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1 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Some Basic Definitions Definition of a factor effect: The change in the mean response when.

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1 1 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Some Basic Definitions Definition of a factor effect: The change in the mean response when the factor is changed from low to high - (Low) + (High) + (High) - (Low) A Two-Factor Factorial Experiment, with the response(y) shown at the corners 52 20 30 40 Factor B Factor A 60 50 40 30 20 10 B+ B- - + Factor A Response B- A Factorial experiment without interaction

2 2 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The Case of Interaction: - (Low) + (High) + (High) - (Low) A Two-Factor Factorial Experiment, with interaction. 12 20 40 50 Factor B Factor A 60 50 40 30 20 10 B+ B- - + Factor A Response B- A Factorial experiment without interaction B+

3 3 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Example 5.1 The Battery Life Experiment Text reference pg. 167 A = Material type; B = Temperature (A quantitative variable) 1.What effects do material type & temperature have on life? 2. Is there a choice of material that would give long life regardless of temperature (a robust product)? Life (in hours) Data for the Battery Design Example Temperature ( 0 F) Material type 1570125 113015534402070 7418080758258 21501881361222570 1591261061155845 313811017412096104 1681601501398260

4 4 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The General Two-Factor Factorial Experiment a levels of factor A; b levels of factor B; n replicates This is a completely randomized design 12…b 1Y111, y112, …,y11n Y121, y122, …,y12n Y1b1, y1b2, …,y1bn 2Y211, y212, …,y21n Y221, y222, …,y22n Y2b1, y2b2, …,y2bn aYa11, ya12, …,ya1n Ya21, ya22, …,ya2n Yab1, yab2, …,yabn General Arrangement for a Two-Factor Factorial Design Factor A Factor B

5 5 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Statistical (effects) model: Other models (means model, regression models) can be useful 12…b 1Y111, y112, …,y11n Y121, y122, …,y12n Y1b1, y1b2, …,y1bn 2Y211, y212, …,y21n Y221, y222, …,y22n Y2b1, y2b2, …,y2bn aYa11, ya12, …,ya1n Ya21, ya22, …,ya2n Yab1, yab2, …,yabn General Arrangement for a Two-Factor Factorial Design Factor A Factor B

6 6 Prof. Indrajit Mukherjee, School of Management, IIT Bombay ANOVA Table – Fixed Effects Case Design-Expert will perform the computations Text gives details of manual computing (ugh!) – see pp. 171 Source of variation Sum of squares Degrees of freedom Mean squareF0F0 Treatments SS A a-1 B treatments SS B b-1 Interactions SS AB (a-1)(b-1) Error SS E ab(n-1) total SS T abn-1 The Analysis of Variance Table for the Two-Factorial, Fixed Effects Model

7 7 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Quantitative and Qualitative Factors Candidate model terms from Design- Expert: Intercept A B B 2 AB B 3 AB 2 A = Material type B = Linear effect of Temperature B 2 = Quadratic effect of Temperature AB = Material type – Temp Linear AB 2 = Material type - Temp Quad B 3 = Cubic effect of Temperature (Aliased)

8 8 Prof. Indrajit Mukherjee, School of Management, IIT Bombay SourceSum of SquaresDFMean SquareF ValueProb>F Model59416.2287427.0311<0.0001 A39118.72139042.6757.82<0.0001 B10683.7225341.867.910.0020 A2A2 76.061 0.110.7398 AB9613.7821157.541.710.0186 A2BA2B7298.6923649.355.400.1991 Residual18230.7527675.210.0106 Lack of Fit00 Pure Errror18230.7527675.21 Cor Total77646.9735 Std.dev25.98R-Squared0.7652 Mean105.53 Adj R-Squared0.6956 C.v24.62 PredR-Squared0.5826 Press32410.22Adeq Precision8.178 Design Expert Output Response: Life in Hours ANOVA for Selected Factorial Model Analysis of variance table [Partial sum of Squares]

9 9 Prof. Indrajit Mukherjee, School of Management, IIT Bombay Regression Model Summary of Results

10 10 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The effective life a cutting tool installed in a numerically controlled machine in Thought to be affected by the cutting speed and the tool angle. Three speeds and three angles are selected and a 32 factorial experiment with two replicates is performed. The coded data are shown in table. The circle numbers in the cells are the cell totals {yij} Table shows a JMP out for this experiment. This is a classical ANOVA, Treating both factors at categorical. Notice that the both design factors tool Angle and speed as well as the angle- speed interaction are significant. Since the factors are quantitative, and both factors have three levels, a second-order model such as Where x1=angle and x2=speed could also be fit to the data. The JMP output for this model is shown in table. Data for Tool Life Experiment Total Average (degrees) Cutting Speed (in/min) 125150175y i.. -2 -3 2 5 1503 0 2 1 4 4 1016 20236 5 11 0 9 2506 y.j. -2121424=y…

11 11 Prof. Indrajit Mukherjee, School of Management, IIT Bombay The second-order model doesn’t look like a very good fit to the data: the value of R2 is only 0.0465 (compared to R2= 0.895 in the categorical variable ANOVA) and the only significant factor is the linear term in speed for which the p value is 0.0731. notice that the mean square for the error in the second-order model fit is 5.5278, considerably larger than it was in the classical categorical variable ANOVA of table. The JMP output in table shows the prediction profiler, a graphical showing the response Variable life as a function of each design factor, angle and speed. The prediction profiler is very useful for optimization. Here it has been set to the levels of angle and speed that result in maximum predicted life. Part of the reason for the relatively fit of the second-order model is that Only one of the four degrees of freedom for interaction and accounted for in this model. In addition to the term, there are three other terms that could be fit to completely account for the four degrees of freedom for interaction, namely and. Summary of Fit Rsquare0.465054 Rsquare Adj0.242159 Root Mean square Error2.351123 Mean of Response1.333333 Observations (or Sum Wgts)18 Analysis of variance sourceDF Sum of squares Mean squaresF ratio Model557.6666711.53332.0864 Error1266.333335.578prob>F C Total171240.1377 Parameter Estimates TermEstimateStd.errort ratioprob>|t| Intercept-85.048683-1.580.139 Angle0.1666670.1357421.230.2431 Speed0.0533330.0271481.960.0731

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