Design of Experiments 2k Designs Catapult Experiment An engineering statistics class ran a catapult experiment to develop a prediction equation for how far a catapult can throw a plastic ball. The class manipulated two factors: how far back the operator draws the arm (angle), measured in degrees, and the height of the pin that supports the rubber band, measured in equally spaced locations. The results follow… L. Wang, Department of Statistics University of South Carolina; Slide 2 Catapult Experiment Angle Height Distance 1 Distance 2 140 2 27 27 180 2 81 67 140 4 67 62 180 4 137 158 Angle Height Distance 1 Distance 2 -1 -1 27 27 1 -1 81 67 -1 1 67 62 1 1 137 158 L. Wang, Department of Statistics University of South Carolina; Slide 3 This is a 2 2 (-1,1) factor (level ) Height Design (1,1) Angle (-1,-1) (1,-1) L. Wang, Department of Statistics University of South Carolina; Slide 4 Response Notation a = Average of responses when A (Angle) is high and B (Height) is low. b = Average of responses when B (Height) is high and A (Angle) is low. ab = average of responses when both A (Angle) and B (Height) are high. (1) = Average of responses when both A (Angle) and B (Height) are low. L. Wang, Department of Statistics University of South Carolina; Slide 5 This is a b 2 2 (-1,1) factor (level ) Height Design ab (1,1) Angle (-1,-1) (1) (1,-1) a L. Wang, Department of Statistics University of South Carolina; Slide 6 Catapult Experiment Angle Height Distance 1 Distance 2 Avg Distance 140 2 27 27 27 180 2 81 67 74 140 4 67 62 64.5 180 4 137 158 147.5 Angle Height Distance 1 Distance 2 Avg Distance -1 -1 27 27 27 1 -1 81 67 74 -1 1 67 62 64.5 1 1 137 158 147.5 L. Wang, Department of Statistics University of South Carolina; Slide 7 Catapult Experiment – Main Effects Angle Height Avg Distance -1 -1 (1) 27 1 -1 a 74 -1 1 b 64.5 1 1 ab 147.5 Effect of Angle = avg response at high level – avg response at low level a ab (1) b 74 147 .5 27 64.5 effect of Angle 65 2 2 2 2 L. Wang, Department of Statistics University of South Carolina; Slide 8 Catapult Experiment – Main Effects Angle Height Avg Distance -1 -1 (1) 27 1 -1 a 74 -1 1 b 64.5 1 1 ab 147.5 Effect of Height = avg response at high level – avg response at low level b ab (1) a 64.5 147 .5 27 74 effect of Height 55.5 2 2 2 2 L. Wang, Department of Statistics University of South Carolina; Slide 9 Interactions If Angle and Height interact, then the effect of Angle depends on the specific level of Height. An interaction plot plots the means of one factor given the levels of the other factor. L. Wang, Department of Statistics University of South Carolina; Slide 10 Interaction Plot 160 A v g ab 140 120 Height(1): 4 100 2 b 80 D i s t 60 20 Height(-1): 2 1 40 a (1) 0 -1 140 -0.8 -0.6 -0.4 -0.2 0 Angle 0.2 0.4 0.6 0.8 1 180 L. Wang, Department of Statistics University of South Carolina; Slide 11 Interaction of A (Angle) and B (Height) 2 1 [ab a] [b (1)] AB interactio n 2 2 (1) ab a b 27 147 .5 74 64.5 AB interactio n 18 2 2 We have a positive interaction which is smaller in size than the main effects (effect of Angle = 65 and effect of Height = 55.5. L. Wang, Department of Statistics University of South Carolina; Slide 12 The model we are fitting is yi 0 1xi1 2 xi 2 12 xi1xi 2 i yi is the distance for the ith test run β0 is the y-intercept β1 is the regression coefficient associated with angle β2 is the regression coefficient associated with height β12 is the regression coefficient associated with angle/height interaction εi is the random error L. Wang, Department of Statistics University of South Carolina; Slide 13 Table of Contrasts Intercept x1 (Angle) x2 (Height) x1x2 1 -1 -1 1 (1) 1 1 -1 -1 a 1 -1 1 -1 b 1 1 1 1 ab This table tells us how to combine the average response for each treatment combination to form the numerator of our estimate of the effect. For two-level factorial designs, the denominator for estimating main effects and interactions will always by one-half of the number of distinct factorial treatment combinations. Ex: 22 = 4, so our denominator is 2. Use total number of distinct treatment combinations as L. Wang, Department of Statistics denominator for the intercept. University of South Carolina; Slide 14 Model Coefficients are Slopes A slope represents the expected change in the response when we increase one factor by one unit while holding the other factor constant. Going from -1 to 1 in a factor represents movement of two units. So: est effect Est regression coef 2 L. Wang, Department of Statistics University of South Carolina; Slide 15 Multiple Linear Regression Dependent Variable: Distance Independent Variables: Angle, Height, Angle*Height Parameter estimates: Variable Estimate Std Err Tstat P-value Intercept 78.25 3.21617 24.330 <0.0001 Angle 32.5 3.21617 10.105 0.0005 Height 27.75 3.21617 8.628 0.001 Angle*Height 9 3.21617 2.798 0.0489 ANOVA table: Source DF SS MS F-stat P-value Model 3 15258.5 5086.1665 61.46425 0.0008 Error 4 331 82.75 Total 7 15589.5 L. Wang, Department of Statistics University of South Carolina; Slide 16 k 2 Full Factorial Designs We will look at every possible combination of the two levels for k factors. Let’s extend our catapult Experiment to include: – Angle: 180, Full – Peg Height: 3, 4 – Stop Position: 3, 5 – Hook Position: 3, 5 Each combination was run twice. L. Wang, Department of Statistics University of South Carolina; Slide 17 Angle Pg Ht Stp Ps Hk Ps Dist Angle Pg Ht Stp Ps Hk Ps Dist -1 -1 -1 -1 363 -1 -1 -1 -1 354 1 -1 -1 -1 401 1 -1 -1 -1 406 -1 1 -1 -1 416 -1 1 -1 -1 460 1 1 -1 -1 470 1 1 -1 -1 490 -1 -1 1 -1 380 -1 -1 1 -1 383 1 -1 1 -1 437 1 -1 1 -1 440 -1 1 1 -1 474 -1 1 1 -1 477 1 1 1 -1 532 1 1 1 -1 558 -1 -1 -1 1 426 -1 -1 -1 1 413 1 -1 -1 1 474 1 -1 -1 1 494 -1 1 -1 1 480 -1 1 -1 1 502 1 1 -1 1 520 1 1 -1 1 555 -1 -1 1 1 446 -1 -1 1 1 467 1 -1 1 1 512 1 -1 1 1 550 -1 1 1 1 480 -1 1 1 1 485 1 1 1 1 580 1 1 L. Wang,1Department of 1 Statistics591 University of South Carolina; Slide 18 Angle Pg Ht Stp Ps Hk Ps Run Avg Dist -1 -1 -1 -1 (1) 363.5 1 -1 -1 -1 a 403.5 -1 1 -1 -1 b 438 1 1 -1 -1 ab 480 -1 -1 1 -1 c 381.5 1 -1 1 -1 ac 438.5 -1 1 1 -1 bc 475.5 1 1 1 -1 abc 545 -1 -1 -1 1 d 419.5 1 -1 -1 1 ad 484 -1 1 -1 1 bd 491 1 1 -1 1 abd 537.5 -1 -1 1 1 cd 456.5 1 -1 1 1 acd 531 -1 1 1 1 bcd 482.5 1 1 1 1 abcd 585.5 L. Wang, Department of Statistics University of South Carolina; Slide 19 4 2 Model: yi 0 1xi1 2 xi 2 3 xi 3 4 xi 4 12 xi1xi 2 13 xi1xi 3 14 xi1xi 4 23 xi 2 xi 3 24 xi 2 xi 4 34 xi 3 xi 4 123 xi1 xi 2 xi 3 124 xi1 xi 2 xi 4 134 xi1xi 3 xi 4 234 xi 2 xi3 xi 4 1234 xi1xi 2 xi3 xi 4 L. Wang, Department of Statistics University of South Carolina; Slide 20 Effects Use Contrast Table for numerator. Denominator is one half the number of distinct combinations. Use total number of distinct combinations as denominator for Intercept. L. Wang, Department of Statistics University of South Carolina; Slide 21 I x1 x2 x3 x4 x1 x1 x1 x2 x2 x3 x1 x1 x1 x2 x1 2 3 4 3 4 4 23 24 34 34 234 Run Avg Dist 1 -1 -1 -1 -1 (1) 363.5 1 1 -1 -1 -1 a 403.5 1 -1 1 -1 -1 b 438 1 1 1 -1 -1 ab 480 1 -1 -1 1 -1 c 381.5 1 1 -1 1 -1 ac 438.5 1 -1 1 1 -1 bc 475.5 1 1 1 1 -1 abc 545 1 -1 -1 -1 1 d 419.5 1 1 -1 -1 1 ad 484 1 -1 1 -1 1 bd 491 1 1 1 -1 1 abd 537.5 1 -1 -1 1 1 cd 456.5 1 1 -1 1 1 acd 531 1 -1 1 1 1 bcd 482.5 1 1 1 1 1 L. Wang, Department of Statistics University of South Carolina; 22 abcd Slide585.5 Fractions of 2k Factorial Designs Used to reduce the total number of treatment combinations while preserving the basic factorial structure. Main effects tend to dominate two-factor interactions, two-factor interactions tend to dominate three-factor interactions, and so on. We will sacrifice the ability to estimate the higher-order interactions in order to reduce the number of treatment combinations. L. Wang, Department of Statistics University of South Carolina; Slide 23 Half Fraction of 23 Design I x1 x2 x3 x1x2 x1x3 x2x3 x1x2x3 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 1 1 1 1 1 1 I x1 x2 x3 x1x2 x1x3 x2x3 x1x2x3 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 L. Wang, Department of Statistics University of South Carolina; Slide 24 Half Fraction of 23 Design I x1 x2 x3 x1x2 x1x3 x2x3 x1x2x3 1 -1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 -1 1 1 1 -1 1 -1 -1 1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 1 1 1 1 1 1 1 I x1 x2 x3 x1x2 x1x3 x2x3 x1x2x3 1 -1 -1 -1 1 1 1 -1 1 1 1 -1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 L. Wang, Department of Statistics University of South Carolina; Slide 25 Aliasing Effects When we alias one effect with another (eg: Aliasing effect of A with effect of BC), we can not distinguish one effect from the other (eg: we can not distinguish effect of A from effect of BC.). Positive half fraction of a 23 design uses x1x2x3 = 1 to select the treatment combinations to be run. Negative half fraction of a 23 design uses x1x2x3 = -1 to select the treatment combinations to be run. L. Wang, Department of Statistics University of South Carolina; Slide 26 Aliasing Effects We say that ABC is the defining interaction. General notation: 23-1 – 2 indicates number of factor levels. – Exponent 3 indicates the number of factors. – Exponent -1 indicates a half (2-1) fraction. – Total number of treatment combination is 23-1 = 4. L. Wang, Department of Statistics University of South Carolina; Slide 27 The Alias Structure ABC as the defining interaction is equated to the intercept, I. Then add each effect to the defining interaction using modulo 2 arithmetic. Eg: A + ABC = BC B + ABC = AC C + ABC = AB L. Wang, Department of Statistics University of South Carolina; Slide 28 The Alias Structure I = ABC A = BC B = AC C = AB AB = C AC = B BC = A or I = ABC A = BC B = AC C = AB yi 0 1xi1 2 xi 2 3 xi 3 i L. Wang, Department of Statistics University of South Carolina; Slide 29 Warning Remember that aliases can not be distinguished from one another, so be careful what you alias. L. Wang, Department of Statistics University of South Carolina; Slide 30