Construction of repeated measurements designs strongly balanced for residual effects

Abstract

Repeated Measurements Designs have been widely used in agriculture, animal husbandry, education, biology, botany and engineering. Balanced or strongly balanced repeated measurements designs are useful to balance out the residual effects. In this article, some new generators and construction procedures are proposed to obtain circular strongly balanced repeated measurements designs in periods of (a) equal sizes, (b) two different sizes, and (c) three different sizes.     

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.     

Some new constructions of circular balanced repeated measurements designs

Repeated measurements designs are widely used in medicine, pharmacology, animal sciences, and psychology. In this article, some infinite series are developed to generate the balanced repeated measurements designs for p (periods) even. For p odd, construction procedures are also described. Catalogues of the proposed designs are also presented for p = 5, 7, 9, when v ≤ 100.     

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.     

Minimal circular strongly balanced repeated measurements designs in periods of three different sizes

Repeated measurements designs (RMD) are widely used in medicine, pharmacology, animal sciences, and psychology. If there is a restriction on the total number of treatments, some experimental units can receive on the total length of time while some experimental units can remain in the trial, then RMD in periods of unequal sizes should be used. In this article, some infinite series are developed to generate the minimal circular strongly balanced RMD in periods of three different sizes p₁, p₂, and p₃, where 2 ≤ p₃ < p₂ ≤ 10.     

Designs balanced for neighbor effects in circular blocks of size six

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 statistics and in combinatorics. In this article, several series of neighbor balanced designs are considered in circular blocks of six units.     

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 extensions of circular balanced and circular strongly balanced repeated measurements designs

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences, and psychology. In this paper the works of Iqbal and Tahir (2009 Iqbal, I., and M. H. Tahir. 2009. Circular strongly balanced repeated measurements designs. Communications in Statistics—Theory and Methods 38:3686–96.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) and Iqbal, Tahir, and Ghazali (2010 Iqbal, I., M. H. Tahir, and S. S. A. Ghazali. 2010. Circular first- and second-order balanced repeated measurements designs. Communications in Statistics—Theory and Methods 39:228–40.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) are generalized for the construction of circular-balanced and circular strongly balanced repeated measurements designs through the method of cyclic shifts for three periods.     

Optimal repeated measurements designs for comparing test treatments with a control

A-optimal and mv optimal repeated measurments designs for comparing serveral test treatments with a control are considered. the models considered are basically of two types: without preperides and the cirular model. It is shown known that some known strongly balanced uniform repeated measurements designs can be modified to obtain optimal designs for this problem. Some other methods of finding optimal designs are also given.     

Nonproper variance balanced designs and optimality

This communication deals with the construction and optimality of non-proper (unequal block sized) variance balanced (VB) designs obtainable under linear homoscedastic normal model. Several methods of construction of non-proper VB designs have been given. Some constructed designs are universally optimal non-proper variance balanced designs.     

Circular balanced repeated measurement designs in periods of three different sizes

Abstract

Repeated measurement designs are widely used in medicine, pharmacology, animal sciences and psychology. If there is a restriction on the total number of treatments, some experimental units can receive on the total length of time while some experimental units can remain in the trial, then repeated measurements designs with unequal period sizes should be used. In this article, some infinite series are developed to generate the minimal balanced repeated measurement designs in periods of three different sizes p₁, p₂ and p₃, where 2?≤?p₃?<?p₂ ≤ 10 and p₂?<?p₁.     

Designs with Repeated Measurements on Any Number of Units over Varying Periods

In the usual repeated measurements designs (RMDs), the subjects are all observed for the same number of periods and the optimum RMDs require specified numbers of subjects, usually depending on the number of treatments to be used. In practice, it is sometimes not feasible to meet these requirements. To overcome this problem, alternative designs are suggested where any number of available subjects may be used and they may be observed for different periods. These designs are based on suitable serially balanced sequences which are shown to be optimal. Moreover, besides the usual direct and residual effects, the model considered has an extra term due to the interaction effect between them. The recommended designs are universally optimal in a very general class.     

Optimal Circular Block Designs for Neighbouring Competition Effects

Competition or interference occurs when the responses to treatments in experimental units are affected by the treatments in neighbouring units. This may contribute to variability in experimental results and lead to substantial losses in efficiency. The study of a competing situation needs designs in which the competing units appear in a predetermined pattern. This paper deals with optimality aspects of circular block designs for studying the competition among treatments applied to neighbouring experimental units. The model considered is a four-way classified model consisting of direct effect of the treatment applied to a particular plot, the effect of those treatments applied to the immediate left and right neighbouring units and the block effect. Conditions have been obtained for the block design to be universally optimal for estimating direct and neighbour effects. Some classes of balanced and strongly balanced complete block designs have been identified to be universally optimal for the estimation of direct, left and right neighbour effects and a list of universally optimal designs for v<20 and r<100 has been prepared.     

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.     

Non binary partially neighbor balanced designs for circular blocks

ABSTRACT

Neighbor designs are recommended for the cases where the performance of treatment is affected by the neighboring treatments as in biometrics and agriculture. In this paper we have constructed two new series of non binary partially neighbor balanced designs for v = 2n and v = 2n+1 number of treatments, respectively. The blocks in the design are non binary and circular but no treatment is ever a neighbor to itself. The designs proposed here are partially balanced in terms of nearest neighbors. No such series are known in the literature.     

A note on circular neighbor balanced designs

ABSTRACT

This paper describes some methods of constructing circular neighbor balanced and circular partially neighbor balanced block designs for estimation of direct and neighbor effects of the treatments. A class of circular neighbor balanced block designs with unequal block sizes is also proposed.     

On totally balanced block designs for competition effects

Competition between neighbouring units in field experiments is a serious source of bias. The study of a competing situation needs construction of an environment in which it can happen and the competing units have to appear in a predetermined pattern. This paper describes methods of constructing incomplete block designs balanced for neighbouring competition effects. The designs obtained are totally balanced in the sense that all the effects, direct and neighbours, are estimated with the same variance. The efficiency of these designs has been computed as compared to a complete block design balanced for neighbours and a catalogue has also been prepared.     

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.     

Circular First- and Second-Order Balanced Repeated Measurements Designs

Sharma (1977 Sharma , V. K. ( 1977 ). Change-over designs with complete balance for first and second order residual effect . Canad. J. Statist. 5 : 121 – 132 .[Crossref] , [Google Scholar]) and Aggarwal et al. (2006 Aggarwal , M. L. , Deng , L.-Y. , Jha , M. K. ( 2006 ). Balanced residual treatment effects designs of first and second order . Statist. Probab. Lett. 76 : 597 – 600 .[Crossref], [Web of Science ®] , [Google Scholar]) considered non circular construction of first- and second-order balanced repeated measurements designs. Sharma et al. (2002 Sharma , V. K. , Varghese , C. , Jaggi , S. ( 2002 ). On optimality of change-over designs balanced for first and second order residual effects . Metron 60 : 153 – 162 . [Google Scholar]) constructed circular first- and second-order balanced repeated measurements designs only for a class with parameters (v, p = 3n, n = v ²) and also showed its universal optimality. In this article, we consider circular construction of first- and second-order balanced repeated measurements designs and strongly balanced repeated measurements designs by using the method of cyclic shifts. Some new circular designs with parameters (v, p, n) for cases p = v, p < v and p > v are given.