다중가설검정을 적용한 새로운 공정관리방안

Abstract

As a fundamental key of process improvement, control chart for statistical process control can be viewed as a repeated hypothesis testing. The null hypothesis is that the process is in-control at the time when a sample was drawn. In many fields of research, p-value is commonly used to report the results of statistical testing of whether the null hypothesis will be rejected or not. The use of p-values in a control scheme allows easy interpretation, incorporating more complex control procedure, and incorporating the effect of multiple testing. When testing multiple hypotheses simultaneously, the overall error rate to be measured becomes much more complicated than single hypothesis testing. The widely used measure for multiple hypotheses testing was the familywise error rate (FWER), which is the probability that one or more type I errors will be made. Since FWER is too conservative in most cases, when using FWER, as the number of hypotheses increases, the power decreases. Therefore an approach to control false discovery rate (FDR), which is the expected proportion of false positives among the rejected hypotheses, was proposed. Controlling false discovery rate is known to have many advantages and to be more powerful than controlling family-wise error rate. The main goal of this research is to develop new control schemes based on the multiple-hypothesis testing procedure for controlling false discovery rate. To this end, two different procedures (Benjamini-Hochberg and Storey procedures) are considered and these are applied to Shewhart X-bar and the EWMA control charts. The false discovery rate in a control scheme can be regarded as the proportion of out-of-control signals that are turned out to be false alarms (or falsely detected). This is different from type I error, which is the proportion of in-control signals turned out to be false alarms. The proposed control schemes are expected to control this quantity by taking advantages of Benjamini-Hochberg and Storey procedures. This work is divided in two parts

combining X-bar chart with FDR control procedure, and combining EWMA with FDR control procedure. In Chapter III, New control schemes based on Shewhart X-bar chart by using the multiple-hypothesis testing procedure are developed under the assumption that the observations are normally distributed. The performance of the proposed control schemes are evaluated and compared with the performance of the original X-bar chart in terms of the average run length and the conditional expected delay. And the relationship between the specified false discovery rate level and the actual false discovery rate is examined. The domination of the proposed control scheme over the X-bar chart is theoretically proved for the 2-span scheme. Chapter IV develops new EWMA control schemes applied the FDR control procedure, as a complement of the proposed schemes of Chapter III. Through numerical simulations, it is shown that the proposed control schemes perform as good as the EWMA or the CUSUM control chart for a small size of shift in the process. And the numerical experiment for the optimal design of the proposed control schemes. In conclusion, the proposed control schemes give a good performance in terms of the average run length as well as the conditional expected delay. And it shows that the proposed control schemes maintain the false discovery rate as similar as the specified level in the process monitoring. The expansion of proposed control scheme for the multivariate data may be possible under the assumption that the observations follow Hotelling’s T2 distribution.