Higher Order Factorial Designs. Estimated Effects: Section 4.3. Main Effects: Definition 5 on page 166.
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1 Higher Order Factorial Designs Estimated Effects: Section 4.3 Main Effects: Definition 5 on page 166. Without A effects, we would fit values with the overall mean. The main effects are how much we need to move from the overall mean to the means for each factor level. Diameter Mean Difference a Fluid Mean Difference b 1 Glycol Water Interaction effects: The predicted values in the plot below are parallel lines if there is no interaction included in the model. The ab ij effects are how much we have to move the parallel line fitted values to fit the means for each treatment group, the difference between the fitted model, lines with + marker, and means with O markers. See definition 6 on page 169. A model with a, b, and ab effects has 4 parameters. o With 4 parameters we can fit 4 means exactly. The degree of non-parallel fit can be measured by o (Glycol Water for Diameter) (Glycol Water for Diameter) o ( ) ( ) = = Overall, the difference of glycol-water fluid effects for and diameters is o We need to nudge the 4 points up or down by ±0.519/4 = ± 0.013, the ab 11 effect. o See definition 6 on page 169.
2 Diameter Fluid Mean Glycol - Water Glycol - Water Difference ab Glycol Water Glycol Water Prediction Expression An important perspective for understanding concepts such as fractional factorials: The interaction effect comparison is a product of main effect comparisons. Diameter Fluid Diameter Fluid Diam*Fluid Mean D*F x Fluid Glycol Water Glycol Water Sum The notation D*F for interactions also comes from df D*F = df D * df F
3 We could fit the factorial model with a regression using these 1, -1 variables. a1 a2 Sum a2 = -a1 b1 b2 Sum b2 = -b1 Sum ab11 = ab21 = ab12 = ab22 = Sum 0 0 ab21 = -ab11 ab12 = -ab11 ab22 = ab11 Diameter Fluid Model Glycol a1 + b1 + ab11 = a1 + b1 + ab Water a1 + b2 + ab12 = a1 b1 ab Glycol a2 + b1 + ab21 = a1 + b1 ab Water a2 + b2 + ab22 = a1 b1 ab11 Diameter Fluid Model. a1*x1 + b1*x2 + ab11*x Glycol a1*(+1) + b1*(+1) + ab11*(+1) Water a1*(+1) + b1*(1) + ab11*(1) Glycol a1*(1) + b1*(+1) + ab11*(1) Water a1*(1) + b1*(1) + ab11*(1) The factorial model is the same as a regression model with X columns of 1's and 1's. Diameter Fluid Diam*Fluid Ln Time Diameter Fluid X1 X2 X3 Y Glycol Glycol Glycol Water Water Water Glycol Glycol Glycol Water Water Water
4 ANOVA Table and R 2 The sum of squares for any effect is the improvement in Error Sum of Squares by adding this effect to the model with all other effects. Fitting a model without the interaction gives Total SS = Model SS = Total SS Error SS = Improvement in error SS by including fluid and diameter effects. See equation 4.27 on page 173 and section 7.4 Fitting a model with all 3 factorial effects gives. The improvement in Error SS by including the diameter*fluid interaction effect in addition to fluid and diameter effects is = SS Fluid*Diameter This increased the R 2 from 97.38% to 99.88%.
5 An effect is statistically significant if it has a large F ratio which comes from a large reduction in error SS from including that effect. In this case all 3 effects are highly significant. But with a significant interaction, we should ignore the fluid and diameter main effects. 2 3 design: 3 factors each with 2 levels We can also fit a model which includes the technician main effects and associated interactions in the model. The 3-way interaction effects are how much we have to nudge the fitted values from a model with no 3-way interaction in order to fit the 8 treatment combination means. The plot below is the fitted values from a model with all 2-way interactions but no 3-way interaction. These fitted values are very close to the means. In the table below, the last column finds the difference between the diameter*fluid measures for tech 1 and the diameter*fluid interaction for fluid 2. [Tech 1 (Glycol Water for Diameter) (Glycol Water for Diameter)] - [Tech 2 (Glycol Water for Diameter) (Glycol Water for Diameter)] = This measures the degree to which the diameter*fluid interaction difference, (Glycol Water for Diameter) (Glycol Water for Diameter)], is not the same for both techs. To move from not quite fitting the means without the 3-way interaction, the fit above with the same 2-way interactions for both techs, to fitting the 8 means exactly, we need to nudge each fitted value by ±0.0520/8 = ± = abc 111. Tech Diam Fluid Tech Diam Fl;uid T*D T*F D*F T*D*F Mean T*D*F * Mean Glycol Water Glycol Water Glycol Water Glycol Water Sum ab Sum/8 To fit a factorial model make sure that the Tech variable with values 1 and 2 has Data Type = Character. Double click the column name as usual to reset variable attributes.
6 Notice: All parameter estimates have the same standard error. The p-values for these estimated effects are the same as the p-values from the ANOVA table. The Sum of Squares for any effect is the improvement in Error SS by including that effect in the model in addition to all other effects. The Tech effects do little for improving R 2 by decreasing error SS and increasing model SS. See section 8.2 and section
7 How are the estimated effects used in practice? See section 4.3, 4.4, and Example 12. Page 195. An Unreplicated 2 4 Design Y = Advance rate of drill A = Load B = Flow rate through the drill C = Rotational speed D = Mud type used with drill Only B, C and D appear to have effects detectable above the noise level. See JMP help for "Pseudo Standard Error" for reference PSE in the 1989 paper by Lenth.
8 Fractional Factorials Going a step further in cutting down the number of runs: A fractional factorial Recall Hahn article ½ Fraction of a 2 5 Design A 2 5 design would take 32 runs to run all treatment combinations. A B C D E ABCDE
9 Using just the combinations with ABCDE = 1 cuts the number of runs down to half as many, 16 runs. Algebraically, ABCDE = 1 A 2 BCDE = A BCDE = A A and BCDE are completely confounded. o The BCDE effect is just another name for, alias of, the A effect. ABCDE = 1 A 2 B 2 CDE = AB CDE = AB AB and CDE are completely confounded. o The BCD effect is just another name for, alias of, the AB effect. A B C D E ABCDE B*C*D*E A - BCDE A*B C*D*E AB - CDE See section 8.3.
10 Problem 19 in Chapter 8 A ½ Fractional Factorial of a 2 5 Design, a design with 2 4 = 16 runs. Data from Kennett and Vogel, Quality Progress, 1991 Y = number of solder faults per 100 solder joints A = Conveyor Speed B = Preheat Temperature C = Solder temperature D = Conveyor Angle E = Flux Concentration Set all variables to be Data Type = Character for ANOVA model.
11 The important effect could be either D = Conveyor Angle or C*E = Solder Temp*Conveyor Angle Usually, the ½ factorial would confound main effects with 4-way interactions, for example confound D with A*B*C*E
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