V2 E3 V1 E2 V2 E2 V1 E1 V1 E1 V2 7.5 11 12 30.5 53 58 Okay Lower Pressure (Vacuum) is better, pick the Red section. 150s220V, 150s127V, 90s220V Okay! Don’t forget to assess the normality Assumption

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

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 31 108.930

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

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 http://www.spss.com/downloads/Papers.cfm?ProductID=00035&Name=SPSS_Base&DLType=Demo Minitab 14, http://www.minitab.com/downloads/