Parametric control charts

Standard control charts are based on the assumption that the observations are normally distributed. In practice, normality often fails and consequently the false alarm rate is seriously in error. Application of a nonparametric approach is only possible with many Phase I observations. Since nowadays such very large sample sizes are usually not available, there is need for an intermediate approach by considering a larger parametric model containing the normal family as a submodel. In this paper control limits are presented in such a larger parametric model, the so called normal power family. Correction terms are derived, taking into account that the parameters are estimated. Simulation results show that the control limits are accurate, not only in the considered parametric family, but also for common distributions outside the parametric family, thus covering a broad class of distributions.     

Minimum control charts

Standard control charts are very sensitive to estimation effects and/or deviations from normality. Hence a program has been carried out to remedy these problems. This is quite adequate in most circumstances, but not in all. In the present paper, the remaining complication is attacked: what to do if a nonparametric approach is indicated, but too few control observations are available for the estimation step? It is shown that grouping the observations during the monitoring phase works well. Surprisingly, rather than using the group averages, it is definitely preferable to compare the minimum for each group to a suitably chosen upper control limit. (And in the two-sided case, also the maximum to an analogous lower control limit.) This ‘minimum control chart’ is demonstrated to be quite attractive: it is easy to explain and to implement. Moreover, while it is truly nonparametric, its power of detection is comparable to that of the customary, normality assuming, charts based on averages.     

Control charts in practice

In the first issue of Significance , Roland Caulcutt suggested that a common feature of many successful companies is their development of a "management-by-fact" culture. He further suggested that such a culture requires that managers focus on processes, and that control charts be used at all levels throughout the organisation. In this follow-up article he explains how the various types of control chart are set up and used in practice.     

Double-sampling control charts for attributes

In this article, we propose a double-sampling (DS) np control chart. We assume that the time interval between samples is fixed. The choice of the design parameters of the proposed chart and also comparisons between charts are based on statistical properties, such as the average number of samples until a signal. The optimal design parameters of the proposed control chart are obtained. During the optimization procedure, constraints are imposed on the in-control average sample size and on the in-control average run length. In this way, required statistical properties can be assured. Varying some input parameters, the proposed DS np chart is compared with the single-sampling np chart, variable sample size np chart, CUSUM np and EWMA np charts. The comparisons are carried out considering the optimal design for each chart. For the ranges of parameters considered, the DS scheme is the fastest one for the detection of increases of 100% or more in the fraction non-conforming and, moreover, the DS np chart is easy to operate.     

An overview of risk-adjusted charts

Summary.  The paper provides an overview of risk-adjusted charts, with examples based on two data sets: the first consisting of outcomes following cardiac surgery and patient factors contributing to the Parsonnet score; the second being age–sex-adjusted death-rates per year under a single general practitioner. Charts presented include the cumulative sum (CUSUM), resetting sequential probability ratio test, the sets method and Shewhart chart. Comparisons between the charts are made. Estimation of the process parameter and two-sided charts are also discussed. The CUSUM is found to be the least efficient, under the average run length (ARL) criterion, of the resetting sequential probability ratio test class of charts, but the ARL criterion is thought not to be sensible for comparisons within that class. An empirical comparison of the sets method and CUSUM, for binary data, shows that the sets method is more efficient when the in-control ARL is small and more efficient for a slightly larger range of in-control ARLs when the change in parameter being tested for is larger. The Shewart p -chart is found to be less efficient than the CUSUM even when the change in parameter being tested for is large.     

Bivariate dispersion quality control charts

A Shewhart procedure is used to simultaneously control the standard deviations of quality characteristics assumed to have a bivariate normal distribution. Following Krishnaiah et al (1963), we use the bivariate chi-square distribution to determine probabilities of out-of-control signals and thus the respective average run lengths (ARLs). Results from an example indicate that for both one-sided and two-sided cases, signals occur only slightly more quickly for changes in the process standard deviations for uncorrected variables than for correlated variables.     

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.     

A class of distribution-free control charts

Summary.  A class of Shewhart-type distribution-free control charts is considered. A key advantage of these charts is that the in-control run length distribution is the same for all continuous process distributions. Exact expressions for the run length distribution and the average run length (ARL) are derived and properties of the charts are studied via evaluations of the run length distribution probabilities and the ARL. Tables are provided for implementation for some typical ARL values and false alarm rates. The charts proposed are preferable from a robustness point of view, have attractive ARL properties and would be particularly useful in situations where one uses a classical Shewhart   X -chart. A numerical illustration is given.     

Runs tests for group control charts

When a process consists of several identical streams that are not highly correlated, an alternative to using separate control charts for each stream is to use a group control chart. Rather than plotting sample means from each stream at any time point, one could plot only the largest and/or smallest sample mean from among all the streams. Using the theory of stochastic processes and majorization together with numerical methods, the properties of a test that signals if r consecutive extreme values come from the same stream are examined. Both one and two-sided cases are considered. Average run lengths (ARL's), the least favorable configuration of the stream (population) means, and sample sizes necessary to have specified in-control and out-of-control ARL's are obtained. A test that signals if r-1 out of r consecutive extreme values come from the same stream is also considered     

Data driven choice of control charts

Standard control charts are often seriously in error when the distributional form of the observations differs from normality. Recently, control charts have been developed for larger parametric families. A third possibility is to apply a suitable (modified version of a) nonparametric control chart. This paper deals with the question when to switch from the control chart based on normality to a parametric control chart, or even to a nonparametric one. This model selection problem is solved by using the estimated model error as yardstick. It is shown that the new combined control chart asymptotically behaves as each of the specific control charts in their own domain. Simulations exhibit that the combined control chart performs very well under a great variety of distributions and hence it is recommended as an omnibus control chart, nicely adapted to the distribution at hand. The combined control chart is illustrated by an application on real data. The new modified nonparametric control chart is an attractive alternative and can be recommended as well.     

Exceedance probabilities for parametric control charts

Common control charts assume normality and known parameters. Quite often, these assumptions are not valid and large relative errors result in the usual performance characteristics such as the false alarm rate or the average run length. A fully nonparametric approach can form an attractive alternative but requires more Phase I observations than usually available. Sufficiently general parametric families then provide realistic intermediate models. In this article, the performance of charts based on such families is considered. Exceedance probabilities of the resulting stochastic performance characteristics during in-control are studied. Corrections are derived to ensure that such probabilities stay within prescribed bounds. Attention is also devoted to the impact of the corrections for an out-of-control process. Simulations are presented both to illustrate and to demonstrate that the approximations obtained are sufficiently accurate for practical usage.     

Bayesian updating in reference centile charts

Reference centile charts, used for monitoring the health of an individual over time (e.g. the weight gain of pregnant women over successive periods in their pregnancy) do not take into account the longitudinal nature of the individual profiles. There is also generally more than one variable which affects the outcome of interest, and information regarding the path of a group of variables over time may prove advantageous in terms of sensitivity. We propose a Bayesian approach to this problem in which the successive deviations of the individual's observations from the mean of the corresponding reference distribution are used to compute updated reference centiles for future measurements on the individual. The univariate and multivariate situations are discussed, and consideration is given to non-normal cross-sectional distributions. The theory is illustrated on data obtained from the records of weight gain and fundal height of pregnant women visiting a clinic at intervals during pregnancy.     

Acceptance control charts for non-normal data

Control charts are one of the most important methods in industrial process control. The acceptance control chart is generally applied in situations when an X¯ chart is used to control the fraction of conforming units produced by the process and where 6-sigma spread of the process is smaller than the spread in the specification limits. Traditionally, when designing control charts, one usually assumes that the data or measurements are normally distributed. However, this assumption may not be true in some processes. In this paper, we use the Burr distribution, which is employed to represent various non-normal distributions, to determine the appropriate control limits or sample size for the acceptance control chart under non-normality. Some numerical examples are given for illustration. From the presented examples, ignoring the effect of non-normality in the data leads to a higher type I or type II error probability.     

Robust dual-CUSUM control charts for contaminated processes

ABSTRACT

In this article, we improve the efficiency of the Dual CUSUM chart (which combines the designs of two CUSUM structures to detect a range of shift) by focusing on its robustness, ability to resist some disturbances in the process environment and violation of basic assumptions. We do that, by proposing some robust estimators for constructing the chart for both contaminated and uncontaminated environments. The average run length is used as the performance evaluation measure of the charts. After comparing the performances of the proposed charts based on the estimators, it is noticed that the tri-mean estimator out-performs others in all ramifications. Next to it in performance is the Hodges-Lehmann and midrange estimators. We substantiated the simulation results of the study by applying the scheme on a real-life data set.     

Control charts for multivariate spatial autoregressive models

This paper deals with spatial detection of changes in model parameters of spatial autoregressive processes. The respective sequential testing problems are formulated. Moreover, we introduce characteristic quantities to monitor means or covariances of multivariate spatial autoregressive processes. Additionally, we also take into account the simultaneous surveillance of the mean vector and the covariance matrix. The aim is to apply control charts, important tools of sequential analysis, to these quantities. The considered control procedures are based on either cumulative sums or exponential smoothing. Further, we illustrate the methodology of statistical process control studying the spectrum of additive colors in a satellite photograph. Via simulation studies, the proposed control procedures are calibrated for a predefined average run length. In addition, we compare the performance of the control procedures considering the out-of-control situation. Eventually, the control charts are applied, and the signals of the different schemes are visualized. The final results are critically discussed.     

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.     

Loss-based optimal control statistics for control charts

This work proposes a means for interconnecting optimal sample statistics with parameters of the process output distribution irrespective of the specific way in which these parameters change during transition to the out-of-control state (jumps, trends, cycles, etc). The approach, based on minimization of the loss incurred by the two types of decision errors, leads to a unique sample statistic and, therefore, to a single control chart. The optimal sample statistics are obtained as a solution of the developed optional boundary equation. The paper demonstrates that, for particular conditions, this equation leads to the same statistics as are obtained through the Neyman-Pearson fundamental lemma. Application examples of the approach when the process output distribution is Gamma and Weibull are given. A special loss function representing out-of-control state detection as a pattern recognition problem is presented.     

Empirical nonparametric control charts for high-quality processes

For attribute data with (very) small failure rates often control charts are used which decide whether to stop or to continue each time r failures have occurred, for some r?1. Because of the small probabilities involved, such charts are very sensitive to estimation effects. This is true in particular if the underlying failure rate varies and hence the distributions involved are not geometric. Such a situation calls for a nonparametric approach, but this may require far more Phase I observations than are typically available in practice. In the present paper it is shown how this obstacle can be effectively overcome by looking not at the sum but rather at the maximum of each group of size r.     

Dynamic sampling plans in on-line control charts

Three simple dynamic sampling plans for detecting the change point are investigated in the discrete-time case. The first is a two-rate sampling CUSUM procedure. The second is a two-rate sampling Shiryayev-Roberts procedure. The third is a periodic sequential testing procedure. Two problems are discussed. First, simple design methods are provided for practical use. Second, a comparison between the three plans is made in the continuous-time case, which shows that by properly choosing the design parameters, the three plans can be made equally efficient in certain senses.