Robust Parameter Designs: an Empirical Study

1. The Problem In off-line quality control, robust parameter design is a valuable tool in determining values of design variables that would minimize losses due to process variation and deviation from target. In this regard, combined-array designs have been suggested to estimate the effects of the design variables on the mean and variation of the quality characteristic under study. In combined array experiments, the levels of the noise factors are considered fixed, and an analysis of the response surface for the mean of the quality characteristic conditional on the noise factors is first conducted. The variances of this conditional (given the values of the noise factors) model are usually assumed to be homogeneous. The results are then used to derive the unconditional variance (treating also the noise factors as random) to be minimized. However, Engel and Huele (1996) demonstrated with a real example in which the residual variances of the conditional model may not be homogeneous, giving rise to an additional component in the unconditional variance to be minimized. They then used a loglink function is used to formulate the dependency of residual variance on a linear function of the control variables. The conditional model is then fitted using iteratively weighted least squares (IWLS) alternating between the mean and variance parameter estimates based effectively on the normal likelihood. The present paper attempts to explore alternative strategy for modeling and parameter estimation. We consider in this paper the application of a class of link functions suggested in Mak (2001) for modeling variances. This class of link functions always yields positive variances and also includes the log-link as a special case. The use of a more general parametric approach in modeling the residual error is examined, and thus permitting the use of a general loss function. We also show how the validity of a variance estimation method can be examined by comparing the predicted variances with the empirical variances. 2. Extended Formulation and Model Fitting Let y be the quality characteristic of interest. Suppose that there are p control variables and q noise variables. The noise variables are assumed to be measurable or controllable in the experimental stage. Denote by x1,..., xp the p control variables and by z1 , ..., zq the q noise variables. As in Engel and Huele (1996), we assume that conditional on z1 , ..., zq , ε θ σ β μ ) , ( ) , , ( w z x y + =