Fractional factorial designs are often used in industry to screen potentially important factors in terms of their impact on the mean response, that is, their location effect. To save resources, these experiments are often unreplicated, in that only one observation is made at each design point. In addition to studying location effects, these designs may also be used to study dispersion effects. One of the difficulties in studying dispersion in these unreplicated designs is that dispersion effects are confounded with pairs of location effects. In this paper, we investigate the situation where a single dispersion effect is identified. We develop a procedure to determine the minimum number of additional runs needed to totally remove the confounding between this dispersion effect and two location effects. For situations where replication is not feasible, we develop a test for a pair of unidentified location effects conditioned on a single dispersion effect being detected. An example from the literature is used to demonstrate these two procedures.Key Words: Confounding, Dispersion, Fractional Factorial Designs, Variance.

*By* **RICHARD N. McGRATH, Bowling Green State University, Bowling
Green, OH 43403**

**Introduction**

SCREENING experiments are used in industry to identify the few factors
that have the largest impact on a response. An assumption of effect sparsity
is often made, where it is assumed that only a few factors produce substantial
effects and the rest are null or of negligible comparative magnitude.
In this paper, we analyze unreplicated 2* ^{k-p}* screening
designs where

Assume that _{},
or, equivalently, _{}, *i*
= 1,..., n. The *n* × *n* effect matrix X has columns x* _{j}*
= (

where _{} and the _{j}
are unknown parameters. This is a reparametrization of a model used by
Cook and Weisberg (1983) and Davidian and Carroll (1987), among others.
Defining _{} = _{}
= _{} As such, * _{d}*
is the measure of the true dispersion effect for column

There are *n*/2 pairs of columns whose interaction
column (component-wise product) appears in column *d*. For *j*
_{} U, there are 0 ≤ *g*
≤ *u*/2 pairs of columns (*j, j*´) such that the
interaction of columns *j* and *j*´ occurs in column *d*,
i.e. *x _{ij}x_{ij}*

where _{} is the sample variance
of the *e _{i}* |

From Equations (1) and (2) we see that the _{}* _{q}*s
can not create spurious dispersion effects as they appear in both

and if a single pair of these location effects, _{}
is active, then

Thus, an observed significant dispersion effect may be the result of one or more pairs of unidentified location effects.

Box and Meyer (1986) and Montgomery (1990), among
others, have studied the sample variances defined by Equations (1) and
(2) in order to detect dispersion effects. Others, such as Bergman and
Hynén (1997) and McGrath and Lin (2002) adapted L by also including
the _{}* _{q}* columns,
so the residuals from the +1 level of the suspected dispersion column
are uncorrelated with those at the -1 level. With this adaptation, under
a normality assumption and the null hypothesis of equal variances,

The rest of this paper is organized as follows. We first discuss an example from the literature where there is the location-dispersion confounding described above. In the next section, the structure of the columns of interest, i.e., those of the identified location effects, the suspected dispersion effect, and the pair of suspected location effects, is analyzed. This structure determines the minimum number of design points that must be replicated to remove the location-dispersion confounding. We then apply the procedure to the example, and show how an additional four runs could be used to remove the locationdispersion confounding. As replication is not always feasible, we then develop a formal test that is performed using the original unreplicated data. Since the test is designed to detect two location effects conditioned on detecting what appears to be a dispersion effect, we also assess its performance under more general conditions. Finally, in the last section we summarize the proposed procedures and briefly discuss the difficulties involved in developing alternative (e.g., likelihood and Bayesian) approaches.

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