The two-period crossover design with baseline measurements

We consider a two-period crossover study in which each patients measured on the response variable at the start as well as at the end of both periods. We examine models in which the carryover effect at the start of the second period may be different from the carryover effect at the end, and in which the correlations between observations decrease as a function of the time between them.

In trials with a relatively short washout period, we recommend that the second baseline measurement not be incorporated into the analysis and that the data be evaluated by analysis of covariance, with the difference between the post-treatment values as the response variable and the first period's baseline value as the covariate. The absence of carryover effects must be assumed.

When the washout period is moderately long (comparable in length to either treatment period), the preferred analysis for a difference between direct treatment effects will again generally be based on the differences between post-treatment values. An analysis based on changes from baseline would, under certain assumptions about the form of the variance-covariance matrix, be preferred only for quite long washout periods and large correlations between observations. Even then, the efficiency of the test for equality of direct effects is improved if the difference between the baseline values is used as the covariate.     

BALANCED CHANGE-OVER DESIGNS FOR AUTOCORRELATED OBSERVATIONS

Some aspects of design of change-over experiments with autocorrelated observations are investigated. Williams' designs (1949, 1950) are seen to be balanced for autocorrelated observations.     

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.     

Equi-information robust designs: Which designs are possible?

The effects of missing observations in a designed experiment are reviewed. Conditions are determined for a design to retain equal information, i.e. the same generalized variance of unknown parameters, when either any single or any pair of observations is lost. Some examples of designs with this property are given. Although there are many designs which retain equal information for the loss of exactly t observations, where t = 1,2,3,…, it is shown that it is not possible to obtain any design which retains equal information when any one and any two and also any three observations are missing.     

Optimal designs for the Michaelis–Menten model with correlated observations

In this paper we investigate the problem of designing experiments for generalized least-squares analysis in the Michaelis–Menten model. We study the structure of exact D-optimal designs in a model with an autoregressive error structure. Explicit results for locally D-optimal designs are derived for the case where two observations can be taken per subject. Additionally standardized maximin D-optimal designs are obtained in this case. The results illustrate the enormous difficulties to find exact optimal designs explicitly for nonlinear regression models with correlated observations.     

Posterior consistency of random effects models for binary data

In longitudinal studies or clustered designs, observations for each subject or cluster are dependent and exhibit intra-correlation. To account for this dependency, we consider Bayesian analysis for conditionally specified models, so-called generalized linear mixed model. In nonlinear mixed models, the maximum likelihood estimator of the regression coefficients is typically a function of the distribution of random effects, and so the misspecified choice of the distribution of random effects can cause bias in the estimator. To avoid the problem of the misspecification of the distribution of random effects, one can resort in nonparametric approaches. We give sufficient conditions for posterior consistency of the distribution of random effects as well as regression coefficients.     

Circular balanced repeated measurements designs

The concept of a circular design is defined and when proper balance for various effects is assumed, its universal optimality is proved over the class of all designs with the same set of parameters, Such designs are shown to minimize the variance of the best linear unbiased estimators of contrasts of residual and direct effects over the class of equireplicated designs. All models assume first order residual effects and are of a circular nature. The proofs are presented in a unified manner for several models at a time. They are based on certain matrix domination which occurs when parameters are eliminated from a linear modelj this latter fact is proved for a general linear model.     

Nonparametric methods in factorial designs

Summary In this paper, we summarize some recent developments in the analysis of nonparametric models where the classical models of ANOVA are generalized in such a way that not only the assumption of normality is relaxed but also the structure of the designs is introduced in a broader framework and also the concept of treatment effects is redefined. The continuity of the distribution functions is not assumed so that not only data from continuous distributions but also data with ties are included in this general setup. In designs with independent observations as well as in repeated measures designs, the hypotheses are formulated by means of the distribution functions. The main results are given in a unified form. Some applications to special designs are considered, where in simple designs, some well known statistics (such as the Kruskal-Wallis statistic and the χ²-statistic for dichotomous data) come out as special cases. The general framework presented here enables the nonparametric analysis of data with continuous distribution functions as well as arbitrary discrete data such as count data, ordered categorical and dichotomous data. Received: October 13, 1999; revised version: June 26, 2000     

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.     

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.     

A geometric characterization of optimal designs for regression models with correlated observations

We consider the problem of optimal design of experiments for random effects models, especially population models, where a small number of correlated observations can be taken on each individual, while the observations corresponding to different individuals are assumed to be uncorrelated. We focus on c-optimal design problems and show that the classical equivalence theorem and the famous geometric characterization of Elfving (1952) from the case of uncorrelated data can be adapted to the problem of selecting optimal sets of observations for the n individual patients. The theory is demonstrated by finding optimal designs for a linear model with correlated observations and a nonlinear random effects population model, which is commonly used in pharmacokinetics.     

Minimum breakdown designs in blocks of size two

In scientific investigations, there are many situations where each two experimental units have to be grouped into a block of size two. For planning such experiments, the variance-based optimality criteria like A-, D- and E-criterion are typically employed to choose efficient designs, if the estimation efficiency of treatment contrasts is primarily concerned. Alternatively, if there are observations which tend to become lost during the experimental period, the robustness criteria against the unavailability of data should be strongly recommended for selecting the planning scheme. In this study, a new criterion, called minimum breakdown criterion, is proposed to quantify the robustness of designs in blocks of size two. Based on the proposed criterion, a new class of robust designs, called minimum breakdown designs, is defined. When various numbers of blocks are missing, the minimum breakdown designs provide the highest probabilities that all the treatment contrasts are estimable. An exhaustive search procedure is proposed to generate such designs. In addition, two classes of uniformly minimum breakdown designs are theoretically verified.     

THE ROBUSTNESS OF BINARY AND NON-BINARY BALANCED NESTED ROW-COLUMN DESIGNS UNDER THE UNAVAILABILITY OF BLOCKS: A COMPARISON

In cases where both exist, the balanced, binary nested row-column designs are known to be inferior to a class of balanced non-binary designs. However, if it is possible for blocks of observations to become unavailable after an experiment has commenced, a binary nested row-column design may possibly be better than a non-binary one. This paper investigates the robustness of binary and non-binary variance-balanced nested row-column designs to the unavailability of one or more blocks of observations. Robustness is measured through the C-matrices of the designs resulting from removing blocks, using optimality criteria such as A-, D-, E- and MV-optimality.     

Non-parametric group sequential designs in randomized clinical trials

This paper examines some non‐parametric group sequential designs applicable for randomized clinical trials, for comparing two continuous treatment effects taking the observations in matched pairs, or applicable in event‐based analysis. Two inverse binomial sampling schemes are considered, of which the second one is an adaptive data‐dependent design. These designs are compared with some fixed sample size competitors. Power and expected sample sizes are calculated for the proposed procedures.     

A comparison of inference procedures in unbalanced split-plot designs

Several procedures for constructing confidence intervals and testing hypotheses about fixed effects in unbalanced split-plot experiments are described in this paper. These procedures can also be used for unbalanced repeated measures experiments when the repeated measures satisfy the Huyhn-Feldt (1970) conditions. A number of these procedures require that the whole plot error mean square has a distribution proportional to a chi-square distribution and that it be independent of estimators of the parameter functions. Often, neither of these conditions are met in unbalanced split-plot experiments. Simulation studies of a small design of eight observations and larger designs with 34 to 48 observations are used to investigate the performance of the different procedures.     

Cost‐efficient higher‐order crossover designs for two‐treatment clinical trials

Higher‐order crossover designs have drawn considerable attention in clinical trials, because of their ability to test direct treatment effects in the presence of carry‐over effects. The important question, when applying higher‐order crossover designs in practice, is how to choose a design with both statistical and cost efficiencies from various alternatives. In this paper, we propose a general cost function and compare five statistically optimal or near‐optimal designs with this cost function for a two‐treatment study under different carry‐over models. Based on our study, to achieve both statistical and cost efficiencies, a four‐period, four‐sequence crossover design is generally recommended under the simple carry‐over or no carry‐over models, and a three‐period, two‐sequence crossover design is generally recommended under the steady‐state carry‐over models. Copyright © 2005 John Wiley & Sons, Ltd.     

Optimum designs in complete two-way layouts

In the usual two-way layout of ANOVA (interactions are admitted) let nij ? 1 be the number of observations for the factor-level combination(i, j). For testing the hypothesis that all main effects of the first factor vanish numbers $\text{n}{}^1\text{ij}$ are given such that the power function of the F-test is uniformly maximized (U-optimality), if one considers only designs (nij) for which the row-sums ni are prescribed. Furthermore, in the (larger) set of all designs for which the total number of observations is given, all D-optimum designs are constructed.     

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.     

Robustness of subset response surface designs to missing observations

Experiments designed to investigate the effect of several factors on a process have wide application in modern industrial and scientific research. Response surface designs allow the researcher to model the effects of the input variables on the response of the process. Missing observations can make the results of a response surface experiment quite misleading, especially in the case of one-off experiments or high cost experiments. Designs robust to missing observations can attract the user since they are comparatively more reliable. Subset designs are studied for their robustness to missing observations in different experimental regions. The robustness of subset designs is also improved for multiple levels by using the minimax loss criterion.     

PARTIALLY EQUINEIGHBOURED DESIGNS

This paper presents further results on a class of designs called equineighboured designs, ED. These designs are intended for field and related experiments, especially whenever there is evidence that observations in the same block are correlated. An ED has the property that every unordered pair of treatments occurs as nearest neighbours equally frequently at each level. Ipinyomi (1986) has defined and shown that ED are balanced designs when neighbouring observations are correlated. He has also presented ED as a continuation of the development of optimal block designs. An ED would often require many times the number of experimental materials needed for the construction of an ordinary balanced incomplete block, BIB, design for the same number of treatments and block sizes. Thus for a relatively large number of treatments and block sizes the required minimum number of blocks may be excessively large for practical use of ED. In this paper we shall define and examine partially equineighboured designs with n concurrences, PED (n), as alternatives where ED are practically unachievable. Particular attention will be given to designs with smaller numbers of blocks and for which only as little balance as possible may be lost.