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 byA-,D-,E-, andG-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), *x*_{1}, *x*_{2}, : :
:, *x _{v}*, have the largest effects on the dependent
variable,

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 *b _{j}*
is the same for all

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 *s _{ij}*,
the sum of cross products between columns

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 *b _{j}* is a factor indicating how much the
variance of

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 *b*_{1} – *b*_{4},
but /2
for the variance of *b*_{5}. Table 1 contains
an equal-precision, two-level saturated design for five factors.
This design has /3
for the variance of *b*_{1} – *b*_{5}
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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