David Birkes

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.

Estimation of Population Pharmacokinetic Parameters Using Destructively Obtained Experimental Data: A Simulation Study of the One-Compartment Open Model

A simulation study of the one-compartment open pharmacokinetic model has been made. The population pharmacokinetic parameters which characterize the population of drug residues over time are assumed to be stochastic. A general theoretical model framework for parameter estimation via the method of extended least squares is presented. Formulas approximating the required mean and variance time functions are developed and subsequently used in the simulation study. The effects of four different designs in four different animal populations are presented. The simulated data are those of the single observation per animal per time point type. The characterizing population pharmacokinetic parameters have been analyzed for bias and reliability in both a naive and second-order mean model. Recommendations for choosing an appropriate sampling design are included.

Obtaining species: sample size considerations

Suppose fish are to be sampled from a stream. A fisheries biologist might ask one of the following three questions: ‘How many fish do I need to catch in order to see all of the species?’, ‘How many fish do I need to catch in order to see all species whose relative frequency is more than 5%?’, or ‘How many fish do I need to catch in order to see a member from each of the species A, B, and C?’. This paper offers a practical solution to such questions by setting a target sample size designed to achieve desired results with known probability. We present three sample size methods, one we call ‘exact’ and the others approximate. Each method is derived under assumed multinomial sampling, and requires (at least approximate) independence of draws and (usually) a large population. The minimum information needed to compute one of the approximate methods is the estimated relative frequency of the rarest species of interest. Total number of species is not needed. Choice of a sample size method depends largely on available computer resources. One approximation (called the ‘Monte Carlo approximation’) gets within ±6 units of exact sample size, but usually requires 20–30 minutes of computer time to compute. The second approximation (called the ‘ratio approximation’) can be computed manually and has relative error under 5% when all species are desired, but can be as much as 50% or more too high when exact sample size is small. Statistically, this problem is an application of the ‘sequential occupancy problem’. Three examples are given which illustrate the calculations so that a reader not interested in technical details can apply our results.

Optimal a × b connected designs with a + bobservations

Abstract Consider the problem of determining optimal a×b two-factor designs with only a+b observations. It is shown that, when a ≠ 3, the A-optimal designs are those constructed by placing a + b−1 observations in a perpendicular pattern so that a observations are at a single level of factor 2 and b observations are at a single level of factor 1, and then adding one more observation. These designs are also optimal under a modification of the E-criterion.

Generalized Likelihood Ratio Tests and Uniformly Most Powerful Tests

Abstract For testing a one-sided hypothesis in a one-parameter family of distributions, it is shown that the generalized likelihood ratio (GLR) test coincides with the uniformly most powerful (UMP) test, assuming certain monotonicity properties for the likelihood function. In particular, the equivalence of GLR tests and UMP tests holds for one-parameter exponential families. In addition, the relationship between GLR and UMPU (UMP unbiased) tests is considered when testing two-sided hypotheses.

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.

Existence of maximum likelihood estimates in normal variance-components models

Abstract Under the usual nonnegativity constraints on the variance components, a maximum likelihood estimate (MLE) of the parameter vector in a normal variance-components model is known to exist. We investigate the question of existence under more general types of constraints and, in particular, under the constraint that requires only that the variance–covariance matrix be positive definite. Attention is restricted to models in which all the possible variance–covariance matrices commute with one another. It is found that in some models, such as all random one-way models with a single group having the largest size and all balanced random two-way models, the likelihood becomes infinite under the positive definiteness constraints, so that no MLE exists. In (practically) all normal balanced mixed-effects classification models, a residual maximum likelihood estimate (REMLE) exists.

The number of minimally connected block designs

Abstract The formula a b −1 b a −1 is derived for the number of connected a × b block designs having the least possible number, a + b − 1, of observations needed for connectedness. It is shown that this formula also arises as the number of labeled bicolored trees with a vertices of one color and b vertices of the other.

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.

DATA STRUCTURE WITH RESPECT TO THE MAIN EFFECTS MODEL: A DISCUSSION MOTIVATED BY A META-ANALYSIS DATA SET

A discussion on data structure relative to the main effects model is motivated by a severely unbalanced meta-analysis data set. This data set is used to highlight the difficulty of assessing data structure when multiple factor data sets are severely unbalanced. Both theoretical results and numerical examples are used to establish that simple approaches to examining data structure using two-way tables provide easily assimilated information about the effect of data unbalance on main effect contrast variances. In addition, notions of balance, proportionality, unbalance, and missing cells with respect to the main effects model are defined in terms of the two-way tables and are related to main effect contrast estimate variances as assessed using the D-optimality criterion.