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# Some New Two-Level Saturated Designs

First-order saturated designs can be orthogonal and have two levels only if the number of design points is a multiple of 4. For other cases, saturated two-level designs have been obtained from balanced incomplete blocks and by computer searches for design matrices of maximal determinant (D-optimal designs). Recently, two-level saturated designs that are efficient for submodels containing only a few of the factors have been developed. Some of these designs do not estimate the effects of all factors with equal precision. In this article, alternative designs that estimate the effects of all factors with equal precision are obtained from partially balanced incomplete block designs. The new designs are compared to all previously known designs by A-, D-, E-, and G-efficiency, and by the average and maximum variance inflation factor.

Keywords: cyclic design, D-optimality, incomplete block design, Plackett-Burman design

by Ronald B. Crosier, U.S. Army Edgewood Chemical Biological Center, Aberdeen Proving Ground, MD 21010-5424

INTRODUCTION

In many experiments the goal is to determine which of v independent variables (experimental factors), x1, x2, : : :, xv, have the largest effects on the dependent variable, y. A common strategy for such problems is to approximate the relationship between y and the process factors by the first-order polynomial where the independent random errors have a mean of zero and variance . The coeffcients of the polynomial in Equation (1) are estimated from data collected during n experimental runs of the process; the estimated coeffcients are then used to select the factors with the largest effects. The settings of the x´s for the n experimental runs are given by an n v design matrix D. To estimate the coeffcients of the polynomial, the design matrix is expanded into a model matrix X that has an additional column which represents the constant term. The estimate b of the coeffcient vector is obtained from the least-squares formula The two-level designs of Plackett and Burman (1946) are widely used to estimate the coeffcients in Equation (1). The Plackett and Burman designs are only available for n 0 mod 4. The usual strategy when v + 1 is not a multiple of 4 is to use the next larger Plackett and Burman design by ignoring 1, 2, or 3 design columns. Sometimes an experimenter wishes to minimize the number of experimental runs for cases where v + 1 0 mod 4.

Two approaches have been used to find saturated designs for the n 0 mod 4 cases. One approach generates saturated designs from balanced incomplete block (BIB) designs. Raghavarao (1959, 1960) and Rao (1966) give designs of this type. The Plackett and Burman designs are also related to BIB designs. This type of design has two attractive properties: the variance of bj is the same for all j and the covariance of bi and bj is the same for any i; j pair. Unfortunately, such designs don´t exist for all n, nor are they necessarily good by other criteria.

The other approach to finding saturated designs is a numerical search for designs that optimize some design criterion (usually the determinant of X´X). Ehlich (1964); Yang (1966, 1968); Mitchell (1974); Moyssiadis and Kounias (1982); and Chadjipantelis, Kounias, and Moyssiadis (1987) follow this approach. Galil and Kiefer (1980) summarized the known results for D-optimal designs. Lin (1993) noted that D-optimal designs do not necessarily have the balance and symmetry of the Plackett and Burman designs.

In the Plackett and Burman designs, each factor appears at its high level and at its low level in half of the experimental runs; Lin (1993) referred to this property as equal occurrence. The Plackett and Burman designs are orthogonal designs, so sij, the sum of cross products between columns i and j of the model matrix, is zero for any i, j pair, i j. For n 0 mod 4, the sij cannot all be zero. Some measure is needed to assess the effect of the nonorthogonality of the designs. For supersaturated designs (which have v > n – 1), Booth and Cox (1962) used average to measure nonorthogonality. In the saturated case there are better measures of nonorthogonality. It is known (in the context of regression analysis) that the multicollinearity of a set of variables cannot be determined by examining their pairwise correlations individually. Hence, other measures, such as the variance inflation factor (VIF), have been developed to judge the multicollinearity of a set of variables. The VIF for bj is a factor indicating how much the variance of bj has been increased by the nonorthogonality of the design. See Montgomery and Peck (1992) for discussion of the VIF. The well-known alphabetic effciencies may be used as a measure of nonorthogonality for first-order saturated designs. Each of these effciencies is based on some quantity: the A-effciency is based on the trace of (X´X)–1, the D-effciency on the determinant of X´X, the E-effciency on the minimum eigenvalue of X´X, and the G-effciency on the maximum prediction variance within the experimental region. To compute the effciency, the obtained quantity must be compared to that from the ideal design, which I consider to be an orthogonal design. Thus, these efficiencies are measures of deviation from orthogonality. A design is said to be optimal if it has the maximum effciency for a design within the class of designs under consideration—here, two-level saturated designs. For n 0 mod 4, optimal designs will have effciencies less than 1. See Box and Draper (1987) for discussion of alphabetic optimality criteria.

Lin (1993) developed saturated designs that have the equal-occurrence (or near equal-occurrence for odd n) property and minimum (for (near) equaloccurrence designs) average . In examining these designs, I noticed that the estimated factor coefficients do not necessarily have the same variance. For example, his design for n = 6 has /3 for the variance of b1b4, but /2 for the variance of b5. Table 1 contains an equal-precision, two-level saturated design for five factors. This design has /3 for the variance of b1b5 and it has the equal-occurrence and minimal average properties. The design in Table 1 is related to a partially balanced incomplete block (PBIB) design in the same manner as the Plackett-Burman designs are related to BIB designs. This relationship, which I call the Plackett-Burman technique, is described in the appendix. I used two types of PBIB designs to construct saturated designs: two associate-class PBIB designs (PBIBD/2´s) and cyclic incomplete block designs. Cyclic incomplete block designs are partially balanced designs, but may have up to v/2 associate classes (John, Wolock, and David (1972)). The effciencies of the saturated designs were used to select the best design. This method is heuristic; there is no relationship between the efficiency of a PBIB design and the effciency of the saturated, two-level design constructed from it—in fact, the method can generate singular designs.

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