Shewhart X bar control charts with estimated control limits are widely used in practice. When the sample size is not fixed,we propose seven statistics to estimate the standard deviation sigma . These estimators are applied to estimate the control limits of Shewhart X bar control chart. The estimated results through simulated computation are given and discussed. Finally, we investigate the performance of the Shewhart X bar control charts based on the seven estimators of sigma via its simulated average run length (ARL).

Even a modest positive autocorrelation results in a considerable increase in the number of false alarms that are produced when applying a CUSUM chart. Knowledge of the process to be controlled allows for suitable adaptation of the CUSUM procedure. If one has to suspect the normality assumption, nonparametric control procedures such as the rank-based CUSUM chart are a practical alternative. The paper reports the results of a simulation study on the robustness (in terms of sensitivity of the ARL) of the rank-based CUSUM chart against serial correlation of the control variable. The results indicate that the rank-based CUSUM chart is less affected by correlation than the observation-based chart: The rank-based CUSUM chart shows a smaller increase in the number of false alarms and a higher decrease in the ARL in the out-of-control case than the the observation-based chart. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

The risk-adjusted Bernoulli cumulative sum (CUSUM) chart developed by Steiner et al. (2000) is an increasingly popular tool for monitoring clinical and surgical performance. In practice, however, use of a fixed control limit for the chart leads to quite variable in-control average run length (ARL) performance for patient populations with different risk score distributions. To overcome this problem, the simulation-based dynamic probability control limits (DPCLs) patient-by-patient for the risk-adjusted Bernoulli CUSUM charts is determined in this study. By maintaining the probability of a false alarm at a constant level conditional on no false alarm for previous observations, the risk-adjusted CUSUM charts with DPCLs have consistent in-control performance at the desired level with approximately geometrically distributed run lengths. Simulation results demonstrate that the proposed method does not rely on any information or assumptions about the patients' risk distributions. The use of DPCLs for risk-adjusted Bernoulli CUSUM charts allows each chart to be designed for the corresponding particular sequence of patients for a surgeon or hospital. The effect of estimation error on performance of risk-adjusted Bernoulli CUSUM chart with DPCLs is also examined. Our simulation results show that the in-control performance of risk-adjusted Bernoulli CUSUM chart with DPCLs is affected by the estimation error. The most influential factors are the specified desired in-control average run length, the Phase I sample size and the overall 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, there is a substantial reduction in 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. These researches are results of joint work with Dr. William H. Woodall (Department of Statistics, Virginia Tech). Moreover, DPCLs are adapted to design the risk-adjusted CUSUM charts for multiresponses developed by Tang et al. (2015). It is shown that the in-control performance of the charts with DPCLs can be controlled for different patient populations because these limits are determined for each specific sequence of patients. Thus, the risk-adjusted CUSUM chart for multiresponses with DPCLs is more practical and should be applied to effectively monitor surgical performance by hospitals and healthcare practitioners. This research is a result of joint work with Dr. William H. Woodall (Department of Statistics, Virginia Tech) and Mr. Justin Loda (Department of Statistics, Virginia Tech). / Ph. D.