The need for process monitoring in industry is ubiquitous. By monitoring process output, process changes may be rapidly detected and problems corrected. However, in many industrial and medical applications, observations are censored due to either inherent limitations or cost/time considerations. For example, when testing breaking strengths or failure times, often a limited stress test is performed and only a small proportion of the true failure strengths or failure times are observed. With highly censored observations, a direct application of traditional monitoring procedures is not appropriate. In this article, Shewhart-type and S control charts based on the conditional expected value weight are suggested for monitoring processes where the censoring occurs at a fixed level. We provide an example to illustrate the application of this methodology.
Keywords: Censored Data, Process Control, Scores
by Stefan H. Steiner and R. Jock Mackay, University of Waterloo, Waterloo, Canada N2L 3G1
In many industrial applications, censored observations are collected for process monitoring purposes. For example, in the manufacture of material for use in the interior trim of an automobile, a vinyl outer layer is glued to an insulating foam backing. The strength of the bond between the layers is an important characteristic. To check the bond strength, a rectangular sample of the material is cut and the force required to break the bond is then measured. A pre-determined maximum force is applied to avoid tearing the foam backing. Most samples do not fail, so it is known only that the bond strength exceeds the pre-determined force. That is, the bond strength data are censored. The process is monitored by selecting samples across the width of the material at a given frequency based on the amount of material produced. The purpose of the monitoring is to ensure that the bond strength does not deteriorate. Deterioration includes decreases in the average strength or increased variability. A second example, which we do not consider in detail here, is the use of plug gauges to monitor hole size. To measure hole diameter, two plugs machined to have diameters at the upper and lower specification of the hole diameter are applied. If the larger plug enters the hole, then the diameter exceeds the upper specification. If the smaller plug does not enter the hole, then the hole size is below the minimum specification. For the purpose of process monitoring, the actual diameter of the few holes that fail are measured. Here all diameters within the specification limits are censored. Similar situations that result in censored data occur in life testing and other areas of application. For simplicity we will always refer to the variable of interest as strength, although it may be any other censored response.
In these examples, a direct application of an and S control chart on the observed strength, where we ignore the censoring, is reasonable if the censoring proportion is not large, say, less than 50%. On the other hand, when the censoring proportion is very high, say, greater than 95%, it is feasible to use a traditional np chart where we record only the number of censored observations. In this article, we propose conditional expected value weight (CEV) control charts appropriate for monitoring processes that produce censored observations. The proposed charts are superior to traditional methods, especially when the censoring proportion lies between 5095%.
This article is organized in the following manner. We first introduce the CEV charting procedure that allows for the rapid detection of deterioration in the process quality when the monitored output is censored. Figures needed to determine control limits are given. The use of the procedure is then illustrated with the first example described above. Finally, we determine the power of the proposed procedure and compare it with more traditional approaches.
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