Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main effects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.
Deep Dive into Recent Developments in Nonregular Fractional Factorial Designs.
Nonregular fractional factorial designs such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditional analysis focuses on main effects only. Hamada and Wu (1992) went beyond the traditional approach and proposed an analysis strategy to demonstrate that some interactions could be entertained and estimated beyond a few significant main effects. Their groundbreaking work stimulated much of the recent developments in design criterion creation, construction and analysis of nonregular designs. This paper reviews important developments in optimality criteria and comparison, including projection properties, generalized resolution, various generalized minimum aberration criteria, optimality results, construction methods and analysis strategies for nonregular designs.
an explanation for the success of the analysis strategy due to Hamada and Wu (1992). Sun and Wu (1993) were the first to coin the term "nonregular designs" when studying statistical properties of Hadamard matrices of order 16. Deng and Tang (1999) and Tang and Deng (1999) introduced the concepts of generalized resolution and generalized minimum aberration for two-level nonregular designs. Xu and Wu (2001) proposed the generalized minimum aberration for mixed-level nonregular designs. Because of the popularity of minimum aberration, the research on nonregular designs has been largely focused on the construction and properties of generalized minimum aberration designs.
Our reference list suggests that keen interest in nonregular designs began in 1999 and continues to this day as evident by the increasing number of scientific papers on nonregular designs in major statistical journals.
Section 2 reviews the data analysis strategies for nonregular designs. Section 3 discusses the geometrical and hidden projection properties of the Plackett-Burman designs and other orthogonal arrays. Section 4 introduces the generalized resolution and generalized minimum aberration and their statistical justifications. Section 5 introduces the minimum moment aberration criterion, another popular criterion for nonregular designs. Section 6 considers uniformity and connections with various optimality criteria. Section 7 reviews construction methods and optimality results. Section 8 gives concluding remarks and future directions.
We begin with a review of a breakthrough approach (Hamada and Wu 1992) by entertaining interactions in Plackett-Burman designs and other orthogonal arrays after identifying a few important main effects. Then we review another strategy proposed by Cheng and Wu (2001) for the dual purposes of factor screening and response surface exploration (or interaction detection) with quantitative factors.
The analysis strategy proposed by Hamada and Wu (1992) consists of three steps.
Step 1. Entertain all the main effects and interactions that are orthogonal to the main effects.
Use standard analysis methods such as ANOVA and half-normal plots to select significant effects.
Step 2. Entertain the significant effects identified in the previous step and the two-factor interactions that consist of at least one significant effect. Identify significant effects using a forward selection regression procedure.
Step 3. Entertain the significant effects identified in the previous step and all the main effects. Identify significant effects using a forward selection regression procedure.
Iterate between Steps 2 and 3 until the selected model stops changing. Note that the traditional analysis of Plackett-Burman or other nonregular designs ends at Step 1.
Hamada and Wu (1992) based their analysis strategy on two empirical principles, effect sparsity and effect heredity (see Wu and Hamada 2000, Section 3.5). Effect sparsity implies that only few main effects and even fewer two-factor interactions are relatively important in a factorial experiment. Effect heredity means that in order for an interaction to be significant, at least one of its parent factors should be significant. Effect heredity excludes models that contain an interaction but none of its parent main effects, which lessens the problem of obtaining uninterpretable models. Hamada and Wu (1992) wrote that the strategy works well when both principles hold and the correlations between partially aliased effects are small to moderate. The effect sparsity suggests that only a few iterations will be required.
Using this procedure, Hamada and Wu (1992) reanalyzed data from three real experiments, a cast fatigue experiment using a 12-run Plackett-Burman design with seven 2-level factors, a blood glucose experiment using an 18-run mixed-level orthogonal array with one 2-level and seven 3-level factors, and a heat exchange experiment using a 12-run Plackett-Burman design with ten 2-level factors. They demonstrated that the traditional main effects analysis was limited and the results were misleading.
For illustration, consider the cast fatigue experiment conducted by Hunter, Hodi and Eager (1982) that used a 12-run Plackett-Burman design to study the effects of seven factors (A-G) on the fatigue life of weld repaired castings. Table 1 gives the data matrix and responses, where
This model has R 2 = 0.89, which is a significant improvement over model (1) in terms of goodness of fit. The identification of F G was not only consistent with the engineering knowledge reported in Hunter, Hodi and Eager (1982) but also provided a sound explanation on the discrepancy of the sign of factor D. The coefficient of D in (1) actually estimates D + 1 3 F G and therefore the sign of D in (1) could be negative even if D had a small positive effect. This experiment was later reanalyzed with other methods by several authors, including Box and Meyer (1993) Hadama and Wu (1992) discussed limitat
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