Approximate Confidence Intervals for Contrasts in the Balanced Three Factor Mixed Model

When conducting a statistical analysis of data from a designed experiment, an investigator is often interested in confidence intervals for contrasts of the fixed effects. If the analysis involves a mixed linear model, exact confidence intervals for contrasts of the fixed effects are not always available. In such cases, confidence intervals with approximate coverage probabilities must be used. As will be shown, this problem may be generalized to that of constructing a confidence interval for the parameter μ, where X is a normal random variable with mean μ and variance ∑ a_qθ_q, where a₁…,a_Q are known constants, U_q = n_qS/θ_q is a chi-squared random variable with n_q degrees of freedom, for each q = 1,…, Q, and X,U₁,…, U_Q are mutually independent. In this paper, we consider the case where Q = 3 and a₃ ≤0.