Reduction of the Effect of Estimation Error on In‐control Performance for Risk‐adjusted Bernoulli CUSUM Chart with Dynamic Probability Control Limits

The in‐control performance of any control chart is highly associated with the accuracy of estimation for the in‐control parameter(s). For the risk‐adjusted Bernoulli cumulative sum (CUSUM) chart with a constant control limit, it had been shown that the estimation error could have a substantial effect on the in‐control performance. In our study, we examine the effect of estimation error on the in‐control performance of the risk‐adjusted Bernoulli CUSUM chart with dynamic probability control limits (DPCLs). Our simulation results show that the in‐control performance of risk‐adjusted Bernoulli CUSUM chart with DPCLs is also affected by the estimation error. The most important factors affecting estimation error are the specified desired in‐control average run length, the Phase I sample size, and the adverse event rate. However, the effect of estimation error is uniformly smaller for the risk‐adjusted Bernoulli CUSUM chart with DPCLs than for the corresponding chart with a constant control limit under various realistic scenarios. In addition, we found a substantial reduction in the mean and variation of the standard deviation of the in‐control run length when DPCLs are used. Therefore, use of DPCLs has yet another advantage when designing a risk‐adjusted Bernoulli CUSUM chart. Copyright © 2016 John Wiley & Sons, Ltd.     

Dynamic Probability Control Limits for Lower and Two‐Sided Risk‐Adjusted Bernoulli CUSUM Charts

Because of its advantages of design, performance, and effectiveness in reducing the effect of patients' prior risks, the risk‐adjusted Bernoulli cumulative sum (CUSUM) chart is widely applied to monitor clinical and surgical outcome performance. In practice, it is beneficial to obtain evidence of improved surgical performance using the lower risk‐adjusted Bernoulli CUSUM charts. However, it had been shown that the in‐control performance of the charts with constant control limits varies considerably for different patient populations. In our study, we apply the dynamic probability control limits (DPCLs) developed for the upper risk‐adjusted Bernoulli CUSUM charts to the lower and two‐sided charts and examine their in‐control performance. The simulation results demonstrate that the in‐control performance of the lower risk‐adjusted Bernoulli CUSUM charts with DPCLs can be controlled for different patient populations, because these limits are determined for each specific sequence of patients. In addition, practitioners could also run upper and lower risk‐adjusted Bernoulli CUSUM charts with DPCLs side by side simultaneously and obtain desired in‐control performance for the two‐sided chart for any particular sequence of patients for a surgeon or hospital.     

Cumulative Sum Control Charts for Monitoring Weibull‐distributed Time Between Events

The Weibull distribution can be used to effectively model many different failure mechanisms due to its inherent flexibility through the appropriate selection of a shape and a scale parameter. In this paper, we evaluate and compare the performance of three cumulative sum (CUSUM) control charts to monitor Weibull‐distributed time‐between‐event observations. The first two methods are the Weibull CUSUM chart and the exponential CUSUM (ECUSUM) chart. The latter is considered in literature to be robust to the assumption of the exponential distribution when observations have a Weibull distribution. For the third CUSUM chart included in this study, an adjustment in the design of the ECUSUM chart is used to account for the true underlying time‐between‐event distribution. This adjustment allows for the adjusted ECUSUM chart to be directly comparable to the Weibull CUSUM chart. By comparing the zero‐state average run length and average time to signal performance of the three charts, the ECUSUM chart is shown to be much less robust to departures from the exponential distribution than was previously claimed in the literature. We demonstrate the advantages of using one of the other two charts, which show surprisingly similar performance. Copyright © 2014 John Wiley & Sons, Ltd.     

Exponential CUSUM Charts with Estimated Control Limits

Exponential CUSUM charts are used in monitoring the occurrence rate of rare events because the interarrival times of events for homogeneous Poisson processes are independent and identically distributed exponential random variables. In these applications, it is assumed that the exponential parameter, i.e. the mean, is known or has been accurately estimated. However, in practice, the in‐control mean is typically unknown and must be estimated to construct the limits for the exponential CUSUM chart. In this article, we investigate the effect of parameter estimation on the run length properties of one‐sided lower exponential CUSUM charts. In addition, analyzing conditional performance measures shows that the effect of estimation error can be significant, affecting both the in‐control average run length and the quick detection of process deterioration. We also provide recommendations regarding phase I sample sizes. This sample size must be quite large for the in‐control chart performance to be close to that for the known parameter case. Finally, we provide an industrial example to highlight the practical implications of estimation error, and to offer advice to practitioners when constructing/analyzing a phase I sample. Copyright © 2013 John Wiley & Sons, Ltd.     

The performance of EWMA median and CUSUM median control charts for a normal process with measurement errors

Measurement error is often occurred in statistical process control. The effect of a linearly covariate error model on the exponentially weighted moving average (EWMA) median and cumulative sum (CUSUM) median charts is investigated. The results indicate that the EWMA median and CUSUM median charts are significantly affected in the presence of measurement errors. We compared the performance of the EWMA median and CUSUM median charts by using Markov chain method in the average run length and the standard deviation of the run length. We concluded that the CUSUM median chart for small shifts and the EWMA median chart for larger shifts are recommended. Two examples are provided to illustrate the application of the EWMA and CUSUM median charts with measurement errors.     

An Evaluation of the Crosier's CUSUM Control Chart with Estimated Parameters

We evaluate the performance of the Crosier's cumulative sum (C‐CUSUM) control chart when the probability distribution parameters of the underlying quality characteristic are estimated from Phase I data. Because the average run length (ARL) under estimated parameters is a random variable, we study the estimation effect on the chart performance in terms of the expected value of the average run length (AARL) and the standard deviation of the average run length (SDARL). Previous evaluations of this control chart were conducted while assuming known process parameters. Using the Markov chain and simulation approaches, we evaluate the in‐control performance of the chart and provide some quantiles for its in‐control ARL distribution under estimated parameters. We also compare the performance of the C‐CUSUM chart to that of the ordinary CUSUM (O‐CUSUM) chart when the process parameters are unknown. Our results show that large number of Phase I samples are required to achieve a quite reasonable performance. Additionally, the performance of the C‐CUSUM chart is found to be superior to that of the O‐CUSUM chart. Finally, we recommend the use of a recently proposed bootstrap procedure in designing the C‐CUSUM chart to guarantee, at a certain probability, that the in‐control ARL will be of at least the desired value using the available amount of Phase I data. Copyright © 2015 John Wiley & Sons, Ltd.     

A New Cumulative Sum Quality Control Scheme for Monitoring the Process Mean

A control chart is a powerful statistical process monitoring tool that is frequently used in many industrial and service organizations to monitor in‐control and out‐of‐control performances of the manufacturing processes. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts have been recognized as potentially powerful tool in quality and management control. These control charts are sensitive to both small and moderate changes in the process. In this paper, we propose a new CUSUM (NCUSUM) quality control scheme for efficiently monitoring the process mean. It is shown that the classical CUSUM control chart is a special case of the proposed controlling scheme. The NCUSUM control chart is compared with some of the recently proposed control charts by using characteristics of the distribution of run length, i.e. average run length, median run length and standard deviation of run length. It is worth mentioning that the NCUSUM control chart detects the random shifts in the process mean substantially quicker than the classical CUSUM, fast initial response‐based CUSUM, adaptive CUSUM with EWMA‐based shift, adaptive EWMA and Shewhart–CUSUM control charts. An illustrative example is given to exemplify the implementation of the proposed quality control scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

A linear‐time method for computing the ANOS of Bernoulli and Poisson CUSUM charts

The average number of observations to signal (ANOS) is an important measure of the effectiveness of Bernoulli or Poisson cumulative sum (CUSUM) control charts. Being able to quickly and accurately calculate an ANOS vector facilitates effective control chart design. We present a linear‐time method for computing the ANOS of Bernoulli CUSUM charts and generalize to Poisson CUSUM charts. This method overcomes computation challenges associated with previously existing methods.     

Increasing the Sensitivity of Cumulative Sum Charts for Location

The cumulative sum (CUSUM) chart is a very effective control charting procedure used for the quick detection of small‐sized and moderate‐sized changes. It can detect small process shifts missed by the Shewhart‐type control chart, which is sensitive mainly to large shifts. To further enhance the sensitivity of the CUSUM control chart at detecting very small process disturbances, this article presents CUSUM control charts based on well‐structured sampling procedures, double ranked set sampling, median‐double ranked set sampling, and double‐median ranked set sampling. These sampling techniques significantly improve the overall performance of the CUSUM chart over the entire process mean shift range, without increasing the false alarm rate. The newly developed control schemes do not only dominate most of the existing charts but are also easy to design and implement as illustrated through an application example of real datasets. The control schemes used for comparison in this study include the conventional CUSUM chart, a fast initial response CUSUM chart, a 2‐CUSUM chart, a 3‐CUSUM chart, a runs rules‐based CUSUM chart, the enhanced adaptive CUSUM chart, the CUSUM chart based on ranked set sampling (RSS), and the single CUSUM and combined Shewhart–CUSUM charts based on median RSS. Copyright © 2014 John Wiley & Sons, Ltd.     

CUSUM charts with controlled conditional performance under estimated parameters

We study the effect of the Phase I estimation error on the cumulative sum (CUSUM) chart. Impractically large amounts of Phase I data are needed to sufficiently reduce the variation in the in-control average run lengths (ARL) between practitioners. To reduce the effect of estimation error on the chart's performance we design the CUSUM chart such that the in-control ARL exceeds a desired value with a specified probability. This is achieved by adjusting the control limits using a bootstrap-based design technique. Such approach does affect the out-of-control performance of the chart; however, we find that this effect is relatively small.     

Improved CUSUM monitoring of Markov counting process with frequent zeros

There has been a growing interest in monitoring processes featuring serial dependence and zero inflation. The phenomenon of excessive zeros often occurs in count time series because of the advancement of quality in manufacturing process. In this study, we propose three control charts, such as the cumulative sum chart with delay rule (CUSUM‐DR), conforming run length (CRL)‐CUSUM chart, and combined Shewhart CRL‐CUSUM chart, to enhance the performance of monitoring Markov counting processes with excessive zeros. Numerical experiments are conducted based on integer‐valued autoregressive time series models, for example, zero‐inflated Poisson INAR and INARCH, to evaluate the performance of the proposed charts designed for the detection of mean increase. A real example is also illustrated to demonstrate the usability of our proposed charts.     

Mixed Cumulative Sum–Exponentially Weighted Moving Average Control Charts: An Efficient Way of Monitoring Process Location

Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) charts are famous statistical tools, to handle special causes and to bring the process back in statistical control. Shewhart charts are useful to detect large shifts, whereas EWMA and CUSUM are more sensitive for small to moderate shifts. In this study, we propose a new control chart, named mixed CUSUM‐EWMA chart, which is used to monitor the location of a process. The performance of the proposed mixed CUSUM‐EWMA control chart is measured through the average run length, extra quadratic loss, relative average run length, and a performance comparison index study. Comparisons are made with some existing charts from the literature. An example with real data is also given for practical considerations. Copyright © 2014 John Wiley & Sons, Ltd.     

The Effect of Parameter Estimation on Upper‐sided Bernoulli Cumulative Sum Charts

The Bernoulli cumulative sum (CUSUM) chart has been shown to be effective for monitoring the rate of nonconforming items in high‐quality processes where the in‐control proportion of nonconforming items (p₀) is low. The implementation of the Bernoulli CUSUM chart is often based on the assumption that the in‐control value p₀ is known; therefore, when p₀ is unknown, accurate estimation is necessary. We recommend using a Bayes estimator to estimate the value of p₀ to incorporate practitioner knowledge and to avoid estimation issues when no nonconforming items are observed in phase I. We also investigate the effects of parameter estimation in phase I on the upper‐sided Bernoulli CUSUM chart by using the expected value of the average number of observations to signal (ANOS) and the standard deviation of the ANOS. It is found that the effects of parameter estimation on the Bernoulli CUSUM chart are more significant than those on the Shewhart‐type geometric chart. The low p₀ values inherent to high‐quality processes imply that a very large, and often unrealistic, sample size may be needed to accurately estimate p₀. A methodology to identify a continuous variable to monitor is highly recommended when the value of p₀ is low and the required phase I sample size is impractically large. Copyright © 2012 John Wiley & Sons, Ltd.     

A Study of Control Chart for Monitoring Exponentially Distributed Characteristics Based On Type‐II Censored Samples

In this paper, control charts for monitoring the exponential type‐II censoring samples are investigated. Such data are very common in many practical inspection scenarios in reliability context when items are replaced in groups after a period of time. The average time to signal, which involves both the number and the time of samples inspected until a signal occurs, is a good criterion to evaluate the performance of control charts. We propose an average time to signal‐unbiased control chart with known parameter and compare the proposed method with the traditional ones. The results indicate the proposed control chart is more sensitive to system deterioration. Then the effects of parameter estimation on the proposed control charts are evaluated. Because the control limits with estimated parameters result in more false alarms, an adjusted control chart with estimated parameters is proposed and the self‐starting control chart based on a sequential sampling scheme is adopted to solve the phase I problem. Finally, two examples are given to illustrate the implementation of the proposed approach. Copyright © 2017 John Wiley & Sons, Ltd.     

Guaranteed in‐control performance of the EWMA chart for monitoring the mean

Research on the performance evaluation and the design of the Phase II EWMA control chart for monitoring the mean, when parameters are estimated, have mainly focused on the marginal in‐control average run‐length (ARL_IN). Recent research has highlighted the high variability in the in‐control performance of these charts. This has led to the recommendation of studying of the conditional in‐control average run‐length (CARL_IN) distribution. We study the performance and the design of the Phase II EWMA chart for the mean, using the CARL_IN distribution and the exceedance probability criterion (EPC). The CARL_IN distribution is approximated by the Markov Chain method and Monte Carlo simulations. Our results show that in‐order to design charts that guarantee a specified EPC, more Phase I data are needed than previously recommended in the literature. A method for adjusting the Phase II EWMA control chart limits, to achieve a specified EPC, for the available amount of data at hand, is presented. This method does not involve bootstrapping and produces results that are about the same as some existing results. Tables and graphs of the adjusted constants are provided. An in‐control and out‐of‐control performance evaluation of the adjusted limits EWMA chart is presented. Results show that, for moderate to large shifts, the performance of the adjusted limits EWMA chart is quite satisfactory. For small shifts, an in‐control and out‐of‐control performance tradeoff can be made to improve performance.     

Using FIR to Improve CUSUM Charts for Monitoring Process Dispersion

Statistical process control deals with monitoring process to detect disturbances in the process. These disturbances may be from the process mean or variance. In this study, we propose some charts that are efficient for detecting early shifts in dispersion parameter, by applying the Fast Initial Response feature. Performance measures such as average run length, standard deviation of the run length, extra quadratic loss, relative average run length, and performance comparison index are used to compare the proposed charts with their existing counterparts, including the Shewhart R chart and the Shewhart S chart with and without warning lines. Others include the CUSUM R chart, the CUSUM S chart, the EWMA of ln S², the CUSUM of ln S², the P_σ CUSUM, the χ CUSUM, and the Change Point (CP) CUSUM charts. The proposed charts do not only detect early shifts in the process dispersion faster, but also have better overall performance than their existing counterparts. Copyright © 2016 John Wiley & Sons, Ltd.     

A comparison of control charts from the viewpoint of change-point estimation

ISO/DIS 7870 has presented the cumulative sum chart, the moving average chart, and the exponentially weighted moving average chart as control charts using accumulated data. In this paper, we compare the three control charts in terms of change-point estimation. We show the probability distribution, the bias and the mean square error of the change-point estimators using a Markov process and Monte Carlo simulation. These control charts have almost equivalent performances based on average run length considerations when parameters of each control chart are set appropriately. However, from the viewpoint of change-point estimation we recommend the CUSUM chart.     

A double exponentially weighted moving average chart based on likelihood ratio for monitoring an inflated Pareto process

In this paper, we present a new chart called a likelihood ratio based double exponentially weighted moving average (LR_DEWMA) chart to monitor the shape parameter of the inflated Pareto process. Three other control charts such as the Shewhart type, the classical cumulative sum (CUSUM), and the likelihood ratio based EWMA (LR_EWMA) charts are also investigated. The performance of the control charts is evaluated by the average run length (ARL) and standard deviation of run lengths (SDRL) computed through the Monte Carlo simulation approach. Moreover, the median run length (MRL) and some other run length (RL) percentiles are also considered in some cases. Different charts have shown the best performance in different cases. In detecting smaller shifts, while the LR_DEWMA chart outperformed the other charts in terms of ARL and MRL, the CUSUM chart has shown the best performance in terms of SDRL and IQR of RLs. The application of the proposed control charts is illustrated using a chromatography analyses data from the food industry.     

A head-to-head comparative study of the conditional performance of control charts based on estimated parameters

Implementation of the Shewhart, CUSUM, and EWMA charts requires estimates of the in-control process parameters. Many researchers have shown that estimation error strongly influences the performance of these charts. However, a given amount of estimation error may differ in effect across charts. Therefore, we perform a pairwise comparison of the effect of estimation error across these charts. We conclude that the Shewhart chart is more strongly affected by estimation error than the CUSUM and EWMA charts. Furthermore, we show that the general belief that the CUSUM and EWMA charts have similar performance no longer holds under estimated parameters.     

CUSUM charts for detecting special causes in integrated process control

This paper investigates control charts for detecting special causes in an ARIMA(0, 1, 1) process that is being adjusted automatically after each observation using a minimum mean‐squared error adjustment policy. It is assumed that the adjustment mechanism is designed to compensate for the inherent variation due to the ARIMA(0, 1, 1) process, but it is desirable to detect and eliminate special causes that occur occasionally and produce additional process variation. It is assumed that these special causes can change the process mean, the process variance, the moving average parameter, or the effect of the adjustment mechanism. Expressions are derived for the process deviation from target for all of these process parameter changes. Numerical results are presented for sustained shifts, transient shifts, and sustained drifts in the process parameters. The objective is to find control charts or combinations of control charts that will be effective for detecting special causes that result in any of these types of parameter changes in any or all of the parameters. CUSUM charts designed for detecting specific parameter changes are considered. It is shown that combinations of CUSUM charts that include a CUSUM chart designed to detect mean shifts and a CUSUM chart of squared deviations from target give good overall performance in detecting a wide range of process changes. Copyright © 2009 John Wiley & Sons, Ltd.