Outline. 1. The Screening Philosophy 2. Robust Products/Processes 3. A Statistical Screening Procedure 4. Some Caveats 5.

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1 Screening to Identify Robust Products Based on Fractional Factorial Experiments Thomas J. Santner Ohio State University Guohua Pan Novartis Pharmaceuticals Corp Outline 1. The Screening Philosophy 2. Robust Products/Processes 3. A Statistical Screening Procedure 4. Some Caveats 5. Conclusions

2 1. The Screening Philosophy Objective: Show how (Gupta) screening using subset selection can be used to identify robust combinations (minimax) of control variables for products/processes having control & noise variables. Simple Screening Screening: Balanced One-way Layout ] µrð. ß5 # ß "Ÿ3Ÿ>ß "Ÿ4Ÿ where. ßáß. and 5 are unknown and 8is given " > NT: Ordered treatment means. ŸâŸ. Ò"Ó # Objective: Find a subset of the > treatments containing the treatment having mean. Ò>Ó (analog of classical optimal design for qualitative factors / "hot spot detection") Confidence Level Guarantee: Given!, "Î> "! "ß determine a set of treatments, say Z œ ZÐ] e", #á>,, fß so that Ò>Ó Tš Ò>Ó ZÐ] l. ß 5 # "!

3 for all. œ a. ßáß. band 5 " > #!Þ

4 Statistical Screening is based on Extremes: 8 " œ" W # œ pooled estimator of 5 # ] œ! ] œ3 >2 sample mean, "Ÿ3Ÿ> NT: ] ŸâŸ ] Ò"Ó Ò>Ó Fact: With 2 an upper! critical point from a certain multivariate > -distribution Z œ œ3à] ] 2 3 Ò>Ó W Ê # 8 is a 100 Ð"! % confidence level screening procedure for treatment associated with. Ò>Ó "yardstick" èëëëëëëëëëéëëëëëëëëëê l l ] 2W É ] Ò>Ó # 8 Ò>Ó

5 Features: Ð+ Analysis Tool: works for any sample size 8 Ð, As for Confidence Intervals, confidence level guarantee required to hold for all. ß 5 #. Width of the yardstick Å W 2WÉ # with the measurement error ( ) 8 œ Æ in the sample size 8 Ð- Unlike "usual" CI solutions, probability of CS depends on. and 5 #, i.e., Tš Ò>Ó ZÐ] l. ß 5 # "! with "! for. having.. Ò>Ó Ò> "Ó (similar issues in confidence interval construction for parameters of discrete distributions) Ð d Loss function approach L(., + ) œ! Š.. Ð Ð 3 + Ò>Ó 3 e Screening followed by selection(a design problem) f Screening relative to a control/standard (hot spot detection) Ð g ] Model? (hierarchical/spatial) Ð h Factorial Experiments: selection/screening for treatment means, interactions

6 2. Robust Products/Processes Multiple Factors Control Ðmanufacturing, engineeringvariables B 7 œðb" ßáßB c Noise Ðenvironmental, field, operatingvariables B / œðb" ßáßB0 Example In biomechanics, design a prosthetic hip to be used in a population of arthritic patients. The performance of the device is measured by the maximum strain produced at the bone/prosthesis boundary, call it ]. Suppose the distribution ] depends on Typical B - prosthesis geometry stiffness (material) Typical B / forces exerted on the prosthesis patient bone quality (bone elasticities) adherence of surgeon to insertion protocol other environmental variables

7 Objective: Select a combination of control variables that is "robust" to the effect of the noise variables Robust in what sense? This talk concerned with applications where it is critical to avoid designs, i.e., B -, for which there exist levels of the noise variables that can cause poor performance. Bottom Line: Designs of Hip prosthesis, heart valve replacements, gas tank mounting systems that perform poorly (disastrously) under certain field conditions should not be used.

8 Example Box and Jones (1992) describe a study whose goal is to improve the taste of a cake mix that consumers bake under conditions that can vary from the directions printed on the cakebox. Manufacturing Conditions: Control factors amounts of Shortening ÐW Flour ÐJ Egg ÐI (used in the mix) Field Conditions: Noise factors baking temperature ÐXà baking time Ð^. (baking directions) Control and Noise factors variables are (treated as qualitative) with two levels: 0 = low level 1 = high level

9 Mean: Response œ] 345, 67 œ taste testrating when ÐWßJßIßXß^œÐ34567 Ö!ß" & (larger taste testratings correspond to better tasting product) ˆ. 345ß67 œ mean response of ] 345, ß67 œ7! W 3 J 4 I 5 X ^ 6 7 ÐWX 36 axß^b Ð!ß! Ð!ß" Ð"ß! Ð"ß" ( Wß F,E). 000ß ß ß ß11 (0,0,0). 001ß ß ß ß11 (0,0,"). 010ß ß ß ß11 (!",,0). 011ß ß ß ß11 (!"",, ). 100ß ß ß10. (",!!, ) 100ß ß ß ß10. ("!",, ) 101ß ß ß ß10. ("",,0) 101ß11 (""",, ) ß00 111ß01 111ß10 111ß11 Maximin robustness 0 œ min , 67 6ß 7 be the worst-case performance of the recipe Ð3ß4ß5

10 Goal: find recipe that maximizes Data 2 &-1 Fractional Factorial Experiment with defining contrast MœWJIX^ Mean Model axß^b Ð!ß!Ð!ß"Ð"ß!Ð"ß" ( Wß F,E) "Þ# "Þ' (0,0,0) "Þ' %Þ% (0,0,") #Þ# 'Þ& (!",,0) %Þ* 'Þ" (!"",, ) "Þ$ "Þ( ("!!,, ) #Þ' #Þ% ("!",, ) $Þ) $Þ& ("",,0) &Þ# 'Þ! (""",, ). 345ß67 œ7! W 3 J 4 I 5 X 6 ^ 7 ÐWX 36 œ e7! W3 X6 ÐWX36f, 3ß6 Ö!ß". -8 Ð3\ ß4\ J4 I5, 4ß5 Ö!ß" c. - Ð3µ d e^ f, 7 Ö!ß" c Ð4 d 7. 8 µ has :œ" aw3bß;œ# ˆ J4, I5 ß=œ" ( X6) ß>œ" ( ^7) Ð3 Ð4ß5 Ð6 Ð7 Ð- œ$ Ð0 œ#

11 7 W X ÐWX 7 W X ÐWX! "!". -8 œ Œ 7 W X ÐWX 7 W X ÐWX! "! " " " "" is # # basis. matrix with no contraints, spanned by the usual

12 3. A Statistical Screening Procedure Suppose that a balanced full or orthogonal fractional factorial experiment has been conducted with each factor at levels 0 and 1 Let Control factors are indexed by 3 Noise factors are indexed by 4 denote the response of interest. ] 34, when B- isat combination 3 B is at combination 4 / Data:. 34, œ mean response of ] 34, ] œ. % 34, 34, 34, where ( 34,) ƒ is the set of control and noise factor combinations at which observations are taken and % 34, are independent RÐ!ß5 # measurement errors.

13 Definition: The worst case performance of the engineering design 3 is 0 min. 3 34, 4 Objective: Choose a set of control factors 03* œ max so that based on the (full factorial or fractional) experiment producing data ] ƒ A Model for. 34, Every mean model for. 34, has 3 types of terms: control variables (# œ: that interact with noise variables (# œ= control variables that have no interactions with noise variables (# œ; noise variables that have no interactions with control variables(# œ> So total # control variables -œ: ; total # noise variables 0œ= >

14 Notation \ \ 1 : \ \ Meaning C ßáßC Control factors that interact w/ Noise factors R1 ßáßR= Noise factors that interact w/ Control factors C~ C~ 1 ßáß ; Control factors that do not interact w/ Noise Factors R ~ ßáßR ~ Noise factors that do not interact w/ Control factors 1 > \ µ : ; - 3 œð3 ß3 Ö!ß" 3 \ Ö!ß" : 3 µ Ö!ß" ; 4 œð4 ß4 Ö!ß" 4 \ Ö!ß" = 4 µ Ö!ß" > \ µ = > 0 Assume. 34, œ.-8ð3\ ß4\. -Ð3µ. 8Ð4µ where. -8 Ð3\ ß4\ œ7! ÐC\ ßáßC\ R\ ßáßR\. µ - Ð3 œ! ÐC ßáßC 3 µ U! ß U-8 " : 1 = 3 4 µ µ " ;. µ 8 Ð4 œ! ÐRµ ßáßRµ 4 µ U " > where U-8ßUßU - 8 denote the sets of main effects and interactions presentin the model \ \

15 The Scree8ing Procedure Notation. 34, œ OLS of for - 3 œ min 3ß4 3 Ö!ß" 4 ordered 03 : 0[1] ŸâŸ 0[ #-] # # W œ moment estimator of Procedure: Select control factor combination 3 if and only if where [ #-] 2Wœ max 04 4 is chosen to satisfy 5 2W TÖBest 3 selected subset l. ß 5 "! ðóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóñóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóò œ CS for all. satisfying model and 5 #!, i.e, inf TÖGWl.ß 5 "! ß 5 #. `! # #

16 where CS denotes the event that the best 3 is in selected subset and "! is given

17 How to Choose 2? The mean model for. satisfies Symmetry Condition f. -8 Ð \ ß \ C3 \ ßáßC3 \ R4 \ ßáßR4 \ 1 (1) 3 4 : whenever an ( ) " : = involving : control factors and = noise factors is present in. -8 Ð3\ ß4\ ß then every other : = factor interaction involving : control factors and = noise factors must also be in. -8 Ð3\ ß4\ Ð ÐCµ ßáßC µ 3 3 " ; ; control factors is present in. - Ð3 µ, then every other interaction involving ; control factors must also be present in. µ - (2). µ - 3 : whenever an interaction among Ð3 (3). µ 8 Ð4 : whenever an interaction ÐRµ ßáßRµ 3 3 " > among > noise factors is present. 8 Ð4µ, ( in then every other interaction involving > noise factors must also be present in. µ 8 Ð4

18 Theorem Suppose that the mean model for. satisfies the symmetry condition f, and every main effect and interaction specified by the model is estimable based on the experiment data in ƒ. Then if there exists a sequence Ö / witheach / a 2: # = matrix satisfying the constraints of the mean model for the interacting control and noise variables, i.e., \ ß \ 7 Ð \ ßáß \ R\ ßáßR\!! C3 C3 4 4 ß U-8 " : 1 = 3 4 / 5 ( 3 4 ) = and is such that then inf.5 ß # lim / 5p_ 5 œ Î0 _ â _ ã ã ã ã Ð Ó 0 _ â _ Ï Ò. / ŒN 2 ; > satisfies 5 5 # : ; = > : = ; > 2 2 œ 2 # Œ 2 # # \ \ " Ò2 - Ó 5 1 TÖGWl. ß 5 œ lim TÖ 0 0 2Wl. ß 5p_ where 1 7 œð"ßáß" T length 7Þ : ; denotes the unit vector of

19 Example (Box and Jones:continued) Recall. -8 Ð3ß6œ7! W3 X ÐWX 6 36 ß36 Ö!ß", puts no restriction on the elments of the matrix and hence every 2 2 Hence model. Take Þ every / matrix satisfies the mean _ / 5 œ Œ p 0 0 Œ 0 0 then the limiting. 5 œ / 5 ŒN 2 # # 1 is Œ 0 _ 0 0 ŒN % # œ ÎÎ Ð Ó Ï Ò Î ÐÐ Ó ÏÏ Ò Î Ð Ï _ _ _ _ Ó _ _ _ _ Ò Î Ð Ó Ó Ï Ò Ò (for large 5, ŒN / 5 # 2 ; > is the "LFC")

20 Î Ð Ï Ó Ò Î Ð Ï Î Ð Ï Î Ð Ï Ó Ò Ó Ò Î Ð Ï _ _ _ _ _ _ _ _ Î Ð Ó Ï Ò Ó Ò Ó Ò Features of the LFC: (a) œ min œ! for all 3ß4ß5 Ö!ß" 6ß7. 345ß67 (b) 0045 is minimum of 2means Ð. 045,00 &. 045, ,00 145,01 145,10 145,11 (c) 0 is minimum of 4means Ð.,.,.,.

Theory of Screening Procedures to Identify Robust Product Designs Using Fractional Factorial Experiments

Theory of Screening Procedures to Identify Robust Product Designs Using Fractional Factorial Experiments Guohua Pan Biostatistics and Statistical Reporting Novartis Pharmaceuticals Corporation East Hanover,