In this article we will discuss the use of two-level factorial and fractional factorial experiments with binary responses (defectives/non-defectives) where the purpose is to reduce the rate of defectives. Contrary to a traditional fixed sample size scheme, we will consider one where each factorial combination is sampled until a fixed number of defectives is observed. The total number of items until that occurs is then used as the response. Such an inverse binomial sampling scheme has several practical and economical benefits that will be discussed. For the design of experiments based on this idea, we provide a methodology for choosing the necessary number of defectives r, to detect a given change in the probability of producing a defective unit with fixed levels of Type I and Type II errors.
Keywords: factorial experiments, inverse binomial sampling, quality improvement, sample size
by Søren Bisgaard, Institute for Technology Management, University of St. Gallen, Switzerland and Ilya Gertsbakh, Ben Gurion University, Beer-Sheva, Israel
Customers of industrial products increasingly expect to receive mass produced items with guaranteed levels of defectives in a range below 100 parts per million (ppm). One example where such stringent requirements are imposed is in the production of cathode-ray tubes (CRT's) for televisions or computer displays. Achieving such low levels of defectives is usually a technological challenge of extraordinary proportions. It frequently requires close collaboration between research, production and the quality engineering department. A potent tool in such efforts is the use of factorial experiments.
In this article we will consider the situation where the only reliable response is a binary defective or nondefective. This is often the case for the production of CRT's, micro-chips, metal castings, plastic moldings and a vast number of other products. For such production processes, low levels of defectives are traditionally achieved with control charts by controlling key process parameters within narrow limits. The hope is that if these input parameters are held at consistent levels, it will guarantee consistency in the quality of the output product.
The success of such an approach is based on at least two assumptions. One is that the right set of parameters is being monitored and controlled. A second assumption is that the nominal settings of the process parameters are targeted at their optimal (or satisfactory) levels so that defectives are produced only because of variability in these process parameters around the nominal values. That both of these assumptions are satisfied, or nearly so, is seldom if ever known in practice. In fact, it would appear safer to assume that they are not. Consequently it ought to be standard practice to experiment with processes to screen out the most important factors influencing the rate of defectives, and when those have been identified, engage in optimization experiments to determine the best, or at least a set of more satisfactory, parameter settings to be used as target values for subsequent process control.
Experimenting as we are suggesting with only a binary response yi = (0, 1) = (non-defective, defective) is inherently diffcult. A binary response has a very low information content. Nevertheless, it is fairly common in industry that factorial experiments with binary responses are set up such that for each of the N factorial trials, a set of n products is run off and the proportion defectives is used as the response in the subsequent analysis. Experiments of this kind have been discussed by Bisgaard and Fuller (1995a) who provided simple tables for determining an appropriate fixed sample size n for each factorial combination in two-level factorial designs. In another article focusing on the analysis of such experiments, Bisgaard and Fuller (1995b) further discussed the use of variance stabilizing transformations. Since this fixed sample size approach provides important clues to how an inverse binomial sampling scheme can be constructed, we will briefly review this material in the next section.
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