Statistical Power for Detecting Single Stratum Shift in a Multi-Strata Production Process

INTRODUCTION Many manufacturing firms still suffer from poor quality of manufactured goods in spite of using quality control charts for over a decade. These firms continue to face challenges in implementing quality programs due to difficulties in correctly applying the statistical process control (SPC) techniques to their processes. Given that a control chart's function is to monitor a production process, selection of an appropriate sampling method is crucial for the charts to function effectively. It is critical that the chart detects changes in process as soon as possible (with a high degree of sensitivity) after they occur, especially for processes that are barely capable of meeting the specifications [Caulcutt, 1995; Evans, 1993]. Osborn (1990) states that an insensitive control chart may miss out in detecting small shifts in a process and jeopardize a company's continuous improvement efforts. Thus, a chosen sampling method can be termed appropriate if it enables control charts to detect process shifts with greater sensitivity without generating excessive false alarms [Osborn, 1990; Wheeler, 1983]. The problem of choosing appropriate sampling method is straight-forward when a production process consists of just one population (stratum). The control charts ([bar.x] and R) would be based on a rational sample selected from successive time periods of production [Grant, 1988; Wadsworth, 1986]. In certain chemical and pharmaceutical applications the production process may consist of multiple fill-heads, thus producing a mix of populations (strata). When applying quality control techniques to monitor such process, the choice of an appropriate sampling method is not so easy. For example, a four cavity machine could be producing four distinctly different populations (strata) and the choice of different sampling methods would affect the sensitivity of detecting a shift in one or more strata. Most often, whichever is "simplest", "most convenient", or "seemingly logical" is used to determine the sampling method [Caulcutt, 1995; Mayer, 1983; Osborn, 1990; Squires, 1982]. In the past, most literature has focused on studying different aspects of process shift for production processes producing only one population. The seminal studies by Scheffe (1949) and King (1952) develop operating characteristic (OC) curves for [bar.x] and R charts, when samples are rational and process standards are given. Olds (1961) investigates power characteristics of control charts for detecting process shifts in the context of rational sampling for both the "no standard" and "standard given" cases. Costa (1997) shows that the [bar.x] chart with variable sample size and sampling intervals is more sensitive than the traditional [bar.x] charts in detecting even moderate shifts in the process. Osborn's (1990) work emphasizes the importance of "statistical power" to a QC practitioner's ability in detecting a particular shift in the process average and laments its lack of understanding among practitioners who utilize process control techniques. He also emphasizes the role of sample size in enhancing the statistical power of control charts. The work by Davis et al. (1993) improvises on the explanation of the "statistical power" of a [bar.x] control chart as used by Osborn (1990). Unlike any previous study, Palm (1990) and Wheeler (1983) present tables of the power function for the [bar.x] chart using multiple detection rules for the mean process shift. The literature focusing on process shifts for production systems producing multiple populations is still limited. Ott and Snee (1973) present three different methods of analysis plots of raw data, residuals methods, and analysis of variance method--to examine fills of individual heads in a multiple fill-head machine. Montgomery (1982) proposes use of group control charts for detecting process shifts when the multiple population streams are not highly correlated. The work by Mortell and Runger (1995) suggests using a pair of control charts, Shewart and CUSUM charts, to monitor multiple stream processes. …