The Estimator Of Dispersion Effect In Unreplicated Factorial Experiments

In design experiments, we traditionally need do replicated experiments at each experiment's point in order to identify and estimate the dispersion effects. However, in the practical considerations, because of the experiments' funds and conditions, we can't do replicated experiments. There has been considerable interest in the use of unreplicated experiments to identify and estimate the dispersion effects,σ_i~2 does not have direct estimation in unreplicated experiments, it's difficult to identify the dispersion effects in this situation .In replicated experiments, we identify the dispersion effects use each point's samples variance as response to built up the dispersion model, which is proposed by Bartlett and Kendall (1946), using the least square method to get the dispersion estimation, and using the half-norm probability plot to identify the active effects. If the model has been established, we use the maximum likelihood or the restricted maximum likelihood to estimate the dispersion effects.In unreplicated experiments, there are Box and Meyer(1986) (BM method) and Bergman and Hynen (1997) (BH method) methods to estimate and identify the dispersion effects. The dispersion effects estimation is based on the ratio of residual square's arithmetic average at the plus or minus level of the dispersion estimated. In the paper of Brenneman and Nair(2001), it has proved that this method has serious structural bias. Other methods are based on the ratio of log-residual square's arithmetic average at the plus or minus level of the dispersion which is to be estimated, such as Harvey (1976) and MH method proposed in Brenneman and Nair (2001). Brenneman and Nair (2001) provide a systematic study of various methods that are commonly used to estimate the dispersion effects. Simulation results are used to prove that MH method is better than others on MSE criteria. However, when the absolute value of residuals is zero or very small after fitting the location model, the log value of the residual will become very large, so the dispersion effects estimation is unreliable. Specially, we can't get the log value of the residual when residual is zero, in this situation we can't use MH method.This paper proposes a new method based on the nature of the distribution of the residual square. First, adding a larger than zero modified-item to the residual square, then taking the log value of the modified residual as the dispersion effects estimation. This method can decrease the estimation's bias, while avoiding the situation that the absolute of residual is zero or very small. The methods in this paper are named corrected MH method(CMH method) and the iteratived corrected MH method (ICMH method). The simulation result shows that CMH in MSE criteria is better than MH method in allmost models. Simulation experiments show that iteratived corrected MH (ICMH methods) can further reduce the estimated bias. It is better on the MSE criteria. Finally, We use this method to analysis the famous Montgomery's (1990) injection mold experiment.