Confounding of Location and Dispersion Effects in Unreplicated Fractional Factorials

When studying both location and dispersion effects in unreplicated fractional factorial designs, a “standard” procedure is to identify location effects using ordinary least squares analysis, fit a model, and then identify dispersion effects by analyzing the residuals. In this paper, we show that if the model in the above procedure does not include all active location effects, then null dispersion effects may be mistakenly identified as active. We derive an exact relationship between location and dispersion effects, and we show that without information in addition to the unreplicated fractional factorial (such as replication) we can not determine whether a dispersion effect or two location effects are active.

[1]  C. Daniel.  Use of Half-Normal Plots in Interpreting Factorial Two-Level Experiments , 1959 .

[2]  S. Goldfeld,et al.  Some Tests for Homoscedasticity , 1965 .

[3]  S. Addelman.  Statistics for experimenters , 1978 .

[4]  George E. P. Box,et al.  Dispersion Effects From Fractional Designs , 1986 .

[5]  R. Daniel Meyer,et al.  An Analysis for Unreplicated Fractional Factorials , 1986 .

[6]  R. Carroll,et al.  Variance Function Estimation , 1987 .

[7]  D. Pregibon,et al.  Analyzing dispersion effects from replicated factorial experiments , 1988 .

[8]  R. Lenth.  Quick and easy analysis of unreplicated factorials , 1989 .

[9]  P. C. Wang.  Tests for dispersion effects from orthogonal arrays , 1989 .

[10]  D. Montgomery.  USING FRACTIONAL FACTORIAL DESIGNS FOR ROBUST PROCESS DEVELOPMENT , 1990 .

[11]  D. Peña,et al.  A simple method to identify significant effects in unreplicated two-level factorial designs , 1992 .

[12]  A. Ferrer,et al.  Small samples estimation of dispersion effects from unreplicated data , 1993 .

[13]  Rafael Romero,et al.  A SIMPLE METHOD TO STUDY DISPERSION EFFECTS FROM NON- NECESSARILY REPLICATED DATA IN INDUSTRIAL CONTEXTS , 1995 .

[14]  B. Bergman,et al.  Dispersion Effects From 7 Unreplicated Designs in the 2 k—p Series , 1997 .

[15]  B. Bergman,et al.  Dispersion effects from unreplicated designs in the 2 k-p series , 1997 .

[16]  Narayanaswamy Balakrishnan,et al.  ANALYZING UNREPLICATED FACTORIAL EXPERIMENTS: A REVIEW WITH SOME NEW PROPOSALS , 1998 .

[17]  Kenny Q. Ye,et al.  A Step-Down Lenth Method for Analyzing Unreplicated Factorial Designs , 2001 .