Analysis of Multi-Factored Experiments: Estimating Effects in Two-Level Factorial Designs, Study notes of Mechanical Engineering

Download Analysis of Multi-Factored Experiments: Estimating Effects in Two-Level Factorial Designs and more Study notes Mechanical Engineering in PDF only on Docsity! 1 ME 311, Mechanical Engineering University of Kentucky ME 311 Experimentation II Spring, 2006 22 and 23 Factorial Experiments ME 311, Mechanical Engineering University of Kentucky Analysis of Multi-Factored Experiments Two-level 2 Factor Factorial Designs ME 311, Mechanical Engineering University of Kentucky The 2k Factorial Design • Special case of the general factorial design; k factors, all at two levels • The two levels are usually called low and high (they could be either quantitative or qualitative) • Very widely used in industrial experimentation • Form a basic “building block” for other very useful experimental designs (DNA) • Special (short-cut) methods for analysis ME 311, Mechanical Engineering University of Kentucky Chemical Process Example A = reactant concentration, B = catalyst amount, y = recovery (response) 2 ME 311, Mechanical Engineering University of Kentucky The Simplest Case: The 22 “-” and “+” denote the low and high levels of a factor, respectively Low and high are arbitrary terms Geometrically, the four runs form the corners of a square Factors can be quantitative or qualitative, although their treatment in the final model will be different ME 311, Mechanical Engineering University of Kentucky Estimating Effects of Each Factor Estimate of the effect of A a1b1 - a0b1 estimate of effect of A at high B a1b0 - a0b0 estimate of effect of A at low B sum/2 estimate of effect of A over all B Estimate of the effect of B a1b1 - a1b0 estimate of effect of B at high A a0b1 - a0b0 estimate of effect of B at high A sum/2 estimate of effect of B over all A ME 311, Mechanical Engineering University of Kentucky Estimate the interaction of A and B a1b1 - a0b1 estimate of effect of A at high B a1b0 - a0b0 estimate of effect of A at low B difference/2 estimate of effect of B on the effect of A Called the interaction of A and B a1b1 - a1b0 estimate of effect of B at high A a0b1 - a0b0 estimate of effect of B at low A difference/2 estimate of the effect of A on the effect of B Called the interaction of B and A Estimating Interaction of Factors ME 311, Mechanical Engineering University of Kentucky • Note that the two differences in the interaction estimate are identical; by definition, the interaction of A and B is the same as the interaction of B and A. In a given experiment one of the two literary statements of interaction may be preferred by the experimenter to the other; but both have the same numerical value. Interactions 5 ME 311, Mechanical Engineering University of Kentucky Estimating effects in three-factor two-level designs (23) Estimate of A (1) a - 1 estimate of A, with B low and C low (2) ab - b estimate of A, with B high and C low (3) ac - c estimate of A, with B low and C high (4) abc - bc estimate of A, with B high and C high = (a+ab+ac+abc - 1-b-c-bc)/4, = (-1+a-b+ab-c+ac-bc+abc)/4 (in Yates’ order) Note that nothing new is added here, the algebra just becomes a little more complex. ME 311, Mechanical Engineering University of Kentucky Estimate of AB Note that interactions are averages. Just as our estimate of A is an average of response to A over all B and all C, so our estimate of AB is an average response to AB over all C. AB = {[(4)-(3)] + [(2) - (1)]}/4 = {1-a-b+ab+c-ac-bc+abc)/4, in Yates’ order or, = [(abc+ab+c+1) - (a+b+ac+bc)]/4 Effect of A with B high - effect of A with B low, all at C high plus effect of A with B high - effect of A with B low, all at C low ME 311, Mechanical Engineering University of Kentucky interaction of A and B, at C high minus interaction of A and B at C low ABC = {[(4) - (3)] - [(2) - (1)]}/4 =(-1+a+b-ab+c-ac-bc+abc)/4, in Yates’ order or, =[abc+a+b+c - (1+ab+ac+bc)]/4 Estimate of ABC ME 311, Mechanical Engineering University of Kentucky This is our first encounter with a three-factor interaction. It measures the impact, on the yield of the nitration process, of interaction AB when C (heel) goes from C absent to C present. Or it measures the impact on yield of interaction AC when B (stirring time) goes from 1/2 hour to 4 hours. Or finally, it measures the impact on yield of interaction BC when A (time of addition of nitric acid) goes from 2 hours to 7 hours. As with two-factor two-level factorial designs, the formation of estimates in three-factor two-level factorial designs can be summarized in a table. 6 ME 311, Mechanical Engineering University of Kentucky Sign Table for a 23 design A B AB C AC BC ABC 1 - - + - + + - a + - - - - + + b - + - - + - + ab + + + - - - - c - - + + - - + ac + - - + + - - bc - + - + - + - abc + + + + + + + ME 311, Mechanical Engineering University of Kentucky Example A = main effect of nitric acid time = 1.25 B = main effect of stirring time = -4.85 AB = interaction of A and B = -0.60 C = main effect of heel = 0.60 AC = interaction of A and C = 0.15 BC = interaction of B and C = 0.45 ABC = interaction of A, B, and C = -0.50 Yield of nitration process discussed earlier: 1 a b ab c ac bc abc Y = 7.2 8.4 2.0 3.0 6.7 9.2 3.4 3.7 NOTE: ac = largest yield; AC = smallest effect ME 311, Mechanical Engineering University of Kentucky We describe several of these estimates, though on later analysis of this example, taking into account the unreliability of estimates based on a small number (eight) of yields, some estimates may turn out to be so small in magnitude as not to contradict the conjecture that the corresponding true effect is zero. The largest estimate is -4.85, the estimate of B; an increase in stirring time, from 1/2 to 4 hours, is associated with a decline in yield. The interaction AB = -0.6; an increase in stirring time from 1/2 to 4 hours reduces the effect of A, whatever it is (A = 1.25), on yield. ME 311, Mechanical Engineering University of Kentucky Or equivalently an increase in nitric acid time from 2 to 7 hours reduces (makes more negative) the already negative effect (B = -485) of stirring time on yield. Finally, ABC = -0.5. Going from no heel to heel, the negative interaction effect AB on yield becomes even more negative. Or going from low to high stirring time, the positive interaction effect AC is reduced. Or going from low to high nitric acid time, the positive interaction effect BC is reduced. All three descriptions of ABC have the same numerical value; but the chemist would select one of them, then say it better. 7 ME 311, Mechanical Engineering University of Kentucky Number and kinds of effects We have already introduced the notation 2k. This means a factor design with each factor at two levels. The number of treatments in an unreplicated 2k design is 2k. The following table shows the number of each kind of effect for each of the six two- level designs shown across the top. ME 311, Mechanical Engineering University of Kentucky 2 2 2 3 2 4 2 5 2 6 2 7 2 3 4 5 6 7 1 3 6 10 15 21 1 4 10 20 35 1 5 15 35 1 6 21 1 7 1 Main effect 2 factor interaction 3 factor interaction 4 factor interaction 5 factor interaction 6 factor interaction 7 factor interaction 3 7 15 31 63 127 In a 2k design, the number of r-factor effects is Ckr = k!/[r!(k-r)!] ME 311, Mechanical Engineering University of Kentucky Notice that the total number of effects estimated in any design is always one less than the number of treatments In a 22 design, there are 22=4 treatments; we estimate 22- 1 = 3 effects. In a 23 design, there are 23=8 treatments; we estimate 23-1 = 7 effects One need not repeat the earlier logic to determine the forms of estimates in 2k designs for higher values of k. A table going up to 25 follows. ME 311, Mechanical Engineering University of Kentucky A B A B C A C B C A B C D A D B D A B D C D A C D B C D A B C D E A E B E A B E C E A C E B C E A B C E D E A D E B D E A B D E C D E A C D E B C D E A B C D E 1 - - + - + + - - + + - + - - + - + + - + - - + + - - + - + + - a + - - - - + + - - + + + + - - - - + + + + - - + + - - - - + + b - + - - + - + - - + + - - + - - + - + + - + - + - + - - + - + ab + + + - - - - - - - - + + + + - - - - + + + + + + + + - - - - c - - + + - - + - + + - - + + - - + + - - + + + + - - + + - - + ac + - - + + - - - - + + - - + + - - + + - - + + + + - - + + - - bc - + - + - + - - + - + - + - + - + - + - + - + + - + - + - + - abc + + + + + + + - - - - - - - - - - - - - - - - + + + + + + + + d - - + - + + - + - - + - + + - - + + - + - - + - + + - + - - + ad + - - - - + + + + - - - - + + - - + + + + - - - - + + + + - - bd - + - - + - + + - + - - + - + - + - + + - - - + - + + - + - abd + + + - - - - + + + + - - - - - - - - + + + + - - - - + + + + cd - - + + - - + + - - + + - - + - + + - - + + - - + + - - + + - acd + - - + + - - + + - - + + - - - - + + - - + + - - + + - - + + bcd - + - + - + - + - + - + - + - - + - + - + - + - + - + - + - + abcd + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - e - - + - + + - - + + - + - - + + - - + - + + - - + + - + - - + ae + - - - - + + - - + + + + - - + + - - - - + + - - + + + + - - be - + - - + - + - + - + + - + - + - + - - + - + + + - + + - + - abe + + + - - - - - - - - + + + + + + + + - - - - - - - - + + + + ce - - + + - - + - + + - - + + - + - - + + - - + - + + - - + + - ace + - - + + - - - - + + - - + + + + - - + + - - - - + + - - + + bce - + - + - + - - + - + - - + + - + - + - + - + - + - + - + abce + + + + + + + - - - - - - - - + + + + + + + + - - - - - - - - de - - + - + + - + - - + - + + - + - - + - + + - + - - + - + + - ade + - - - - + + + + - - - - + + + + - - - - + + + + - - - - + + bde - + - - + - + + - + - - + - + + - + - - + - + + - + - - + - + abde + + + - - - - + + + + - - - - + + + + - - - - + + + + - - - - cde - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + acde + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - bcde - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - abcde + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 22 2 3 24 2 5 T r e a t m e n t s E f f e c t s