Bernoulli Cumulative Sum Chart Research Articles

Improved implementation of the risk-adjusted Bernoulli CUSUM chart to monitor surgical outcome quality.

The traditional implementation of the risk-adjusted Bernoulli cumulative sum (CUSUM) chart for monitoring surgical outcome quality requires waiting a pre-specified period of time after surgery before incorporating patient outcome information. We propose a simple but powerful implementation of the risk-adjusted Bernoulli CUSUM chart that incorporates outcome information as soon as it is available, rather than waiting a pre-specified period of time after surgery. A simulation study is presented that compares the performance of the traditional implementation of the risk-adjusted Bernoulli CUSUM chart to our improved implementation. We show that incorporating patient outcome information as soon as it is available leads to quicker detection of process deterioration. Deterioration of surgical performance could be detected much sooner using our proposed implementation, which could lead to the earlier identification of problems.

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.

Monitoring foreclosure rates with a spatially risk-adjusted Bernoulli CUSUM chart for concurrent observations

ABSTRACTFrequently in process monitoring, situations arise in which the order that events occur cannot be distinguished, motivating the need to accommodate multiple observations occurring at the same time, or concurrent observations. The risk-adjusted Bernoulli cumulative sum (CUSUM) control chart can be used to monitor the rate of an adverse event by fitting a risk-adjustment model, followed by a likelihood ratio-based scoring method that produces a statistic that can be monitored. In our paper, we develop a risk-adjusted Bernoulli CUSUM control chart for concurrent observations. Furthermore, we adopt a novel approach that uses a combined mixture model and kernel density estimation approach in order to perform risk-adjustment with regard to spatial location. Our proposed method allows for monitoring binary outcomes through time with multiple observations at each time point, where the chart is spatially adjusted for each Bernoulli observation's estimated probability of the adverse event. A simulation study is presented to assess the performance of the proposed monitoring scheme. We apply our method using data from Wayne County, Michigan between 2005 and 2014 to monitor the rate of foreclosure as a percentage of all housing transactions.

A Generalized Likelihood Ratio Chart for Monitoring Bernoulli Processes

This paper considers the problem of monitoring the proportion p of nonconforming items when a continuous stream of Bernoulli observations is available and the objective is to effectively detect a wide range of increases in p. The proposed control chart is based on a generalized likelihood ratio (GLR) statistic obtained from a moving window of past Bernoulli observations. The Phase II performance of this chart in detecting sustained increases in p is evaluated using the steady state average number of observations to signal. Comparisons of the Bernoulli GLR chart to the Shewhart CCC‐r chart, the Bernoulli cumulative sum chart, and the Bernoulli exponentially weighted moving average chart show that the overall performance of the Bernoulli GLR chart is better than its competitors. In addition, methods are provided for designing the Bernoulli GLR chart so that this chart can be easily applied in practice. Copyright © 2012 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.

On the Equivalence of the Bernoulli and Geometric CUSUM Charts

The Bernoulli cumulative sum (CUSUM) chart has been shown to have good properties in detecting an increase in the nonconforming rate, p, based on the average number of observations to signal (ANOS). The geometric CUSUM chart has also been proposed, in particular for high-quality processes where p is very small. The geometric CUSUM chart has very good properties in a zero-state analysis because of a built-in headstart feature. In steady-state performance analyses, often a shift in the process has been assumed to occur immediately after a nonconforming item has been found. In this case, the performance of the geometric CUSUM chart is said to be better than that of the Bernoulli CUSUM chart. We explore the case where a process shift may occur at any time. We show that the steady-state properties of the Bernoulli CUSUM and geometric CUSUM charts are the same because the charts can be designed to be mathematically equivalent.

A Note on the ARL of Two-Sided Bernoulli-Based CUSUM Control Charts

In this article, we examine the average run length (ARL) properties of two types of two-sided Bernoulli cumulative sum (CUSUM) control charts. In particular, we give the conditions under which the ARL of the two-sided Bernoulli CUSUM chart can be determined exactly from the ARLs of the corresponding upper and lower Bernoulli CUSUM charts. This ARL relationship is established for both the standard industrial application and the risk-adjusted health-care monitoring application.

Monitoring surgical and medical outcomes: the Bernoulli cumulative SUM chart. A novel application to assess clinical interventions

Background: Monitoring clinical interventions is an increasing requirement in current clinical practice. The standard CUSUM (cumulative sum) charts are used for this purpose. However, they are difficult to use in...

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 inp, but the Bernoulli CUSUM chart is better for detecting large shifts /np. 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

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.

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 control chart being considered is a cumulative sum (CUSUM) chart based on the Bernoulli observations corresponding to the inspection of the individual items. Bernoulli CUSUM charts can be constructed to detect increases in p, decreases in p, or both increases and decreases in p. The properties of the Bernoulli CUSUM chart are evaluated using exact Markov chain methods and by using a corrected diffusion theory approximation. The corrected diffusion theory approximation provides a relatively simple method of designing the chart for practical applications. It is shown that the Bernoulli CUSUM chart will detect changes in p substantially faster than the traditional approach of grouping items into samples and applying a Shewhart p-chart. The Bernoulli CUSUM chart is also better than grouping items into samples and applying a CUSUM chart to the sample statistics. The Bernoulli CUSUM chart is equivalent to a geometric CUSUM chart which is based on counting the number of items in the second category that occur between items in the first category.