Minimal circular balanced repeated measurement designs in unequal period sizes

Abstract

Repeated measurement designs (RMDs) are widely used in medicine, pharmacology, animal sciences and psychology. In these fields, there are several situations where these designs should be used in periods of different sizes. With the use of RMD, residual effects or carry over effects may arise and balanced RMDs are solution to this problem. In this article, therefore, some infinite series are developed through method of cyclic shifts to obtain circular balanced repeated measurements designs in periods of two different sizes.     

Some new construction of circular weakly balanced repeated measurements designs in periods of two different sizes

Abstract

Balanced repeated measurements designs (RMDs) balance out the residual effects. Williams Latin square designs work as minimal combinatorial balanced as well as variance balanced for RMDs for p (period sizes) = v (number of treatments). If minimal balanced RMDs cannot be constructed for the situations where p must be less than v then weakly balanced RMDs should be preferred. In this article, some generators are developed to generate circular weakly balanced RMDs in periods of two different sizes. To obtain the proposed designs, some construction procedures are also described for some of the cases where we could not develop generators.     

Optimal Designs for the Carryover Model with Random Interactions Between Subjects and Treatments

The paper considers a model for crossover designs with carryover effects and a random interaction between treatments and subjects. Under this model, two observations of the same treatment on the same subject are positively correlated and therefore provide less information than two observations of the same treatment on different subjects. The introduction of the interaction makes the determination of optimal designs much harder than is the case for the traditional model. Generalising the results of Bludowsky's thesis, the present paper uses Kushner's method to determine optimal approximate designs. We restrict attention to the case where the number of periods is less than or equal to the number of treatments. We determine the optimal designs in the important special cases that the number of periods is 3, 4 or 5. It turns out that the optimal designs depend on the variance of the random interactions and in most cases are not binary. However, we can show that neighbour balanced binary designs are highly efficient, regardless of the number of periods and of the size of the variance of the interaction effects.     

Optimal treatment allocation and study duration for trials with discrete-time survival endpoints

Studies on event occurrence may be conducted in experiments, where one or more treatment groups are compared to a control group. Most of the randomized trials are designed with equally sized groups, but this design is not always the best one. The statistical power of the study may be larger with unequal sample sizes, and researchers may want to place more participants in one group relative to the other due to resource constraints or costs. The optimal designs for discrete-time survival endpoints in trials with two groups, where different proportions of subjects in the experimental group are taken into account, can be studied using the generalized linear model. Applying a cost function, the optimal combination of the number of subjects and periods in the study and the optimal allocation ratio can be found. It is observed that the ratio of the recruitment costs in both groups, the ratio of the recruitment cost in the control group to the cost of obtaining a measurement, the size of the treatment effect, and the shape of the survival distribution have the greatest influence on the optimal design.     

Efficient circular neighbour designs for spatial interference model

We consider circular block designs for field-trials when there are two-sided spatial interference between neighbouring plots of the same blocks. The parameter of interest is total effects that is the sum of direct effect of treatment and neighbour effects, which correspond to the use of a single treatment in the whole field. We determine universally optimal approximate designs. When the number of blocks may be large, we propose efficient exact designs generated by a single sequence of treatment. We also give efficiency factors of the usual binary block neighbour balanced designs which can be used when the number of blocks is small.     

Cross-over Designs in the Presence of Higher Order Carry-overs

In cross-over experiments, where different treatments are applied successively to the same experimental unit over a number of time periods, it is often expected that a treatment has a carry-over effect in one or more periods following its period of application. The effect of interaction between the treatments in the successive periods may also affect the response. However, it seems that all systematic studies of the optimality properties of cross-over designs have been done under models where carry-over effects are assumed to persist for only one subsequent period. This paper proposes a model which allows for the possible presence of carry-over effects up to k subsequent periods, together with all the interactions between treatments applied at k + 1 successive periods. This model allows the practitioner to choose k for any experiment according to the requirements of that particular experiment. Under this model, the cross-over designs are studied and the class of optimal designs is obtained. A method of constructing these optimal designs is also given.     

Construction of circular partially balanced-Repeated measurement designs using cyclic shifts

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences and psychology. These designs balance out the residual effects. The situations where balanced repeated measurements designs require a large number of the subjects, partially-balanced repeated measurements designs should be used. In this paper some infinite series are developed which provide circular partially-balanced repeated measurement designs for p (periods) even. Catalogues of circular partially-balanced repeated measurement designs are also presented for v (treatments) ≤ 100 with p = 5, 7 & 9.     

Optimal crossover designs when carryover effects are proportional to direct effects

Crossover designs are used for a variety of different applications. While these designs have a number of attractive features, they also induce a number of special problems and concerns. One of these is the possible presence of carryover effects. Even with the use of washout periods, which are for many applications widely accepted as an indispensable component, the effect of a treatment from a previous period may not be completely eliminated. A model that has recently received renewed attention in the literature is the model in which first-order carryover effects are assumed to be proportional to direct treatment effects. Under this model, assuming that the constant of proportionality is known, we identify optimal and efficient designs for the direct effects for different values of the constant of proportionality. We also consider the implication of these results for the case that the constant of proportionality is not known.     

Applications and Implementations of Continuous Robust Designs

In the literature concerning the construction of robust optimal designs, many resulting designs turn out to have densities. In practice, an exact design should tell the experimenter what the support points are and how many subjects should be allocated to each of these points. In particular, we consider a practical situation in which the number of support points allowed is constrained. We discuss an intuitive approach, which motivates a new implementation scheme that minimizes the loss function based on the Kolmogorov and Smirnov distance between an exact design and the optimal design having a density. We present three examples to illustrate the application and implementation of a robust design constructed: one for a nonlinear dose-response experiment and the other two for general linear regression. Additionally, we perform some simulation studies to compare the efficiencies of the exact designs obtained by our optimal implementation with those by other commonly used implementation methods.     

An optimal selection of two-level regular single arrays for robust parameter experiments

Efforts have been made in the literature to find optimal single arrays which work best for the robust parameter experiments. However, examples show that in many cases the optimal designs obtained by the existing criteria cloud not attain the maximum number of clear interested effects for robust parameter experiments. In this paper, through a similar way of Zhang et al. (2008) (ZLZA, in short), an aliasing pattern to measure the confounding between the interested effects and other effects for the case of robust parameter designs is introduced. A new criterion for selecting optimal two-level regular single arrays is proposed. In the consideration of the criterion, two rank-orders of effects are suggested: one is based on the interest of experimenters and the other is under the usual effect hierarchy principle. The optimal designs are tabulated in the appendix.     

The construction of nearly balanced and nearly strongly balanced uniform cross-over designs

In this paper we consider the class of uniform cross-over designs. Existing results on the universal optimality of uniform cross-over designs are reviewed and a general method of construction is described. The constructed designs fall into four families, which include the balanced and strongly balanced designs as special cases: the remaining designs we refer to as nearly strongly balanced, a term first introduced by Kunert (Ann. Statist. 11 (1983)), and nearly balanced. The nearly strongly balanced and nearly balanced designs form an important family of uniform cross-over designs which provide designs where balanced or strongly balanced designs do not exist. The construction method can be easily generalized for any number of periods and subjects, as long as they are both a multiple of the number of treatments. Some illustrative examples are included.     

Assessing robustness of crossover designsto subjects dropping out

In some crossover experiments, particularly in medical applications, subjects may fail to complete their sequences of treatments for reasons unconnected with the treatments received. A method is described of assessing the robustness of a planned crossover design, with more than two periods, to subjects leaving the study prematurely. The method involves computing measures of efficiency for every possible design that can result, and is therefore very computationally intensive. Summaries of these measures are used to choose between competing designs. The computational problem is reduced to a manageable size by a software implementation of Polya theory. The method is applied to comparing designs for crossover studies involving four treatments and four periods. Designs are identified that are more robust to subjects dropping out in the final period than those currently favoured in medical and clinical trials.     

Optimal adaptive designs for acute oral toxicity assessment

Acute oral toxicity studies are used to assess the toxicity to experimental animals of a single dose of the substance under investigation, assigning the substance to one of a number of classes. Animal welfare concerns have led to the development of three adaptive designs as alternatives to the traditional fixed sample design. These lead to reductions in the number of animals required in total and in the number exposed to lethal doses. In this paper, we show how designs can be constructed to optimise utility functions combining the need to classify correctly with the desire to use a small number of animals. Constrained optimal designs are also obtained in which no animal is exposed to a dose higher than that at which a death has been observed. The optimal designs lead to the correct classification with high probability whilst reducing the expected number of animal deaths relative to existing adaptive designs.     

Asymptotics in multivariate linear models with optimal experimental designs

The conditions guarantying consistency and asymptotic normality of Least-Square estimators as well as the consistency of the usual tests for linear hypotheses in multivariate linear models are shown to be valid whenever almost optimal experimental designs are used.     

AN ALGORITHM FOR THE CONSTRUCTION OF OPTIMAL OR NEAR-OPTIMAL CHANGE-OVER DESIGNS

This paper presents an algorithm for the construction of optimal or near optimal change-over designs for arbitrary numbers of treatments, periods and units. Previous research on optimality has been either theoretical or has resulted in limited tabulations of small optimal designs. The algorithm consists of a number of steps:first find an optimal direct treatment effects design, ignoring residual effects, and then optimise this class of designs with respect to residual effects. Poor designs are avoided by judicious application of the (M, S)-optimality criterion, and modifications of it, to appropriate matrices. The performance of the algorithm is illustrated by examples.     

The effect of small sample on optimal designs for logistic regression models

Asymptotic methods are commonly used in statistical inference for unknown parameters in binary data models. These methods are based on large sample theory, a condition which may be in conflict with small sample size and hence leads to poor results in the optimal designs theory. In this paper, we apply the second order expansions of the maximum likelihood estimator and derive a matrix formula for the mean square error (MSE) to obtain more precise optimal designs based on the MSE. Numerical results indicate the new optimal designs are more efficient than the optimal designs based on the information matrix.     

Robust designs for nearly linear regression

In this paper we seek designs and estimators which are optimal in some sense for multivariate linear regression on cubes and simplexes when the true regression function is unknown. More precisely, we assume that the unknown true regression function is the sum of a linear part plus some contamination orthogonal to the set of all linear functions in the L₂ norm with respect to Lebesgue measure. The contamination is assumed bounded in absolute value and it is shown that the usual designs for multivariate linear regression on cubes and simplices and the usual least squares estimators minimize the supremum over all possible contaminations of the expected mean square error. Additional results for extrapolation and interpolation, among other things, are discussed. For suitable loss functions optimal designs are found to have support on the extreme points of our design space.     

Nearly optimal orthogonally blocked desings for a quadratic mixture model with q components

In experiments with mixtures involving process variables, orthogonal block designs may be used to allow estimation of the parameters of the mixture components independently of estimation of the parameters of the process variables. In the class of orthogonally blocked designs based on pairs of suitably chosen Latin squares, the optimal designs consist primarily of binary blends of the mixture components, regardless of how many ingredients are available for the mixture. This paper considers ways of modifying these optimal designs so that some or all of the runs used in the experiment include a minimum proportion of each mixture ingredient. The designs considered are nearly optimal in the sense that the experimental points are chosen to follow ridges of maxima in the optimality criteria. Specific designs are discussed for mixtures involving three and four components and distinctions are identified for different designs with the same optimality properties. The ideas presented for these specific designs are readily extended to mixtures with q>4 components.     

Optimal designs for estimating the slope of a regression

In the common linear model with quantitative predictors we consider the problem of designing experiments for estimating the slope of the expected response in a regression. We discuss locally optimal designs, where the experimenter is only interested in the slope at a particular point, and standardized minimax optimal designs, which could be used if precise estimation of the slope over a given region is required. General results on the number of support points of locally optimal designs are derived if the regression functions form a Chebyshev system. For polynomial regression and Fourier regression models of arbitrary degree the optimal designs for estimating the slope of the regression are determined explicitly for many cases of practical interest.     

Aligned rank statistics for repeated measurement models with orthonormal design, employing a Chernoff–Savage approach

In this paper, aligned rank statistics are considered for testing hypotheses regarding the location in repeated measurement designs, where the design matrix for each set of measurements is orthonormal. Such a design may, for instance, be used when testing for linearity. It turns out that the centered design matrix is not of full rank, and therefore it does not quite satisfy the usual conditions in the literature. The number of degrees of freedom of the limiting chi-square distribution of the test statistic under the null hypothesis, however, is not affected, unless rather special hypotheses are tested. An independent derivation of this limiting distribution is given, using the Chernoff–Savage approach. In passing, it is observed that independence of the choice of aligner, which in the location problem is well-known to be due to cancellation, may in scale problems occur as a result of the type of score function suitable for scale tests. A possible extension to multivariate data is briefly indicated.