EE290H F03 Spanos & Poolla Design of Experiments in Semiconductor Manufacturing Costas J. Spanos Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720, U.S.A. tel (510) 643 6776, fax (510) 642 2739 email [email protected] http://bcam.eecs.berkeley.edu Lecture 5: Comparison of Treatments and ANOVA 1 EE290H F03 Spanos & Poolla Lecture 5: Comparison of Treatments and ANOVA 2 EE290H F03 Spanos & Poolla Design of Experiments • Comparison of Treatments – which recipe works the best? • Simple Factorial Experiments – to explore impact of few variables • Fractional Factorial Experiments – to explore impact of many variables • Regression Analysis – to create analytical expressions that “model” process behavior • Response Surface Methods – to visualize process performance over a range of input parameter values Lecture 5: Comparison of Treatments and ANOVA 3 EE290H F03 Spanos & Poolla Design of Experiments • Objectives: – Compare Methods – Deduce Dependence – Create Models to Predict Effects • Problems: – Experimental Error – Confusion of Correlation with Causation – Complexity of the Effects we study Lecture 5: Comparison of Treatments and ANOVA 4 EE290H F03 Spanos & Poolla Problems Solved • Compare Recipes – Choose the recipe that gives the best results – Organize experiments to facilitate the analysis – Use experimental results to build process models – Use models to optimize the process Lecture 5: Comparison of Treatments and ANOVA 5 EE290H F03 Spanos & Poolla Comparison of Treatments • Internal and External References • The Importance of Independence • Blocking and Randomization • Analysis of Variance Lecture 5: Comparison of Treatments and ANOVA 6 EE290H F03 Spanos & Poolla The BIG Question in comparison of treatments: • How does a process compare with other processes? – Is it the same? – Is it different? – How can we tell? Lecture 5: Comparison of Treatments and ANOVA 7 EE290H F03 Spanos & Poolla Using an External Reference to make a Decision • An external reference can be used to decide whether a new observation is different than a group of old observations. • Example: Create a comparison procedure for lot yield monitoring. Do it without "statistics". • Use “external reference data“ (historical data from the same process, but not from the same experiment): Lecture 5: Comparison of Treatments and ANOVA 8 EE290H F03 Spanos & Poolla Example: Using an External Reference To compare the difference between the average of successive groups of ten lots, I build the histogram from the reference data: • Each newpoint can then be judged on the basis of the reference data. • The only assumption here is that the reference data is relevant to my test! Lecture 5: Comparison of Treatments and ANOVA 9 EE290H F03 Spanos & Poolla Using an Internal Reference... • We could generate an "internal" reference distribution from the very data we are comparing. • Sampling must be random, so that the data is independently distributed. • Independence would allow us to use statistics such as the arithmetic average or the sum of squares. • Internal references are based on Randomization. Lecture 5: Comparison of Treatments and ANOVA 10 EE290H F03 Spanos & Poolla Randomization Example • Is recipe A different than recipe B? 660 A A B e p e y t650 T a R pe h ci tc e E R 640 B 630 620 630 640 650 660 0 2 4 6 8 10 12 Etch Rate Sample Lecture 5: Comparison of Treatments and ANOVA 11 EE290H F03 Spanos & Poolla Randomization Example - cont. • There are many ways to decide this... 1. External reference distribution (based on old data.) 2. Assumed, approximate external reference distr. (such as student-t, normal, etc). 3. Internal reference distribution. 4. "Distribution free" tests. • Options 2, 3 and 4 depend on the assumption that the samples are independently distributed. Lecture 5: Comparison of Treatments and ANOVA 12 EE290H F03 Spanos & Poolla Randomization Example - cont. • If there was no difference between A and B, then let me assume that I just have one out of the 10!/5!5! (252) possible arrangements of labels A and B. • I use the data to calculate the differences in means for all the combinations: Lecture 5: Comparison of Treatments and ANOVA 13 EE290H F03 Spanos & Poolla The Origin of the student-t Distribution The student-t distribution was, in fact, defined to approximate such randomized distributions, when the “parent” distribution is normal! m m (y -y ) - ( - ) B A A B t = 0 1 1 s + n n A B • For the etch example, t = 0.44 and Pr (t > t ) = 0.34 0 0 • Randomized Distribution = 0.33 Lecture 5: Comparison of Treatments and ANOVA 14 EE290H F03 Spanos & Poolla Example in Blocking • Compare recipes A and B on five machines. • If there are inherent differences from one machine to the other, what scheme would you use? Random Blocked A A A B A B A B A B A B A B B A B B B A Lecture 5: Comparison of Treatments and ANOVA 15 EE290H F03 Spanos & Poolla Example in Blocking - cont. • With the blocked scheme, we could calculate the A-B difference for each machine. • The machine-to-machine average of these differences could be randomized. – d – d – d – d – d 1 2 3 4 5 d = 5 d - d ~ t n-1 s / n d IInn ggeenneerraal,l, rraannddoommizizeewwhhaatt yyoouu ddoonn't't kknnooww aannddbblolocckkwwhhaatt yyoouu ddoo kknnooww.. Lecture 5: Comparison of Treatments and ANOVA 16 EE290H F03 Spanos & Poolla Analysis of Variance e p y T D e p ci C e R B A 610 620 630 640 650 660 Etch Rate YYoouurr QQuueessttioionn:: AArree tthhee ffoouurr ttrreeaattmmeennttss tthhee s saammeeoorr nnoott?? TThhee SSttaattisistticiciaiann's's QQuueessttioionn:: AArree tthhee ddisisccrreeppaannccieiess bbeettwweeeenn tthhee ggrroouuppss ggrreeaatteerr tthhaann tthhee vvaarriaiattioionn wwitithhinineeaacchh ggrroouupp?? Lecture 5: Comparison of Treatments and ANOVA 17 EE290H F03 Spanos & Poolla Calculations for our Example i=1 i=2 i=3 i=4 i=5 Avg s2 n (yt-y)2 t t 1: 650 648 632 645 641 643.20 202.804 25.00 2: 645 650 638 643 640 643.20 86.80 4 25.00 3: 623 628 630 620 618 623.80 104.804 207.36 4: 645 640 648 642 638 642.60 63.20 4 19.36 s2 = R s2= T s2 T = s2 R Lecture 5: Comparison of Treatments and ANOVA 18 EE290H F03 Spanos & Poolla Variation Within Treatment Groups First, lets assume that all groups have the same spread. Lets also assume that each group is normally distributed. The following is used to estimate their common s: n S = S t (y -y)2 s2= St t tj t t n -1 j=1 t s2 = n 1s12+n 2s22+...+ n ksk2 = SR = SR R n 1+n 2 +...+ n k N - k n R • This is an estimate of the unknown, within group s - square. • It is called the within treatment mean square Lecture 5: Comparison of Treatments and ANOVA 19 EE290H F03 Spanos & Poolla Variation Between Treatment Groups • Let us now form Ho by assuming that all the groups have the same mean. • Assuming that there are no real differences between groups, a second estimate of s 2 would be: T k S n(y -y)2 t t s2 = t=1 = ST T n k -1 T This is the between treatment mean square IIff aalll l tthhee ttrreeaattmmeennttss aarree tthhee ssaammee,, tthheenn tthhee wwitithhinin aanndd bbeettwweeeennttrreeaattmmeenntt mmeeaann ssqquuaarreess aarree eessttimimaattiningg tthhee ssaammee nnuummbbeerr!! Lecture 5: Comparison of Treatments and ANOVA 20