Justus Seely

Sufficient conditions for orthogonal designs in mixed linear models

This paper revisits the concepts of orthogonal block structure and orthogonal design in mixed linear models. The primary tool used in investigating these concepts is that of a commutative quadratic subspace. The idea of an error orthogonal design is introduced. These designs are less restrictive than orthogonal designs, but retain almost all of the nice properties. Easy-to-check conditions are given that insure a model has an orthogonal or error orthogonal design. The results given are most useful for those mixed models with a nested covariance structure or those whose covariance structure comes from random main effects.

Some Remarks on Exact Confidence Intervals for Positive Linear Combinations of Variance Components

Abstract A technique given by Burdick and Sielken (1978) for determining an exact confidence interval for a positive linear combination of the two variance components in a random one-way model is examined from a different viewpoint. It is found that their technique can sometimes be improved on with respect to degrees of freedom and with respect to the particular linear combinations for which an exact confidence interval can be obtained. In addition, our development is applicable to any mixed model having two variance components and to some special cases involving three or more variance components.

Estimability and Linear Hypotheses

Abstract A proposition is given which provides an easily justified reason as to why attention should be confined to estimable parametric vectors when formulating linear hypotheses. The possibility of justifying ones linear estimation effort on the estimable parametric functions via an identifiability condition is also mentioned.

Symmetrically distributed and unbiased estimators in linear models

Let Y be distributed symmetrically about Xβ. Natural generalizations of odd location statistics, say T‘Y’, and even location-free statistics, say W‘Y’, that were used by Hogg ‘1960, 1967)’ are introduced. We show that T‘Y’ is distributed symmetrically about β and thus E[T‘Y’] = β and that each element of T‘Y’ is uncorrelated with each element of W‘Y’. Applications of this result are made to R-estiraators and the result is extended to a multivariate linear model situation.

Some sufficient conditions for establishing (M, S)-optimality

Abstract Two sufficient conditions are given for an incomplete block design to be ( M,S - optimal. For binary designs the conditions are (i) that the elements in each row, excluding the diagonal element, of the association matrix differ by at most one, and (ii) that the off-diagonal elements of the block characteristic matrix differ by at most one. It is also shown how the conditions can be utilized for nonbinary designs and that for blocks of size two the sufficient condition in terms of the association matrix can be attained.

An Efficient Estimator of the Mean in a Two-Stage Nested Model

To estimate the mean in a two-stage nested model one would usually use a weighted average of the group sample means with the weights summing to one. Among all those with nonrandom weights we find the estimator that maximizes the minimum possible efficiency. It is found to perform very well in comparison with other estimators that have been proposed.

A note on the Satterthwaite confidence interval for a variance

When using a Satterthwaite chi-squared approximation, it is generally thought that the approximation is satisfactory when it is applied to a positive linear combination of mean squares. In this note, we describe how the Williams - Tukey idea for getting a confidence interval for the among groups variance in a random one-way model can be incorporated into Satterthwaite’s procedure for getting a confidence interval for a variance. This adjusted Satterthwaite procedure insures that his chi-squared approximation is always applied to positive linear combinations of mean squares. A small simulation is included which suggests that the adjustment to the Satterthwaite procedure is effective.

When can random effects be treated as fixed effects for computing a test statistic for a linear hypothesis

The Idea of treating the random effects as fixed for constructing a test for a linear hypothesis (of fixed effects) in a mixed linear model is considered in this paper. The paper examines when such a test statistic can be computed and what are its distributional properties with respect to the actual mixed model.

A modified harmonic mean test procedure for variance components

A complete class of tests of variance components is characterized within the class of tests statistics of the form of a ratio of a linear combination of chi-squared random variables to an independent chi-squared random variable. This result is used in the context of general unbalanced mixed models to show that the harmonic mean method results in an inadmissible test of the random treatment effects. The harmonic mean procedure is then modified in such a way that the modified test uniformly dominates the original test. Two competitive tests are the LMP (locally most powerful) and Walds tests, which have optimal power properties against small and large alternatives, respectively. A Monte Carlo simulation study reveals that the modified test outperforms both the LMP and Walds tests in badly unbalanced designs and that it is a viable alternative in less unbalanced designs.

Characterizing sums of squares by their distributions

Abstract In a linear model under normality it is shown that the error sum of squares is characterized by its distribution. Two proofs are presented, one using the almost-sure uniqueness of uniformly minimum variance unbiased estimators and the other using linear algebra. Two illustrations of how this characterization can be used are given.