Abstract: Nonregularfractionalfactorialdesigns such as Plackett-Burman designs and other orthogonal arrays are widely used in various screening experiments for their run size economy and flexibility. The traditionalanalysis 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 developmentsin optimality criteria, constructionand analysis of nonregular designs. This paper reviews important developments in nonregular designs, including projection properties, generalized resolution, generalized minimum aberration criteria, optimality results, construction methods and analysis strategies.

Since the penalized likelihood function of the smoothly clipped absolute deviation (SCAD) penalty is highly non-linear and has many local optima, finding a local solution to achieve the so-called oracle property is an open problem. We propose an iterative algorithm, called the OEM algorithm, to fill this gap. The development of the algorithm draws direct impetus from a missing-data problem arising in design of experiments with an orthogonal complete matrix. In each iteration, the algorithm imputes the missing data based on the current estimates of the parameters and updates a closed-form solution associated with the complete data. By introducing a procedure called active orthogonization, we make the algorithm broadly applicable to problems with arbitrary regression matrices. In addition to the SCAD penalty, the proposed algorithm works for other penalties like the MCP, lasso and nonnegative garrote. Convergence and convergence rate of the algorithm are examined. The algorithm has several unique theoretical properties. For the SCAD and MCP penalties, an OEM sequence can achieve the oracle property after sufficient iterations. For various penalties, an OEM sequence converges to a point having grouping coherence for fully aliased regression matrices. For computing the ordinary least squares estimator with a singular regression matrix, an OEM sequence converges to the Moore-Penrose generalized inverse-based least squares estimator.