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Technometrics

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Statistical Process Control for Latent QualityCharacteristics Using the Up-and-Down Test

Dongdong Xiang, Fugee Tsung & Xiaolong Pu

To cite this article: Dongdong Xiang, Fugee Tsung & Xiaolong Pu (2017) Statistical ProcessControl for Latent Quality Characteristics Using the Up-and-Down Test, Technometrics, 59:4,496-507, DOI: 10.1080/00401706.2016.1273139

To link to this article: https://doi.org/10.1080/00401706.2016.1273139

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Accepted author version posted online: 20Dec 2016.Published online: 25 May 2017.

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TECHNOMETRICS, VOL. , NO. , –https://doi.org/./..

Statistical Process Control for Latent Quality Characteristics Using theUp-and-Down Test

Dongdong Xianga, Fugee Tsungb, and Xiaolong Puc

aSchool of Statistics, East China Normal University, Shanghai, China; bDepartment of Industrial Engineering and Logistic Management, Hong KongUniversity of Science and Technology, Hong Kong; cSchool of Statistics, East China Normal University, Shanghai, China

ARTICLE HISTORYReceived June Revised December

KEYWORDSBinary data; Censored data;Control chart; Latent qualitycharacteristics; Latentvariable; SPC; Up-and-downtest

ABSTRACTIn many applications, the quality characteristic of a product is continuous but unobservable, for example,the critical electric voltage of electro-explosive devices. It is often important to monitor a manufacturingprocess of a productwith such latent quality characteristic. Existing approaches all involve specifying a fixedstimulus level and testing products under that level to collect a sequence of response outcomes (zeros orones). Appropriate control charts are then applied to the collected binary data sequence. However, theseapproaches offer limited performance. Moreover, the collected dataset provides little information for trou-bleshootingwhenanout-of-control signal is triggered. Toovercome these limitations, this article introducesthe up-and-down test for collecting data and proposes a new control chart based on this test. Numericalstudies show that the proposed chart is able to detect any shifts effectively and is robust inmany situations.Finally, an example involving realmanufacturingdata is given todemonstrate theuseof our proposed chart.

1. Introduction

In many applications, the quality characteristic of a product iscontinuous but unobservable. One example of this is the criticalelectric voltage of electro-explosive devices (EEDs) fabricatedusing integrated circuit techniques. EEDs are used in automotiveairbags, rocket ignition systems, and various ordnance systemsto ignite energetic compounds (see Zhang et al. 2012). A criti-cal electric voltage is associated with each EED, but it cannot bemeasured. It is extremely important to monitor the productionof EEDs, because early detection of abnormalities in the pro-duction process not only reduces costs but also avoids unpleas-ant consequences. For instance, if the process suffers from anupward shift in mean, then some of the EEDs produced in thisprocess will not respond to the prescribed electric voltage. Onthe other hand, if the process experiences a downward shift inmean, then some of the EEDs produced will have a much lowercritical electric voltage in which case the safety of operationmay be compromised. There are many other examples of prod-ucts whose quality characteristics must be monitored carefullyduring the production process to avoid losses, for example, thestrength of materials. For expositional convenience, we refer tothese quality characteristics (denoted byY ) as latent variables orlatent quality characteristics, and their cumulative distributionfunction (cdf), denoted by F0(·), as latent distribution.

Control charts are designed with the purpose of monitoringa process in mind (see Montgomery 2009; Qiu 2014). Sucha chart will trigger a signal if its charting statistic crosses thecontrol limit. Its performance is often measured by the averagerun length (ARL), which is the average number of observa-tions needed for the chart to trigger a signal. There are twotypes of ARL in the literature: the in-control ARL (IC ARL) and

CONTACT Dongdong Xiang terryxdd@.comSupplementary materials for this article are available online. Please go to http://www.tandfonline.com/r/TECH.

the out-of-control ARL (OC ARL). The IC ARL (or ARL0for short) is often prespecified at a given level. The charthaving a shorter OC ARL (or ARL1 for short) offers a betterperformance in detecting changes in an IC distribution.

To our best knowledge, few papers provide a systematic dis-cussion on how to control the distribution F0(·) of latent qualitycharacteristics, which requires an appropriate test plan to col-lect information on the products as well as an efficient controlchart to monitor the potential changes in production process. Inpractice, quality engineers often set an acceptable stimulus levelx, sample products from the production line at equally spacedtimes, and test the products under the fixed stimulus level x.If x is at or above the critical stimulus level of the product, theproduct will respond (denoted by a = 1); otherwise, it will not(denoted by a = 0). Note that nomatter what the outcome is, theproduct cannot be reused. Due to the destructive nature of thetest, batch data are often unacceptable in such application. Thus,in this article, we will focus on cases where there is only onebinary observation a at each time point. When the process is incontrol, the response outcomes, denoted by {a1, a2, . . . , . . .}, areindependent and identically distributed binary variables havinga Binomial(1, p0) distribution, where p0 is the response prob-ability that the latent variable is lower than the fixed stimuluslevel x, that is, p0 = F0(x). Then, quality engineers monitor thebinary sequence and try to detect potential changes in p0 tocontrol F0(·). With respect to monitoring binary data, varioustypes of charting schemes have been developed in the litera-ture, including those by Marcucci (1985), Acosta-Mejia (1999),Reynolds and Stoumbos (1999, 2000), and Gadre and Rattihalli(2005).

© American Statistical Association and the American Society for Quality

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TECHNOMETRICS 497

Table . ARL1 values of the P-CUSUM chart in cases when the IC distribution isN(0, 1), the specified stimulus level x = −0.5, 0.25, 0, 0.25, or ., the assumed ARL0 valueequals or , and the IC distribution shifts to N(δ, 1) for δ = −1.0,−0.5, 0.25, 0.25, 0.5, or ..

δ

ARL0 = 200 ARL0 = 370x − . − . − . . . . − . − . − . . . .−. . . . . . . . . . .−. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . .

The approach described above is not efficient and has the fol-lowing limitations: (1) the ARL1 values may be extremely largeunder some shifts if the specified stimulus level x is not appropri-ately chosen. As an illustration, we compute some ARL1 valuesof the CUSUM chart based on the Pearson’s chi-square testingstatistic (denoted as P-CUSUM chart whose charting statistic isdefined in (5)) in the cases when the IC distribution is the stan-dard normal distribution N(0, 1), the specified stimulus levelx = −0.5, 0.25, 0, 0.25, or 0.5, the assumed ARL0 value equals200 or 370, and the IC distribution shifts to N(δ, 1), whereδ = −1.0,−0.5, 0.25, 0.25, 0.5 or 1.0. The results are shown inTable 1. From this table, the above conclusion can be drawn. (2)It cannot trigger a signal in cases when the latent distributionF0(x) has a shift that does not change the IC response probabil-ity p0. This occurs, for instance, when the stimulus level x is setto zero, and F0(x) changes from an N(0, 1) to a t distribution.(3) The collected data have a structure that is too simplistic toprovide sufficient information for finding the root causes of anOC signal.

These limitations call for a new approach that not onlyresponds rapidly to changes in F0(x), but also collects better pro-cess data for a quicker recovery in the event of a process failure.The sequential sensitivity tests in the statistics literature are onesuch approach. In the tests, the sampled products are evaluatedat designed stimulus levels xn’s which change over time, so thatmore information on the latent distribution can be obtained andmore effective inferences can be made. Research on sensitivitytests for binary response data has a rich history, going back to thework of Dixon andMood (1948) and the sequential approxima-tion scheme of Robbins andMonro (1951). There has also been aconsiderable volume of recent work, includingWu (1985), Chaoand Fuh (1999, 2001), Joseph (2004), Dror and Steinberg (20062008), Joseph, Tian, andWu (2007) andWu and Tian (2014). Inthis article, we introduce the up-and-down test, which is one ofthe sensitivity tests, for collecting online data. This test is effi-cient and easy to implement. We describe it in more detail inSection 2.

Although the up-and-down test provides sufficient informa-tion on the latent process distribution and is easy to implement,it does bring other challenges to our monitoring problem. Oneof these challenges is that the IC response probability F0(xn) isno longer fixed. Instead, it changes with the stimulus level xn.Thus, existing control charts for binary data cannot be applieddirectly. To overcome this challenge, we propose a control chartconsisting of two steps. In the first step, the time-varying ICresponse probability F0(xn) under stimulus level xn is estimatedfrom the collected IC data. In the second step, a charting

statistic that accumulates information from online observationsusing the Pearson chi-square test is constructed. The numericalresults show that our proposed chart performs favorably invarious cases considered. Since it does not require any priorknowledge of the latent distribution, the proposed control chartis extremely practical.

The rest of this article is organized as follows. In Section 2, theup-and-down test is described. The estimation of the responseprobabilities F0(xn) from an IC dataset and the constructionof the control chart are described in Section 3. In Section 4,numerical studies are presented. Finally, an example involvingreal manufacturing data is given to demonstrate the use of ourproposed chart.

2. The Up-and-Down Test

As we mentioned in Section 1, sequential sensitivity tests areuseful tools for collecting process data. To improve the efficiencyof inference (i.e., to overcome the third limitation described inSection 1), most of the sensitivity tests use a complicated andtime-consuming search algorithm to determine the next stimu-lus level at each time point. Such time-consuming test plans areoften unacceptable for onlinemonitoring of the production pro-cess. An efficient, fast, and easy to implement test plan is oftenpreferred in practice. In this article, we suggest one such sensi-tivity test for collecting online data.

Let us discuss the first and second limitations described inSection 1 in more detail. By examining the results in Table 1, wecan conclude that the P-CUSUM chart performs very well incases when the specified stimulus levels are consistent with theshift directions. This implies that a desirable test plan shouldbe able to track potential shifts to some extent and should nottest products at a fixed stimulus level. The second limitationreveals that we should choose the stimulus levels so that controlcharts can react to any changes in F0(x). Unfortunately, it is verydifficult to determine the stimulus levels because the potentialchanges in F0(x) are usually unknown in advance. But we canovercome this limitation by using multiple stimulus levels. Aslong as the number of stimulus levels is chosen sufficientlylarge, most changes in F0(x) can be identified. Consideringthat we have no prior knowledge about F0(x), these stimuluslevels will be chosen to be equally spaced. More specifically, wesample a product from the production line and test it under aninitial stimulus level x1. If x1 is at or above the product’s criticalstimulus level, the product will respond (denoted by a1 = 1);otherwise, it will not (denoted by a1 = 0). Then, the (n + 1)th(n ≥ 1) randomly sampled product is tested under the stimulus

498 D. XIANG, F. TSUNG, AND Z. PU

Figure . The last observations from an EED study. The symbols “x” and “o” indi-cate a response and a nonresponse from the tested EED to the corresponding stim-ulus level, respectively.

level xn+1 = xn − d if an = 1 or under xn+1 = xn + d if an = 0.Here, d is a positive real number. This is the so-called up-and-down test (Dixon and Mood 1948). Figure 1 presents the last80 observations from an EED study carried out by the Institu-tion of Initiators and Pyrotechnics in China. For confidentialityreasons, the raw stimulus levels have been linearly transformed.It should be abundantly clear from Figure 1, where the name“the up-and-down test” comes from.

It is easy to check that the chosen stimulus levels will increaseor decrease if the process suffers a positive or negative meanshift, and that the stimulus levels further away from the medianare more likely to be selected due to an increasing shift in vari-ance. That is, the up-and-down test indeed has the ability totrack the shift directions. Therefore, we can expect the up-and-down test to perform well in process monitoring, as will beshown in Section 4. As one of the existing sequential sensitivitytests, the up-and-down test exhibits an inference efficiency levelof 70%ormore compared to the bestD- orA-optimal design andthis is achieved without any prior knowledge of related param-eters (Chao and Fuh 1999, 2001). These are the major reasonswhywe suggest using the up-and-down test for collecting onlinedata. Besides the up-and-down test has two other merits. First,in this test, the sampled products are evaluated under almosta finite set of stimulus levels. This ensures enough products aretested under these stimulus levels so that their IC response prob-abilities can be accurately estimated. Second, the stimulus levelscan be easily determined, unlike in the sensitivity tests based oncomplicated search algorithms.

3. The Proposed Control Chart

Statistical process control usually involves two phases (Qiuand Li 2011). In phase I, a set of process data is collected andanalyzed in a retrospective study to examine if any irregularpattern has occurred. Upon repeated investigations and adjust-ments, a clean set of data (i.e., IC data) is obtained. The ICdata are mainly used to estimate the IC model. In phase II, themajor goal is to detect changes in the IC distribution online.In this article, we focus on phase II process monitoring witha latent quality characteristic. Like some of the existing works(e.g., Chakraborti, van der Laan, and Bakir 2001; Qiu andLi 2011; Zou and Tsung 2011), we do not assume the latentIC distribution to be known. Instead, we assume that an ICdataset has been collected sequentially using the up-and-downtest, denoted by {(x∗1, a∗1 ), (x∗2, a∗2 ), . . . , (x∗n, a∗n), . . . , (x∗N, a∗N )}where x∗1 is an initial stimulus level satisfying 0 < F0(x∗1 ) < 1,

x∗n denotes the stimulus level at the nth time point, a∗n is thecorresponding response under x∗n, N is the IC sample size(N ≥ 2), and |x∗n − x∗n−1| is the step length d > 0.

In the IC dataset {(x∗1, x∗2, . . . , x∗n, . . . , x∗N )}, some of thestimulus levels are equal. Let s be the number of different stimu-lus levels whose IC response probabilities satisfy 0 < F0(x∗i ) < 1for i = 1, . . . ,N, and denote the corresponding stimulus lev-els in increasing order as y1 < y2 < · · · < ys. Then, s = (ys −y1)/d + 1. Let us present it with an example. In Figure 1 weassume that the IC response probabilities satisfy F0(4) = 1 andthat the probabilities under the remaining stimulus levels belongto (0, 1). Therefore, among the 80 stimulus levels, there are6 ones under which the IC response probabilities satisfy 0 <F0(x∗i ) < 1 in this up-and-down test. Then, the correspondingstimulus levels y1 < y2 < · · · < ys in increasing order are −2,−1, . . . , and 3, respectively. Also, we can check that in this cases = (3 − (−2))/1 + 1 = 6. Moreover, we denote pi = F0(yi) asthe IC response probability under the stimulus level yi. Then, theIC distribution of response ai is Binomial(1, pi) for the stimu-lus level yi and 0 < pi < 1. In practice, however, the IC responseprobabilities are often unknown and have to be estimated fromthe IC data, for example, using the maximum likelihood esti-mation (MLE). The resulting estimators are still denoted as pi,i = 1, . . . , s, throughout this article unless stated otherwise.

Next, we develop a control chart for detecting changes in theprocess. Let (x1, a1), (x2, a2), . . . , (xn, an), . . . be the phase IIsequence, collected from the production line using the up-and-down test with x1 = x∗1 and d = |x∗n − x∗n−1|. To monitor thechanges in F0(·), we consider using the Pearson’s chi-square teststatistic for the reason that this statistic can result in an omnibuscontrol chart. At the nth time point, the Pearson’s chi-square teststatistic is defined as

X2(n) = (an − F0(xn))2

F0(xn)+ (1 − an − (1 − F0(xn)))

2

1 − F0(xn)= (an − F0(xn))

2

F0(xn)(1 − F0(xn)) . (1)

Note that in the up-and-down test, a phase II stimulus levelxn that does not belong to {y1, y2, . . . , ys} may sometimes beapplied to a product. Since its response probability cannot beestimated from the IC data in such cases or it is equal to 0or 1, the statistic defined by (1) at such stimulus level is notwell defined. In such situations, we slightly modify the up-and-down test and set the stimulus level to the previous value, thatis, xn = xn−1.

From (1), it can be checked that the distribution of X2(n)across different time points is not fixed. This may invalidate thecharts for fixed binomial/multinomial IC distributions, espe-cially in cases when F0(xn) is close to 0 or 1. Even if its controllimit is adjusted properly, the ARL1 values would still be severelycompromised. To overcome this issue, we suggest taking thefollowing steps. First, the entire process is divided into s sub-processes according to the stimulus levels {y1, y2, . . . , ys} and acharting statistic is then constructed for each subprocess witha restarting mechanism. Second, the charting statistics are cal-ibrated by dividing them by their corresponding control limitsand their summation is used as the final charting statistic.Math-ematically, these steps—when incorporated with the efficient

TECHNOMETRICS 499

control chart proposed by Qiu (2008) and Qiu and Li (2011)—can be described as follows. For i = 1, . . . , s, let Gi,0 = Pi,0 = 0be 2 × 1 vectors, and⎧⎪⎪⎪⎨⎪⎪⎪⎩

Gi,n = Gi,n−1,Pi,n = Pi,n−1, if xn �= yi;Gi,n = Pi,n = 0, if Ci,n ≤ kiandxn = yi;Gi,n = (Gi,n−1 + g(n))(Ci,n − ki)/Ci,n, if Ci,n > kiandxn = yi;Pi,n = (Pi,n−1 + pi)(Ci,n − ki)/Ci,n, if Ci,n > kiandxn = yi;

(2)

where

Ci,n = {(Gi,n−1 − Pi,n−1) + (g(n) − pi)}′(diag(Pi,n−1 + pi))−1×{(Gi,n−1 − Pi,n−1) + (g(n) − pi)}, (3)

pi = (pi, 1 − pi)′, g(n) = (an, 1 − an)′, ki’s are the referencevalues of charting schemes for subprocesses and diag(A)denotes a diagonalmatrix whose diagonal elements are the sameas those of A. Then, a shift in the production process is signaledif

Un =s∑

i=1ui,n/h(pi) > L (4)

where L > 0 is a control limit,

ui,n = (Gi,n − Pi,n)′(diag(Pi,n))−1(Gi,n − Pi,n) (5)is the charting statistic for the ith subprocess and h(pi) is itscontrol limit. When ki = 0, for all i = 1, . . . , s, ui,n in (5) is thePearson’s chi-square test statistic (1). This statistic is well-knownto have an approximate χ2 distribution with degrees of free-dom 1 when n is sufficiently large. This implies that the chartingscheme should work well when the process is in control. On theother hand, if the process experiences a shift, the values of theui,n’s will tend to be large. A large value ofUn defined by (4) indi-cates a possible violation. It should be pointed out that, at eachtime point, only one of the ui,n’s will be accumulated and dividedby its control limit. Hence, the computation speed of our pro-posed charting statistic is quite fast. Roughly, the computationtime on average for each simulation run is about 0.01 sec whenARL0 is specified to be 200.

In a similar way, we can construct two one-sided CUSUMcharts (see Qiu 2014, p. 156) to detect upward and downwardshifts, respectively. Then, we combine them together to monitorthe process. It alarmswhen any one of the charts triggers a signal.Due to some optimal properties of the CUSUM chart, one mayexpect that the combined CUSUM chart will inherit the merits.Unfortunately, our experience based on extensive simulationstells us that the combinedCUSUMchart would not performbet-ter than our proposed chart. Instead, they performs comparablywell. This is not surprising to us. Indeed, the combined CUSUMchart with given reference values is optimal for certain shift size.However, it is very difficult for us to specify the optimal referencevalues for the combinedCUSUMchart as we often have no priorknowledge about shift size even the shift type. Besides we shouldset 2s design parameters for the combined CUSUM chart beforewe can use the chart formonitoring. But, for our proposed chart,the number of design parameters is halved. Thus, we recom-mend our proposed chart for use in practice. In what follows, wegive several practical guidelines on the design parameters below.

On calculation of the control limit L: Given ki and h(pi),i = 1, . . . , s, the control limit L can be computed by simulationsas follows. First, choose an initial value for the control limit andgenerate observations using the modified up-and-down test.Note that the response outcomes can be generated from the ICbinomial distribution Binomial(1, pi) when the current stim-ulus level is yi. Second, collect data sequentially until the chartsignals and record the run length. Third, repeat the above stepsfor a large number of runs (e.g., 10,000 runs) and average therun lengths. If the real ARL0 is smaller than the nominal ARL0,then a larger value is required for the control limit. Otherwise,a smaller one should be chosen. Through a bisection search,the control limit can be determined. Computation of L doesnot require any information about the latent distribution of theprocess. It only depends on the ARL0, pi, ki and h(pi), for i =1, . . . , s. In this sense, the proposed chart is distribution-free.

Because our proposed charting statistic is a discrete randomvariable, some nominal ARL0 values cannot be reached. Thisis common when monitoring discrete data (see Hawkins andOlwell 1998; Qiu and Li 2011;Mukherjee andChakraborti 2012;Qiu 2014). In practice, it is appropriate to choose a nominalARL0 that can be reached. In this article, however, we follow thesuggestion of Qiu and Li (2011) so that any given nominal ARL0can be reached. Let g(n) = (an + r1, 1 − an + r2) where r1 andr2 are generated from a normal distribution with mean 0 andvariance σ 2. Qiu and Li (2011) pointed out that ARL1 is littleaffected by this minor modification as long as a small σ is cho-sen. Thus, in the later numerical study, we will set σ = 0.01 forthe investigation of our proposed chart.

On choice of reference value ki: As mentioned before, ki isthe reference value for the charting scheme of the ith subpro-cess when the IC distribution is Binomial(1, pi). In themultino-mial case, Qiu and Li (2011) investigated the OC performancein detail. Roughly speaking, a larger ki is sensitive to large shifts,while a smaller ki reacts quickly to small shifts. In the binomialcase, the conclusion is similar. Thus, we forgo a demonstrationof the OC performance to save space.

Besides the OC performance, we investigate the IC perfor-mance as well. As recognized in the literature (e.g., Zhou et al.2012; Shen et al. 2013), it is insufficient to summarize the IC per-formance only by the ARL0. As an alternative, we will evaluatethe IC performance in this article using theARL0, the 10th, 50th,and 90th percentiles of the run length distribution and the ICstandard deviation of the run length (denoted by SDRL0) rec-ommended by Zhou et al. (2012).

We compute the real ARL0 and SDRL0 through 500,000 sim-ulation runs when the response probability pi = 0.1, 0.3, or 0.5,the nominal ARL0 = 200 or 370, and the reference value ki isbetween 0.005 and 0.15 in steps of 0.005. Since the results forpi > 0.5 are the same as those for 1 − pi, we only consider thecase of pi < 0.5 here. To illustrate the influence of ki on the ICperformance, we first plot the ratio SDRL0/ARL0 in Figure 2.From this figure, it can be seen that the ratio SDRL0/ARL0decreases quickly as pi tends to 0.5 with ki fixed. When pi isfixed, the ratio decreases as ki increases, and tends to a con-stant. That is, a small ki results in a large SDRL0. Usually, a con-trol chart with a large SDRL0 will give many false alarms in itsearly runs, which renders it useless in practice. In this sense, alarger ki should be chosen for a given pi and ARL0. However,

500 D. XIANG, F. TSUNG, AND Z. PU

Figure . The ratio SDRL0/ARL0 in cases when the response probability pi = 0.1, ., or ., the nominal ARL0 = 200 or , and the reference value ki is between .and . in steps of ..

a larger ki will affect the OC performance under small shifts.To strike a balance, we suggest choosing a ki so that the rela-tionship SDRL0/ARL0 ≈ 1 holds as in many existing chartingschemes in the literature (e.g., Han and Tsung 2006; Zou andTsung 2011; Shen et al. 2013; Qiu 2014). The IC performancefor pi = 0.1, 0.3, and 0.5 is shown in Table S.1 in a Supplemen-tary File. From that table, we can see that the summary statistics(i.e., ARL0, SDRL0, the 10th, 50th, and 90th percentiles of therun length distribution) are very close to the characteristics ofa geometric distribution with mean ARL0, as shown in the rowlabeled standard values.

Finally, to facilitate the implementation of our proposedchart, we calculate some h(pi)’s and ki’s according to the sugges-tions described in the subsection “On choice of reference valueki.” These results will be useful in our simulations and applica-tion, and are tabulated in the supplementary file (see Tables S.2and S.3).

On choice of stimulus levels: It is easy to check that the cho-sen stimulus levels in the up-and-down test rely heavily on theinitial stimulus level x∗1 and the step length d (recommendationsgiven in Section 4.1). When both of them are fixed, the stimuluslevels are completely determined. However, considering the factthat pi is estimated from the IC data, we do not suggest usingthe stimulus levels that have extremely large or extremely smallresponse probabilities. The reason is that in the up-and-downtest, few products are tested under these stimulus levels. Thisindicates that these pi’s cannot be accurately estimated undera fixed N (e.g., N = 2000). Based on extensive simulations, werecommend using the stimulus levels, whose response probabil-ities lie between 0.065 and 0.935 in practice.

4. Numerical Studies

In this section, we present some simulation results regarding thenumerical performance of the proposed chart. First, we investi-gate the effect of the initial stimulus level x∗1 and the step lengthd on the proposed chart. Then, we study the influence of the ICsample sizeN on the proposed chart. Finally, we briefly describesome competing test plans or charts, and compare them withthe proposed chart under various situations. Throughout thissection, all ARL values are obtained from 10,000 replications

unless indicated otherwise. In addition, we focus on the steady-state ARL values of the charts (Hawkins and Olwell 1998), andassume that the shift occurs at τ = 40. When computing theARL, any simulation runs in which a signal is raised before the(τ + 1)th observation are discarded. The IC distribution is cho-sen to be the standardized version with mean 0 and variance 1from one of the following three distributions,N(0, 1), t(3), andLN(0, 1), representing the standard normal distribution (nor-mal), the Student’s distribution with degree 3 (heavy tailed), andthe log normal distribution (skewed), respectively.

4.1. Investigation of the Initial Stimulus Level x∗1 and StepLength d

As recognized in the literature (Choi 1990; Voelkel 1999), theanalysis of data collected in the up-and-down test (e.g., to esti-mate the 50th percentile) relies heavily on the initial stimuluslevel x∗1 and the step length d. A well-chosen x∗1 and d will usu-ally result in a satisfactory estimation. Similarly, our proposedchart depend on x∗1 as well as d. Through extensive simulations,we find that when x∗1 is chosen as the population mean of theIC distribution and d is fixed, our proposed chart gives the bestoverall performance. But its superiority is not significant (notreported here) if d is chosen to be small (i.e., d < 0.5). The rea-son is that in the up-and-down test, all of the stimulus levelsuniformly take values in the support set of the latent IC dis-tribution. When d is not too large, the stimulus levels underdifferent x∗1 are very close. Therefore, we can take any valuein the support set {x : 0 < F0(x) < 1}, as well as the estimatedpopulation mean of the IC distribution, as the initial stimuluslevel.

Next, we investigate the effect of the step length d under afixed x∗1 . To do so, we consider the cases when the nominalARL0 = 200, the initial stimulus level x∗1 = 0 (i.e., mean of thelatent IC distribution) and the step length d is 0.25, 0.5, or 1 rep-resenting a small, medium, or large step length. Thus, the stim-ulus levels here are integer multiples of d. Moreover, we assumethe response probabilities under the stimulus levels are knownexactly instead of estimated. First, we discard the stimulus lev-els whose response probabilities are smaller than 0.065 or largerthan 0.935. In some situations (e.g., in our log normal case when

TECHNOMETRICS 501

Figure . ARL values of the proposed chart in cases when d = 0.25, ., or , the mean (plots (a)–(c)) or standard deviation (plots (d)–(f )) is shifted and the IC continuousdistribution is N(0, 1) (the first row), t(3) (the second row) or logN(0, 1) (the third row).

d = 1), all of the remaining response probabilities may be eitherlarger or smaller than 0.5. Then, the proposed chart may haveextremely large ARL1 values under some shift sizes, as in tra-ditional approaches. This implies the chosen d is too large tobe used. Hence, in the log normal case, we only consider thestep lengths d = 0.25 and 0.5. Second, we compute the controllimits for different d’s based on the response probabilities of theremaining stimulus levels, as described in Section 2. Third, wecompute the steady-state ARL1 values for the following cases:(1) when the size δ of the shift in mean changes from −1.0 to1.5 in steps of 0.25, and (2) when the size δ of the shift in stan-dard deviation changes from 1 to 4 in steps of 0.5, as shown in

Figure 3. From the figure, we can see that (1) a small d is sen-sitive to small mean shifts in all considered cases, and a larged reacts quickly to large mean shifts; (2) the proposed chartperforms best in detecting increasing shifts in variance when dis large. But such improvement increase slightly as d > 0.5. Inpractice, we usually do not know in advance what the shift sizeand the shift type are. Therefore, we recommend using amoder-ate d = 0.5. For some processes, their shift types may be regard-less of the mean and variance, and have their special patterns.In this scenario, we can search for the optimal step length d’s inthe sense that the proposed chart under it has the best overallperformance.

502 D. XIANG, F. TSUNG, AND Z. PU

Table . ARL0 values and their standard errors (in parentheses) of theproposed chart in caseswhen the standardized IC distribution isN(0, 1), t(3),or logN(0, 1),N = 500,, , , , , or ,, d = 0.25, ., or and the nominal ARL0 = 200.

d = 0.25 d = 0.5 d = 1.0N N(0, 1) t(3) logN(0, 1) N(0, 1) t(3) logN(0, 1) N(0, 1) t(3)

(.) (.) (.) (.) (.) () (.) (.), (.) (.) (.) (.) (.) (.) (.) (.), (.) (.) (.) (.) (.) (.) (.) (.), (.) (.) (.) (.) (.) (.) (.) (.), (.) (.) (.) (.) (.) (.) (.) (.), (.) (.) (.) (.) (.) (.) (.) (.), (.) (.) (.) (.) (.) (.) (.) (.)

4.2. Investigation of the IC Sample Size N

As discussed before, an IC dataset is required to estimate theresponse probabilities. Obviously, the larger the IC sample size,the better the performance of the proposed chart. But howlarge should the IC sample size be? To answer this question,we conduct a numerical comparison and set N = 500, 1000,2000, 3000, 5000, 7000, or 10,000 and ARL0 = 200. In thiscomparison study, other settings are the same as in the abovenumerical example. It consists of three steps. First, for each dand IC distribution, we search for the control limit by simula-tions based on the stimulus levels whose response probabilitiesare known and lie between 0.065 and 0.935. Second, we generateN observations from a latent IC distribution and estimate theresponse probabilities of all stimulus levels from the generateddataset. Then, we discard the stimulus levels whose responseprobabilities are smaller than 0.065 or larger than 0.935, andcompute the charting statistics based these estimated responseprobabilities. Finally, we compute the charting statistics basedon these estimated response probabilities, instead of the onesused in Step 1, and obtain the IC run length value of the pro-posed control chart using the control limit found in Step 1. Werepeat Steps 2 and 3, including the generation of IC data andthe computation of actual run length values 10,000 times. Theaveraged actual ARL0 values and the corresponding standarderrors are presented in Table 2. From Table 2, it can be seenthat: (i) the difference between the actual ARL0 value and thenominal one is within 10% when N ≥ 2000. When N becomeslarge, the actual ARL0 approaches to the nominal one. (ii)The larger d is, the smaller the IC sample size required, and(iii) as IC the sample size increases, the standard errors in theparentheses tend to be smaller. These conclusions are consistentwith our expectations. In practice, N ≥ 5000 is enough whenARL0 = 200. For other nominal values (e.g., ARL0 < 1000),N ≥ 5000 is also sufficient. Otherwise, one would need todetermine N by conducting a similar comparison study.

4.3. Performance Studies

In this subsection, we investigate the performance of our pro-posed control chart in three parts. In the first part, various casesare illustrated to show the IC performance of our proposedchart. The advantages of the up-and-down test in OC perfor-mance is investigated in the second part, where three test plansare compared under several different types of changes. Finally,we compare our proposed chart with alternative methods wherethe up-and-down test is chosen as the test plan.

... The IC Performance of the Proposed ChartTo illustrate the IC performance of our proposed control chart,we consider the following cases in which the nominal ARL0 =200, initial stimulus level x∗1 = 0, the step length d = 0.25, 0.5or 1 (d = 0.25 or 0.5 in the log normal case), and the IC samplesize N is fixed at 5000. We then compute the actual ARL0 valuesand standard errors, the 10th, 50th, and 90th percentiles (labeledQ(0.1), Q(0.5), and Q(0.9), respectively) of the run length dis-tribution, which are shown in Table 3, under three standardizeddistributions. From the table, it can be seen that the actual valuesof summary statistics are very close to the nominal ones shownin the last row, which is as expected. Results for the case whenthe nominal ARL0 = 370 are provided in the supplementary file(see Table S.4) and the conclusion is very similar.

... The Advantages of the Up-and-Down TestIn this subsection, we show the advantages of the up-and-downtest in the OC performance of our proposed chart by comparingit with two other test plans. The first plan is the one describedin the Introduction. Because there is only one stimulus levelinvolved, we call it the SINGLE plan hereafter. It is easy tocheck that our proposed can be applied to this test plan directly.For simplicity, we call the chart based on the SINGLE plan theSINGLE chart. The second plan is quite similar to the up-and-down test, in which the next stimulus level is chosen randomlyfrom the stimulus levels set {y1, y2, . . . , ys}. That is, the nextstimulus level is completely independent of the current stim-ulus level. For expositional convenience, we refer to it as theRANDOM plan. The resulting chart is called the RANDOMchart for short.

In this study, we select three different stimulus levels (see thesecond row of Tables 4–6) for the SINGLE plan and set d = 0.5

Table . IC performance of the proposed chart summarized by ARL0 , SDRL0 , andthe th (labeled Q(0.1)), th (labeled Q(0.5)), and th (labeled Q(0.9)) per-centiles in cases when d = 0.25, 0.5, or , the nominal ARL0 = 200, and the stan-dardized IC distribution N(0, 1), t(3), or logN(0, 1). The standard values of thesesummary statistics are shown in the last row.

Distribution d ARL0 SDRL0 Q(0.1) Q(0.5) Q(0.9)

N(0, 1) . . .

t(3) . . .

logN(0, 1) . .

Standard values

TECHNOMETRICS 503

Table . ARL1 values and their standard errors (in parentheses) of the SINGLEplan, the RANDOMplan and theup-and-downplan (labeledUPDOWN) in caseswhend = 0.5,the nominal ARL0 = 200 and the standardized IC distribution N(0, 1).

SINGLE

Shift types Shift size x0 = −0.5 x0 = 0 x0 = 0.5 RANDOM UPDOWN(δ1 ,) (−,) .(.) .(.) .(.) .(.) .(.)

(−.,) .(.) .(.) .(.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)

(,δ2) (,.) .(.) (.) .(.) .(.) .(.)(,) .(.) (.) .(.) .(.) .(.)(,.) .(.) (.) .(.) .(.) .(.)(,) .(.) (.) .(.) .(.) .(.)(,) .(.) (.) .(.) .(.) .(.)

(δ3, δ4) (−,.) .(.) .(.) .(.) .(.) .(.)(−.,) .(.) .(.) (.) .(.) .(.)(.,.) (.) .(.) .(.) .(.) .(.)(,) (.) .(.) .(.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)

(δ5 ,) (.,) .(.) (.) (.) (.) (.)(,) .(.) (.) (.) .(.) .(.)(.,) .(.) (.) (.) .(.) .(.)(,) .(.) (.) (.) .(.) .(.)(,) .(.) (.) (.) .(.) .(.)

RMI . . . . .

for the other two plans. Then, we keep ARL0 = 200 and com-pute the ARL1 values under the following types of shifts:(a) Shifts in mean from 0 to δ1; denoted as (δ1,1) for short.(b) Shifts in standard deviation from 0 to δ2; denoted as (0,δ2)

for short.(c) Shifts in both mean and standard deviation from 0 to δ3

and δ4, respectively; denoted as (δ3, δ4) for short.(d) Shifts in the shape of the density function, which changes

from Y to its counterpart of δ5YIY≤0 +YIY>0, where IY≤0is the indicator function, equaling 1 when Y ≤ 0, and 0otherwise; denoted as (δ5, 0).

The shift sizes δi for i = 1, . . . , 5 are shown in the secondcolumn of Tables 4–6. To evaluate the overall performance of thefive control charts, alongwithARL1 we also compute the relative

mean index (RMI) values, shown in the last row of Tables 4–6.As suggested by Han and Tsung (2006), the RMI of a controlchart is defined as

RMI = 1T

T∑i=1

ARLδi − MARLδiMARLδi

,

where T is the number of shifts, ARLδi is the ARL1 value of thegiven control chart under shift δi (labeled in the second col-umn of Tables 4–6), and MARLδi is the smallest ARL1 amongall ARL1 values of the five charts under shift δi. Accordingto this index, a chart with a smaller RMI value is consideredto have better overall performance. From the results shown inTables 4– 6, we draw three conclusions. First, the SINGLE plan

Table . ARL1 values and their standard errors (in parentheses) of the SINGLEplan, theRANDOMplan and theup-and-downplan (labeledUPDOWN ) in caseswhend = 0.5,the nominal ARL0 = 200 and the standardized IC distribution t(3).

SINGLE

Shift types Shift size x0 = −0.5 x0 = 0 x0 = 0.5 RANDOM UPDOWN(δ1 ,) (−,) .(.) .(.) .(.) .(.) .(.)

(−.,) .(.) .(.) (.) .(.) .(.)(.,) (.) .(.) .(.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)

(,δ2) (,.) .(.) (.) .(.) .(.) .(.)(,) .(.) (.) .(.) .(.) .(.)(,.) .(.) (.) .(.) .(.) .(.)(,) .(.) (.) .(.) .(.) .(.)(,) .(.) (.) .(.) .(.) .(.)

(δ3, δ4) (−,.) .(.) .(.) (.) .(.) .(.)(−.,) .(.) .(.) (.) .(.) .(.)(.,.) (.) .(.) .(.) .(.) .(.)(,) (.) .(.) .(.) .(.) .(.)(.,) (.) .(.) .(.) .(.) .(.)

(δ5 ,) (.,) .(.) (.) (.) (.) (.)(,) .(.) (.) (.) .(.) .(.)(.,) .(.) (.) (.) .(.) .(.)(,) .(.) (.) (.) .(.) .(.)(,) .(.) (.) (.) .(.) .(.)

RMI . . . . .

504 D. XIANG, F. TSUNG, AND Z. PU

Table . ARL1 values and their standard errors (in parentheses) of the SINGLEplan, the RANDOMplan and theup-and-downplan (labeledUPDOWN) in caseswhend = 0.5,the nominal ARL0 = 200 and the standardized IC distribution logN(0, 1).

SINGLE

Shift types Shift size x0 = −0.5 x0 = 0 x0 = 0.5 RANDOM UPDOWN(δ1 ,) (−,) .(.) .(.) (.) .(.) .(.)

(−.,) .(.) .(.) (.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)

(,δ2) (,.) .(.) (.) (.) .(.) .(.)(,) .(.) (.) .(.) .(.) .(.)(,.) .(.) (.) .(.) .(.) .(.)(,) .(.) (.) .(.) .(.) .(.)(,) .(.) (.) .(.) .(.) .(.)

(δ3, δ4) (−,.) .(.) .(.) (.) .(.) .(.)(−.,) .(.) (.) (.) .(.) .(.)(.,.) .(.) .(.) .(.) .(.) .(.)(,) .(.) .(.) .(.) .(.) .(.)(.,) .(.) .(.) .(.) .(.) .(.)

(δ5 ,) (.,) .(.) (.) (.) .(.) .(.)(,) .(.) (.) (.) .(.) .(.)(.,) .(.) (.) (.) .(.) .(.)(,) .(.) (.) (.) .(.) .(.)(,) .(.) (.) (.) .(.) .(.)

RMI . . . . .

relies heavily on the stimulus level. Even if the stimulus level ischosen appropriately, the SINGLE chart can only perform wellunder its near shifts in mean. But in some situations (e.g., inthe log normal case), the ARL1 values of the SINGLE chart areextremely large. Second, with multiple stimulus levels, our pro-posed control chart outperforms the RANDOM chart in mostcases, especially in the cases when the mean of the IC distri-bution is shifted. This is not surprising because the stimuluslevels of the RANDOM test plan are chosen independent of

Table . ARL1 values and their standard errors (in parentheses) of the PROPOSEDchart, the PROPOSEDM chart, the EWMA-CS chart, and the CUSUM-CS chart whend = 0.5, the nominal ARL0 = 200 and the latent IC distribution N(0, 1), t(3) orlogN(0, 1).

Distribution δ1 PROPOSED PROPOSEDM EWMA-CS CUSUM-CS

N(0, 1) − . .(.) .(.) .(.) .(.)− . .(.) .(.) .(.) .(.)− . .(.) .(.) .(.) .(.)− . .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.)

RMI . . . .t(3) − . .(.) .(.) .(.) .(.)

− . .(.) .(.) .(.) .(.)− . .(.) .(.) .(.) .(.)− . .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.. .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.)

RMI . . . .logN(0, 1) − . .(.) .(.) (.) .(.)

− . .(.) .(.) (.) .(.)− . .(.) .(.) (.) .(.)− . .(.) .(.) (.) (.). .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.). .(.) .(.) .(.) .(.)

RMI . . . .

the shift directions, while those of the up-and-down test planare designed to track the directions of the shifts. Third, theRMI values suggest that in all cases considered, the proposedchart is the best performer among the five charts. In summary,the up-and-down test can significantly improve the monitoringefficiency.

... Comparison of the OC Performance Under theUp-and-Down Test

In this subsection, we show the efficiency of the proposed chartby comparing it with alternative methods based on the up-and-down test. For conciseness, we only aim to detect shifts in themean in this study. Results on other types of shifts are omittedbut can be computed similarly.

Recalling the construction of the proposed chart, we firstdivide the entire process into s subprocesses. Then, we sumthe charting statistics for the subprocesses with calibrations toobtain the final charting statistic. In this subsection, we denotethe proposed chart by PROPOSED. Besides the proposed chart,an alternative procedure is to maximize these charting statisticsdefined by (2)–(5). Specifically, the charting statistic in this caseis max1≤i≤s ui,n/h(pi). We refer to the maximized version as thePROPOSEDM chart and compare it with the proposed chart inthis subsection.

The collected data can also be regarded as censored data,in which case control charts for monitoring censored datacan be used for our purpose. Thus, we adapt the exponentialweight moving average (EWMA) chart for the censored data,proposed by Zhang and Chen (2004), and the CUSUM chartfor the censored data, proposed by Dickinson et al. (2014), tosolve our problem. For expositional convenience, we refer to theextended charts for censored data as the EWMA-CS chart andthe CUSUM-CS chart. The two charts rely on the assumptionthat the IC and/or OC distributions are known. In this study, weassume the latent IC distribution to be normal for both charts.

TECHNOMETRICS 505

The main idea of the EWMA-CS chart is to replace the cen-sored observation by a conditional expectation value (CEV).Precisely, the EWMA-CS chart can be constructed by replacingthe nth stimulus level xn with

CEV(xn) = (1 − an)E0(Z|Z ≥ xn) + anE0(Z|Z ≤ xn)= (1 − an)

∫z≥xn

zφ((z − μ0)/σ0)1 − �((xn − μ0)/σ0)dz

+ an∫z≤xn

zφ((z − μ0)/σ0)�((xn − μ0)/σ0)dz,

whereZ is the latent variable (i.e., critical stimulus level),E0(Z|·)is the conditional expected value under the latent IC distributionN(μ0, σ 20 ), and φ(·) is the probability density function (pdf) ofthe standard normal distribution. The charting statistic of theEWMA-CS chart for detecting mean shift is defined as

vn = (1 − λ)vn−1 + λCEV(xn),where v0 = 0, λ ∈ (0, 1] is a weighting parameter. A mean shiftis signaled if

|vn| ≥ LEwhere LE > 0 is the control limit of the EWMA-CS chart.

The CUSUM-CS chart combines a lower-side chart and anupper-side chart. Its charting statistics for the mean shifts aredefined as

Qn = max{Q+n ,Q−n }, (6)where Q+n = max

(0,Q+n−1 + ln(μ1, μ0)

), Q−n =

max(0,Q−n−1 + ln(−μ1, μ0)

)with Q+0 = Q−0 = 0, μ0 and

μ1 are respectively the IC and OC mean, and ln(μ,μ0) followsthe log-likelihood function

ln(μ,μ0) = an log �((xn − μ)/σ0)�(xn − μ0)/σ0)

+(1 − an) log (1 − �((xn − μ)/σ0)(1 − �((xn − μ0)/σ0)) .

A mean shift is signaled if

Qn > LC,

where LC is a control limit chosen to achieve a desired ARL0.In what follows, we compare the OC performance under the

mean shift when the shift size δ1 ranges from−1.0 to 1.0 in stepsof 0.25, d = 0.5, and ARL0 = 200. For a fair comparison, thecontrol limits of the EWMA-CS chart and the CUSUM-CS chartare calibrated so that the actual ARL0 values can reach the nom-inal ones. In the CUSUM-CS chart, the OCmeanμ1 is assumedto be 1, while the weighting parameter λ is set to the optimal val-ues for detecting the shift δ1 = 1. All of the ARL1 values for theabove four charts and their RMI values are presented in Table 7.

FromTable 7, we canmake three conclusions. First, when the ICdistribution is symmetric, the proposed chart performs slightlyworse than the CUSUM-CS chart and the EWMA-CS chart atδ1 = 1. This is not surprising since the design parameters ofthe three charts are optimal in the sense that the ARL1 valueat δ1 = 1 is minimized. But the proposed chart outperformsthe other three charts for small shifts (cf. δ1 ≤ 0.5). Second,under the skewed log normal distribution, when themean of theIC distribution shifts downward, the proportion of CEV(xn)’sclose to zero becomes higher. This means the absolute valuesof the charting statistics vn under the current downward shiftare smaller than before. This explains why the EWMA-CS chartdistorts the ARL1 values when detecting downward shifts. For asimilar reason, the CUSUM-CS chart distorts the ARL1 valuesunder a downward shift δ = −0.25. But our proposed methodis very robust and still performs very well in the skewed casecompared with the distribution-specified control chart. Third,by the RMI, our proposed method performs best overall whenthe latent IC distribution is nonnormal.

In summary, our proposed chart is very sensitive to smallshifts and have a good overall performance even when the formof the latent distribution is incorrectly specified. Results forthe case when the nominal ARL0 = 370 are provided in thesupplementary file (see Table S.4) and the conclusion is verysimilar.

5. An Application

In this section, we apply our proposed chart to the dataset fromthe EED study mentioned before. This dataset was collected inan up-and-down test. It consists of two subsets of observations.The first subset of 11,452 observations are historical data col-lected from hundreds of batches of historical EEDs . The secondsubset contains 42 observations sampled from another type ofthe EED process. Although the EED study was not conductedfor monitoring purpose, it can be used for illustrating the use ofour proposed chart.

We view the historical dataset as the IC dataset and the sec-ond dataset as theOCdataset.We summarize the IC dataset intoa matrix. The first row of the matrix shows the electric voltagesused in the EED study. The number of EEDs tested under eachelectric voltage is shown in the second row. In the third row, werecord the responses from the tested EEDs. The last row showsthe MLE of the response probabilities. For expositional conve-nience, we denote the number of EEDs tested at and the num-ber of EEDs responding to the ith stimulus level by Ni and ni,respectively. The Pearson’s chi-square test (cf. Hogg, McKean,and Craig 2005) for normality gives a p-value lower than 10−5when it is applied to the IC dataset. This means that the IC latentdistribution is significantly different from a normal distribution.The 42 OC observations taken at the 39th time point are pre-sented in Figure 1.

⎛⎜⎜⎝

−6 −5 −4 −3 −2 −1 0 1 2 3 4 51 10 75 398 1211 2484 3196 2454 1164 370 79 100 1 9 66 332 879 1605 1591 863 301 69 10

0.000 0.100 0.120 0.166 0.274 0.354 0.502 0.648 0.741 0.814 0.873 1.000

⎞⎟⎟⎠

506 D. XIANG, F. TSUNG, AND Z. PU

Figure . Charting statistics of the proposed chart (plot (a)), the PROPOSEDM chart (plot (b)), the EWMA-CS chart (plot (c)), and the CUSUM-CS chart (plot (d)) when theyare applied to the dataset from the EED study with ARL0 = 200.

Next, we apply our proposed charting scheme to the80 observations in Figure 1. We first eliminate the stimuluslevels −6 and 5 because their response probabilities do not liebetween 0.0635 and 0.935, and keep the remaining stimuluslevels and their response probabilities. We then compute thecharting statistics un/L sequentially for the 80 observationsand present them in Figure 4(a). We also compute the chartingstatistic maxsi=1 ui,n/LM and present that in Figure 4(b). Here,LM is the control limit of the PROPOSEDM chart. In addition,the EWMA-CS chart, and the CUSUM-CS chart are applied tothe 80 observations as well. For these charts, the IC mean μ0and variance σ0 are obtained by minimizing

5∑i=−6

{ni log�

(i − μ0

σ0

)+ (Ni − ni) log�

(i − μ0

σ0

)}.

The IC mean μ0 and the IC variance σ0 are estimated tobe −0.0173 and 3.0673, respectively. The OC mean for theCUSUM-CS chart is specified as μ1 = 2, while the weightingparameter in the EWMA-CS chart is λ = 0.2. We then calculatethe quantities Qn/LE and Tn/LC based on the 80 observationsand plot them in Figures 4(c) and 4(d), respectively. All ofthe control limits of the above four charts are computed from10,000 simulation runs so that ARL0 = 200.

FromFigure 4, it can be seen that both the proposed chart andthe PROPOSEDMchart (Figures 4(a) and 4(b)) cross the controllimits at the same time point of 53. Therefore, the proposed chartgives a signal correctly in this example. The EWMA-CS chart(Figure 4(c)) and the CUSUM-CS chart (Figure 4(d)), however,fail to trigger a signal. The failuremay be related to an incorrectlyspecified assumption on the latent IC distribution.

6. Concluding Remarks

In this article, we have described a novel approach to moni-tor a process with a latent quality characteristic in cases whenthe process distribution cannot be determined beforehand. Thisapproach integrates the up-and-down test and a control chart formonitoring latent quality characteristics. We have also providedguidelines on how to choose the design parameters of our pro-posed chart. The approach not only detects exceptions in a pro-cess mean efficiently, but also provides a comprehensive datasetfor deducing the root causes of exceptions.

Despite these advances, a number of issues remain to beaddressed in our future research. For instance, our proposedmethod is able to detect shifts in the mean and increases orlarge decreases in variability (not reported in this article), butit is not sensitive to small decreases in variability (e.g., shiftsfrom σ0 = 1 to σ1 = 0.9) when d ≥ 0.25. This may reduce the

TECHNOMETRICS 507

efficiency in detecting mean shifts. How to strike a balance isleft for future research. Also, when successive observations arecorrelated, some of the parametric time series models might beappropriate in certain situations, although time series modelingis generally challenging when the observations are ordinal (seeApley and Tsung 2002, Han and Tsung 2009). Since the up-and-down test is a destructive one, it is desirable to develop a chartingscheme with a variable sampling interval (see Zou, Wang, andTsung 2008). Such a charting scheme would reduce the numberof products that needs to be tested.

Acknowledgments

The authors thank the Editor, the Associate Editor, and two refereesfor many constructive comments and suggestions, which have greatlyimproved the quality of the article. This work was supported by NationalScience Fund of China (No. 11501209, 11571113, 11271135, 71402133,71602155), the Postdoctoral Science Foundation of China (2015M570348),Shanghai Rising Star Program (16QA1401700), the Fundamental ResearchFunds for the Central Universities and the 111 Project (B14019), TheInternational Postdoctoral Exchange Fellowship Program (20160089),Program of Shanghai Subject Chief Scientist (14XD1401600), RGC Gen-eral Research Fund (619913) and the Project of Shanghai Universitiesto enhance the competition and innovation “collaborative innovation ofmodern statistical methods and theory.”

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Abstract1.Introduction2.The Up-and-Down Test3.The Proposed Control Chart4.Numerical Studies4.1.Investigation of the Initial Stimulus Level and Step Length 4.2.Investigation of the IC Sample Size 4.3.Performance Studies

5.An Application6.Concluding RemarksAcknowledgmentsReferences