Phase I Monitoring of Nonlinear Profiles
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1 Phase I Montorng of Nonlnear Profles James D. Wllams Wllam H. Woodall Jeffrey B. Brch May, 003 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY
2 Scenaro Profle Montorng Montor a process or product whose qualty cannot be assessed by a sngle qualty characterstc Measure across some contnuum (a sequence of tme, space, etc.) producng a profle Varous profle shapes: Lnear Profles: (Kang and Albn (000), Km, Mahmoud, and Woodall (003), Mahmoud and Woodall (003)) Nonlnear Profles: (Brll (00)) Very lttle work has been done to address montorng nonlnear profles (Woodall, et. al. (003)) J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY
3 Profle Montorng Path forward Brll s (00) method Suggest two more methods Illustrate methods wth nonlnear profle data Recommendatons J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 3
4 Example : Vertcal Densty Profle (VDP) Board A from Walker and Wrght (00, JQT) J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 4
5 Example : Dose-Response Profle of a Drug Response Dose J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 5
6 Phase I Analyss: Hstorcal Data Response n Nonlnear Profle Sample y y L y,,, n y y L y,,, n M m M M M M y y L y m, m, m, n J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 6
7 Bref Intro to Nonlnear Regresson Models Smple Case: One Y and one X y = f ( x, ) + ε =, K, n where y s the th response f ( x, ) s an approprate nonlnear functon x s the th regressor varable value s the p vector of parameters to estmate ε s the th resdual error J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 7
8 Bref Intro to Nonlnear Regresson Models obtaned teratvely for each sample ^ Var ( ) ( ) = σ D D = C where D s the estmated dervatve matrx used n the estmaton of the nonlnear regresson parameters J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 8
9 Parameter Estmates from Hstorcal Data Sample M m Parameter p M,, m, M,, m, L L L M, p, p m, p J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 9
10 How to Montor Nonlnear Profles Ideally, montor each parameter ndependently Problem: parameter estmates are correlated n nonlnear regresson Cannot montor each parameter separately, so use a multvarate T control chart to montor the parameters smultaneously J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 0
11 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY Multvarate T Control Chart Statstc General form of the statstc: T m,, = K = S T S s the covarance matrx of parameter estmates = = = = m p m m m m m,,, M = p,,, M
12 Three Choces for S Method : Sample Covarance Matrx (Brll, 00) S m = m = Pros: Easy to calculate Wdely used and easly understood Cons: Greatly affected by shfts n mean vector Results n low power for the T control chart J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY
13 Three Choces for S Method : Successve Dfferences (Holmes and Mergen, 993) Let V = v v v v m = + =, K, m Then S = V V ( m ) Pros: Lke movng range wth ndvdual observatons Not effected by shfts n the mean vector Hgh power Cons: Less statstcal theory developed to date J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 3
14 Three Choces for S Method 3: Intra-Profle Poolng ^ Var = σ D D = ( ) ( ) For each of the m samples: C Then S 3 = m m = C Pros: Uses nformaton from nonlnear regresson estmaton Cons: Does not account for profle-to-profle common cause varablty J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 4
15 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 5 Three Choces for T Three formulatons of the statstc: T = S, T Method : Sample Covarance Matrx = S, T Method : Successve Dfferences = S 3, 3 T Method 3: Intra-Profle Poolng
16 Upper Control Lmts Method : Sample Covarance Matrx T m p m p ~ Beta, ( m ) As dscussed by Sullvan and Woodall (996) UCL ( m ) B = α, p /,( m p )/ m J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 6
17 Upper Control Lmts Method : Successve Dfferences Approxmately T where m p f p ~ Beta, ( m ) f = ( m ) 3m 4 For more nformaton, see Scholz and Tosch (994) UCL ( m ) B = α, p /,( f p )/ m J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 7
18 Upper Control Lmts Method 3: Intra-Profle Poolng We thnk that approxmately T 3 m( m p) p( m )( m + ) ~ F ( p, m p) UCL 3 m( m p) = F α, p, m p p( m )( m + ) Control lmts are best approxmatons so far J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 8
19 Illustraton: VDP Data VDP of 4 Partcle Boards J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 9
20 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 0 Nonlnear Functon to Model VDP Data Use a bathtub functon to model each board from the VDP data + + > + = d x c d x a d x c d x a x f b b ) ( ) ( ), ( = d c b b a a x s the th regressor varable value where determne the wdth of the bathtub determne the flatness of the bathtub s the bottom of the bathtub s the center of the bathtub
21 Nonlnear Functon to Model VDP Data Board # from Walker and Wrght (00, JQT) VDP Nonlnear ft J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY
22 Nonlnear Functon to Model VDP Data Estmated nonlnear profle of Board # f ( x 0.33) ( x, ) = ( x ) x x > Estmate profle for each board Calculate S, S, and S 3. Calculate T,, T and T 3 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY
23 T Control Chart Board #5 Board #8 T UCL = 9.6 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 3
24 T Control Chart Board #5 Board #8 T UCL = 4. J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 4
25 T and T Control Charts T T T T UCL = 4. UCL = 9.6 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 5
26 Board 5 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 6
27 Board 8 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 7
28 T 3 Control Chart Board #3 Board #6 Accordng to our UCL, all of the boards are out-of-control UCL J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 8
29 Boards 3 and 6 Board 3 Board 6 J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 9
30 Conclusons Method (sample covarance matrx) does not take nto account the sequental samplng structure of the data: The overall probablty of detectng a shft n the mean vector wll decrease (See Sullvan and Woodall, 996) Should not be used Method (successve dfferences) accounts for the sequental samplng scheme, and gves a more robust estmate of the covarance matrx In the VDP example, both Methods and gave same result because No apparent shft n the mean vector There were only about two outlers J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 30
31 Conclusons Method 3 (ntra-profle poolng) should be used when there s no profle-to-profle common cause varablty Comparson of the three methods: Method assumes all varablty s due to common cause Method 3 assumes that no varablty s due to common cause Method s somewhere n the mddle Issue: Montorng parameters versus montorng the ftted curves J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 3
32 References Brll, R. V. (00). A Case Study for Control Chartng a Product Qualty Measure That s a Contnuous Functon Over Tme. Presentaton at the 45 th Annual Fall Techncal Conference, Toronto, Ontaro. Holmes, D. S., and Mergen, A. E. (993). Improvng the Performance of the T Control Chart. Qualty Engneerng 5, pp Km, K., Mahmoud, M. A., and Woodall, W. H. (003). On The Montorng of Lnear Profles. To appear n the Journal of Qualty Technology. Mahmoud, M. A., and Woodall, W. H. (003), Phase I Montorng of Lnear Profles wth Calbraton Applcatons, Submtted to Technometrcs. Scholz, F. W., and Tosch, T. J. (994), Small Sample Un- and Multvarate Control Charts for Means. Proceedngs of the Amercan Statstcal Assocaton, Qualty and Productvty Secton. Sullvan, J. H., and Woodall, W. H. (996), A Comparson of Multvarate Qualty Control Charts for Indvdual Observatons. Journal of Qualty Technology 8, pp Walker, E., and Wrght, S. P. (00). Comparng Curves Usng Addtve Models. Journal of Qualty Technology 34, pp Woodall, W. H., Sptzner, D. J., Montgomery, D. C., and Gupta, S. (003). Usng Control Charts to Montor Process and Product Profles. Submtted to the Journal of Qualty Technology. J.D. Wllams, Bll Woodall, Jeff Brch, Vrgna Tech 003 Qualty & Productvty Research Conference, Yorktown Heghts, NY 3
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