. példa J.J. Pignatiello, J.S. Ramberg: J. Quality Technology, 17 198-06 (1985) kézbentartható -1 1 A: high heat temperature ( 0 F) 1840 1880 B: heating time (s) 3 5 C: transfer time (s) 10 1 D: hold down time (s) 3 zaj -1 1 E: quench oil temperature ( 0 F) 130-150 150-170 1 A B C D E 1-1 -1-1 -1-1 7.78 7.78 7.81 1-1 -1 1-1 8.15 8.18 7.88 3-1 1-1 1-1 7.50 7.56 7.50 4 1 1-1 -1-1 7.59 7.56 7.75 5-1 -1 1 1-1 7.54 8.00 7.88 6 1-1 1-1 -1 7.69 8.09 8.06 7-1 1 1-1 -1 7.56 7.5 7.44 8 1 1 1 1-1 7.56 7.81 7.69 9-1 -1-1 -1 1 7.50 7.5 7.1 10 1-1 -1 1 1 7.88 7.88 7.44 11-1 1-1 1 1 7.50 7.56 7.50 1 1 1-1 -1 1 7.63 7.75 7.56 13-1 -1 1 1 1 7.3 7.44 7.44 14 1-1 1-1 1 7.56 7.69 7.6 15-1 1 1-1 1 7.18 7.18 7.5 16 1 1 1 1 1 7.81 7.50 7.59
Wu megoldása szerint az ABCD minden kombinációjánál (8) egyetlen szórás van E-1 E1 A B C D y1 y y3 y1 y y3 atlag lnsq 1-1 -1-1 -1 7.78 7.78 7.81 7.5 7.5 7.1 7.540 -.408 1-1 -1 1 8.15 8.18 7.88 7.88 7.88 7.44 7.90 -.649 3-1 1-1 1 7.5 7.56 7.5 7.5 7.56 7.5 7.50-6.949 4 1 1-1 -1 7.59 7.56 7.75 7.63 7.75 7.56 7.640-4.838 5-1 -1 1 1 7.54 8 7.88 7.3 7.44 7.44 7.603 -.600 6 1-1 1-1 7.69 8.09 8.06 7.56 7.69 7.6 7.785 -.939 7-1 1 1-1 7.56 7.5 7.44 7.18 7.18 7.5 7.355-3.515 8 1 1 1 1 7.56 7.81 7.69 7.81 7.5 7.59 7.660-4.058 3 Effect Estimates; Var.:atlag; R-sqr=1. (Pignatiello_ahogyWu) **(4-1) design DV: atlag: =mean(v5:v10) Effect Coeff. Mean/Interc. ()B 1 by 7.6565 7.6565 0.4083 0.1104-0.163750-81875 -49583-479 9150 4565-9583 -1479 0150 0065-917 -11458 A B ()B 1 by yˆ = 7.65 + 0.11x 8x + 0. 046x Confounding of Effects 1 D 3*4 *4 *3 4
Probability Plot; Var.:atlag; R-sqr=1. **(4-1) design DV: atlag: =mean(v5:v10) Expected Half-Normal Values (Half-Normal Plot).5.0.85 ()B.65.45 1by.5.05-5 0 5 0.10 0.15 0.30 Pareto Chart of Effects; Variable: atlag **(4-1) design DV: atlag: =mean(v5:v10).40833 - Interactions - Main effects and other effects Effects (Absolute Values) ()B -.16375.0915 -.049583 1by -.09583 -.0917.0015 0 5 0.10 0.15 0.30 Effect Estimate (Absolute Value) 5 Effect Estimates; Var.:lnsq; R-sqr=1. (Pignatiello_ahogyWu) **(4-1) design DV: lnsq: =log(stdev(v5:v10)^) Effect Coeff. Mean/Interc. ()B 1 by -3.74439-3.74439 0.4651 0.135 -.1917-9563 0.9385 0.4664-0.63879-0.31940 3696 0.6848-0.68795-0.34398 1.17406 8703 ln sˆ = 3.744 1. 096x B 6
Probability Plot; Var.:lnsq; R-sqr=1. **(4-1) design DV: lnsq: =log(stdev(v5:v10)^) Expected Half-Normal Values (Half-Normal Plot).5.0.45 1by.5.05 0. 0.4 0.6 0.8 1. 1.4 1.6 1.8.0..4 ()B - Interactions - Main effects and other effects Effects (Absolute Values) ()B.85.65 Pareto Chart of Effects; Variable: lnsq **(4-1) design DV: lnsq: =log(stdev(v5:v10)^) 1.174059 -.1917.93846 -.687953 -.638794 1by.536958.465089 0. 0.4 0.6 0.8 1. 1.4 1.6 1.8.0..4 Effect Estimate (Absolute Value) 7 Válasszuk a B=-1 szintet, hogy a szórás kisebb legyen! Ezt rögzítve B = 1 [ 7.65 ( 8) ] + 0.11xA + 46xD = 7.543 + 0.11xA 0. xd yˆ = + 046 Az elvárás 8 inch, ez nem teljesíthető. Megjegyzés: az ingadozásnak két komponense van, ezt nem vettük figyelembe. Javaslat: külső szórás (amit a zaj-faktor okoz), belső szórás, ami egy zaj-szinten belül nyilvánul meg. 8
Effect Estimates; Var.:lnsq_atlag; R-sqr=1. **(4-1) design DV: lnsq_atlag: =log(stdev(v13:v14)^) Effect Coeff. Mean/Interc. ()B 1 by 3.37309 3.37309 1510 07605-54513 -756 55116 7558-00578 -0089-4089 -1045-19793 -09897-6047 -3036 ()B 1 by Confounding of Effects 1 3*4 *4 *3 ln sˆ = 3.37 7x + 8x 0. 030x átlag B C A x D vagy ln sˆ = 3.37 7x + 8x 0. 030x átlag B C B x C A hierarchia-szabály a másodikat valószínűsíti. Eszerint B+, C-, a kölcsönhatás ugyanakkora, nem nyerhetünk. 9 Expected Half-Normal Values (Half-Normal Plot).5.0 Probability Plot; Var.:lnsq_atlag; R-sqr=1. **(4-1) design DV: lnsq_atlag: =log(stdev(v13:v14)^) 1by.45.5.05-1 0 1 3 4 5 6 7 - Interactions - Main effects and other effects Effects (Absolute Values) ()B.85.65 Pareto Chart of Effects; Variable: lnsq_atlag **(4-1) design DV: lnsq_atlag: =log(stdev(v13:v14)^).0551164 -.06047 ()B -.054513 1by -.04089 -.019793.015103 -.000578 0 1 3 4 5 6 7 Effect Estimate (Absolute Value) 10
Effect Estimates; Var.:lnsqbelso; R-sqr=1. **(4-1) design DV: lnsqbelso: =mean(v15:v16) Effect Coeff. Mean/Interc. ()B 1 by -4.80944-4.80944 1.6471 0.8356-1517 -0759 0.4915 0.456 0.6603 0.3310 0.44807 0.403-0.66844-0.334 0.398 0.11199 ()B 1 by Confounding of Effects 1 3*4 *4 *3 ln sˆ = 3.37 7x + 8x 0. 030x átlag ln sˆ = 4.809 + 0. 84 belsső x A B C num log (-3.37+3)= num log (-3.40)=36 num log (-4.809+08)=036 36+036=40 B x C 11 Probability Plot; Var.:lnsqbelso; R-sqr=1. **(4-1) design DV: lnsqbelso: =mean(v15:v16) Expected Half-Normal Values (Half-Normal Plot).5.0 ()B.65 Pareto Chart of Effects; Variable: lnsqbelso.45 **(4-1) design DV: lnsqbelso: =mean(v15:v16) 1by.5.05 0. 0.4 0.6 0.8 1. 1.4 1.6 1.8 - Interactions - Main effects and other effects ()B Effects (Absolute Values).85-1517 1.64711 -.66844.660314.491477 1by.4480651.3987 0. 0.4 0.6 0.8 1. 1.4 1.6 1.8 Effect Estimate (Absolute Value) 1
. példa J. Quinlan ("Product Improvement by Application of Taguchi Methods," in American Supplier Institute News (special symposium ed.), Dearborn, MI: American Supplier Institute, pp. 11-16. 1985) G. E. P. Box, Signal-to-Noise Ratios, Performance Criteria, and Transformations, Technometrics, 30 1-17 (1988) 13 H: liner tension H D: liner line speed D EL: liner die B B: liner outside diameter A EJ: melt temperature EL=H*D F: coating material EJ=H*B N: liner temperature EI=H*A A: braid tension F=-B*D EI: wire braid type EE=D*A EE: liner material EC=B*A M: cooling method N=H*D*B EC: screen pack M=H*D*A K: coating die type K=H*B*A G: wire diameter G=B*D*A EO: line speed EO=H*D*B*A 14
Effect Estimates; Var.:YAV; R-sqr=1. (QUINLAN) 15 factors at two levels DV: YAV Effect Coeff. Mean/Interc. 0.91719 0.91719 (1)H 6188 31094 ()D 60938 30469 (3)B -55938-7969 (4)A 6781 33906 (5)EL 7188 13594 (6)EJ -00938-00469 (7)EI 4813 1406 (8)F -74063-37031 (9)EE 0.45313 0.1656 (10)EC 89688 44844 (11)N 0656 0381 (1)M 7813 13906 (13)K 68438 3419 (14)G -0.140313-70156 (15)EO 05937 0969 15 Confounding of Effects (QUINLAN) 1 3 4 5 6 7 (1)H *5 3*6 4*7 8*11 9*1 10*13 14*15 ()D 1*5 3*8 4*9 6*11 7*1 10*14 13*15 (3)B 1*6 *8 4*10 5*11 7*13 9*14 1*15 (4)A 1*7 *9 3*10 5*1 6*13 8*14 11*15 (5)EL 1* 3*11 4*1 6*8 7*9 10*15 13*14 (6)EJ 1*3 *11 4*13 5*8 7*10 9*15 1*14 (7)EI 1*4 *1 3*13 5*9 6*10 8*15 11*14 (8)F 1*11 *3 4*14 5*6 7*15 9*10 1*13 (9)EE 1*1 *4 3*14 5*7 6*15 8*10 11*13 (10)EC 1*13 *14 3*4 5*15 6*7 8*9 11*1 (11)N 1*8 *6 3*5 4*15 7*14 9*13 10*1 (1)M 1*9 *7 3*15 4*5 6*14 8*13 10*11 (13)K 1*10 *15 3*7 4*6 5*14 8*1 9*11 (14)G 1*15 *10 3*9 4*8 5*13 6*1 7*11 (15)EO 1*14 *13 3*1 4*11 5*10 6*9 7*8 16
Probability Plot; Var.:YAV; R-sqr=1. 15 factors at two levels DV: YAV Expected Half-Normal Values (Half-Normal Plot).5.0 (10)EC (14)G (8)F (13)K.75 Pareto Chart of Effects; Variable: YAV (1)H (4)A.65 15 factors at two levels (11)N (5)EL (1)M (7)EI (3)B ()D DV: YAV.45 (9)EE.5 (15)EO -.140313 (6)EJ (10)EC.05.0896875-5 0 5 0.10 0.15 0.30 -.074063 - Interactions - Main effects and other (13)K effects Effects (Absolute Values) (1)H (3)B (1)M (11)N (9)EE.006565.0059375 (6)EJ -.000938.07815.071875.85 -.055938.04815.0684375.067815.061875.0609375.45315 0 5 0.10 0.15 0.30 Effect Estimate (Absolute Value) 17 Probability Plot; Var.:YAV; R-sqr=1. 15 factors at two levels DV: YAV Expected Normal Value.5.0 - - - -.0 -.5 (14)G (3)B (8)F (10)EC (13)K (1)H (4)A (11)N (5)EL (1)M (7)EI ()D (15)EO (6)EJ (9)EE.75.55.35.15.05.01 - -0. -0.1 0.1 0. 0.3 - Interactions - Main effects and other effects Effects Effect Estimates; Var.:YAV; R-sqr=.68371; Adj:.63506 (QUINLAN) 15 factors at two levels; MS Residual=.0113679 DV: YAV Effect Std.Err. t(13) p Coeff. Mean/Interc. 0.91719 6655 10.94419 00000 0.91719 (9)EE=D*A 0.45313 53310 4.60160 00496 0.1656 (14)G=B*D*A -0.140313 53310 -.6300 0709-70156 18
Effect Estimates; Var.:YAV; R-sqr=.75746; Adj:.6696 (QUINLAN) 15 factors at two levels; MS Residual=.010304 DV: YAV Effect Std.Err. t(11) p Coeff. Mean/Interc. 0.91719 5375 11.4963 00000 0.91719 (3)B -55938 50750-1.101 0.9390-7969 (8)F=-B*D -74063 50750-1.45935 0.1748-37031 (9)EE=D*A 0.45313 50750 4.83371 0054 0.1656 (14)G=B*D*A -0.140313 50750 -.76476 18399-70156 s y = s within 4 = 01303 4 = 3.575 10 4 ez vetendő össze az MSResidual 103 értékével, ez azt mutatja, hogy az újrabeállítások varianciája sokkal nagyobb, mint az ismétléseké. 19 Effect Estimates; Var.:LNS; R-sqr=1. (QUINLAN) 15 factors at two levels DV: LNS: =log(szorasn)/ Effect Coeff. Mean/Interc. -3.71601-3.71601 (1)H -0.445-0.11 ()D -0.39751-0.19876 (3)B 0.19353 9676 (4)A 0.15196 7598 (5)EL -9909-4955 (6)EJ 6834 3417 (7)EI -0.1360-6630 (8)F -5456-0.778 (9)EE -6180-3090 (10)EC -0.11410-5705 (11)N 0.33053 0.1657 (1)M -0.45853-0.96 (13)K 345 1713 (14)G 0.36895 0.18448 (15)EO 0.30855 0.1547 s within = 01303 swithin = 01303 = 361 s 4 s 01303 4 361 = within 4 = = 3.575 10 within = 18 0
Probability Plot; Var.:LNS; R-sqr=1. 15 factors at two levels DV: LNS: =log(szorasn)/ Expected Half-Normal Values (Half-Normal Plot).5.0 (1)M (1)H.85 ()D (14)G.75 (11)N (15)EO.65 (4)A (3)B (7)EI (10)EC.45 (6)EJ (8)F.5 (9)EE (13)K.05-0.1 0.1 0. 0.3 0.4 (1)H 0.6 0.7 - Interactions - Main effects and other effects Effects (Absolute Values) (14)G (15)EO (4)A (10)EC (6)EJ (13)K (8)F.0683377 -.061801.034514 -.13598 -.114105 -.09909 Pareto Chart of Effects; Variable: LNS 15 factors at two levels DV: LNS: =log(szorasn)/.151965.193579.3305339.3085487 -.397514.3689538 -.445 -.45859 -.554556 0.1 0. 0.3 0.4 0.6 0.7 Effect Estimate (Absolute Value) 1 LNS -.6 -.8 - -3. -3.4-3.6-3.8-4.0-4. -4.4-4.6-4.8-5.0-5. -5.4 Scatterplot of LNS against LNAV QUINLAN 3v*16c LNS = -4.018-0.144*x -5.6 -.6 -.4 -. -.0-1.8-1.6-1.4-1. - -0.8-0.6-0.4 LNAV
Expected Normal Value.5.0 - - - -.0 -.5 (8)F (1)H ()D (1)M Probability Plot; Var.:LNS; R-sqr=1. 15 factors at two levels DV: LNS: =log(szorasn)/ (9)EE (13)K (6)EJ (4)A(3)B (5)EL (10)EC (7)EI (14)G (11)N (15)EO - -0.7-0.6 - -0.4-0.3-0. -0.1 0.1 0. 0.3 0.4 - Interactions - Main effects and other effects Effects.75.55.35.15.05.01 3 3. példa J. S. Space, A. M. Opio, B. Nickerson, H. Jiang, M. Dumont, M. Berry: Validation of a dissolution method with HPLC analysis for lasofoxifene tartrate low dose tablets, J. Pharmaceutical and Biomedical Analysis 44 (007) 1064 1071 Robusztusság 4
Space_JPBA_44_1064.sta 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 5 Temperatur e (0C) 6 ph 7 Flow rate (ml/min) 8 Organic (acetonit rile) (%) 9 Retention time =.5 and=4.5 (min) 10 Peak efficienc y =1000 11 Peak asymmetry =.0 45.8 0.4 33 5.6 5180 1.1 45 3. 0.6 33 3.7 4879 1.1 45.8 0.4 37 3.1 590 1 45 3. 0.6 37. 461 1.1 35.8 0.4 33 6.5 5640 1 35.8 0.4 37 3.5 5117 1.1 35 3. 0.6 33 4.4 4780 1 35 3. 0.6 37.5 46 1.1 45.8 0.6 37.1 4579 1.1 45.8 0.6 33 3.7 4691 1.1 45 3. 0.4 33 5.7 5506 1.1 45 3. 0.4 37 3.3 554 1.1 35 3. 0.4 33 6.6 509 1.1 35 3. 0.4 37 3.6 519 1.1 35.8 0.6 33 4.4 4667 1 35.8 0.6 37.4 4453 1.1 Robusztusság 5 Mean/Interc. (1)Temperature ( 0C) ()ph (3)Flow rate (ml/min) (4)Organic (acetonitrile) (%) 1 by by 3 by 4 3 by 4 1**3 1**4 1*3*4 *3*4 Effect Estimates; Var.:Retention time =.5 and=4.5 (min); R-sqr=998; Adj:971 **(4-0) design; MS Residual=.00065 DV: Retention time =.5 and=4.5 (min) Effect Std.Err. t(1) p -95.% +95.% Coeff. Cnf.Limt Cnf.Limt 36 06 63 010 3.877 4.036 36-6 15-45.0 141-0.71-0.404-0.81 88 15 7.0 903-71 0.46 44-63 15-15.0 051-1.71-1.404-0.781 -.38 15-179.0 036 -.396 -.079-1.119 1 15 000-0.146 0.171 06 6 15 5.0 0.157-96 0.1 31 0.37 15 19.0 335 79 0.396 0.119-37 15-48 -0.196 0.11-19 37 15 48-0.11 0.196 19 0.487 15 39.0 163 0.39 0.646 0.44-1 15-000 -0.171 0.146-06 1 15 000-0.146 0.171 06-37 15-48 -0.196 0.11-19 13 15 000-0.146 0.171 06 Robusztusság 6
Probability Plot; Var.:Retention time =.5 and=4.5 (min); R-sqr=998; Adj:971 **(4-0) design; MS Residual=.00065 DV: Retention time =.5 and=4.5 (min) Expected Half-Normal Values (Half-Normal Plot).5.0 ()ph 1*3*4 by3 by4 *3*4 1**4 1**3 1by (1)Temperature (0C) 3by4 (3)Flow rate (ml/min) (4)Organic (acetonitrile) (%).05 -.0.5 - Interactions - Main effects and other effects Effects (Absolute Values).85.75.65.45.5 Robusztusság 7 Mean/Interc. (1)Temperature (0C) (3)Flow rate (ml/min) (4)Organic (acetonitrile) (%) 3 by 4 Regr. Coefficients; Var.:Retention time =.5 and=4.5 (min); R-sqr=843; Adj:739 **(4-0) design; MS Residual=.00565 DV: Retention time =.5 and=4.5 (min) Regressn Std.Err. t(9) p -95.% +95.% Coeff. Cnf.Limt Cnf.Limt 68.894 688 1.483 0000 61.639 76.148-03 6835-7.361 0004-0.658-0.348-5.969 3.617-14.66 0000-61.141-44.796-1.644 8894-18.48 0000-1.845-1.443 6 3750 1.667 0.1993-0.147 1 0187 6.333 0014 08 16 1.19 9375 100 0000 07 1.431 Yˆ = 68.9 03 T 5.97 flow 1.644 org% + Hibaterjedési törvény: + 1 T org% 1.19 flow org% nem új kísérlet, csak redukált modell σ Yˆ r j= 1 Yˆ x j σ x j Yˆ T = 03+ 1 org% Robusztusság 8
Mean/Interc. (1)Temperature (0C) (3)Flow rate (ml/min) (4)Organic (acetonitrile) (%) 3 by 4 Regr. Coefficients; Var.:Retention time =.5 and=4.5 (min); R-sqr=843; Adj:739 **(4-0) design; MS Residual=.00565 DV: Retention time =.5 and=4.5 (min) Regressn Std.Err. t(9) p -95.% +95.% Coeff. Cnf.Limt Cnf.Limt 68.894 688 1.483 0000 61.639 76.148-03 6835-7.361 0004-0.658-0.348-5.969 3.617-14.66 0000-61.141-44.796-1.644 8894-18.48 0000-1.845-1.443 6 3750 1.667 0.1993-0.147 1 0187 6.333 0014 08 16 1.19 9375 100 0000 07 1.431 Yˆ = 68.9 03 T 5.97 flow 1.644 org% + + 1 T org% + 1.19 flow org% Ha pl. a hőmérsékletet ±1 0 C pontossággal, az áramot ±5mL/min pontossággal, az acetonitril-koncentrációt % pontossággal tudjuk tartani, σ T =77T=77 0 C, σ flow =77 5=8mL/min σ org% =77 =0.8% Robusztusság 9 Yˆ = 68.9 03 T 5.97 flow 1.644 org% + + 1 T org% + 1.19 flow org% Ha pl. a hőmérsékletet ±1 0 C pontossággal, az áramot ±5mL/min pontossággal, az acetonitril-koncentrációt % pontossággal tudjuk tartani, σ T =77 1=77 0 C, a közepes beállításoknál: σ flow =77 5=8mL/min T=40, flow=, org%=35 σ org% =77 =0.8% σ tr = ( 03 + 1 35) 77 + ( 5.97 + 1.19 35) 8 + + =.9 10 3 ( 1.644 + 1 40 + 1.19 ) 0.8 = + 83 + 4 = 0.110 tr = 0.110 77 = 7 Robusztusság 30
σ tr Optimalizálás: Yˆ = 68.9 03 T 5.97 flow 1.644 org% + + 1 T org% + 1.19 flow org% = ( 03 + 1 org% ) 77 + ( 5.97 + 1.19 org% ) 8 + org% max (de 4% alatt) org% max (de 43 % alatt) ( 1.644 + 1 T + 1.19 ) 0.8 + flow T max flow max σ tr T: 50, flow: 0.8, org%: 4 = 3.33 10 7 +.46 10 3 + 3.71 10 4 =.83 10 3 Tr = 9 a 0.11 helyett, a 7 helyett Robusztusság 31