Runs rules schemes for monitoring process variability

To increase the sensitivity of Shewhart control charts in detecting small process shifts sensitizing rules based on runs and scans are often used in practice. Shewhart control charts supplemented with runs rules for detecting shifts in process variance have not received as much attention as their counterparts for detecting shifts in process mean. In this article, we examine the performance of simple runs rules schemes for monitoring increases and/or decreases in process variance based on the sample standard deviation. We introduce one-sided S charts that overcome the weakness of high false-alarm rates when runs rules are added to a Shewhart control chart. The average run length performance and design aspects of the charts are studied thoroughly. The performance of associated two-sided control schemes is investigated as well.     

Design of adaptive s control charts

The Shewhart s chart has been widely used to monitor the standard deviation of a process. However, the main disadvantage of an s chart is its slowness to signal small increases in the variability. In this paper, ideas of adaptive control charts are extended to the Shewhart s chart for improving the efficiency in signalling increases in the standard deviation. A Markov chain model is applied to evaluate its performances and compares its performances with combined double sampling and variable sampling intervals s chart, variable parameters (VP) R chart, exponentially weighted moving average and Cusum charts. The statistical performances show that the VP s chart is more sensitive to increases in standard deviation.     

Analysis of means: a generalized approach using R

Papers on the analysis of means (ANOM) have been circulating in the quality control literature for decades, routinely describing it as a statistical stand-alone concept. Therefore, we clarify that ANOM should rather be regarded as a special case of a much more universal approach known as multiple contrast tests (MCTs). Perceiving ANOM as a grand-mean-type MCT paves the way for implementing it in the open-source software R. We give a brief tutorial on how to exploit R's versatility and introduce the R package ANOM for drawing the familiar decision charts. Beyond that, we illustrate two practical aspects of data analysis with ANOM: firstly, we compare merits and drawbacks of ANOM-type MCTs and ANOVA F-test and assess their respective statistical powers, and secondly, we show that the benefit of using critical values from multivariate t-distributions for ANOM instead of simple Bonferroni quantiles is oftentimes negligible.     

Combined Shewhart–EWMA control charts with estimated parameters

Shewhart and EWMA control charts can be suitably combined to obtain a simple monitoring scheme sensitive to both large and small shifts in the process mean. So far, the performance of the combined Shewhart–EWMA (CSEWMA) has been investigated under the assumption that the process parameters are known. However, parameters are often estimated from reference Phase I samples. Since chart performances may be even largely affected by estimation errors, we study the behaviour of the CSEWMA with estimated parameters in both in- and out-of-control situations. Comparisons with standard Shewhart and EWMA charts are presented. Recommendations are given for Phase I sample size requirements necessary to achieve desired in-control performance.     

Improved R and s control charts for monitoring the process variance

The Shewhart R control chart and s control chart are widely used to monitor shifts in the process spread. One fact is that the distributions of the range and sample standard deviation are highly skewed. Therefore, the R chart and s chart neither provide an in-control average run length (ARL) of approximately 370 nor guarantee the desired type I error of 0.0027. Another disadvantage of these two charts is their failure in detecting an improvement in the process variability. In order to overcome these shortcomings, we propose the improved R chart (IRC) and s chart (ISC) with accurate approximation of the control limits by using cumulative distribution functions of the sample range and standard deviation. Simulation studies show that the IRC and ISC perform very well. We also compare the type II error risks and ARLs of the IRC and ISC and found that the s chart is generally more efficient than the R chart. Examples are given to illustrate the use of the developed charts.     

The effect of parameter estimation on phase II monitoring of poisson regression profiles

ABSTRACT

The effect of parameters estimation on profile monitoring methods has only been studied by a few researchers and only the assumption of a normal response variable has been tackled. However, in some practical situation, the normality assumption is violated and the response variable follows a discrete distribution such as Poisson. In this paper, we evaluate the effect of parameters estimation on the Phase II monitoring of Poisson regression profiles by considering two control charts, namely the Hotelling’s T² and the multivariate exponentially weighted moving average (MEWMA) charts. Simulation studies in terms of the average run length (ARL) and the standard deviation of the run length (SDRL) are carried out to assess the effect of estimated parameters on the performance of Phase II monitoring approaches. The results reveal that both in-control and out-of-control performances of these charts are adversely affected when the regression parameters are estimated.     

Conditional design of the CUSUM median chart for the process position when process parameters are unknown

In practice, different practitioners will use different Phase I samples to estimate the process parameters, which will lead to different Phase II control chart's performance. Researches refer to this variability as between-practitioners-variability of control charts. Since between-practitioners-variability is important in the design of the CUSUM median chart with estimated process parameters, the standard deviation of average run length (SDARL) will be used to study its properties. It is shown that the CUSUM median chart requires a larger amount of Phase I samples to sufficiently reduce the variation in the in-control ARL of the CUSUM median chart. Considering the limitation of the amount of the Phase I samples, a bootstrap approach is also used here to adjust the control limits of the CUSUM median chart. Comparisons are made for the CUSUM and Shewhart median charts with estimated parameters when using the adjusted- and unadjusted control limits and some conclusions are made.     

A new approach for monitoring process variance

ABSTRACT

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having non-normal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used in such circumstances. In this paper, we propose a new variance chart based on a simple statistic to monitor process variance shifts. We explore the sampling properties of the new monitoring statistic and calculate the average run lengths (ARLs) of the proposed variance chart. Furthermore, an arcsine transformed exponentially weighted moving average (EWMA) chart is proposed because the ARLs of this modified chart are more intuitive and reasonable than those of the variance chart. We compare the out-of-control variance detection performance of the proposed variance chart with that of the non-parametric Mood variance (NP-M) chart with runs rules, developed by Zombade and Ghute [Nonparametric control chart for variability using runs rules. Experiment. 2014;24(4):1683–1691], and the nonparametric likelihood ratio-based distribution-free exponential weighted moving average (NLE) chart and the combination of traditional exponential weighted moving average (EWMA) mean and EWMA variance (CEW) control chart proposed by Zou and Tsung [Likelihood ratio-based distribution-free EWMA control charts. J Qual Technol. 2010;42(2):174–196] by considering cases in which the critical quality characteristic has a normal, a double exponential or a uniform distribution. Comparison results showed that the proposed chart performs better than the NP-M with runs rules, and the NLE and CEW control charts. A numerical example of service times with a right-skewed distribution from a service system of a bank branch in Taiwan is used to illustrate the application of the proposed variance chart and of the arcsine transformed EWMA chart and to compare them with three existing variance (or standard deviation) charts. The proposed charts show better detection performance than those three existing variance charts in monitoring and detecting shifts in the process variance.     

Synthetic exponential control charts with unknown parameter

The existing synthetic exponential control charts are based on the assumption of known in-control parameter. However, the in-control parameter has to be estimated from a Phase I dataset. In this article, we use the exact probability distribution, especially the percentiles, mean, and standard deviation of the conditional average run length (ARL) to evaluate the effect of parameter estimation on the performance of the Phase II synthetic exponential charts. This approach accounts for the variability in the conditional ARL values of the synthetic chart obtained by different practitioners. Since parameter estimation results in more false alarms than expected, we develop an exact method to design the adjusted synthetic charts with desired conditional in-control performance. Results of known and unknown in-control parameter cases show that the control limit of the conforming run length sub-chart of the synthetic chart should be as small as possible.     

Nonparametric quality control charts based on the sign statistic

Nonparametric control chart are presented for the problem of detecting changes in the process median (or mean), or changes in the process variability when samples are taken at regular time intervals. The proposed procedures are based on sign-test statistics computed for each sample, and are used in Shewhart and cumulative sum control charts. When the process is in control the run length distributions for the proposed nonparametric control charts do not depend on the distribution of the observations. An additional advantage of the non-parametric control charts is that the variance of the process does not need to be established in order to set up a control chart for the mean. Comparisons with the corresponding parametric control charts are presented. It is also shown that curtailed sampling plans can considerably reduce the expected number of observations used in the Shewhart control schemes based on the sign statistic.     

Zone control charts with estimated parameters for detecting prespecified changes in linear profiles

ABSTRACT

In profile monitoring, control charts are proposed to detect unanticipated changes, and it is usually assumed that the in-control parameters are known. However, due to the characteristics of a system or process, the prespecified changes would appear in the process. Moreover, in most applications, the in-control parameters are usually unknown. To overcome these issues, we develop the zone control charts with estimated parameters to detect small shifts of these prespecified changes. The effects of estimation error have been investigated on the performance of the proposed charts. To account for the practitioner-to-practitioner variability, the expected average run length (ARL) and the standard deviation of the average run length (SDARL) is used as the performance metrics. Our results show that the estimation error results in the significant variation in the ARL distribution. Furthermore, in order to adequately reduce the variability, more phase I samples are required in terms of the SDARL metric than that in terms of the expected ARL metric. In addition, more observations on each sampled profile are suggested to improve the charts' performance, especially for small phase I sample sizes. Finally, an illustrative example is given to show the performance of the proposed zone control charts.     

The effect of Phase I sample size on the run length performance of control charts for autocorrelated data

Traditional control charts assume independence of observations obtained from the monitored process. However, if the observations are autocorrelated, these charts often do not perform as intended by the design requirements. Recently, several control charts have been proposed to deal with autocorrelated observations. The residual chart, modified Shewhart chart, EWMAST chart, and ARMA chart are such charts widely used for monitoring the occurrence of assignable causes in a process when the process exhibits inherent autocorrelation. Besides autocorrelation, one other issue is the unknown values of true process parameters to be used in the control chart design, which are often estimated from a reference sample of in-control observations. Performances of the above-mentioned control charts for autocorrelated processes are significantly affected by the sample size used in a Phase I study to estimate the control chart parameters. In this study, we investigate the effect of Phase I sample size on the run length performance of these four charts for monitoring the changes in the mean of an autocorrelated process, namely an AR(1) process. A discussion of the practical implications of the results and suggestions on the sample size requirements for effective process monitoring are provided.     

A Multivariate Synthetic Control Chart for Monitoring the Process Mean Vector of Skewed Populations Using Weighted Standard Deviations

This article proposes a multivariate synthetic control chart for skewed populations based on the weighted standard deviation method. The proposed chart incorporates the weighted standard deviation method into the standard multivariate synthetic control chart. The standard multivariate synthetic chart consists of the Hotelling's T ² chart and the conforming run length chart. The weighted standard deviation method adjusts the variance–covariance matrix of the quality characteristics and approximates the probability density function using several multivariate normal distributions. The proposed chart reduces to the standard multivariate synthetic chart when the underlying distribution is symmetric. In general, the simulation results show that the proposed chart performs better than the existing multivariate charts for skewed populations and the standard T ² chart, in terms of false alarm rates as well as moderate and large mean shift detection rates based on the various degrees of skewnesses.     

Robustness to non-normality and autocorrelation of individuals control charts

This paper studies the effects of non-normality and autocorrelation on the performances of various individuals control charts for monitoring the process mean and/or variance. The traditional Shewhart X chart and moving range (MR) chart are investigated as well as several types of exponentially weighted moving average (EWMA) charts and combinations of control charts involving these EWMA charts. It is shown that the combination of the X and MR charts will not detect small and moderate parameter shifts as fast as combinations involving the EWMA charts, and that the performana of the X and MR charts is very sensitive to the normality assumption. It is also shown that certain combinations of EWMA charts can be designed to be robust to non-normality and very effective at detecting small and moderate shifts in the process mean and/or variance. Although autocorrelation can have a significant effect on the in-control performances of these combinations of EWMA charts, their relative out-of-control performances under independence are generally maintained for low to moderate levels of autocorrelation.     

Comparison of Robust Estimators of Standard Deviation in Normal Distributions Within the Context of Quality Control

Pitman criterion is used in simulation to determine the “closer” estimator of the standard deviation among selected choices. The initial simulation utilizes a standard normal distribution from which samples are taken of specific sizes. Popular and commonly used estimators of standard deviation are compared with the known population standard deviation in this study. Closeness criterion is calculated for each comparison and sample size. A secondary simulation applies the findings to variables control charts, in order to verify the ability of each estimator to identify out-of-control conditions.     

The product <Emphasis Type="Italic">t</Emphasis> density distribution arising from the product of two Student’s <Emphasis Type="Italic">t</Emphasis> PDFs

The Student’s t distribution has become increasingly prominent and is considered as a competitor to the normal distribution. Motivated by real examples in Physics, decision sciences and Bayesian statistics, a new t distribution is introduced by taking the product of two Student’s t pdfs. Various structural properties of this distribution are derived, including its cdf, moments, mean deviation about the mean, mean deviation about the median, entropy, asymptotic distribution of the extreme order statistics, maximum likelihood estimates and the Fisher information matrix. Finally, an application to a Bayesian testing problem is illustrated.     

An Improved Distribution-free EWMA Mean Chart

Control charts are effective tools for signal detection in both manufacturing processes and service processes. Much service data come from a process with variables having nonnormal or unknown distributions. The commonly used Shewhart variable control charts, which depend heavily on the normality assumption, should not be properly used here. In this article, we propose an improved asymmetric EWMA mean chart based on a simple statistic to monitor process mean shift. We explored the sampling properties of the new monitoring statistic and calculated the average run lengths of the proposed asymmetric EWMA mean chart. We recommend the proposed improved asymmetric EWMA mean chart because the average run lengths of the modified charts are more accurate and reasonable than those of the five existed mean charts. A numerical example of service times with a right skewed distribution from a service system of a bank branch is used to illustrate the application of the improved asymmetric EWMA mean chart and to compare it with the five existing mean charts. The proposed chart showed better detection performance than those of the five existing mean charts in monitoring and detecting shifts in the process mean.     

Performance of the zone control chart

Average run lengths of the zone control chart are presented, The performance of this chart is compared with that of several Shewhart charts with and without runs rules, It is shown that the standard zone control chart has performance similar to some even simpler charts and a much higher false alarm rate than the Shewhart chart with all of the common runs rules. It is also shown that a slightly modified zone control chart outperforms the Shewhart chart with the common runs rules.     

EWMA charts: ARL considerations in case of changes in location and scale

Widely spread tools within the area of Statistical Process Control are control charts of various designs. Control chart applications are used to keep process parameters (e.g., mean \(\mu \) , standard deviation \(\sigma \) or percent defective \(p\) ) under surveillance so that a certain level of process quality can be assured. Well-established schemes such as exponentially weighted moving average charts (EWMA), cumulative sum charts or the classical Shewhart charts are frequently treated in theory and practice. Since Shewhart introduced a \(p\) chart (for attribute data), the question of controlling the percent defective was rarely a subject of an analysis, while several extensions were made using more advanced schemes (e.g., EWMA) to monitor effects on parameter deteriorations. Here, performance comparisons between a newly designed EWMA \(p\) control chart for application to continuous types of data, \(p=f(\mu ,\sigma )\) , and popular EWMA designs ( \(\bar{X}\) , \(\bar{X}\) - \(S^2\) ) are presented. Thus, isolines of the average run length are introduced for each scheme taking both changes in mean and standard deviation into account. Adequate extensions of the classical EWMA designs are used to make these specific comparisons feasible. The results presented are computed by using numerical methods.