Monitoring Two-Stage Processes with Binomial Data ...

Monitoring Two-Stage Processes with Binomial Data Using Generalized Linear Model-Based Control Charts Amirhosein Amiri*1, Arthur B. Yeh2 and Ali Asgari1 1 2

Department of Industrial Engineering, Shahed University, Tehran, Iran

Department of Applied Statistics and Operations Research, Bowling Green State University, Bowling Green, OH 43403, USA

Abstract In this study, we propose a control chart for monitoring two-stage processes whose quality characteristic to be monitored in the second stage follows a binomial distribution. The proposed control chart is based on the deviance residual which essentially the generalized log-likelihood ratio statistic is obtained from the generalized linear model. To establish the relationship between the first- and second-stage quality characteristics, we propose using a new link function in a generalized linear model framework. The performance of the proposed control chart with the new link function is compared with that under the traditional logit link function in terms of the average run length criterion. In addition, the performance of the proposed control chart is compared with the chart designed based on the original residuals under the new link function as well as the traditional npchart applied for monitoring the binomial quality characteristic in the second stage. Keywords: Average run length; Binomial data; Deviance residual; Generalized linear model; Two-stage process. 1. Introduction

In modern manufacturing environments, many products are produced through a series of dependent process stages. Due to the cascade property in most of these processes, it is not suitable to monitor each stage using a different Shewhart chart. Further, using multivariate or multi-attribute control charts such as Hotelling T 2 -chart to monitor all stages simultaneously could also be problematic, because when an out-of-control signal is detected, it is difficult to identify which stage of the process is out-of-control (Wade and Woodall, [1]). Another approach to monitoring this kind of processes is to use the cause selecting chart (CSC) introduced by Zhang [2]. The CSC is similar to the regression control chart of Mandel [3]. Hawkins [3,4] extended Mandel’s method and proposed control charts based on regression adjusted variables. Wade and Woodall [1] introduced a CSC with prediction limits to improve its statistical performance. Lucas and Saccucci [6] proposed using the exponential weighted moving average (EWMA) chart to monitor residuals in the second stage. Hauck et al. [7], extending the regression adjusted variables studied in Hawkins [5], proposed control charts for monitoring multistage processes with multi-variables in each stage. Shu et al. [8] considered a two dependent process with parameters uncertainties and multiple quality characteristics in the first stage and one quality * Corresponding author. E-mail addresses: [email protected] (A. Amiri), [email protected] (A.B. Yeh ), [email protected] (A. Asgari).

characteristic in the second stage. They extended the cause selecting control chart (CSC) and proposed the multiple cause-selecting control chart (MCSC) using prediction limits when the output variable is a function of multiple input variables and the parameters have uncertainty. The prediction limits are developed using two types of procedures: the least squares method and principal component regression. Finally, they showed the effectiveness of the proposed methods in terms of false-alarm rate. Niaki and Davoodi [9] considered a multivariate-multistage manufacturing process and used multivariate time series to relate the stages. They considered a three-stage process with two correlated variables in each stage and proposed monitoring such a process using artificial neural networks. Amiri et al. [10] considered a two dependent process with a normal variable in each stage. They used MLE approach to estimate step change point in the mean of multistage process in Phase II and showed suitable performance of the proposed estimator through simulation studies. Yang [11] proposed economic design model using a renewal theory approach for the multiple dependent sub processes. They used the optimal individual Y control chart and the multiple CSC chart for monitoring the process with k+1 sub processes. They assumed one assignable cause can occur in each stage and there are k+1 assignable causes. The optimal design parameters of the proposed control charts are obtained by minimizing the cost model. Although the design parameters of a control chart are usually assumed fixed in traditional SPC techniques, there are some schemes known as adaptive control chart in which these parameters (namely sample size, sampling interval and control limits) are varying. In some twodependent process monitoring works, researchers assumed the design parameters of the cause selecting control chart are not fixed. In this field of research, Yang and Su [12] assumed the sample size in a two dependent process are variable and proposed variable sample size (VSS) scheme to monitor the mean of quality characteristics in a two dependent process. They used the adjusted average time to signal (AATS) to investigate the performance of the proposed VSS CSC. Yang and Su [13] considered the variable sampling interval (VSI) control chart for a two dependent process and then proposed a VSI scheme to monitor the mean of quality characteristics in a two dependent process. They showed the performance of their proposed VSI CSC in terms of the AATS criterion. Moreover, Yang and Su [14] considered both VSS and VSI properties and designed the variable sample size and sampling interval (VSSI) CSC scheme for monitoring means of two dependent processes. They also used the AATS criterion to evaluate the performance of their proposed control charts. Yang and Chen [15] proposed the variable sampling interval (VSI) scheme to monitor the means and the variances in two dependent process steps. They also considered the AATS criterion by Markov chain approach to measure the performance of proposed VSI control charts. Then, they showed the application of the proposed schemes in a real case. Furthermore, the performance of the VSI control charts and the fixed sampling interval control charts are compared via the numerical analysis and the results showed the suitable performance of their proposed methods. In most of the afore-mentioned work, it is typically assumed that the quality characteristics follow normal distributions. In practice, however, quality characteristics at different stages do not always follow normal distributions. Skinner et al. [16] assumed that the response variable, typically the quality characteristic at the last stage, follows a Poisson 2

distribution. They used a generalized linear model (GLM) to obtain the deviance residual by a log-likelihood ratio statistic. They showed that the distribution of the deviance residual is approximately normal and designed a control chart based on this statistic. Jearkpaporn et al. [17] assumed that the response quality characteristic follows a Gamma distribution. The model relating the response quality characteristic and quality variables in previous stages is a log link function. Their procedure is based on the residual that is obtained similarly to that of Skinner et al. [16]. Jearkpaporn et al. [18] considered a two-stage process and assumed the response variable follows a Gamma distribution. Under the premise that there are outliers in historical data, they applied robust GLM for the Gamma response variable and developed a control charting scheme based on the robust GLM scheme. Jearkpaporn et al. [19] considered a three-stage process and assumed that there are three mixture correlated variables in each stage which follow Gamma, normal and Poisson distributions, respectively They obtained the deviance residuals for each variable and monitored each variable separately by a deviance residual based control chart. Aghaie et al. [20] considered processes with two dependent stages, where the first-stage and second-stage variables follow Bernoulli and Poisson distributions, respectively. The authors presented a new control chart to monitor the process based on the generalized Poisson distribution. Asgari et al. [21] proposed a standardized residual statistics based on a new link function when the response variable in the second stage follows a Poisson distribution. Then, they used shewhart and EWMA control charts to monitor the proposed statistic. Yang and Yeh [22] proposed using the CSC to monitor two dependent process stages with a binary variable in each stage. They assumed that the paired data can only be obtained at the end of the second stage and that the paired data follow a bivariate binary distribution. They expressed the relationship between the response and the predictor variables by an arcsine transformed model, then derived the CSC based on the deviance residual thus obtained. In practice, there are manufacturing processes which can be considered as two-stage processes whose response quality characteristic follows a binomial distribution. For example, consider a drilling process in which an alloying procedure is performed on the surface of the part, before the drilling process begins, to differentiate the surface properties of the part. The thickness of the alloy coat on the surface is always of interest whose distribution typically follows a normal distribution. On the other hand, the thickness of alloy coat can also impact the number of nonconforming drilled holes on the surface of the part. Therefore, the alloying procedure and drilling process could be considered as first and second stages, respectively. Another example can be found in the circuit board manufacturing process. In the first stage, the property of the manufactured board such as the thickness or the strength of the board, which typically follows a normal distribution, is an important quality characteristic. In the second stage, after mounting the devices on the board, the number of incomplete mounted devices on the board, obviously influenced by the thickness or the strength of the board, follows a binomial distribution. Yang and Yeh [22] also discussed an example related to the paint defects of a ceiling fan cover. There are two types of defect, one is the number of patty defect which is considered as the quality

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characteristic of the first stage, and the other is the number of poor covering which is considered as the quality characteristic of the second stage. In this paper, we consider the problem of monitoring a two-stage process whose quality characteristic in the second stage follows a binomial distribution. We propose a control chart which is based essentially on the deviance residual calculated from the generalized log-likelihood ratio (GLR) statistic. The proposed chart is called the deviance residual (DR) control chart. In addition, we propose a new link function to establish the relationship between the second-stage (response) and the first-stage (predictor) variables. The performance of the proposed control chart with the proposed link function is compared with the traditional logit link function under three different scenarios for nonconforming probability in terms of the average run length (ARL) criterion. The results of the simulation studies show that the new link function applied in the proposed DR control chart leads to better performance of the control chart in detecting any step shifts in the process. Specifically, the logit link function leads to poor performance of the DR control chart in small shifts. Our focus in this work is on Phase II monitoring, and therefore, we assume that the model parameters of the assumed generalized linear model are known or can be reasonably estimated based on Phase I data. The performance of the proposed DR control chart with the new link function is also compared with the conventional -chart which is applied only to the response variable, as well as the control chart designed using just the original residuals. The rest of the paper is organized as follows. In Section 2, we derive the deviance residual based on the proposed link function in a GLM framework. The proposed control chart to monitor two-stage processes is then presented. Section 3 is devoted to performance evaluation and comparison. We further investigate, in Section 4, the effect on the chart performance when other link functions are used. The impact on the chart performance when using original residuals, rather than the deviance residual, is being examined in Section 5. In Section 6, the proposed chart is applied to a real example discussed in Yang and Yeh [22]. Section 7 concludes the paper with some additional remarks and discussion of potential future research along the same line. 2. The Proposed Control Charting Mechanism

In this section, we first give a brief discussion of the GLM which utilizes our proposed link function. Based on the GLM model, a generalized likelihood ratio test is developed and the deviance residual is derived. The proposed control chart is then constructed based on the derived deviance residual. 2.1. The GLM framework When the response variable of a process is non-normal, the GLM is suitable for modeling the response variable, especially if its distribution belongs to the exponential family of distributions. The GLM links the mean of the response variable, μ, to some linear combination of the predictor variables, x '  , through a link function , 4

x '   w (  ),

(1)

where x is a vector of predictor variables and β is a vector of model parameters. The mean of the response variable is given by the inverse of the link function, w 1 (x ' ) . For example, if one uses  p  the logit link function, w( p )  log   for a binomial response, the parameter of this response, 1  p   namely p (0  p  1) , is given by

e x ' p . 1  e x '

(2)

Therefore, the mean of the binomial response,   np , is given by

 e x '  n x ' 1 e

 . 

(3)

In this paper, we consider a response variable which can be modeled by a binomial distribution. The distribution function of a binomial distribution can be written as

n f ( y )    p y (1  p)n  y , y  0,1, 2,..., n,  y where

(4)

is the binomial variable. The prevalent link function for a binary response is the logit

link function described earlier. For example, Yeh et al [23] proposed and studied a Phase I control chart for monitoring a profile whose response is binary. They used a logit link function to link the binary response and the predictor variables. In the current paper, we propose a new link function in the GLM setup and use it to derive the proposed chart to monitor two-stage processes whose quality characteristic in the second stage follows a binomial distribution. This link function is defines as

x '  w ( p ) 

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p , 1 p

(5)

where p is the nonconforming probability of the response quality characteristic, x is a vector of incoming variables and β is a vector of model parameters. Through the inverse of the link function in (5), can be obtained by

 ( x ' ) 2  p  w (x ' )   . 2  1  (x ' )  1

(6)

Hence, the mean of the binomial response variable   E ( y ) , is equal to

 (x ' )2    n w 1 (x ' )   n  . 2  1  ( x '  ) 

(7)

where n is sample size. Therefore, the binomial response y can be expressed as

    0  1 x  2   y ~ B  n, p    . 2  1     x      0 1  

(8)

2.2. The deviance residual and the proposed chart In this subsection, we develop the generalized likelihood ratio test for testing that H 0 :   0 v.s. H1 :   0 . Skinner et al. [16] assumed that the response variable follows a Poisson distribution and used the generalized likelihood ratio test to obtain the deviance residual to detect mean changes. Jearkpaporn et al. [17] assumed that the response variable follows a Gamma distribution and also derived a deviance residual based procedure. Since we assume in this paper that the response variable follows a binomial distribution, the likelihood function for the response variable is written as

n  L( y )    p y (1  p )n y .  y

(9)

The GLR statistic for testing H 0 :   0 v.s. H1 :   0 , or equivalently H 0 : p  p0 v.s.

H1 : p  p0 since   np , is equal to

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ln[ L( y, n, p1 )]  ln[ L( y , n, p0 )]  n    n    ln   p1 y (1  p1 )n y   ln   p0 y (1  p0 )n  y   y    y    ylnp1  (n  y )ln(1  p1 )  ylnp0  (n  y )ln(1  p0 )

(10)

p   1  p1   yln  1   (n  y )ln  .  p0   1  p0  Now, replace

by w1 ( x '  ) and

by y/n. After multiplying by two, talking square root, and

then multiplying by the sign of the difference between expected and actual values, the deviance residual (DR) for the binomial response variable is equal to: 1 2    y   y       n   1 n     DR  sign [ y  0 ] 2  yln  1    (n  y )ln    , 1 w ( x '  ) 1  w ( x '  )                 

(11)

Where 0 is equal to n[w 1 (x ' ) ]. The proposed chart will be based on the deviance residual calculated using equation (11). When the Phase II monitoring begins, at each sampling period, a sample of items will be collected and the total number of non-conforming items, y, is obtained from the sample. The plotting statistic will be calculated based on (11) and plotted on the control chart against sampling sequence. In theory, the control limits and the center line of the proposed chart are equal to

UCL   DR  L   DR CL   DR

(12)

LCL   DR  L   DR , 2  Var ( DR) are the population mean and variance of DR, Where DR  E( DR) and  DR

respectively. Nevertheless, the distribution of DR is difficult to obtain, especially for small to moderate sample sizes. Therefore, Monte Carlo simulations were used in this paper to 2 approximate the  DR ,  DR   DR as well as the multiplier L. Note that the multiplier L depends

on the sample size n, as well as the desired in-control performance of the proposed chart. In Table 1, the approximated values of L are given for various sample sizes under in-control ARL (ARL0)

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of approximately 200. The estimated  DR and  DR are also given in the Table 1 for each sample size. When the sample size (n) changes, the estimated values of  DR and  DR will change. As a result, the value of L changes as well. However, the changes in UCL and LCL are not significant. Also note that the estimated values of  DR ,  DR and, consequently, L depend on the true parameter values of the GLM as well.

The Monte Carlo simulation is conducted as follows: 1. First, 1,000 observations of normal input variable (x) are generated. 2. The non-conforming probability, p , is obtained from equation (6) for each observation of x. 3. The Binomial response variable is generated for each observation generated in the step 1 by Eq. (8) with a defined sample size (n) 4. The deviance residual (DR) for each observation is obtained by Eq. (11). 2  Var ( DR) ) are obtained 5. Then, the mean and variance of DR ( DR  E( DR) and  DR also the center line (CL) is equal to DR  E( DR) . 6. The control limit parameter (L) and control limits of the DR control are determined to achieve ARL0 of 200 in 10,000 times repeats. We used this procedure for different sample sizes (n) of Binomial distribution and the results are summarized in Table 1. Insert Table 1 about here 2.3. Normality tests on DR statistic To determining whether the deviance residual of binomial data is normal or non-normal, we used normality test on DRs by Matlab software. The normality test is performed based on 1000 observations which were simulated with the model

and

where y is

binomial response variable and n is equal to 5, 16, 40 and 80. Also, x follows standardized normal distribution. The normality test is performed in confidence level of (1-α) equal to 0.95 and the corresponding normal probability plots are illustrated in Figures 1(a-d). Insert Figure 1 about here Based on P-Values (in all of probability plots for different sample sizes (n)) that are greater than α, the distribution of the DRs is approximately normal in this condition. Furthermore, based on the Figures 1(a-d), the distribution of DR statistic is closed to the standardized normal distribution as the sample size increases.

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Performance Evaluations and Comparisons

2.4. Simulation settings In this section, we evaluate the performance of proposed chart in terms of the ARL and compare that with the conventional np -chart which is applied only to the binomial quality characteristic in the second stage. In essence, the conventional np -chart does not take into account of the possible relationship between the continuous quality characteristic in the first stage and the binomial quality characteristic in the second stage. In the simulation set-up, the quality characteristic of the first stage is assumed to follow a normal distribution with mean equal to 2 and unit variance. In the second stage, the quality characteristic is assumed to follow a binomial distribution with the number of trials equal to 20 ( n  20 ). The non-conforming probability, p , is obtained from equation (6), where the model parameters  0 and 1 are assumed to be equal to 1 and -0.25, respectively, i.e., when the process is in-control, the non-conforming probability can be calculated as

 (1  0.25x )2  p0   , x ~ N (2,1) 2  1  (1  0.25x )  When the process is out-of-control due to model parameters, we assume that  0'   0   and 1'  1   ,  ,   0 and that

 (  0'  1' x) 2  p . ' ' 2  1  ( 0  1 x)  For each simulation, a value of the variable x is first generated from N (2,1) . The non-conforming probability is calculated based on the above formula and an observation y is generated from B (20, p ). Based on the generated x and y, the DR in equation (11) is obtained, with n = 20 and 0  20  p0 . This procedure is repeated 100,000 times and the mean and standard deviation of the DR statistic are estimated. We then set a value for L to obtain the desired ARL0. A simulated run length can be obtained by repeatedly generating the x and y values, and computing the plotting statistic DR, until the first out-of-control signal shows up on the control chart. The simulated run length is equal to the total number of samples generated until the first out-ofcontrol signal shows up. This simulation run is repeated 10,000 times and the average of the simulated run lengths is reported as the ARL.

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The ARL of the competing np-chart is obtained similarly. We set the ARL0 to be approximately 200, for which the upper and lower control limits are 2.7269 and -2.9031, respectively, for the proposed DR based chart, and 13.2 and 0, respectively, for the competing np-chart. 2.5. Detecting changes in model parameters The simulated ARL’s of the proposed DR based chart and the competing np-chart when only 0 shifts are summarized in Table 2. Note that the numbers in the parentheses are the standard errors of the simulations. The results summarized in Table 2 indicate that the DR based control chart outperforms the np-chart when changes in the non-conforming probability are only due to shifts in  0 . Also note that when  0 decreases, the np-chart essentially produces out-of-control ARL’s larger than the ARL0, rendering it practically ineffective in detecting changes that only occur in  0 . Summarized in Table 3 are the simulated ARL’s when only 1 changes. The results indicate that the DR based control chart is more effective than the np-chart in detecting changes in p that are due to changes occurring in 1 only. Further, comparing the results summarized in Table 3 with their counterparts in Table 2, they indicate that the proposed DR based control chart is more effective in detecting changes in p that are due to changes in 1 than in  0 . For example, when the upward magnitude is equal to, say, .1 downward shift in  0 or in 1 , the simulated ARL’s for each case are equal to 37.3 and 5.4, respectively. We further simulated the ARL’s of the DR based chart and the np-chart when  0 and 1 both change, and the results are summarized in Table 4. We specifically considered cases when  0 and 1 change in opposite directions of the same magnitude. The opposite shifts with the same magnitude in  0 and 1 sometimes mask the effect of the shifts and could potentially lead to poor performance of the control chart. The simulated ARL’s clearly indicate that the DR based chart is more effective than np-chart in detecting changes in p that are due to changes in both  0 and 1 , except when  0 increases and 1 decreases by small amounts ( 0.025) . Nevertheless, the np-chart is not a viable option for practical applications since it could potentially produce out-of-control ARL’s that are much larger than ARL0. Insert Table 2 about here Insert Table 3 about here Insert Table 4 about here

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2.6. The sensitivity of the chart performance due to changes in x The main goal of the CSC and model-based control chart in multistage processes is to separate the effects of pervious stages from the current stage. Therefore, when these charts are used to monitor the second stage quality characteristic of a two-stage process, change in the mean of the first-stage quality characteristic, x , should not greatly impact the chart performance, i.e., the ARL of the chart should stay relatively unchanged, otherwise it might lead to incorrect conclusion. While keeping all the other parameters at their in-control values, we simulated the ARL’s of the DR based chart and the np-chart when x changes to x  or x   . The results are summarized in Table 5. As can be seen from Table 5, the ARL0 of both the DR based chart and the

-chart seems to

increase as x shifts upward and vice versa. However, the magnitudes of change in ARL0 are much smaller for the DR based chart as compared to those of the np-chart. Further, for the DR based chart, the ARL0 is less affected when x shifts downward than when it shifts upward.

Insert Table 5 about here

(x ' )t 2.7. The chart performance for different values of in p  1  (x ' )t Our proposed link function in (5), and the inverse of it in (6), is essentially a special case

(x ' )t by letting t  2 . Note that has to be an even number, otherwise it may result in 1  (x ' )t a negative . The question naturally arises as to whether is the best choice for the form of the link of p 

function, especially in the context of developing the DR based control chart for monitoring the second-stage quality characteristics of a two-stage process. For various changes in either  0 or 1 but not both, we simulated the ARL’s of the DR based chart under t  2,4,8,16,40 and

. The

results are listed in Table 6 (for changes in  0 ) and Table 7 (for changes in 1 ). In general, larger t results in better performance under large positive shifts in 1 , while smaller t tends to do better under small positive shifts in this parameter. When a negative shift occurs in 1 , setting t  2 is the best selection. Given a specific change magnitude in  0 or 1 , setting t  2 may not produce the best result. However, the other choices of all have some inherent problems that may render them less useful in practice. For instance, setting t  4,8 and 16, the chart may produce out-of-control ARL’s which are larger than in some out-ofcontrol cases considered. Setting t  16, 40 and 80, while the chart does not suffer the same

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problem, however, the chart’s out-of-control ARL converges very slowly, especially when there is a negative shift in  0 and that the shift magnitude is large. Insert Table 6 about here Insert Table 7 about here 3. Comparison Between the Proposed Link Function and the Logit Link Function

As mentioned earlier, the prevalent link function used in the literature to model a binary response is the logit link function. We proposed in this current work to use a different link function as described in (5). As expected, the type of link function used to obtain the DR statistic is likely to affect the control chart whose construction is based primarily on the DR statistic thus obtained. Shown in Figures 2(a) and 2(b) are the plots of p versus x (-10< x < 10) for the logit and the proposed link functions. These Figures show that when the response variable is binomial the shape of the response function based on our proposed link function is a reverse normal shaped. However, this shape for the logistic link function is S-shaped or reverse S-shaped depending on the sign of the slope. It can be seen from the Figures 2(a) and 2(b) that for same values of x and regression parameters, values of p in logit link function and proposed link function can be different. For example, in Figure 2, when the range of x values is between -4 and -10, the range of p in logit and the proposed link functions is between (0 - 0.3) and (0.3 - 0.95), respectively. Hence, in this section, we investigate the performances of logit and proposed link functions in three different ranges of p in logit link function. Insert Figure 2(a) about here Insert Figure 2(b) about here

We simulated and compared the ARL’s of the based chart when the statistic was obtained under the proposed link function and the logit link function. The comparisons were grouped into 3 scenarios, depending on the value of the nonconforming probability, p, in logit link function. These scenarios are: (I) 0.6  p  1 ; (II) 0  p  0.4 ; and (III) 0.4  p  0.6 . In all simulations, the quality characteristic of first stage is normally distributed with mean equal to 2 and variance equal to 1. The number of trials (sample size) for the second-stage binomial quality characteristic is assumed to be 20. 3.1. Scenario I Under this scenario,

is between 0.6 and 1, in logit link function and the model parameters

 0 and 1 are assumed to be 1 and 0.5, respectively. The ARL’s were simulated for the

based

chart under the proposed link as well as the logit link functions. In this scenario, we considered 12

the cases when either  0 or 1 but not both shifts upward. The results are summarized in Table 8. It is evident that using logit link function will lead the DR based control chart to produce larger ARL’s than ARL0, especially when the shift magnitudes are small. This is a very undesirable outcome to have for any control chart to have practical implications. The proposed link function in (5), on the other hand, does enable the DR based chart to produce reasonably good results. Insert Table 8 about here 3.2. Scenario II In this scenario, p is between 0 and 0.4, in logit link function. The values of  0 and 1 are assumed to be equal to -1 and -0.5, respectively. We considered the cases when either  0 or 1 but not both shifts downward. The simulated ARL’s are summarized in Table 9. Utilizing the logit link function to construct the based chart is not to be recommended, as indicated by the larger than out-of-control ARL values. 3.3.

Scenario III Under this scenario,

is between 0.4 and 0.6, in logit link function and the values of  0 and

1 are assumed to be equal to -1 and 0.5, respectively. We considered the cases when either only  0 or only 1 changes. The simulated ARL’s for changes in  0 only are summarized in Table

10. From Table 10, the results clearly show that the based control chart using the proposed link function in (5) outperforms the chart that uses logit link function. Note that the ARL’s for changes in only 1 were also simulated (not reported here). Similar results with Table 10 was obtained in our simulation studies and the based control chart using the proposed link function in (5) outperformed the control chart that uses logit link function, too. Insert Table 9 about here Insert Table 10 about here 4. Comparison Between the Deviance Residual and the Original Residual

Montgomery et al. [24] mentioned that the ordinary or raw residuals from the GLM are just the differences between the observed and fitted values. However, it is generally recommended that deviance residuals be used. In this section, under the same simulation settings as described in Section 3.1, we use the raw residual, R  y  ( np 0 ) , instead of the deviance residual, to construct the proposed control chart. The control limits and the center line of the R statistic are equal to

13

UCL   R  LR   R CL   R

(13)

LCL   R  LR   R , where  R and  R2 are the mean and variance of R, respectively, which are approximated by Monte Carlo simulations as discussed in Section 2.2. Moreover, the simulation procedure to determine the control limits in (13) is similar to the simulation procedure discussed in section 3. The in-control values of  0 and 1 are assumed to be 1 and 0.5, respectively. And the upper and lower control limits for the R statistic are estimated to be 5.3057 and -5.3103, respectively, to achieve an ARL0 of approximately 200. Also the upper and lower control limits for the DR statistics are 2.9273 and -2.7968 respectively which are reported in Table 1. Figures 3 and 4 illustrate the simulated ARL’s for the DR and R based control charts, assuming only either  0 or 1 shifts. Both the upward and downward shifts for  0 and 1 are considered in Figures 3 and 4. The results in Figures 3 and 4 show that the deviance residual control chart outperforms the raw residual control chart when only  0 or only 1 shifts upward. However, the R based control chart performs better when only  0 or only 1 shifts downward. Nevertheless, the R based chart should not be recommended since it could produce ARL values which are larger than ARL0, especially when there is a small upward shift in 1 . Insert Figure 3 about here Insert Figure 4 about here 5. An Example

In this section, we apply the proposed DR based chart to a real example to demonstrate how it can be used in practical applications. The example, first considered in Yang and Yeh [22], concerns the paint defect of a ceiling fan cover. The paint defect includes two types of defect, part defect and poor covering. The variable considered as the second-stage quality characteristic, the total number of poor covering (y), follows a binomial distribution. On the other hand, the variable considered as the first-stage quality characteristic, the number of part defect (x), also follows a binomial distribution. The authors assumed that the paired data (x,y) follow a bivariate binomial distribution with ( p x , p y ,  , n ) . A total of 24 samples were collected which are listed in Table 11. In this work, we assume that the quality characteristics in two stages are related under a GLM framework which utilizes the proposed link function (5). Note that, even though this example may not fit perfectly under our premise, since we assume that the first-stage quality characteristic follows a normal distribution, it still provides a good practical example to demonstrate how the principles of our proposed control charting mechanism can be extended to 14

slightly different model settings. These principles include (1) modelling the relationship between the quality characteristics in the first stage and second stage by a GLM which uses our proposed link function, and (2) developing the control chart for monitoring the second-stage binomial quality characteristic based on the statistic. For this example, since the regression parameters  0 and 1 in Equation (6) are unknown, we have to estimate the parameters with a historical dataset. Further, the historical dataset for this example is not reported in Yang and Yeh [22]. Hence, for illustrated purpose, we use the first 14 samples as the Phase I data, assuming that they are coming from an in-control process, and estimate  0 and 1 based on the maximum likelihood estimation (MLE). The likelihood function for these fourteen samples is equal to:

14 n  l ( 0 , 1 )     p( xi ) yi [1  p( xi )]n  yi , i 1  yi 

where

p (x i ) 

(  0  1x i ) 2 and 1  (  0  1x i )2

n  100 .

(14)

Taking the logarithm of l (β 0 , β1 ) , denoted

by L(β 0 , β1 ) , and taking the derivative of L(β 0 , β1 ) with respect to  0 and 1 and setting them to 0, one obtains the following

L( 0 , 1 ) n  2   yi   0 i 1   0  1 xi

n  2( 0  1 xi )  n  0,   2 i 1 1  (  0  1 xi )  n L( 0 , 1 ) n  2 xi  2 xi (  0  1 xi )   yi   n  0.   2 1 i 1 i 1 1  (  0  1 xi )  0  1 xi 

(15) (16)

Note that the equations (15) and (16) are nonlinear, and therefore the estimated values of  0 and 1 are obtained by an iterative procedure. The resulting estimated values of ˆ0 and ˆ1 are equal

to 0.31451 and 0.00178, respectively. Note that, while not reported here, we had also tried to obtain ˆ0 and ˆ1 by minimizing the mean square errors and the resulting estimates are almost identical to the estimated values obtained using MLE. Insert Table 11 about here After estimating the regression parameters using the first fourteen samples and treating them as the in-control values, and, we determined the control limits under which ARL0  200 based on Monte-Carlo simulations. The UCL and LCL are approximately equal to 2.83 and -2.95, respectively. The established DR based control chart is then used to monitor the remaining samples from sample 15 to 24. The np-chart whose control charting parameters are estimated 15

using the first 14 samples is also used to monitor samples 15 to 24, but based only on the secondstage observations, the y'’s. Further, to demonstrate how the DR based chart is less affected by changes in the x variable. We simulated a pair of (x,y), by first letting x  15 and y was then generated based on a binomial distribution with and calculated from the link function (5) with  0 and 1 being replaced by their MLE’s. The generated y was equal to 20. This generated pair of observations was added as sample 25. The DR based chart for samples 15 to 25 is shown in Figure 5, while the corresponding np-chart is shown in Figure 6. As seen from Figures 5 and 6, both charts indicated that the process was in control from samples 15 to 24, as reflected in the second-stage quality characteristic. However, the generated sample 25 remained in control on the based chart, while it showed up as an out-of-control signal on the corresponding np-chart. As discussed earlier, the model based control chart such as the proposed DR based chart is more suitable in separating the effects from previous stages and the current stages since the model accounts for such possibilities. On the other hand, the np-chart only looks at the outcome of the current stage, and yet when it signals, it could lead to misleading diagnostics. Insert Figure 5 about here Insert Figure 6 about here 6. Conclusions and future researches

In this paper, we considered the problem of monitoring the second-stage quality characteristic, which follows a binomial distribution, of a two-stage process. We also proposed a new link function which is to be used in the GLM that links the first-stage and the second-stage quality characteristics. The proposed chart is constructed based essentially on the DR statistic obtained from the generalized log-likelihood ratio test. Through simulations, we demonstrated that the DR based chart outperforms the conventional np-chart which only relies on the observations obtained from the second stage. It was also shown that the proposed link function, when used to construct the DR based chart, performs better than using the prevalent logit link function. Moreover, we compared the ARL performance of the DR based chart and the R based chart, where the former was shown to have a better performance. The proposed DR based chart was applied to a real data set to demonstrate how it can be used in real applications. The proposed link function in (5) seems to perform reasonably well in our GLM setting. It would be worthwhile to investigate whether such a success can be replicated in different GLM settings such as two-stage processes in which the second-stage quality characteristic follows a Poisson distribution. In a recent work, Yeh et al [23] proposed Phase I control charts for monitoring profiles with binary response. They used logit link function to estimate the profile parameters which are then used to construct the Phase I control charts. It would be worthy to study whether using the proposed link function in (5) can improve the performance of the Phase I control charts studied in Yeh et al [23]. In modern manufacturing processes, many of the socalled high-yield processes have extremely low defective rates. These processes are usually 16

monitored by high quality control charts such as the cumulative count of conforming (CCC) control charts or the EWMA charts studied in Yeh et al [25]. Therefore, developing effective control charting mechanisms to monitor high-yield multistage processes could be a future research direction worthy of pursuing. Acknowledgement: The authors are thankful to the anonymous referees for constructive comments which led to improvement in the paper.

References [1] Wade M.R., Woodall W.H. (1993). A review and analysis of cause-selecting control charts. Journal of Quality Technology, 25(3), 161-169. [2] Zhang G.X. (1984). A new type of control charts and theory of diagnosis with control charts. World Quality Congress Transactions, 3, 175-185. [3] Mandel B.J. (1969). The regression control chart. Journal of Quality Technology, 1(1), 1-9. [4] Hawkins D.M. (1991). Multivariate Quality Control Based on Regression Adjusted Variables. Technimetrics, 33(1), 61-75. [5] Hawkins D.M. (1993). Regression adjustment for variables in multivariate quality control. Journal of Quality Technology, 25(3), 170-182. [6] Lucas, J.M. and Saccucci, M.S. (1990). Exponentially weighted moving average control schemes: properties and enhancements. Technometrics, 32(1), 1–29. [7] Hauck D.J., Runger G.C. (1999). Montgomery D.C., Multivariate statistical process monitoring and diagnosis with grouped regression-adjusted variables. Communications in Statistics, Simulation and Computation, 28(2), 309-328. [8] Shu L., Tsung F., Kapur K.C. (2004). Design of multiple cause-selecting charts for multistage processes with model uncertainty. Quality Engineering, 16(3), 437-450. [9] Niaki S.T.A and Davoodi M. (2009). Designing a multivariate-multistage quality control system using artificial neural networks. International Journal of Production Research, 47(1), 251-271. [10] Amiri A., Zolfaghari S., Asgari A. Identifying the Time of a Step Change in the Mean of a TwoStage Process. Proceedings of the IEEE International Conference on Industrial Engineering and Engineering Management (IEEM), Hong Kong, 10-13 Dec. 2012.

[11] Yang S.F. (1998). Optimal process control for multiple dependent subprocesses. Quality and Reliability Engineering International 14(5), 347-355. [12] Yang S.F., Su H.C. (2006). Controlling-dependent process steps using variable sample size control charts. Applied Stochastic Models in Business and Industry, 22(5-6), 503-517. [13] Yang S.F., Su H.C. (2007a). Adaptive sampling interval cause-selecting control charts. International Journal of Advanced Manufacturing Technology 31(11), 1169-1180. [14] Yang S.F., Su H.C. (2007b). Adaptive control schemes for two dependent process steps. Journal of Loss Prevention in the Process Industries, 20(1), 15-25. [15] Yang, S. and Chen, Y. (2011). Monitoring and diagnosing dependent process steps using VSI control charts, Journal of Statistical Planning and Inference, 141(5), 1808-1816.

17

[16] Skinner K.R., Montgomery D.C., Runger G.C. (2003). Process monitoring for multiple count data using generalizd-linear model-based control charts. International Journal of Production Research, 41(6), 1167-1180. [17] Jearkpaporn D., Montgomery D.C., Runger G.C., Borror C.M. (2003). Process monitoring for correlated gamma-distributed data using generalized-linear model-based control chart. Quality and Reliability Engineering International, 19(6), 477-491. [18] Jearkpaporn D., Montgomery D.C., Runger G.C., Borror C.M. (2005). Model-based process monitoring using robust generalized linear models. International Journal of Production Research, 43(7), 1337-1354. [19] Jearkpaporn D., Borror C.M., Runger G.C., Montgomery D.C. (2007). Process monitoring for mean shifts for multiple stage processes. International Journal of Production Research, 45(23), 5547-5570. [20] Aghaie A., Samimi Y., Asadzadeh S. (2010). Monitoring and diagnosing a two-stage production process with attribute characteristics. Iranian Journal of Operation Research, 2(1), 1-16. [21] Asgari, A., Amiri, A., Niaki, S. T. A., (2014). A new link function in GLM-based control charts to improve monitoring of two-stage processes with Poisson response", Published online in The International Journal of Advanced Manufacturing Technology, 72(9-12), 1243-1256. [22] Yang S.F., Yeh J.T. (2011). Using cause selecting control charts to monitor dependent process stages with attributes data. Expert System with Applications, 38(1), 667-672. [23] Yeh A.B., Huwang L., Li Y.M. (2009). Profile monitoring for a binary response. IIE Transactions on Quality and Reliability Engineering, 41(11), 931–941. [24] Montgomery D.C., Peck E.A., Vining G.G., Introduction to linear regression analysis. Third Edition, John Wiley and Sons, Inc. [25] Yeh A.B., McGrath, R.N., Sembower, M.A., Shen, Q. (2008). EWMA control charts for monitoring high-yield processes based on non-transformed observations. International Journal of Production Research, 46, 5679–5699.

18

Table 1. The approximated values of L and  DR and  DR for various n to achieve ARL0 of 200, when and are 1 and 0.5, respectively.

n

 DR

 DR

L

ARL0

UCL

LCL

3

0.1735

1.0622

2.82

199.7

3.1689

-2.8919

5

0.1475

1.0726

2.591

199.7

2.9255

-2.6305

8

0.118

1.0671

2.570

200.1

2.8604

-2.6244

12

0.0895

1.0571

2.620

200.8

2.8591

-2.6801

16

0.0784

1.0437

2.710

199.9

2.8964

-2.7396

20

0.0653

1.0315

2.775

200.01

2.9273

-2.7968

25

0.0538

1.0269

2.820

200.8

2.9394

-2.8318

30

0.0496

1.0157

2.850

200.8

2.9446

-2.8448

40

0.04

1.0128

2.835

200.04

2.9113

-2.8313

50

0.0365

1.0068

2.830

200.2

2.8857

-2.8127

70

0.0284

1.0053

2.815

200.2

2.8583

-2.8015

100

0.0243

1.0025

2.8133

199.5

2.8446

-2.7960

19

Table 2. The simulated ARL’s when only

-chart

DR

shifts.

-chart

DR

0

200.1(2.01)

200.01(2)

0

200.1(2.01)

200.01(2)

0.025

163.9(1.6)

146.2(1.5)

0.05

315.7(3.1)

167.1(1.6)

0.05

132.9(1.3)

93.4(0.9)

0.1

>>400

82.4(0.8)

0.1

88.8(0.8)

37.3(0.3)

0.15

>>400

39.9(0.4)

0.25

30.6(0.3)

5.7(0.05)

0.2

>>400

21.05(0.2)

0.5

7.9(0.7)

1.7(0.01)

0.3

>>400

7.4(0.07)

0.5

>>400

2.2(0.02)

0.75

>>400

1.9(0.01)

Table 3. The simulated ARL’s when only

-chart

DR

changes.

-chart

DR

0

200.1 (2.01)

200.01(2)

0

200.1(2.01)

200.01(2)

0.005

202.04(2)

173.6(1.7)

0.015

204.8(2)

185.2(1.8)

0.01

201.3(2)

142.7(1.4)

0.025

207.9(2.1)

129.5(1.3)

0.02

199.4(1.9)

84.1(0.8)

0.05

206.4(2.05)

51.3(0.5)

0.03

195.1(1.9)

50.5(0.5)

0.1

199.6(1.9)

16.5(0.16)

0.05

186.4(1.8)

20.6(0.2)

0.2

139.2(1.4)

4.4(0.04)

0.1

148.1(1.5)

5.4(0.05)

0.5

7.9(0.07)

1.8(0.01)

0.15

93(0.9)

2.9(0.02)

0.25

17.4(0.17)

1.8(0.01)

20

Table 4. The simulated ARL’s when

and

both change simultaneously.

& -chart

& DR

DR

-chart

0

200.1(2.01)

200.01(2)

0

200.1(2)

200.01(2)

0.02

172.7(1.7)

180.8(1.8)

0.005

209.3(2)

185.3(1.6)

0.03

160.4(1.6)

150.1(1.5)

0.01

218.4(2.3)

165.7(1.6)

0.05

142.2(1.4)

84.2(0.8)

0.02

235.3(2.3)

120.3(1.2)

0.075

118.4(1.2)

47.1(0.5)

0.03

260.3(2.6)

86.3(0.8)

0.1

102.7(1.03)

30.04(0.3)

0.04

284.7(2.8)

59.6(0.6)

0.25

48.8(0.5)

8.6(0.08)

0.05

309.8(3)

42.2(0.4)

0.5

13.7(0.13)

2.9(0.02)

0.07

374.6(3.7)

23.2(0.2)

1

3.4(0.03)

1.8(0.01)

0.1

494.5(5)

11.7(0.1)

0.25

493.2(4.8)

3(0.02)

0.5

7.9(0.07)

1.8(0.01)

Table 5. The simulated ARL’s when

changes to

-chart

DR

-chart

.

DR

0

200.1 (2.01)

200.01(2)

0

200.1 (2.01)

200.01(2)

0.1

271.3 (2.7)

207.9 (2.1)

0.1

156.6 (1.5)

197.1 (1.9)

0.25

410.6 (4.1)

212.7(2.1)

0.25

107.3 (1.1)

190.9(1.9)

0.5

878.1 (8.7)

235.1 (2.4)

0.5

59.1(0.6)

185.8 (1.8)

0.75

1934.2(19.1)

251.9 (2.5)

0.75

33.2(0.3)

182.7(1.8)

1

4593(45.6)

283.2 (2.8)

1

20.1 (0.2)

179.3 (1.8)

21

Table 6. The simulated ARL’s under different when only

0

200.01(2)

199.9(2)

199.9(2)

0.01

180.1(1.8)

159(1.6)

149.8(1.4)

146.9(1.4) 113.6(1.1)

80.7(0.2)

0.025 146.2(1.5)

111.6(1.1)

91.5(0.9)

67.3(0.7)

36.5(0.4)

39.5(0.4)

0.05

93.4(0.9)

59.6(0.6)

41.2(0.4)

24(0.2)

18.1(0.2)

26.9(0.2)

0.075

57.2(0.6)

33.5(0.03)

20.3(0.2)

11.7(0.1)

13.3(0.1)

20.7(0.2)

0.1

37.3(0.3)

19.8(0.2)

10.9(0.1)

7.2(0.1)

10.7(0.1)

15.9(0.1)

0.2

9.3(0.09)

3.9(0.03)

2.5(0.02)

3.03(0.02)

5.4(0.05)

7.4(0.07)

0.25

5.7(0.05)

2.4(0.02)

1.8(0.01)

2.4(0.02)

4.1(0.04)

5.3(0.05)

0.5

1.7(0.01)

1.05(0.002)

1.1(0.003)

1.3(0.006)

1.6(0.01)

1.8(0.01)

0

200.01(2)

199.5(2)

199.9(2)

200(2)

200.7(2)

199.5(2)

0.01

196.2(2)

244.7(2.4)

241.7(2.4)

0.025 192.1(1.9)

274.7(2.7)

283.4(2.8)

192(1.9)

66.7(0.7)

55.7(0.5)

0.05

167.1(1.6)

247.9(2.4)

209.6(2)

80.9(0.8)

35.5(0.4)

43.9(0.4)

0.1

82.4(0.8)

109.1(1.1)

63.4(0.6)

26.3(0.3)

29.1(0.3)

38.6(0.4)

0.15

39.9(0.4)

47(0.5)

25.9(0.3)

19.2(0.2)

28(0.3)

35.2(0.3)

0.2

21.05(0.2)

23.3(0.2)

15.6(0.1)

17.6(0.2)

26.3(0.2)

33(0.3)

0.3

7.4(0.07)

8.5(0.1)

10.7(0.1)

17(0.2)

26.5(0.2)

32(0.3)

0.5

2.2(0.02)

3.9(0.03)

9.2(0.1)

16.5(0.2)

26.2(0.2)

31.8(0.3)

0.75

1.9(0.01)

2.8(0.02)

5.9(0.05)

13.4(0.1)

23.5(0.2)

30.2(0.3)

22

200(2)

changes.

200.7(2)

233.3(2.3) 144.4(1.5)

199.5(2)

98.8(1)

Table 7. The simulated ARL’s under different when only

199.9(2)

200(2)

200.7(2)

199.5(2)

0.005 173.6(1.7) 154.9(1.5)

165.5(1.7)

177.5(1.8)

184.4(1.9)

190.2(1.9)

0.01

142.7(1.4) 119.3(1.2)

128.2(1.3)

134.3(1.3)

161.9(1.6)

178.8(1.8)

0.02

84.1(0.8)

72.9(0.7)

84.5(0.8)

98.6(1)

137.4(1.4)

164.1(1.6)

0.03

50.5(0.5)

44.4(0.4)

54.9(0.5)

72.5(0.7)

113.7(1.3)

146.4(1.5)

0.05

20.6(0.2)

18(0.2)

24.3(0.2)

37.5(0.4)

74.1(0.7)

109.2(1.1)

0.1

5.4(0.05)

3.5(0.03)

4.4(0.04)

8.1(0.07)

24.7(0.2)

49.2(0.5)

0.15

2.9(0.02)

1.7(0.01)

1.6(0.01)

2.5(0.02)

9.2(0.1)

22.3(0.2)

0.25

1.8(0.01)

1.2(0.005) 1.07(0.003) 1.03(0.002) 1.01(0.001)

0

200.01(2)

0

200.01(2)

199.5(2)

changes.

199.9(2)

200(2)

200.7(2)

199.5(2)

0.015 185.2(1.8) 272.4(2.7)

318.3(3.2)

255.3(2.5)

213.3(2.1)

209.8(2.1)

155.6(1.5) 277.2(2.7)

359.3(3.6)

276.2(2.7)

223.2(2.2)

211.2(2.1)

0.025 129.5(1.3) 265.1(2.7)

386.1(3.9)

293.4(2.9)

228.1(2.3)

213(2.1)

0.02

199.5(2)

1.004(na)

0.035

87.7(0.9)

215.8(2.2)

417.2(4.2)

326.2(3.3)

227.8(2.3)

214.7(2.2)

0.05

51.3(0.5)

134.1(1.3)

370(3.6)

336.4(3.4)

225.4(2.2)

200.4(2)

0.1

16.5(0.16)

31.1(0.3)

117.1(1.2)

189.9(1.9)

159.7(1.6)

161.8(1.6)

0.2

4.4(0.04)

8(0.07)

15.5(0.1)

29.9(0.3)

44.2(0.4)

53.9(0.5)

0.5

1.8(0.01)

1.9(0.1)

2.3(0.02)

2.7(0.02)

3.2(0.03)

3.5(0.03)

23

Table 8. The simulated ARL’s of the based chart under scenario I for the proposed and logit link functions. +δ

+δ

δ

logit link

proposed link

δ

logit link

proposed link

0

200.6(2)

200.01(2)

0

200.6(2)

200.01(2)

0.15

252.2(2.5)

152.3(1.5)

0.1

312.6(3)

165.2(1.7)

0.25

237.8(2.3)

102.7(1)

0.25

337.2(3.3)

83.5(0.8)

0.5

136.3(1.4)

39.4(0.4)

0.5

213.3(2.1)

32(0.3)

0.75

75.7(0.8)

20.3(0.19)

0.75

127.4(1.3)

17.2(0.1)

1

44.6(0.4)

12.6(0.1)

1

80.1(0.8)

11.3(0.1)

1.5

21(0.2)

7(0.06)

2

27(0.3)

4.6(0.04)

3

8.7(0.08)

3.4 (0.02)

3.5

13.1(0.1)

2.9(0.02)

4

7.1(0.06)

2.8 (0.02)

Table 9. The simulated ARL’s of the based chart under scenario II for the proposed and logit link functions. -δ

-δ

δ

logit link

proposed link

δ

logit link

proposed link

0

200.6(2)

200.01(2)

0

200.6(2)

200.01(2)

0.15

252.2(2.5)

152.3(1.5)

0.1

312.6(3)

165.2(1.7)

0.25

237.8(2.3)

102.7(1)

0.25

337.2(3.3)

83.5(0.8)

0.5

136.3(1.4)

39.4(0.4)

0.5

213.3(2.1)

32(0.3)

0.75

75.7(0.8)

20.3(0.19)

0.75

127.4(1.3)

17.2(0.1)

1

44.6(0.4)

12.6(0.1)

1

80.1(0.8)

11.3(0.1)

1.5

21(0.2)

7(0.06)

2

27(0.3)

4.6(0.04)

3

8.7(0.08)

3.4 (0.02)

3.5

13.1(0.1)

2.9(0.02)

4

7.1(0.06)

2.8 (0.02) 24

Table 10. The simulated ARL’s of the based chart under scenario III for the proposed and logit link functions when only shifts. +δ

-δ

δ

logit link

proposed link

0

200.6(2)

0.05

δ

logit link

proposed link

200.5(2)

200.6(2)

200.5(2)

172.3(1.7)

101.7(1)

213.5(2.1)

100.9(1)

0.15

110.5(1.1)

17.5(0.2)

187.3(1.8)

16.7(0.1)

0.25

68.9(0.7)

5.8(0.05)

122.3(1.2)

5.5(0.05)

0.5

20.9(0.2)

1.7(0.01)

36(0.3)

1.5(0.01)

Table 11. Paint defect data Sample number

Sample number 1

100

1

9

13

100

3

8

2

100

3

8

14

100

2

11

3

100

3

10

15

100

2

7

4

100

3

7

16

100

2

6

5

100

1

3

17

100

5

10

6

100

1

6

18

100

1

7

7

100

2

7

19

100

2

8

8

100

4

8

20

100

10

15

9

100

3

8

21

100

3

10

10

100

3

14

22

100

2

10

11

100

2

15

23

100

3

8

12

100

1

5

24

100

2

10

25

0.999

0.99 0.98

Mean(DR)=0.078 Std(DR)=1.0437 N=1000 P-Value=0.51

0.997 0.99 0.98

0.95

0.95

0.90

0.90

0.75

0.75 Probability

Probability

0.999

Mean(DR)=0.147 Std(DR)=1.072 N=1000 P-Value=0.19

0.997

0.50

0.50

0.25

0.25

0.10

0.10

0.05

0.05

0.02 0.01

0.02 0.01

0.003

0.003

0.001

0.001 -3

-2

-1

0

1

2

3

-3

-2

-1

DR

Figure 1(a). Normal probability plot of the DR statistic when n=5

0 DR

1

2

3

Figure 1(b). Normal probability plot of the DR statistic when n=16

0.999

Mean(DR)=0.025 Std(DR)=1.0032 N=1000 P-Value=0.84

0.997 0.99 0.98 0.95 0.90

Probability

0.75

0.50

0.25 0.10 0.05 0.02 0.01 0.003 0.001 -3

Figure 1(c). Normal probability plot of the DR statistic when n=40

-2

-1

0 DR

1

3

Figure 1(d). Normal probability plot of the DR statistic when n=80

Figure 1. Normal probability plot of the DR statistic when n=5,16,40 and 80

26

2

1 0.9 0.8 0.7

p

0.6 0.5 0.4 0.3 0.2

logit proposed link function

0.1 0 -10

-8

-6

-4

Figure 2(a). The plots of x versus

-2

0 x

2

4

6

8

10

for logit and proposed link functions when regression

parameters are 1 and 0.5. 1 0.9 0.8 0.7

p

0.6 0.5 0.4 0.3 0.2

logit proposed link function

0.1 0 -10

-8

-6

-4

Figure 2(b). The plots of x versus

-2

0 x

2

4

6

8

10

for logit and proposed link functions when regression

parameters are 1 and -0.5.

27

Figure 3. The ARL comparison for DR and R based control charts when only β0 shifts (the horizontal axis is the shift magnitude, δ, and the vertical axis is the ARL).

Figure 4. The ARL comparison for DR and R based control charts when β1 shifts (the horizontal axis is the shift magnitude, λ, and the vertical axis is the ARL).

28

Figure 5. The

Figure 6. The

based chart for samples 15 to 24 and a generated 25th sample.

-chart for samples 15 to 24 and a generated 25th sample.

29