Let &thetacrc;m×1 = (&thetacrc;i) ∼ Nm(θ,Σ) have an m-variate normal distribution, where Σ = A′Aσ2, A′A is a known, nondiagonal positive definite matrix, and σ is unknown. The objective is to construct an exact confidence interval for each effect θi, the ith component of θ. For a saturated design, there are no error degrees of freedom from which to compute an independent estimator of the error variance component σ2. However, under effect sparsity, the smaller effect estimates can be used to provide comparable information with which to construct confidence intervals for the effects. Voss provided a method for the construction of exact individual confidence intervals for each effect θi in the analysis of orthogonal saturated factorial experiments, for which the covariance matrix Σ is diagonal. We extend his results to the case of saturated designs, for which Σ is not diagonal. Such nonorthogonality can be planned, in order to keep the number of observations small, or it may be the unplanned consequence of lost observations. In the latter case, A is a random matrix, so our results are conditional.
