Zachary G. Stoumbos

The State of Statistical Process Control as We Proceed into the 21st Century

Ascher, H., and Feingold, H. (1984), Repairable Systems Reliability, New York Marcel Dekker. Barlow, R. E., and Proschan, F. (1975), Statistical Theoiy of Reliability and Life Testing, New York: Holt, Rinehart and Winston. Becker, R. A., Clark, L. A., and Lambert, D. (1998), “Events Defined by Duration and Severity, With an Application to Network Reliability” (with discussion), Technometrics, 40, 177-194. Berman, M., and Turner, T. R. (1992), “Approximating Point Process Likelihoods With GLIM,” Applied Statistics, 41, 31-38. Cox, D. R. (1972), “Regression Models and Life Tables” (with discussion), Journal of the Royal Statistical Society, Ser. B, 34, 187-220. Cox, D. R., and Lewis, P. A. W. (1966), The Statistical Analysis of Series of Events, London: Methuen. Crowder, M. C., Kimber, A. C., Smith, R. L., and Sweeting, T. J. (1991), Statistical Analysis of Reliability Data, London: Chapman and Hall. Dalal, S. R., and McIntosh, A. A. (1994), “When to Stop Testing for Large Software Systems With Changing Code,” IEEE Transactions on Software Engineering, 20, 318-323. Duchesne, T., and Lawless, J. (2000), “Alternative Time Scales and Failure Time Models,” Lifetime Data Analysis, 6, 157-179. Hoyland, A., and Rausand, M. (1994), System Reliability Theory, New York Wiley. 205-247.

Monitoring the Process Mean and Variance Using Individual Observations and Variable Sampling Intervals

In this paper we investigate control charts for monitoring a process to detect changes in the mean and/or variance of a normal quality variable when an individual observation is taken at each sampling point. The traditional X chart and moving range (MR) chart are evaluated. Also evaluated are the exponentially weighted moving average (EWMA) chart of the observations and the EWMA chart of the squared deviations of the observations from the target. 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. The ability of charts to diagnose the type of parameter shift produced by a special cause is also investigated. It is shown that combinations involving the EWMA charts are just as effective at diagnosing the type of parameter shift as the traditional combination of the X and MR charts. The effect of adding the variable sampling interval (VSI) feature is also evaluated for some of the combinations of charts. The VSI feature allows the sampling interval to be varied as a function of the values of the statistics being plotted. It is shown that adding the VSI feature to the combinations of charts results in very substantial reductions in the expected time required to detect shifts in process parameters.

A CUSUM Chart for Monitoring a Proportion When Inspecting Continuously

A control chart is considered for the problem of monitoring a process when all items from the process are inspected and classified into one of two categories. The objective is to detect changes in the proportion, p, of items in the first category. The c..

Robustness to Non-normality of the Multivariate EWMA Control Chart

We investigate the effects of non-normality on the statistical performance of the multivariate exponentially weighted moving average (MEWMA) control chart, and its special case, the Hotellings chi-squared chart, when applied to individual observations to monitor the mean vector of a multivariate process variable. We show that the chi-squared chart is highly sensitive to non-normality. We argue that the performance is most sensitive to departures from multivariate normality with individual observations (subgroups of size one). We show that with individual observations, and therefore, by extension, with subgroups of any size, the MEWMA chart can be designed to be robust to non-normality and very effective at detecting process shifts of any size or direction, even for highly skewed and extremely heavy-tailed multivariate distributions.

Control charts and the efficient allocation of sampling resources

Control charts for monitoring the process mean μ and process standard deviation σ are often based on samples of n > 1 observations, but in many applications individual observations are used (n = 1). In this article we investigate the question of whether it is better, from the perspective of statistical performance, to use n = 1 or n > 1. We assume that the sampling rate in terms of the number of observations per unit time is fixed, so using n = 1 means that samples can be taken more frequently than when n > 1. The best choice for n depends on the type of control chart being used, so we consider Shewhart, exponentially weighted moving average (EWMA), and cumulative sum (CUSUM) charts. For each type of control chart we investigate a combination of two charts, one chart designed to monitor μ and the other designed to monitor σ. Most control chart comparisons in the literature assume that a special cause produces a sustained shift in a process parameter that lasts until the shift is detected. We also consider transient shifts in process parameters, which are of a short duration, and drifts in which a parameter moves away from its in-control value at a constant rate. We evaluate control chart combinations using the expected detection time for the various types of process changes and a quadratic loss function. When a signal is generated, it is important to know which parameters have changed, so the ability of control chart combinations to correctly indicate the type of parameter change is also evaluated. Our overall conclusion is that it is best to take samples of n = 1 observations and use an EWMA or CUSUM chart combination. The Shewhart chart combination with the best overall performance is based on n > 1, but this combination is inferior to the EWMA and CUSUM chart combinations on almost all performance characteristics (the exception being simplicity). This conclusion seems to contradict the conventional wisdom about some of the advantages and disadvantages of EWMA and CUSUM charts relative to Shewhart charts.

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.

A general approach to modeling CUSUM charts for a proportion

This paper considers two CUmulative SUM (CUSUM) charts for monitoring a process when items from the process are inspected and classified into one of two categories, namely defective or non-defective. The purpose of this type of process monitoring is to detect changes in the proportion p of items in the first category. The first CUSUM chart considered is based on the binomial variables resulting from counting the total number of defective items in samples of n items. A point is plotted on this binomial CUSUM chart after n items have been inspected. The second CUSUM chart considered is based on the Bernoulli observations corresponding to the inspection of the individual items in the samples. A point is plotted on this Bernoulli CUSUM chart after each individual inspection, without waiting until the end of a sample. The main objective of the paper is to evaluate the statistical properties of these two CUSUM charts under a general model for process sampling and for the occurrence of special causes that change the value of p. This model applies to situations in which there are inspection periods when n items are inspected and non-inspection periods when no inspection is done. This model assumes that there is a positive time between individual inspection results, and that a change in p can occur anywhere within an inspection period or a non-inspection period. This includes the possibility that a shift can occur during the time that a sample of n items is being taken. This model is more general and often more realistic than the simpler model usually used to evaluate properties of control charts. Under our model, it is shown that there is little difference between the binomial CUSUM chart and the Bernoulli CUSUM chart, in terms of the expected time required to detect small and moderate shifts in p, but the Bernoulli CUSUM chart is better for detecting large shifts in p. It is shown that it is best to choose a relatively small sample size when applying the CUSUM charts. As expected, the CUSUM charts are substantially faster than the traditional Shewhart p-chart for detecting small shifts in p. But, surprisingly, the CUSUM charts are also better than the p-chart for detecting large shifts in p.

Comparisons of Some Exponentially Weighted Moving Average Control Charts for Monitoring the Process Mean and Variance

An exponentially weighted moving average (EWMA) control chart for monitoring the process mean μ may be slow to detect large shifts in μ when the EWMA tuning parameter λ is small. An additional problem, sometimes called the inertia problem, is that the EWMA statistic may be in a disadvantageous position on the wrong side of the target when a shift in μ occurs, which may significantly delay detection of a shift in μ. Options for improving the performance of the EWMA chart include using the EWMA chart in combination with a Shewhart chart or in combination with an EWMA chart based on squared deviations from target. The EWMA chart based on squared deviations from target is designed to detect increases in the process standard deviation σ, but it is also very effective for detecting large shifts inμ. Capizzi and Masarotto recently proposed the option of an adaptive EWMA control chart in which λ is a function of the data. With the adaptive feature, the EWMA chart behaves like a standard EWMA chart when the current observation is close to the previous EWMA statistic, and like a Shewhart chart otherwise. Here we extend the use of the adaptive feature to EWMA charts based on squared deviations from target, and also consider an alternate way of defining the adaptive feature. We discuss performance measures that we believe are appropriate for assessing the effects of inertia, and compare the performance of various charts and combinations of charts. Standard practice is to simultaneously monitor both μ and σ, so we consider control chart performance when the objective is to detect small or large changes in μ or increases in σ. We find that combinations of EWMA control charts that include a chart based on squared deviations from target give good overall performance whether or not these charts have the adaptive feature.

The SPRT chart for monitoring a proportion

A control chart based on applying a sequential probability ratio test (SPRT) at each sampling point is considered for the problem of monitoring a process proportion p. This SPRT chart can be applied in situations in which items are inspected one by one, and the results of each inspection can be conveniently recorded before the next item is inspected. Some corrected diffusion theory approximations are given for the statistical properties of the SPRT and the SPRT chart. These approximations are very accurate and provide a simple method for designing an SPRT chart for practical applications. The sample size for the SPRT chart at a particular sampling time depends on the observations at that time, but the chart can be designed to have a specified average sampling rate when the process is in control. When there is a small shift in p, the average sampling rate per unit time will increase, but for a large shift in p the average sampling rate will decrease. For a given in-control average sampling rate and a given false alarm rate, the SPRT chart will detect changes in p much faster than the standard p-chart, which has traditionally been used for monitoring p. The SPRT chart will also detect changes in p much faster than the CUSUM chart for p. Thus, the SPRT chart can be used in place of traditional control charts to provide faster detection of changes in p or to reduce the sampling effort required to provide a given detection capability.

Control Charts Applying a Sequential Test at Fixed Sampling Intervals

Traditional control charts take samples from the process using a fixed sample size and fixed sampling interval. The sampling rate in a control chart, however, can be varied depending on what is observed from the process in order to reduce the average ti..