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# Chapter 13 notes BME 3721

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This 10 page Class Notes was uploaded by Kathleen Quijada on Saturday March 21, 2015. The Class Notes belongs to BME 3721 at Florida International University taught by Wei-Chiang Lin in Spring2015. Since its upload, it has received 67 views. For similar materials see Data Evaluation Principals in Biomedical Sciences at Florida International University.

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Date Created: 03/21/15

EXCtmpLa Mm 5 39i ex i 06m 131 What is Experimental Design Chapter 13 Example 131 Characterizing a Process SPC has been applied to a soldering process Through ucharts and Pareto analysis statistical control has been established and the number of defective solder ioints has been reduced to M The average bm solder joints I may still be too large Desired to reduce the defects level more toiiows a p0 i 58M dist quot w l il LL 0m l V 2quot 7 ne p ctram lntrodugggiir SistiEscilisginouaiiiy Introduccgzitrgt EttEtiiEsCtiiicianuality u p v2 cituh o in lm 194 13 1 What is Experimental Design 131 What is Experimental Design Objectives of Experimental Design Determine which variables x39s are more in uential on the Note since the process is in statistical control not obvious response y what machine adjustments will be necessary There are several Determine where to set the influential x39s so that y is near the variables that may affect the occurrence of defects nominal requirement 1 H Mt clam i I z W J gulf w M ismbpj gwmig Determine where to set the influential x39s so that variability is small quot Determine where to set the influential x39s so that the effects of the gw 1 A designed experiment involving these factors could help quotC quotquot quotab39e Variab39es z 3 9 minimized determine which factors could help significantly reduce a g defects Screening experiment i39xeePEh chewi ii cArerfhjctitj tme CW 31 as althj fa ab 39 i 39i I CL Le J 39quotiwduc on 0 Statistical Quality m b QLQ 1 5 Introduction to Statistical Quality A C 1 A U 391 Control 6th Edition Control 6th Edition cL La 4 13 1 What is Experimental Design 132 Guidelines for Designing Experiments Results of Experimental Design used early in process CLCh I Wig1 9 a bif VCt W 39 Improved yield r t v Reduced variability and closem39cs ggpominal Q t W Lu 1 development var i W15 Procedure for designing an experiment Recognition of and statement of the problem Choice on Selection of the response variab e y 1 Reduced development time 2 3 4 Performing the experiment 5 6 6 low Reduced overall costs 9mduo w r Data analysis Conclusions and recommendations quot 1438 makeup prevexperimental planning 2 and 3 often done simultaneously or in rev x order introduction to Statistical Quality introduction to Statistical Quality Control 6th Edition Control 6th Edition Con39wszom am act Seven plot There S P at low Lottl 5 11113 when Hm an 414 f tuj futVt 0w 132 Factorial Experiments 133 Factorial Experiments 0 if 5 07 12 fit When there are several factors of interest in an I 39 Ln it39d75 experiment a factorial design should be used 5 l 3 4 1 A complete trial or replicate of the experiment for all 3 possible combinations of the levels of the factors are investigated quot 39 quot A amp B are independent A St B are dependent 39 AVOID onefactorat a time approach gacmfiak v 139 my Foams hood 0 Wm t WW M z mow 0k Uniqo owe39239th Butt A I n 39 n Ierdu gzyrgl tg gnauamy In W in Hm All facts bombl cmc g cm Stage din chatty com9 mimefun tam W rs 13 3 Factorial Experiments Main effect is the change in response y produced by a 13339 FaCtonal EXpenmentS I r change m the level of a primary factor x D T39OLCJILD f5 39 An interaction is present among factors if a change in the I V levels of one factor influences the effect of another Stat39St39cal Analys39s Factor B b levels factor 1 2 b Consider an experiment with two factors A and B 1 ymy 2 ym y122 ym y 2 x un Vtzn Ytbn Main effect ofA 39 2 y y y y y y Main effect of B h 0 n 1 S FactorA m 222 y m Interaction effect ofAB O O K 3 levels O 4 Vail yn22 yabZ Introduction to Statistical Quality Introduction to Statistical Quality Control 6th Edition Cilantrol 6th Edition SD 2 The experiment is replicated n times It limit lawIr t Loo twat 133 Factorial Experiments 2 J wl Q L YWS In both guresAanchavetwo IeveIs ieand and all Factorial Experiments possible combinations ofA and B are investigated b lowM39s on Statistical Analysis I i Q g 3 Completely randomized design with two factors A and B 1 E The model IS i1323a x s 3 w gm yak uz j r 9 50 123tb k l23n p overall mean II effect of the ith level of factor A 4 Bi effect of the jth level of factor B ppm 135 A mum my Figure 126 factorial expurinmnl MS effect of the interaction between A and B w anili tutu Eacite 333 iii331mm s random error component when f3 ONJf tO39f39 footagealong gl M lcww maletab to H4 eszhR below it woota39be QO D 35 v at farth H K m Lavinc Mactwat Pr f och WWWU A amp B are A amp B are independent dependent oatquot T GPY39CJC m f39J 5L 39 LU vn m X39F rfzwf 41 en W616 Introduction to Statistical Quality Control 6th Edition 774 Kg This isilit scmmmhdm MC all this Outcomes anovct 15 0t l Potltsis tsFr re bKIPDva iv lg 5t Par39l ldul rl g 1 M quotll hpoff n Fmt jjdf I J JONL v 13 3 Factorial Experiments 13 3 Factorial Experiments Q l Ce a a Statistical Analysis Statistical Analysis 7 c quotMacAW Mti Table 123 Tliu Anulvsi ol uriuncc Table for n T A39wlilctnr int39iui39inL Fixed Elkch l1ll l 3 average 39T quot L W quot 39 T T quot39 ML E 2 Jet 1 quot m I l l I l i 1 Im 39 I 581 115 A 584 i 1 Mb TM 39 17 a 2 gt i 39 7 33 313 column 3 LI LXI ijt m J l 2 i h 1 SS I MS T T 5 mr quot r 39 n a 55 Us ll2u 39 5539 3 A L can 21 vii l l q I HILTKHUII AL 1 it i 5 u D I n My h n Error 55 um I my 73 i V l u i Ii J quot39t lm 545 I r 1 grand i221 if U 1in l ld H y r quotquot1 r a 2 Introduction to Statistical Quality Introduction to Statistical Quality TY b Q m K m Control 6th Edition Control 6th Edition elimi Sysmloels Cinema mblc QJOtrceJ lULL Sum o i m 5 61C Que om ML 11quot fame i0 le 15 Fbr l lxQ 2le 5 an 13 3 Factorial Experiments lmyg 5 pm MC g 133 Factorial Experiments t Statistical Analysis Total corrected sum of squares SST decomposition illn L 539 Yb V 39 A A SST A 39 531 T 39 SSE quot 39 Main effects 5 u I x I i I T Y t 39 a 1 v 2 v v c 21x 392 quotm w y I 2i 0quotquot M 39v39 V m 13921 M II 55quot t2 hi In 12 air I u I r I quot i 39 r7 2 1 391 m 553 z 2 39 e 113 I quot 21 2 ill liu 2 quotn quothquot m Y i 539 l I quot1 Interaction 397 I u i h v i l l39 2 2 E Nili SSE 1 5 E 3 7 quot A 3 35 ll9i 39 I F 39lj l39quot l my quot l I l 39 l Error L Aquot 39 39 Wquot quot quot m 397 quot T A 39 T Wl 55L 5s 5t 55 v 55 1210 Y t 390 Y i M 7 7 7 introduction to Statistical Quality introduction to Statistical Quality Control 6th Edition Control 6th Edition aggwbwm 5 3 r W 13 3 Factorial Experiments gtg 00 13 3 Factorial Experiments 0d a it WOOL t b C Lemquot 5 l We I 39i quotma SIS TB Statistical Analysis lges39igg ailzgj lm Le Sl If 11ft u Residuals are important in accessing model accuracy m 9 Q S Wo gne et dEmeosition notation SST 88A 853 SSAB SSE The correspondinq egreesp r o Es composition is abn1 81 D4 31b1 abn1 c Residual vs factor plot or Normal plot of residual can be used to validate the accuracy of the model a 0191 if L V3 W FCQHY 9m ithisiirLM bit it Catalysis He Firtil Cl ih Ci ccura mow 39 The residuals from a twofactor factorial design are Introduction to Statistical Quality p k g TE i to 4 Control 6th Edition gun st 2k Factorial Design The 2k Design Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design 2k is the notation used to indicate that a certain experimental design has k factors of interest each at two levels The simplest design involves two factors replicates A and B k2 and n interested in the main effect ofA the main effect of B and the interaction between A and B v Effects are calculated by Average Response y at high level Average Response at the low level y j A large effect would indicate a significant factor or interaction The 22 Design 22 design two factorsA and B each at two levels Notation Introduction to Statistical Quality Control 6th Edition 13 4 2k Factorial Design Low 1 1 High 14 1 Introduction to Statistical Quality Control 6th Edition 13 4 2k Factorial Design The 22 Design Let the letters 1 a b and ab represent the sum of all n Estimate of main effectA Effect Estimate observations taken at the design points W aab hl 2n 2n 1 2 quot27 aab h l Introduction to Statistical Quality Control 61h Edition 13 4 2quot Factorial Design The 22 Design Estimate of main effect B Effect Estimate 1 i ab at 1 Estimate of interaction AB LLB 11 a l 2n AB abl quotath 2n 2n 1 abt a b 2n introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design The 22 Design For the previous effects formulas the quantities in brackets are called contrasts For example ContrastA a ab b 1 Contrast coefficients are either 1 or 1 The contrasts are used to calculate the sum of squares for the factors and interaction contrast2 SS Ms2 n2 contrast coe tcren is Introduction to Statistical Quality Control 6th Edition 13 4 2k Factorial Design 134 2k Factorial Design Tulilr If Annlxmul Yurith t4 1 illilit lllt39l lixpctirnfixi 117 2 11 The 22 Design Him mlT 11m 30 1 The sum of squares forA B and AB are Si ccll or 1 35113 43 1n tt39i39 l Stimuli 503 LIKE 39 I11 39 2 lftz39nr ii if aabb 1 11 15 55A Z 1 gt1 1 4n 7 7 739 7 7 77 7 7 7 SS W 55 L 7i153931 tl26i B 411 SS abt a b2 T SS 2 W 411 Introduction to Statistical Quality Introduction to Statistical Quality Control 6th Edition Control 6th Edition 13 4 2k Factorial Design 13 4 2k Factorial Design The 22 Design Regression Model A regression model could be t to data from a factorial design I lllhlc39 11 5 l 11 21111 tln Rwzlr I t l39ltilt39l li factors R1111 If ihrulmn Tutti i I 11 182 13 129 111 at q 7 3 J M 139 397quot 3 7 3quot quotM B0 is the grand average of all observations J II 39 HA Ill ME 59 X FactorA 1 1 4 a 410 43 363 391 1611 x Factor Bl 39 quot quot and each coef cient 31 is 12 of effect estimate introduction to Statistical Quality introduction to Statistical Quality Control 6th Edition Control 6th Edition 13 4 2k Factorial De51gn 13 4 2k Factorial DeSIgn Regression Model l xl quot 3 0 I I7 till m 1331 Residual plots are used to access the adequacy of the model T 51 39 9017 1 ll 597 644 a 1604 7 44 7 onCe agaInV I B 37 lb er ab 1 l 1 ReSIduals are calculated usmg the fitted regressmn model J 7 l 9quot 39 I I w 603 The residual plots versus the factor levels interactions predicted 24 l 1 v Ll 644 M8 77 754 values and a normal probability plot are all useful in determining I the adequacy of the model and satisfaction of assumptions AB T 11 l IN1 2n 397 1111644 e 24 61 5971 8 s 71 introduction to Statistical Quality introduction to Statistical Quality Control 6th Edition Control 6th Edition 13 4 2k Factorial Design Analysis Procedure for Factorial Designs QWPWN Estimate the factor effects Form preliminary model Test for signi cance of factor effects Analyze residuals Re ne model if necessary Interpret results Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design The 2k Design for k 2 3 When k 2 2 you could have a single replicate but some assumptions need to be made Can39t estimate all interactions For k 3 the main effects and interactions of interest are A B C AB AC BC ABC and 1 The sum of the outputs y are represented by a b c ab ac bc abc Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design The 2k Design for k 2 3 Effect Estimates AzyA 21 ZLaFabacabcb c bc 1 n B 73 B 1 babbcabc a c ac 1 4n C f y 4ilcacbcabca b ab 1 n introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design The 2k Design for k 2 3 Effect Estimates 1 ABzym yAB Elam Iabcc a b bc ac ACJ7 51 ZLMClabcb ac ab bc quot n BCJ7W BC ibclabca b c abHac Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design The 2k Design for k 2 3 Effect Estimate for ABC ABC ABCI jJBC iabcabcab ac bc I In general the effects can be estimated using The sum of squares for any effect is Introduction to I I I I I I I ll 13 4 2k Factorial Design The 2quot Design for k 2 3 m A ss 1 SSA B 85B 1 SS8 0 ssc 1 SSc AB ssAB 1 855 Ac SSAC 1 ssAC BC ssBC 1 ssEC ABC 5330 1 ssABC Error SSE 8n1 SSEIBn1 Total ssT 8n1 8818n1 Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design Regression Model A regression model could be t to data from a factorial design y 30 lxl ax 163x BIZ39X39IX Z Bxlxit 1623x2x3 5123x1x2 x3 6 where 3 is the grand average of all observations x1 Factor A x2 Factor B x3 Factor 0 and each coef cient 3 is 12 of effect estimate Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design Other Methods for Judging the Significance of Effects The Standard Error of any effect estimate in a 2k design is amp 112 quot 62 the error mean square SSE from ANOVA seE cI 2 Two standard deviation limits on any estimated effect is Effect Estimatei 2scetfcct Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design Other Methods for Judging the Significance of Effects EffectEstimatc i 2sceffect This interval is an approximate 95 con dence interval on the estimated effect Interpretation is simple If zero is contained within the 95 confidence interval then that effect is essentially zero and the corresponding factor is not significant at the a 005 level Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design A Single Replicate of the 2k Design When the number of experiments increases the number of effects that can be estimated also increases In most situations the sparsity of effects principle applies For a large number of factors say k gt 5 it is common practice to run only a single replicate of the 2k design and pool or combine the higher order interactions as estimate of error Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design Nitride etch process on a singlewafer plasma etcher There are four factors of interest The response is etch rate for silicon nitride A single replicate is used I tesign Fact or Gilp Pressure C 7 Flow Po 1 39I a I Lux cl can in 39iorri ISCCM W Iu39 I 180 15 125 2395 High 3 I10 550 200 325 Introduction to Statistical Quality Control 6th Edition 13 4 2k Factorial Design 39I39ablt tilI III 3 Damn in tin 5 mt FIVE pcl tltltnl r 39 A wet de Run tram ilrimurm ugh I39lw w nt hmcn Armin 39f V I I 550 2 I 1 In 3 l I not i I 1 n5ltl t 7 I nt t I m 7 1 an I l I I I I I 8 I 1 IO I II I l l I I I t I it I I I I I I I I it 39I I l I I5 I I 1 I I Introduction to Statistical Quality Control 6th Edition 13 4 2k Factorial Design Estimates 1517661 COMES f 112 A 101 625 AD 153625 B l 625 B0 0625 AB 7875 ABD 4125 C 7375 CD 2125 AC 24875 ACD 5625 BC 43875 BCD 25375 ABC 45625 ABCD 40125 D 306125 Introduction to Statistical Quality Control 6th Edition 134 2k Factorial Design Liter as etch late til Annan Introduction to Statistical Quality Control 6th Edition 13 4 2k Factorial Design Normal probability plot reveals thatA D and AD appear to be signi cant To be sure that other main factors or two factor interactions are not signi cant pool the three and four factor interactions to form the error mean square If the normal probability plot had indicated that any of these interactions were important they would not be included in the error term Introduction to Statistical Quality Control 6th Edition Table 1347 Source of Variation i I C I A iC A if 11D 739 Error l ulul 13 4 2quot Factorial Design ntlux 5 l1ltlfix39 tn r39nt Plattn litth EAS k39rtzzivn39 Sum of Degrt vs of Squares i rccilnn l Mum SHEth 41310503 4131056 ltl ti I Itlit39 21756 LI LSh39i 3748501 l V i lltlf 343063 1 348016 39 Stlt3 l lJTSJKI 9440256 I 9940250 77llllilt13 l 7700053 505 l 156 ul lim v quotinquot 063 m I sm I 5 5 3037363 1 f93l 38 dip Introduction to Statistical Quality Control 6th Edition 13 4 2k Factorial Design Factors A D and the interaction AD are significant The tted regression model for this experiment where x1 represents A x2 represents D Introduction to Statistical Quality Control 6th Edition 13 4 2k Factorial Design Addition of Center Points to the 2k Design 0 So far the assumption of linearity in the factor effects has been made 2k works well when the linearity assumption holds only approximately 2k design will support the main effects and interactions model providing some protection against curvature Introduction to Statistical Quality Control 6th Edition ci 8 I we E l 157 134 2k Factorial Design 13 4 2k Factorial Design Addition of Center Points to the 2k Design Addition of Center Points to the 2k Design There may be situations where a secondorder 39 When Gamer POMS are added to the design the mode is appropriate model can estlmated k V Buz 1xl ZZ3 Jx xl Z 11xj 5 Consider the case of k 2 a model including J 1 If39 second0rder effects is where the Bjj are pure quadratic effects y 3 31 2x1 392xlxl t lxl 32le 8 The test for curvature actually tests the hypothesis The model cannot be fitted using a 22 design in k order to t a quadratic model all factors must be NH 2 z 0 run at least three levels 39 246139 Introduction to Statistical Quality introduction to statistical Quality Control 6th Edition Control 6th Edition t Y V i Q0 ltQJQ 13 4 2k Factorial Design 13 4 2k Factorial Design i CDVQ 39 t I k I 39lihl m m l39 nl 2 i linilllltlif39 m PM Rm a Addition of Center Pornts to the 2 DeSIgn W curt A 39l W Center points can provide not only some protection against curvature if the center points are replicated then an independent estimate of experimental error can be obtained Center points consist of n replicates run at the XI 0 i 1 2i k If Addition of center points does not have an impact on the usual H effect estimates in a 2k design n Assume thek factors are quantitative in order to have a center or middle level of the factor W n l l I I l I Center points can be added to the standard 2k design I l 7 I 7 l l l l I l I l l l l l t 0 7 introduction to Statistical Quality lily 1 Control 6th Edition uullnui out umun 13 4 2k Factorial Design 134 2k Factorial Design Camila l l 9 Addition of Center Points to the 2k Design l V0 Addition of Center Points to the 2k Design ll Reconsider the plasma etch experiment from 0 Sum of squares for pure quadratic curvature 1 example 138 Four center points n 4 have been degree of freedom added to n quotcw if v quot3 j The design with the responses given in previous i f x SS j y Averages yF 7760625 y3 75275 mlrptrl Curvature Sum of Squares where n number of factorial design points and nc C r if V number of observations at the center point 55mmmic j and 7 are average of runs at nf factorial points 39 quotC and average of he runs at the center points 47760625 752752 respectively 4 17391 Introduction to Statistical Quality Introduction to Statistical Quality Control 6th Edition Control 6th Edition 739 tiger mmt to to em 0 CXPPh Pfqu W V 134 2k Factorial Design w Sam Q Q Addition of Center Points to the 2k Design I a b An estimate of exgerimental error pure error can be obtained by calculating the sample variance of the center points 20 23 75275 62 2 1392quot 3 31227 introduction to Statistical Quality Control 6th Edition 13 4 2quot Factorial Design Addition of Center Points to the 2k Design The F test for curvature is given by MS cur valm39c Fquot MS residual where MSresidual ssmiduwldf with SSresidual a combination of sum of squares for pure error and sum of squares for lack of fit See Minitab output next slide introduction to Statistical Quality Control 6th Edition 13 4 2k Factorial Desi n 39 g mint quotlab rmt 1711 1 39iil l39Ii l Estimated litroots and CoefficienCu a flair Erch Rate coded f n M g units Turm Ellec Cour 3F Lonf Y r39 i L ConeAunt no a 100 11 000 V l l O J 20 20 3998 001 0 6 o A 10162 81 1a a O n 163 081 1050 003 0033 c 737 359 1020 056 0727 D 10612 15mins 1020 1501 Loon n E 739 39 1020 039 n7n9 A C 2L88 1gt 1 1o 20 427 1125 An 15363 It 8 10 23 753 ll0U0 on 153 1 9 1n 70 215 uue 5 0 065 U 71 10 zn 39UD H970 E D P13 I 06 1011 Ll 39IU I 97 r r 7 Jr a2au 04 o 17 AndLyI of V I 39th for Etch coded units snu r DF SL 5 r r Hlin tru a 116389 110339 101097 525 u 000 2 way Ilanl39aLliunu 6 104345 104845 173 1050 D 002 Curvuuvu 1 9 1739 173939 105 033 R 1duql Err39ui 8 13310 1521 66f 0 a k 0 fit r 1mm mus 3965 196 0308 rule Error 3 101 l otdl 19 introduction to Statistical Quality Control 6th Edition

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