Bayesian application to the two-stage near-saturated experimental design method with dispersion effects

Abstract

Experimental designs are critical in the quality improvement of manufacturing processes and to the development of new processes. Experimental methods are often used to identify the most important controllable variables to the product, and the relationship of the product response to the controllable variables can be determined by experimental data. Traditionally, regression analysis is used to establish an empirical mean model for the product response, and competing experimental designs are evaluated with regard to estimating coefficients in the regression model. However, most traditional procedures involve the assumption of homogeneous process variance, which in many processes is inappropriate. A two-stage experimental design procedure developed by Mays and Myers [Mays, D.P. and Myers, R.H., 1997, Design and analysis for a two-level factorial experiment in the presence of variance heterogeneity. Computational Statistics and Data Analysis, 26, 219–233.] has proven to be beneficial in many heterogeneous variance situations. In the procedure, the first stage estimates the heterogeneous variance structure, and the second stage augments the first stage design to create an efficient design for estimating the mean. However, the first stage variance estimation is not reliable for small first stage experiment sizes, and hence a modified Bayesian two-stage procedure has been proposed for such situations [Mays, D.P. and Myers, R.H., 1996, Bayesian approach for the design and analysis of a two level factorial experiment in the presence of dispersion effects. Communications in Statistics—Theory and Methods, 25, 1409–1428.]. This study applies the Bayesian two-stage procedure to Koshal designs, hybrid designs, and small composite designs and examines the variance estimation in the first stage in order to find the optimal number of replicates to make at each design location. The Bayesian procedure is compared with the non-Bayesian two-stage procedure and with two competing one-stage equal replicate procedures. The effects of the various variance information on the procedures are also investigated.