A Mixed-Effect Model for Analyzing Experiments with Multistage Processes

Transcription

1 Qualit Technolog & Quantitative Management Vol., No. 4, pp. 49-, 4 QTQM ICAQM 4 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes Kaibo Wang * and Chenxu Dai Department of Industrial Engineering, Tsinghua Universit, Beijing, China (Received October, accepted Jul ) Abstract: In industrial practice, most products are produced b processes that involve multiple stages. In studing multistage processes via designed experiments, some practitioners treat them as single-stage processes and follo the usual factorial designs or split-plot designs. In this paper, through an analsis of the error transmission mechanism, e propose a mixed-effect model for analzing experiments ith multistage processes. Based on an analsis of simulated and real experimental data, e find that different conclusions about factor significance ma be dran if the data are analzed differentl. In addition, the mixed-effect model can help separate errors at different stages and hence provide more information on process improvement. Keords: Design of experiments, mixed-effect model, multistage process.. Introduction D esign of experiments (DOE) is an efficient and idel used technique for process investigation, data collection and model building. In a controlled experiment, the process to be studied is treated as a black box; controllable factors, such as x and x, that ma affect the process output are identified first and then are changed manuall according to a design matrix. Figure (a) shos a diagram that represents the general scenario of experimental design: all of the factors are treated equall, and the impacts of the factors are applied to the process simultaneousl. Hoever, in some processes, the factors are phsicall positioned at different stages, as Figure (b) shos. Compared to the process in Figure (a), the process in Figure (b) has to distinct features: first, there is a distinct difference in terms of the time or location of the factors. In Figure (b), factor x functions first at stage, hile the second factor, x, functions at a later stage, after the effect of x on ', hich is unobservable in practice. Second, a ne interaction structure is presented in the process. With the variation propagating from upstream stages to donstream stages, the error term from stage, ', enters stage and ma interact ith x, hile it is phsicall impossible for the disturbance at stage,, to move backard to interact ith x. The process in Figure (b) is a tpical multistage manufacturing sstem (MMS). An MMS is a process that consists of more than one stage (or ork station). In an MMS, each product moves through multiple manufacturing stages, there are controllable factors at each stage, and the accumulated effect of the factors is embedded in the final qualit of the product. Because it is difficult to finish a product in a single operation, nearl all products * Corresponding author. kbang@tsinghua.edu.cn

2 49 Wang and Dai are produced b an MMS. For example, a semiconductor manufacturing process for producing integrated circuits (ICs) usuall involves hundreds of stages (Mee and Bates []). A process for making a car also has dozens of stages (Shi []). Wu and Hamada [8] introduced a soap bar example consisting of to sub-processes: mixing and forming. The output of a multistage process is not a simple summation of all the individual stages; rather, control actions taken at later stages ma interfere ith the results of actions performed at earlier stages (Zhong et al. []). x x ' ' x x (a) (b) Figure. General model for a single-stage and to-stage process. A considerable amount of research has been carried out on the modeling, monitoring and control of MMSs. Jin and Shi [6] developed a state-space model to characterize the variation propagation of a sheet metal assembl process in automobile production. Yao and Gao [9] suggested that a multistage process ma be divided into blocks for model building. Li and Zhou [8] presented a robust variation source identification method for qualit improvement in manufacturing processes, assuming a linear relationship beteen the variation sources. Gaver et al. [] studied the reliabilit groth of a multistage sstem. Shi and Zhou [6] presented a surve of recent research into the modeling, monitoring, diagnosis, and control of multistage processes. Zi et al. [] presented examples on monitoring a multistage process. Hoever, research on the design and analsis of experiments for multistage processes is still limited.. Process loop 4. Process parameter. Processing time. Processing speed 8. Processing time 9. Temperature S S S S4 Viscosit. Ingredient A. Ingredient B 6. Process speed Figure. The multistage nature of the coffee cream-making process introduced b Schoen et al. [].

3 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes 49 To stud a multistage process, it is preferable that each stage is studied individuall if the intermediate response variables are observable, so that the phsical mechanism of the engineering process is easier to understand. If such intermediate observations are difficult or impossible, the tpical current practice is to treat the multiple stages as a single process. For example, Schoen et al. [] studied a process for making coffee cream. The hole process consists of four consecutive stages. Nine controllable factors are identified in total, ith to factors at the first stage, three at the second stage, one at the third stage and to at the last stage, as shon in Figure. The authors classified these factors as being eas to change or hard to change and then used a split-plot design to stud the process. As another example, Schoen [4] introduced a ood construction experiment involving 6 factors. The construction process has six stages: saing, immersing, pretreatment, gluing, press, and storage. Again, the author treated the six-step process as a single stage and ignored the successive effects of the factors. Saing Immersing Pretreatment Wood Moisture content Pretreatment Gluing Pressing Storage Strength Glue Pressure Temperature Figure. The multistage nature of the ood construction experiment introduced b Schoen [4]. In a later section of this paper, e examine a afer-rinsing process. The process is designed to improve the smoothness of lapped afers in semiconductor manufacturing. In this process, each afer moves through several tanks sequentiall. Each tank has its on factors, such as solution densit and time. The afers cannot be measured until the are treated in all tanks. A designed experiment is needed to stud this process and optimize the settings of each tank. Conventionall, the tanks could be treated as one big tank ith the factors of all the tanks being treated as equal. Hoever, this approach obviousl ignores the multistage nature of the process and ma reduce the efficienc of the data analsis. The experiment is introduced in greater detail later. We also sho the differences if the same experiment is analzed in different as. Because the multistage nature of processes is idel observed, the design and analsis of experiments for multistage processes become important in practice. To optimize a multistage process, it is critical to collect data from the process efficientl and analze the data appropriatel. DOE is an important a to help identif significant factors in a process and construct statistical models to express the relationships beteen process output and input variables. In the literature, some researchers have noticed the unique nature of multistage processes and have proposed as to design and run experiments more efficientl at loer

4 494 Wang and Dai costs. For example, Miller [] presented a laundr example ith to stages: clothes need to go through ashing and dring machines. The ashing and dring steps obviousl happen in sequence. Miller proposed the use of a strip-lot design (Mee and Bates [] called this a to-a split-unit design) to reduce the resources needed for the experiment. Mee and Bates [] presented an example of IC fabrication: afers move through man manufacturing steps in their production. The authors suggested the use of multi-a split-lot designs. The name comes from the fact that in semiconductor manufacturing, afers are processed in lots (batches). A split-lot design can be considered a tpical design ith incomplete blocks in hich some main effects are confounded ith blocking effects (see Section. of Mee [9]). Instead of using a fixed lot, Mee and Bates [] suggested removal of the restriction of separate replicates so that a higher degree of randomization is achieved. Yuangai et al. [] proposed a robust parameter design method for a multistage process. Hoever, the did not provide details regarding ho the experiment should be analzed; rather, the suggested methods for analzing linear models for full-factorial split-plot designs. Butler [] suggested that split-lot designs are potentiall useful for multistage processes. In such a design, each stage has a split-plot structure. This has the same implication as the multi-a split-unit design proposed b Taguchi []. Butler [] also mentioned that a split-lot design ith to stages is equivalent to a strip-plot design as described b Miller [] and provided guidelines for constructing to-level split-lot fractional factorial designs for multistage processes. Hoever, these studies onl focused on the design of experiments for multistage processes; the did not emphasize the analsis of the experiments. Alternativel, Schoen et al. [] treated the experiment for a multistage process as a split-plot design. A split-plot experiment (hich is referred to as a split-unit experiment in chapter 9 of Ran []) is a blocked experiment in hich blocks are formed ith hard-to-change factors. Complete randomization is onl implemented ithin subplots. If an experiment involves factors that are hard to change, it is a natural choice to adjust the hard-to-change factors less frequentl (Mee and Bates []). In a split-plot design, hardto-change factors are treated as hole-plot factors. The levels of a hole-plot factor are first randomized, and then the levels of the split-plot factors (eas-to-change factors) are randomized under each level of the hole-plot factor (Jones and Nachtsheim []). As Anbari and Lucas [] noted, split-plot designs are idel used because of their accurac, efficienc and lo cost. The purposes of this paper are to investigate the variation propagation mechanism in experiments for multistage processes and to propose a mixed-effect model for analzing data collected from multistage processes. The remainder of this paper is organized as follos. Section establishes a mixed-effect model for experiments ith multistage processes. In Section, different model strategies for the experiments are studied. In Section 4, a comparison of the different modeling strategies are carried out based on simulated and real data. Lastl, Section concludes ith suggestions for future research.. A Mixed-Effect Model for Characterizing the Output of a Multistage Process To better understand the data collected from DOE for a multistage process, e first tr to capture the behavior of the process using a statistical model. For simplicit, e focus on a to-stage process that is similar to the one shon in Figure (b). There are S factors at the first stage and S factors at the second stage. The intermediate invisible output of the first stage, ', is given as follos:

5 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes 49 S ' a b x b' x x ', () i i ii ' i i ' i i, i ' : S ; ii ' here a, b i, and b ' ii ' are the intercept and coefficients of the factors and their interactions. The first stage is assumed to have a normall distributed disturbance, ' ~ N (, ). In Equation (), each interaction effect can be observed as a ne factor (for example, e can define x xx ). Because the focus of this paper is the stud of the multistage nature of a process, to simplif the equations and their explanation, in the folloing, e hide all interaction effects of the factors at the same stage from the model. Hoever, these effects could easil be added to the model, and all derivations still hold. In a multistage process, the output of the upstream stages becomes the input of the donstream stages. In the to-stage process, the output of the second stage is therefore given b the folloing expression: S S a b x k ' c x ', () j j j j j j here k is the magnificent coefficient of ' on and c i is the coefficient of the interaction beteen the factors at the second stage, x i, and the output of the first stage, '. Again, the second stage itself is also affected b a normall distributed disturbance,, and ~ N (, ). Substituting Equation () for ' in Equation (), e obtain the folloing expression: S S S S S ( ka a ) k b i x i b j ac j xi c j x j b i x i k ' c j x j ' i j j i j S S S S S x x x x x, i i j j ij i j j j i j j i j () here reflect the effects of the factors and interactions; is the stage-specific random effect, hich is rooted in the transmission of the error at stage to stage and onl appears in a multistage model; j is the random effect due to the interaction beteen the factors at the second stage and the propagated errors from the first stage, and N i N N ~ (, ), ~ (, ), ~ (, ). i Equation () shos the components of the output of a multistage process. First, the output has several fixed effects, including the intercept and the main effects. Second, the output is also affected b the random effects in the sstem. Specificall, is the effect of the error passing through from the first stage, and is the interaction beteen the same error passing through from the first stage and the factors at the second stage. Third, the disturbance of the second stage is the global error shon directl in. Equation () can be observed as a special form of a mixed-effect model. The general form of the mixed-effect model is as follos (Fox [4]): i Xβ i i Zi τi ε i, τ ~ N (, ψ ), i q ε ~ N (, Λ ). i n i i

6 496 Wang and Dai In a multistage experiment, the special structure of the Z matrix is determined b the design matrix and the run order. In the folloing section, e examine several specific design scenarios and analze the data based on the mixed-effect model. If the multistage process is rongl treated as a single-stage process, the model for the output ould be expressed as follos: S S S S x x x x '', (4) i i j j ij i j i j j i here '' represented the effect of the to disturbance sources in the process. Compared ith Equation (), the random effects are missing, hich ma lead to an inaccurate estimation of the effects and erroneous conclusions about the factor significance. Simulated and real examples are presented later to sho the difference beteen these models. If the design is treated as a split-plot design, the model ould be expressed as follos (see Jones and Nachtsheim []): S S S S x x x x '', () i i j j ij i j i j j i here is a random effect corresponding to the hole plot error in the split-plot design, rooted in the transmitted error of stage to stage in Equation (). Obviousl, this formulation does not consider the interaction effect beteen the transmitted error and the factors at stage. The formulation in Equation () is also different from the model presented b Miller [], in hich the transmission and propagation of errors ere also missing. In the folloing section, e appl the above formulation to different DOE scenarios and stud ho the experiments should be analzed if the multistage nature is to be appropriatel considered.. Modeling Strategies of Experiments for Multistage Processes When dealing ith experiments for multistage processes, the unique error transmission mechanism makes the modeling of the experiment different. In practice, some practitioners ma ignore the multistage nature of the process and treat it as a tpical factorial design. Some ma analze the experiment as a split-plot experiment. In the folloing section, for illustration purposes, e stud a simple process ith to stages. Different experimental design schemes are applied to the process. For each design, e then appl different modeling strategies and compare their differences... Unreplicated Experiments ith To Factors In the first scenario, e assume that in the to-stage process, each stage has a single controllable factor. Therefore, the process has to factors in total. Four experimental runs are used to stud this process. The experiment could be treated in four different as: (a) ith a simple randomized factorial design, (b) ith a randomized factorial design ith consideration of the multistage nature, (c) ith a split-plot design or (d) ith a split-plot design ith consideration of the multistage nature. We ill sho the model for each a of treating the experiment and compare the differences.

7 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes Modeling as a Single-Stage Randomized Factorial Design In practice, some practitioners ma simpl treat an experiment for a multistage process as the ould a factorial experiment for a single-stage process. In such a case, an unreplicated full factorial design ould be modeled as follos: x x x x x x x x. (6) x x x x 4 x x x x 4 The covariance matrix of is expressed as follos: V. In this modeling frameork, the to-stage process is treated as a black box on the hole. The errors of the to stages are not separable.... Modeling as a Multistage Randomized Design Folloing the multistage model in Equation (), if the multistage nature is considered for the completel randomized experiment, the observations of the four runs are modeled in the folloing a: () x x x x x x x x x x x x 4 x x x x x x, x 4 x 4 4 (8) and ij ~ N (, ), i,; j,...,4, ~ (, ),,...,4, i i N i here represents the random effect of the transmitted error from stage to stage and represents the random effect of the interactions beteen the stage- error and stage- factors. If the experiment is treated as completel randomized, an identit matrix should be used in front of the vector. Compared to Equation (6), the fixed effects of the factors are the same, hile Equation (8) gives a better explanation of the random errors of the experiment. With the coded factor levels, xi, xi, the random effects are confounded. The covariance matrix of the observational vector becomes the folloing:

8 498 Wang and Dai V. (9) here total. The variation of each observation is the same. The random effect cannot be separated from the disturbance due to the a the experiment is conducted. Therefore, if the experiment is carried out in a totall random order, the modeling strategies ith and ithout consideration of the multistage nature are the same.... Modeling as a Single-Stage Split-Plot Design When doing experiments for a multistage process, it is common and natural to treat the factors of one stage as hard-to-change factors to minimize costs (Mee and Bates []). In such a case, the experiment ould have a split-plot structure. Table. A split-plot design. Run Order x x 4 Suppose the to-stage process is inaccuratel considered as a single-stage process and the experiment is conducted folloing the split-plot design shon in Table, i.e., factor x is the hole-plot factor, and x is randomized under x. The output of the experiment is then modeled as follos: x x x x x x x x, x x x x () 4 x x x x 4 and ~ N (, ), ~ N(, ) here is the hole-plot random effect and is the experiment error of the hole sstem. Because the first to runs are carried out under the same and unchanged setting of x, these to runs share a common random effect; similarl, the second to runs share another random effect. In this design, the covariance matrix of the output becomes the folloing:

9 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes 499 V. () The split-plot model does not take the interaction beteen the first-stage error and the second-stage factors into account. Due to the restricted randomization, the random effect and the random error are no separable. Section 4 present example experiments and shos ho to analze such experiments...4. Modeling as a Multistage Split-Plot Design The formulation in Equation () does not consider the multistage nature of the experiment. If the multistage nature is taken into consideration, the output should be modeled as follos: x x x x x x x x x x x x x x x, 4 x x x x x 4 () here represents the random effect of the hole plot and represents the random effect beteen the factors at the second stage and the transmitted error from the first stage. In Equation (), the random effects and are no separable because the design matrices in front of and are not identical. The above analsis shos that for the same experiment, if the assumption about the a the data ere collected changes, the model changes accordingl. The model is considered correct onl if the assumption and model structure correctl reflect the a the experiment as carried out... Unreplicated Experiments ith Three Factors In this section, e ill consider a multistage process ith three factors. In a single-stage process, an increase in the factor numbers does not change the model structure, but this is no longer true in a multistage process. Suppose the number of factors is increased from to, and an unreplicated design is carried out. In the folloing, e sho ho this experiment shall be modeled.... Modeling as a Randomized Factorial Design If the experiment is considered a full factorial design ith a single-stage process, hen the number of factors increases, the output of the experiment is modeled in a manner similar to that shon in Equation (6):

10 Wang and Dai x x x x x x x x x 4 x x x 4. x x x 6 x x x 6 x x x 8 x x x 8 () The effects of all factors can be estimated as usual.... Modeling as a Randomized Multistage Design If the experiment is modeled ith its multistage nature taken into consideration, the model structure depends on the stage in hich the third factor appears. If the factor is added to the first stage, the model becomes the folloing: x x x x x x x x x 4 x x x 4 x x x 6 x x x 6 x x x 8 x x x 8 x x- x x x x- 6 6 x x- 8 8 (4) Compared ith Equation (8), this model structure is the same, except that the dimension becomes higher. If the third factor appears at the second stage, the number of random effects increases because the interactions beteen the first-stage error and the factors at the second stage increase. The output no can be expressed as follos:

11 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes x x x x x x x x x 4 x x x 4 x x x 6 x x x 6 x x x 8 x x x 8 x x x- 8 8 x+ x- x x x- 4 x- x x x+ 6 x- x x x- 8 Therefore, the modeling structure of a multistage process, as opposed to that of a single-stage process, depends on the location of the factors, because the multistage structure of the process makes the roles of the factors different. The factors at a donstream stage interact ith the errors transmitted from the upstream stages, making the relationships among the factors complicated.... Modeling as a Split-Plot Design The split-plot design is also sensitive to ne factors. If a ne factor is added to the first stage, the number of hole-plot blocks ill increase. The output of the experiment is no modeled as follos: x x x x x x x x x 4 x x x 4 x x x, (6) 6 x x x 4 6 x x x 8 x x x 8 and the covariance matrix of becomes the folloing: V. ()

12 Wang and Dai We see in Equation () that there are more blocks than in the matrix in Equation (). These blocks correspond to the hole-plot random effects in the experiment. In the split-plot design, if the ne factor is added to the second stage, the number of hole-plots does not change, so, the output is expressed as follos: x x x x x x x x x 4 x x x 4. x x x 6 x x x 6 x x x 8 x x x 8 (8) The covariance matrix no becomes the folloing: V. (9) A ne factor at the second stage does not change the number of blocks, but it increases the number of runs in each block. This again shos that an experiment ith a multistage process is rather different from an experiment ith a single-stage process. If the multistage process is rongl treated as a single-stage process, erroneous conclusions ma be dran from the analsis...4. Modeling as a Multistage Split-Plot Design For the split-plot design, if the third factor appears at the first stage, the experiment ill be modeled as follos: x x x x x x x x x x x x 4 x x x x 4. x x x x () 6 x x x 4 x 4 6 x x x x 8 x x x x 8

13 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes Compared ith Equation (), e can see that the number of blocks has increased, hile the number of runs in each block remains the same. If the ne factor appears at the second stage, as in the case of a split-plot design, the number of hole-plot blocks remains the same, but the number of random effects of the interactions beteen the ne factors and the first-stage error increases: x x x x x x x x x+ x x x x x x 4 x x x x + x x x x x + x +.() 6 x x x x x 6 x x x x x + 8 x x x x x 8.. Replicated Experiments ith To Factors Replication is a basic means of improving estimation accurac in experimental design. In this section, e stud a case in hich the experiment is carried out ith replicates. We ant to see ho different the models ma become. The process still has to stages, ith one factor at each stage. If the experiment is totall randomized ithin each replicate and is modeled as a factorial design, the model structure is similar to that shon in Equation (6), except that more observations are included. If the experiment is modeled as a multistage design, the model structure is similar to that shon in Equation (8). In both cases, in a manner similar to that discussed in Section.., onl the main effects can be estimated. Therefore, in the folloing section, e onl focus on the cases in hich the split-plot design is used.... Modeling as a Split-Plot Design Replication in a split-plot design adds another blocking factor to the model. The structure of the experiment is shon in Table. The model of the split-plot design ith replication is a slightl different from that shon in Equation (). With the blocking factor B being considered, the output of the experiment is given as follos: x x x x x x 4 B x x 4 B. x x 6 x x 4 6 x x x x 8 8 () No, e see a ne factor block in the model, and more blocks in the matrix correspond to the random effects.

14 4 Wang and Dai Table. A replicated split-plot design. Replicate A Replicate B Factors x x x x 4... Modeling as a Multistage Split-Plot Design If the multistage nature of the process is considered, the replication in the design increases the number of hole plots and adds the effect of blocking into the model: x x x x x x 4 B x x B x x 6 x x x x 8 x x x x x x + 4. x 4 x 4 6 x x 8 We see that compared to Equation (), there is an extra blocking effect, hile the metrics corresponding to the random effects have larger dimensions. 4. A Comparison of Different Modeling Strategies In this section, e first appl the different modeling techniques to a set of simulated data and compare their performance. Then, a real case stud is presented using data from a afer fabrication process. 4.. A Simulated Example We first use simulated data to see hether the model is able to identif the factors and the sources of variation correctl. We assume a to-stage sstem ith one factor at each stage. The experiment has 8 blocks and runs in total. We assume that the first stage of the simulated process is dominated b the folloing model: ' x ', ()

15 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes here ' ~ N(,) is the experiment error. A random effect is also added to ' to simulate the blocking effect. Stage is assumed to follo the folloing model form: x ' ' x, here ~ N(,), hich is equivalent to the folloing: The simulated data are shon in Table. x x x x x ' '. Table. A simulated experiment. Run order Replicate x ' x The folloing R code is used to prepare the data for further analsis: response=c(4.8,-.,.,-.9,4.6,-9.,.6,-.,4.9,-8.68,6.6,-.4, 44.,-.,.4,-.9,4.,-9.,.8,-9.8,.,-.9,.,-.,4.96,-., 8.6,-8.,8.6,-9.98,9.,-9.86)

16 6 Wang and Dai block=c(,,,,,,,,,,,,4,4,4,4,,,,,6,6,6,6,,,,,8,8,8,8) x=c(,,-,-,,,-,-,,,-,-,,,-,-,,,-,-,,,-,-,,,-,-,,,-,-) x=c(,-,,-,,-,,-,,-,,-,,-,,-,,-,,-,,-,,-,,-,,-,,-,,-) simdata=data.frame(response=response, x=x,x=x,block=block) In the folloing section, three different models are used to analze the data Modeling as a Randomized Factorial Design If the experiment is treated as a randomized factorial design, the data are analzed as follos: model=lm(response~ x*x, data= simdata) summar(model) The output of the above code is shon in Figure 4. The estimated intercept and coefficients of x and x and the interaction are close to the true parameters used in the simulation (the parameters are,, and, respectivel, in the simulation). Hoever, the model can onl estimate the total error as.. Call: lm(formula = response ~ x * x, data = test) Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) <e-6 x <e-6 x <e-6 x:x <e-6 Residual standard error:. on 8 degrees of freedom Multiple R-squared:.99, Adjusted R-squared:.99 F-statistic: 8 on and 8 DF, p-value: <.e-6 Figure 4. Model summar hen the experiment is analzed as a factorial design. Linear mixed-effect model fit b REML Data: test AIC BIC loglik Random effects: Formula: ~ block (Intercept) StdDev:.86e- Formula: ~ x %in% block (Intercept) Residual StdDev:.64e-.969 Fixed effects: response ~ x * x Value Std.Error DF t-value p-value (Intercept) x x x:x Figure. Model summar hen the experiment is analzed as a split-plot design.

17 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes 4... Modeling as a Split-Plot Design If the same experiment is treated as a split-plot design, the folloing code can be used to analze the data: librar(nlme) model=lme(response~x*x,data=simdata, random=list(block=pddiag(~),x=pddiag(~))) summar(model) The restricted maximum likelihood (REML) algorithm is used for this model. The output is given in Figure. The estimates of the intercept and coefficients are nearl the same as those shon in Figure 4. All the factors are identified. One significant difference is that the split-plot design can separate and estimate the error of each single stage (the hole-plot factor and the sub-plot factor). Hoever, the estimate of the first-stage error is much smaller than its true value, although the second-stage error is estimated to be., hich is quite close to the value in the factorial design in Figure Modeling as a Multistage Split-Plot Design Alternativel, if the multistage nature of the process is taken into consideration, e can analze the same dataset as follos: librar(nlme) model=lme(response~x*x,data=simdata, random=list(block=pddiag(~),x=pddiag(~x))) summar(model) The results are shon in Figure 6. Once again, the estimates of the coefficients are nearl the same. The block does not affect the response either. Hoever, it is clearl that this model can separate the errors from both stages more precisel. Linear mixed-effect model fit b REML Data: test AIC BIC loglik Random effects: Formula: ~ block (Intercept) StdDev:.86 Formula: ~x x %in% block Structure: Diagonal (Intercept) x Residual StdDev: Fixed effects: response ~ x * x Value Std.Error DF t-value p-value (Intercept) x x x:x Figure 6. Model summar hen the experiment is analzed as a multistage split-plot design.

18 8 Wang and Dai 4.. Case Stud We next use the afer rinsing process as an example. In the silicon chip production sstem, the cleanness and roughness of the surface could significantl affect the qualit and failure rate of the final product. A multi-cavit rinsing device is used to remove the grain, metal contamination, organic contamination and oxidation film from the afers (Cad and Varadarajan []). The process has to stages, ith different solutions used at each stage. The densit of the solution and the time spent at each stage ma affect the rinsing effect. Therefore, in this experiment, four factors, the solution densit at stage ( x ), the rinsing time at stage ( x ), the solution densit at stage ( x ) and the rinsing time at stage ( x 4 ) are studied. The response variable is the cleanness of the finished afer, hich is measured b the amount of metal ion left on the afer surface. Considering the limitations of time and cost, 6 runs ere conducted in the experiment. As illustrated in the previous section, e ma analze the experiment as a factorial design, a split-plot design, or a multistage split-plot design. The data and the R code used to analze the experiment are shon in Appendix A, and the summar of the three models are presented in Appendix B. If the experiment is analzed as a single-stage factorial design, after removing all insignificant to-factor interactions, e noticed that factors x and x 4 are significant in this model (using. here and later); if the experiment is analzed as a split-plot design, onl the factor x 4 is significant. If the experiment is analzed as a multistage split-plot design, both x and x 4 are significant. In addition, this model can estimate all the random effects and sho that the variabilit of the first stage is small and the variabilit of the second stage is much larger. This piece of information is important to further process diagnosis and improvement.. Conclusions Multistage processes are idel observed in industrial processes. When designed experiments are conducted ith a multistage process, it is crucial to consider the nature of the data generation process in the analsis of the data. In this paper, e develop a mixed-effect model for analzing experiments ith multistage processes. Compared ith cases in hich an experiment is rongl analzed as a factorial design or a single-stage split-plot design, the proposed model can identif the random effects caused b the transmission of errors from upstream stages. If the experiment is correctl designed, the mixed-effect model can also separate the errors associated ith the multiple stages and provide information for further process improvement. Performance studies based on simulated data and real data sho that if the same experiment is modeled in different as, different conclusions about factor significance ma be dran. Because multistage processes are idel observed in practice, e believe that experimental design strategies for such processes, ith the consideration of real constraints, deserve more attention in future research. Acknoledgements We greatl thank the Ex-Editor-In-Chief, Professor Gemai Chen, for his valuable suggestions, hich have helped us improve this paper greatl. This ork as supported b the National Natural Science Foundation of China under grant and Tsinghua Universit Initiative Scientific Research Program.

19 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes 9 References. Anbari, F. T. and Lucas, J. M. (8). Designing and running super-efficient experiments: Optimum blocking ith one hard-to-change factor. Journal of Qualit Technolog, 4(), -4.. Butler, N. A. (4). Construction of to-level split-lot fractional factorial designs for multistage processes. Technometrics, 46(4), Cad, W. and Varadarajan, M. (996). RCA clean replacement. Journal of the Electrochemical Societ, 4, Fox, J. (). An R and S-Plus Cmpanion to Aplied Rgression, Sage Publications, Inc.. Gaver, D. P., Jacobs, P. A., Glazebrook, K. D. and Seglie, E. A. (). On a ne stochastic usage model (Non-time-homogeneous Poisson) for testing a multi-stage sstem to promote reliabilit groth. Qualit Technolog and Quantitative Management, 4(), Jin, J. and Shi, J. (999). State space modeling of sheet metal assembl for dimensional control. Journal of Manufacturing Science and Engineering,, 6.. Jones, B. and Nachtsheim, C. J. (9). Split-plot designs: hat, h, and ho. Journal of Qualit Technolog, 4(4), Li, Z. and Zhou, S. (6). Robust method of multiple variation sources identification in manufacturing processes for qualit improvement. Journal of Manufacturing Science and Engineering, 8, Mee, R. W. (9). A Comprehensive Guide to Factorial To-Level Experimentation, Springer, Dordrecht.. Mee, R. W. and Bates, R. L. (998). Split-lot designs: Experiments for multistage batch processes. Technometrics, 4(), -4.. Miller, A. (99). Strip-plot configurations of fractional factorials. Technometrics, 9(), -6.. Ran, T. P. (). Modern Experimental Design. John Wile & Sons, Ne Jerse.. Schoen, E., Jones, B. and Goos, P. (). A split-plot experiment ith factordependent hole-plot sizes. Journal of Qualit Technolog, 4(), Schoen, E. D. (). Optimum designs versus orthogonal arras for main effects and to-factor interactions. Journal of Qualit Technolog, 4(), Shi, J. (6). Stream of Variation Modeling and Analsis for Multistage Manufacturing Processes. CRC Press, Talor & Francis. 6. Shi, J. and Zhou, S. (9). Qualit control and improvement for multistage sstems: A surve. IIE Transactions, 4(9), Taguchi, G. (98). Sstem of Experimental Design: Engineering Methods to Optimize Qualit and Minimize Cost. White Plains, UniPub, Ne York.. 8. Wu, C. F. J. and Hamada, M. (9). Experiments: Planning, Analsis and Parameter Design Optimization, nd edition. Wile, Ne York. 9. Yao, Y. and Gao, F. (9). A surve on multistage/multiphase statistical modeling methods for batch processes. Annual Revies in Control, (), -8.. Yuangai, C., Nembhard, H. B., Haes, G. and Adair, J. H. (). Robust parameter design for multiple-stage nanomanufacturing. IIE Transactions, 44(), Zhong, J., Liu, J. and Shi, J. (). Predictive control considering model uncertaint for variation reduction in multistage assembl processes. IEEE Transactions on Automation Science and Engineering, (4), 4-.. Zi, X., Zou, C., Zhou, Q. and Wang, J. (). A directional multivariate sign EWMA control chart. Qualit Technolog and Quantitative Management, (), -.

20 Wang and Dai Appendix A: R Code for Analzing the Wafer Rinse Example #real data response=c(.,.,-.6,.4,-.,-.6,.89,-.,.,-.,.6,.64,.,.4,.4,-.) x=c(-,-,-,-,,,,,-,-,-,-,,,,) x=c(-,-,-,-,-,-,-,-,,,,,,,,) x=c(-,,,-,,-,-,,-,,,-,,-,-,) x4=c(-,-,,,,,-,-,,,-,-,-,-,,) aferdata=data.frame(response=response, x=x,x=x,x=x,x4=x4) #analze as a factorial design model=lm(response~x+x+x+x4+x:x+x:x4+x:x+x:x4,data=aferdata) #remove insignificant to-factor interactions model=lm(response~x+x+x+x4,data=aferdata) summar(model) #analze as a split-plot design librar(nlme) K=x+*x model=lme(response~x+x+x+x4+x:x+x:x4+x:x+x:x4,data=aferdata, random=~ K) #remove insignificant to-factor interactions model=lme(response~x+x+x+x4,data=aferdata, random=~ K) summar(model) #analze as a multistage split-plot design librar(nlme) K=x+*x model=lme(response~x+x+x+x4+x:x+x:x4+x:x+x:x4,data=aferdata, random=list(k=pddiag(~x+x4))) #remove insignificant to-factor interactions model=lme(response~x+x+x+x4,data=aferdata, random=list(k=pddiag(~x+x4))) summar(model) Appendix B: Model Summar Generated from the Wafer Rinse Example #summar of model Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) x x x x #summar of model Linear mixed-effects model fit b REML Data: aferdata AIC BIC loglik Random effects: Formula: ~ K (Intercept) Residual StdDev:.68e-.984 Fixed effects: response ~ x + x + x + x4 Value Std.Error DF t-value p-value (Intercept) x x x x #summar of model Linear mixed-effects model fit b REML Data: aferdata AIC BIC loglik Random effects: Formula: ~x + x4 K Structure: Diagonal (Intercept) x x4 Residual StdDev:.469e Fixed effects: response ~ x + x + x + x4 Value Std.Error DF t-value p-value (Intercept) x x x x

21 A Mixed-Effect Model for Analzing Experiments ith Multistage Processes Authors Biographies: Kaibo Wang is an Associate Professor in the Department of Industrial Engineering, Tsinghua Universit, Beijing, China. He received his B.S. and M.S. degrees in Mechatronics from Xi an Jiaotong Universit, Xi an, China, and his Ph.D. in Industrial Engineering and Engineering Management from the Hong Kong Universit of Science and Technolog, Hong Kong. He has published papers in journals such as Journal of Qualit Technolog, IIE Transactions, Qualit and Reliabilit Engineering International, International Journal of Production Research, and others. His research is devoted to statistical qualit control and data-driven complex sstem modeling, monitoring, diagnosis, and control, ith a special emphasis on the integration of engineering knoledge and statistical theories for solving problems from real industr. He is a member of ASQ, INFORMS and IIE. Chenxu Dai received his B.S. and M.S. degrees in Industrial Engineering from Tsinghua Universit, Beijing, China, in and, respectivel. His research focuses on the statistical modeling, monitoring and control of engineering sstems.