Minimal balanced repeated measurements designs

Experimental designs in which treatments are applied to the experimental units, one at a time, in sequences over a number of periods, have been used in several scientific investigations and are known as repeated measurements designs. Besides direct effects, these designs allow estimation of residual effects of treatments along with adjustment for them. Assuming the existence of first-order residual effects of treatments, Hedayat & Afsarinejad (1975) gave a method of constructing minimal balanced repeated measurements [RM(v,n,p)] design for v treatments using n=2v experimental units for p [=(v+1)/2] periods when v is a prime or prime power. Here, a general method of construction of these designs for all odd v has been given along with an outline for their analysis. In terms of variances of estimated elementary contrasts between treatment effects (direct and residual), these designs are seen to be partially variance balanced based on the circular association scheme.     

Optimal and efficient crossover designs for comparing test treatments to a control treatment under various models

We study the optimality, efficiency, and robustness of crossover designs for comparing several test treatments to a control treatment. Since A-optimality is a natural criterion in this context, we establish lower bounds for the trace of the inverse of the information matrix for the test treatments versus control comparisons under various models. These bounds are then used to obtain lower bounds for efficiencies of a design under these models. Two algorithms, both guided by these efficiencies and results from optimal design theory, are proposed for obtaining efficient designs under the various models.     

Nonparametric multiple comparison procedures for unbalanced one-way factorial designs

In this paper, we present several nonparametric multiple comparison (MC) procedures for unbalanced one-way factorial designs. The nonparametric hypotheses are formulated by using normalized distribution functions and the comparisons are carried out on the basis of the relative treatment effects. The proposed test statistics take the form of linear pseudo rank statistics and the asymptotic joint distribution of the pseudo rank statistics for testing treatments versus control satisfies the multivariate totally positive of order two condition irrespective of the correlations among the rank statistics. Therefore, in the context of MCs of treatments versus control, the nonparametric Simes test is validated for the global testing of the intersection hypothesis. For simultaneous testing of individual hypotheses, the nonparametric Hochberg stepup procedure strongly controls the familywise type I error rate asymptotically. With regard to all pairwise comparisons, we generalize various single-step and stagewise procedures to perform comparisons on the relative treatment effects. To further compare with normal theory counterparts, the asymptotic relative efficiencies of the nonparametric MC procedures with respect to the parametric MC procedures are derived under a sequence of Pitman alternatives in a nonparametric location shift model for unbalanced one-way layouts. Monte Carlo simulations are conducted to demonstrate the validity and power of the proposed nonparametric MC procedures.     

Some New Residual Treatment Effects Designs for Comparing Test Treatments with a Control

Pigeon & Raghavarao (1987) introduced control balanced residual treatment effects designs for the situation where one treatment is a control or standard and is to be compared with the v test treatments, and they have also given methods of construction of control balanced residual treatment effects designs and have investigated their efficiencies. In this paper we have developed some new families of control balanced residual treatment effects designs, which are Schur-optimal.     

A-optimal diallel crosses for test versus control comparisons

A-optimality of block designs for control versus test comparisons in diallel crosses is investigated. A sufficient condition for designs to be A-optimal is derived. Type S₀ designs are defined and A-optimal type S₀ designs are characterized. A lower bound to the A-efficiency of type S₀ designs is also given. Using the lower bound to A-efficiency, type S₀ designs are shown to yield efficient designs for test versus control comparisons.     

Resistant BTIB desings

Hedayat and John (1974) defined the concept of a resistant BIB design as one that remains variance-balanced upon loss of all the observations on a treatment. In this paper we consider the designs with supplemented balance (Pearce (1960)), or BTIB designs (Bechhofer and Tamhane (1981)). We investigate a subclass of these designs to determine which of these retain their ‘balance’ upon loss of all observations on a test treatment. We study the properties of such designs, give various methods of construction and examine their efficiencies with respect to the A-criterion.     

A measure of aliasing and applications to two-level fractional factorials

This paper examines some properties of a measure of aliasing proposed by Hedayat, Raktoe, and Federer (1974). It is shown that in the case of balanced orthogonal designs with no repeated treatments, minimizing the alias measure is equivalent to minimizing tr Yar(ψ). Lower bounds are found for fixed eigenvalues of the design matrix. These results are applied to two-level fractional factorials to show that in certain cases classical fractional-factorial designs yield minimal solutions for the alias measure.     

Row-column designs for comparing treatments with a control

This paper considers the problem of the design and analysis of experiments for comparing several treatments with a control when heterogeneity is to be eliminated in two directions. A class of row-column designs which are balanced for treatment vs. control comparisons (referred to as the balanced treatment vs. control row-column or BTCRC designs) is proposed. These designs are analogs of the so-called BTIB designs proposed by Bechhofer and Tamhane (Technometrics 23 (1981) 45–57) for eliminating heterogeneity in one direction. Some methods of analysis and construction of these designs are given. A measure of efficiency of BTCRC designs in terms of the A-optimality criterion is derived and illustrated by several examples.     

Multiple comparisons with a control in repeated measures incomplete block designs using R-estimators

New multiple comparison with a control (MCC) procedures are developed in repeated measures incomplete block design settings based on R-estimates. It is assumed that the errors within each subject are exchangeable random variables. The R-estimators of the treatment effects are obtained by minimizing a sum of Jaeckel (1972)-type dispersion functions. Based on the R-estimators, Dunnett-type multiple comparison procedures are developed for comparing test-treatments with a control-treatment. Under exchangeable errors, it is demonstrated that for Cox-type designs, the new procedures are more efficient than the existing nonparametric procedures. The new MCC procedures are applied to a data set in a clinical trial which consists of patients with reversible obstructive pulmonary disease.     

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.     

Bayes A-optimal designs for comparing test treatments with a control

The problem of comparing v test treatments simultaneously with a control treatment when k, v ⩾ 3 is considered. Following the work of Majumdar (1992), we use exact design theory to derive Bayes A-optimal block designs and optimal Г-minimax designs for a more general prior assumption for the one-way elimination of heterogeneity model. Examples of robust optimal designs, highly efficient designs, and the comparisons of the approximate optimal designs that are derived by our methods and by some other existing rounding-off schemes when using Owen's procedure are also provided.     

Constructions of Residual Treatment Effects Designs for Comparing Test Treatments with a Control

This article deals with residual treatment effects designs for the purpose of comparing v test treatments with a control treatment when the number of periods is no larger than v + 1. Control balanced residual treatment effects designs, which are Schur-optimal, are considered. Some methods of their construction are given.     

RCBDs with a control

The randomized complete block designs, RCBDs, are among the most popular of block designs for comparing a set of experimental treatments. The question of this design's effectiveness when one of the treatments is a control is examined here. Optimality ranges are established for the RBCD in terms of the strength of interest in control comparisons. It is found that if the control treatment is of secondary interest, the RCBD, when not best, is typically near best. This is not so when comparisons with the control are of greater interest than those among the other treatments.     

Comparing more than one treatment to more than one control in incomplete blocks

The class of balanced treatment incomplete block designs is generalized to allow for comparison of v₁ test treatments and v₂ control treatments. The generalized class is equivalent to the class of balanced bipartite block designs considered by Jaggi, Gupta, and Parsad. Some results on design construction and A-optimality are given for small values of v₁ and v₂. Algorithms are developed for computing simultaneous confidence bounds for all test treatment versus control contrasts.     

Universally Optimal Designs Balanced for Neighbor Effects

The performance of a treatment is affected by the treatments applied to its adjacent plots, especially in the experiments of agriculture, horticulture, forestry, serology and industry. Neighbor designs ensure that treatment comparisons are least affected by neighbor effects, therefore, this is a rich field of investigation. In this paper, criterion for construction of universally optimal neighbor balanced designs is discussed.     

E- and R-optimality of block designs for treatment-control comparisons

Abstract

We study optimal block designs for comparing a set of test treatments with a control treatment. We provide the class of all E-optimal approximate block designs, which is characterized by simple linear constraints. Based on this characterization, we obtain a class of E-optimal exact designs for unequal block sizes. In the studied model, we provide a statistical interpretation for wide classes of E-optimal designs. Moreover, we show that all approximate A-optimal designs and a large class of A-optimal exact designs for treatment-control comparisons are also R-optimal. This reinforces the observation that A-optimal designs perform well even for rectangular confidence regions.     

Multiple comparisons for an unbalanced a × b design under mixed models with interaction

Consider a two-way factorial experiment involving a “treatment” factor A with fixed effects, a “blocking” factor B with random effects, and interaction effects perhaps non-negligible. The degree of balance required for multiple comparison procedures to be applicable for the comparison of the treatment effects using ordinary least-squares estimates is investigated. For main effects to be estimated independently of MSAB, a sufficient condition is that the design consist of identical blocks, a strong condition of proportional frequencies. Surprisingly, under this condition of proportional frequencies, MSAB does not provide an appropriate variance estimate for inferences on each treatment contrast, even though the statistics F = MSA/MSAB is appropriate for testing equality of the treatment effects. In short, when factor B is random, standard methods of multiple comparisons apply using the interaction mean square MSAB as a variance estimator only when the treatment-block incidences n_n are constant. Nevertheless, for designs with identical blocks, appropriate variance estimates can be identified to allow for conservative or approximate multiple comparisons. This is illustrated for certain treatment balanced designs for comparisons with a control.     

Optimal change-over designs for correlated observations

Design implications of an autoregressive model for change-over experiments are investigated. In this model, the residual effect due to the previous treatment is assumed to be proportional to the response in the previous period. In addition, the errors from the same experimental subject are assumed to be correlated according to a first-order autoregressive model. Models with fixed and random subject effects are discussed separately. An attempt has been made to identify and construct optimal or nearly optimal designs in various situations. Empirical conclusions of Taka and Armitage [Commun. Statist. Theor. Meth. (1983)12, 865-876] regarding the efficiency of some designs have also been confirmed.     

Nonparametric multiple comparison procedures for unbalanced two-way layouts

In this paper, we consider nonparametric multiple comparison procedures for unbalanced two-way factorial designs under a pure nonparametric framework. For multiple comparisons of treatments versus a control concerning the main effects or the simple factor effects, the limiting distribution of the associated rank statistics is proven to satisfy the multivariate totally positive of order two condition. Hence, asymptotically the proposed Hochberg procedure strongly controls the familywise type I error rate for the simultaneous testing of the individual hypotheses. In addition, we propose to employ Shaffer's modified version of Holm's stepdown procedure to perform simultaneous tests on all pairwise comparisons regarding the main or simple factor effects and to perform simultaneous tests on all interaction effects. The logical constraints in the corresponding hypothesis families are utilized to sharpen the rejective thresholds and improve the power of the tests.     

Neighbor-Balanced Bipartite Block Designs

This article deals with the neighbor-balanced block design setting when there are two disjoint sets of treatments, one set consisting of test treatments and the other of control treatments. The interest here is to estimate the contrasts pertaining to test treatments vs. control treatments (with respect to direct and neighbors) with as high precision as possible. Some series of neighbor-balanced block designs for comparing a set of test treatments to a set of control treatments have been developed. The designs obtained are totally balanced in the sense that all the contrasts among test treatments for direct and neighbor effects are estimated with same variance and all the contrasts pertaining to test vs. control for direct and neighbor effects are estimated with the same variance.