This article demonstrates how Bayesian methods for making inferences about the number, or proportion, of nonconforming units in a quality control setting can be implemented using simulation techniques. Random draws from the posterior distribution of the target quantities of interest are used to construct the needed inferences. The same random draws from the posterior distribution are also used to perform model checking and sensitivity analysis. The methods are illustrated using two examples. The methods can be generalized to more complicated situations which are illustrated by discussing the extensions of both the examples.
Keywords: acceptance sampling, Bayesian methods, posterior distribution, quality control, sensitivity analysis, variable sampling
by T. E. Raghunathan, University of Michigan, Ann Arbor, MI 48106-1248
A Bayesian data analysis essentially involves: (1) setting up a probability model that posits a distribution for observables (measurements or attributes) conditional on unobservables (parameters) where functions of some or all of the unobservables and observables (target quantities) is the object of inference, (2) a prior distribution that summarizes a priori uncertainty about the likely values of the parameters, and (3) computing and interpreting the posterior distribution of the target quantities of interest. Specifically, suppose that (x | ) is the density function of the measurements, x, conditional on the parameter, , and () denotes the prior density function of , then the posterior density function of is given by
The above equation is the essential ingredient for making inferences about the target quantity of interest, (), a known function of . For instance, suppose that the objective is to compute the probability that a scalar function, (), belongs to a subset A in light of the information provided by the measurements x. Here one would simply compute the integral
over all possible values of that result in () being in A. The recent books by Gelman et al. (1995) and Carlin and Louis (1996) discuss many issues concerning Bayes and empirical Bayes analysis.
In many practical situations, computation of the above integral can be diffcult. However, if we had draws from the posterior distribution of , then the target quantities (or some functions of them) of interest can be easily approximated. For instance, if we could draw a random sample, , i = 1, 2, . . . , M from the posterior distribution of given in Equation (1), then the above integral can be approximated as
where I[C] is an indicator function which takes the value 1 if the condition, C, is satisfied and zero otherwise. These same draws can also be used to draw inferences about some other function of . The recent increase in the availability of computational resources and the development of computational techniques have led to great advances in the application of Bayesian methods to complicated problems in various disciplines (Gilks, Richardson, and Spiegelhalter (1996)).
In solving practical problems, the choice of a model for x can be made based on the combination of subject matter information and empirical information provided by the measurements themselves. However, the prior distribution for needs to be formulated based on the prior knowledge. This is usually a diffcult task because such prior knowledge may not be readily available. In such situations, usually a "noninformative" prior distribution is used. The basic idea behind formulating such a prior distribution is that it should be flat so that the likelihood (the density, (x | ), evaluated at the observed value of x) plays a dominant role in the construction of the posterior density given in Equation (1). Jeffreys (1961) formulated such prior distributions based on certain invariance arguments; often these are not proper density functions as they are not integrable functions of . Although this does not create problems in many situations, caution must be used in using these priors (DuMouchel and Waternaux (1992)). When data from prior studies are available then one can formulate a proper prior distribution. Examples of these types of situations in a quality control setting are considered, for example, in Hahn and Raghunathan (1988). Nevertheless, it is possible that both (x | ) and () could be misspecified. Hence, an important component of any Bayesian analysis is model checking and exploring the extent to which the inferences about () are sensitive to the modest departure from the assumed distributions (x | ) and (). The use of simulation techniques also facilitates performing model checking and the sensitivity analyses. We will demonstrate how one can perform model checking and sensitivity analyses while using the simulation methods to draw inferences under a specific model.
To illustrate the basic ideas, we use two common problems in quality control. The first example, discussed in Hahn and Meeker (1990), involves estimating the proportion of conforming units in a manufacturing process when the specification involves two variables. We show how a Bayesian method implemented through simulations leads to non-asymptotic point and interval estimates for the proportion of nonconforming units, assuming a bivariate normal distribution for these two variables. We also explore sensitivity to model assumptions by drawing inferences under robust models. The details of the analysis are given in the second section.
The second example, discussed in the third section, is concerned with drawing inferences about the number of nonconforming units in the accepted lots after acceptance sampling. Specifically, we obtain point and interval estimates of the mean number of nonconforming units among the units that have not been inspected in the accepted lots. We use a hierarchical model for the data given in Hahn (1986). We also explore sensitivity to the assumed model for the random effects in the hierarchical model. Finally, the fourth section concludes the paper with a discussion and possible extensions for both problems.
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