A theorem for selecting oa-based latin hypercubes using a distance criterion

The maximin distance criterion is used for the selection of an OA-based Latin hypercube. For the case in which the underlying orthogonal array is a full factorial design without replication, we construct an OA-based Latin hypercube that reaches the same distance as its parent array.     

Systematic latin squares

In order to properly utilize restricted randomization in the selection of t × t Latin squares it is necessary to have some idea of the various types of systematic Latin squares that should be removed from the admissible sets. The best known systematic squares are the diagonal squares and the Knut Vik squares. When t is not a prime number there are various other types of diagonal and balanced Latin squares. Eleven types of 4 × 4 Latin squares, each of them being systematic, are identified, displayed, and their properties indicated. Eight types of systematic 6 × 6 Latin squares are also identified and displayed. The effect of removing systematic squares from the admissible sets of Latin squares is discussed. Recommendations are made on when a restricted randomization procedure is to be preferred to a full randomization procedure in the selection of a random t × t Latin square.     

Low-order spatially-distinct latin squares

The order three to five spatially-distinct Latin squares, and the order three to six spatially-distinct Latin square treatment designs are listed. Some statistical results are given. Designs for 4, 5 and 6 treatments that were found previously to be robust to a linear by linear interacrion are shown to be optimal. Designs with good neighbour balanced are also considered.     

Indivisible partitions of latin squares

In a latin square of order n, a k-plex is a selection of kn entries in which each row, column and symbol occurs k times. A 1-plex is also called a transversal. An indivisible k-plex is one that contains no c-plex for 0<c<k. For orders n∉{2,6}, existence of latin squares with a partition into 1-plexes was famously shown in 1960 by Bose, Shrikhande and Parker. A main result of this paper is that, if k divides n and 1<k<n then there exists a latin square of order n with a partition into indivisible k-plexes.Define κ(n) to be the largest integer k such that some latin square of order n contains an indivisible k-plex. We report on extensive computations of indivisible plexes and partitions in latin squares of order at most 9. We determine κ(n) exactly for n≤8 and find that κ(9)∈{6,7}. Up to order 8 we count all indivisible partitions in each species.For each group table of order n≤8 we report the number of indivisible plexes and indivisible partitions. For group tables of order 9 we give the number of indivisible plexes and identify which types of indivisible partitions occur. We will also report on computations which show that the latin squares of order 9 satisfy a conjecture that every latin square of order n has a set of ⌊n/2⌋ disjoint 2-plexes.By extending an argument used by Mann, we show that for all n≥5 there is some k∈{1,2,3,4} for which there exists a latin square of order n that has k disjoint transversals and a disjoint (n−k)-plex that contains no c-plex for any odd c.     

Generation of space-filling uniform designs in unit hypercubes

In this paper a new discrepancy measure of uniformity for uniform designs (UDs) in a unit cube is presented. Alternative measures of uniformity based on distance criteria which can be applied to higher dimensions are also discussed. The good lattice point (glp) method was used to construct the uniform designs. Two approaches (generator equivalence and projection) of reducing the computational cost of the glp method are proposed and discussed. Two examples are presented in this paper.     

A study on algorithms for optimization of Latin hypercubes

A crucial component in the statistical simulation of a computationally expensive model is a good design of experiments. In this paper we compare the efficiency of the columnwise–pairwise (CP) and genetic algorithms for the optimization of Latin hypercubes (LH) for the purpose of sampling in statistical investigations. The performed experiments indicate, among other results, that CP methods are most efficient for small and medium size LH, while an adopted genetic algorithm performs better for large LH.Two optimality criteria suggested in the literature are evaluated with respect to statistical properties and efficiency. The obtained results lead us to favor a criterion based on the physical analogy of minimization of forces between charged particles suggested in Audze and Eglais (1977. Problems Dyn. Strength 35, 104–107) over a ‘maximin distance’ criterion from Johnson et al. (1990. J. Statist. Plann. Inference 26, 131–148).     

Optimum properties of latin square designs and a matrix inequality

The paper deals with uniform and D-optimality of designs in the two-way elimination of heterogeneities. It is shown that designs which are optimum for the hypothesis that all treatment effects are equal are optimum for some other hypotheses, too. The Proof is based on a new matrix- and determinantal inequality.     

Estimation of parameters in latin squares and graeco-latin squares with missing observations

We show that if in an additive model with p-2 MOLS if one omits up to p-1 observations from the same row, the same column or which correspond to the same letter in any of the squares all effects are estimable. On the other hand with only two missing observations not from the same row, the same column or corresponding to the same letter in any of the squares one degree of freedom is lost for each set of effects.     

An easy method of constructing latin square designs balanced for the immediate residual and other order effects

Bradley (1958) proposed a very simple procedure for constructing latin square designs to counterbalance the immediate sequential effect for an even number of treatments. When the number of treatments is odd, balance in a single latin square is not possible. In the present note we have developed an analogous method for the construction of such designs which may be used for an even or odd number of treatments. A proof has also been offered to assure the general validity of the procedure.     

Optimal foldover designs

An obvious strategy for obtaining a Doptimal foldover design for p factors at two levels each in 2N runs is to fold a Doptimal main effects plan. We show that this strategy works except when N = 4t + 2 and s is even In that case there are two different classes of D-optimal main effects plans with N runs that have the same determinant. However folding them gives two different values foi the D-optimality criteiion One set of designs is D-optimal The other is not.     

Optimal Bayesian Randomization

Randomization is a puzzle for Bayesians. The intuitive need for randomization is clear, but there is a standard result that Bayesians need not randomize. In this paper we propose a model in which randomization is a strictly optimal procedure. The most important aspect of our model is that there are several parties who make different decisions and observe different data. The result also sheds light on the ethical considerations involving randomization in a clinical trial.     

Optimal few-stage designs

Optimal designs are presented for experiments in which sampling is carried out in stages. There are two Bernoulli populations and it is assumed that the outcomes of the previous stage are available before the sampling design for the next stage is determined. At each stage, the design specifies the number of observations to be taken and the relative proportion to be sampled from each population. Of particular interest are 2- and 3-stage designs.To illustrate that the designs can be used for experiments of useful sample sizes, they are applied to estimation and optimization problems. Results indicate that, for problems of moderate size, published asymptotic analyses do not always represent the true behavior of the optimal stage sizes, and efficiency may be lost if the analytical results are used instead of the true optimal allocation.The exactly optimal few stage designs discussed here are generated computationally, and the examples presented indicate the ease with which this approach can be used to solve problems that present analytical difficulties. The algorithms described are flexible and provide for the accurate representation of important characteristics of the problem.     

Optimal Whitening and Decorrelation

Whitening, or sphering, is a common preprocessing step in statistical analysis to transform random variables to orthogonality. However, due to rotational freedom there are infinitely many possible whitening procedures. Consequently, there is a diverse range of sphering methods in use, for example, based on principal component analysis (PCA), Cholesky matrix decomposition, and zero-phase component analysis (ZCA), among others. Here, we provide an overview of the underlying theory and discuss five natural whitening procedures. Subsequently, we demonstrate that investigating the cross-covariance and the cross-correlation matrix between sphered and original variables allows to break the rotational invariance and to identify optimal whitening transformations. As a result we recommend two particular approaches: ZCA-cor whitening to produce sphered variables that are maximally similar to the original variables, and PCA-cor whitening to obtain sphered variables that maximally compress the original variables.     

Optimal Bayesian two-phase designs

In this paper we present a Bayesian decision theoretic approach to the two-phase design problem. The solution of such sequential decision problems is usually difficult to obtain because of their reliance on preposterior analysis. In overcoming this problem, we adopt the Mont-Carlo-based approach of Müller and Parmigiani (1995) and develop optimal Bayesian designs for two-phase screening tests. A rather attractive feature of the Monte-Carlo approach is that it facilitates the preposterior analysis by replacing it with a sequence of scatter plot smoothing/regression techniques and optimization of the corresponding fitted surfaces. The method is illustrated for depression in adolescents using data from past studies.