A Nonparametric Scheme for Monitoring a Process Output with a Block Effect

This paper proposes a distribution-free (or nonparametric) control scheme to monitor a process output that contains two special causes of variation called “block or batch” effects and “treatment or position” effects. The scheme properties (control limits, false alarm rate, and in-control average run length) stay the same under any assumed continuous probability distribution. For moderate sample sizes, these properties can be computed exactly from available tables without the need to estimate the mean or variance of the process. The proposed monitoring scheme requires ranking the observations within blocks and using the method of analysis of means by ranks. The paper includes an illustrative example concerning the grinding process of silicon wafers used in integrated circuits production. 1. Introduction In statistical process control, there are instances in which the process output contains block effects component in addition to the treatment effects component that is to be controlled. In manufacturing integrated circuits on silicon wafers, Roes and Does ([1, Table 1]) reported data bearing two effects: the batch (or block) effect and the position (or treatment) effect of the wafer under the grinder. To take account of both effects, they proposed a Shewhart-type control chart that is based on a two-way analysis of variance (ANOVA) model for controlling the process treatment mean and for controlling certain linear contrasts of the wafer positions. In manufacturing paper or plastic films, Palm and DeAmico [2] reported that the raw manufacturing material was formed into a continuous sheet (web) which would then be wound up into rolls at the end of the production line. Because of nonuniformities in the product along the cross-direction perpendicular to the direction of the travel of the web, samples were taken from each roll at different cross-directional positions (e.g., the middle, the front or operator side, and the back or motor side.) Palm and DeAmico [2] modeled the data as a two-way ANOVA setup where rolls served as blocks and cross-directional positions served as treatments. To account for the block effects, they suggested performing periodic cross-directional studies that are based on the analysis of main effects (ANOME) when monitoring the silicon oxide on computer chips, Yashchin [3] reported that from each lot, a sample of wafers is selected and then several measurements are made on each of the selected wafers. Based on a nested random effects model, he developed a cumulative sum (CUSUM) control chart for monitoring the process variance