Unequal Block Sizes Research Articles

Second type second order slope rotatable designs utilizing symmetrical unequal block arrangements with two unequal block sizes

Kim and Ko developed second order slope rotatable designs (SOSRD) of second type utilizing central composite designs (CCD), in which two numbers are used to represent the positions of star points. In this paper, we suggest second type SOSRD utilizing symmetrical unequal block arrangements (SUBA) with two unequal block sizes. In some cases, the suggested method may develop designs containing fewer design points than second type SOSRD obtained utilizing CCD, pairwise balanced designs (PBD) and balanced incomplete block designs (BIBD). The first order partial derivative's for estimated second order response variance for factors 6≤v≤15 is also obtained.

Construction of Third Order Slope Rotatable Designs Using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

Construction of Third Order Slope Rotatable Designs Using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

Optimal test-control block designs with unequal block sizes and correlated errors

The purpose of the present article is to introduce theoretical and algorithmic approaches to address the problem of finding optimal test-control incomplete block designs with unequal block sizes where intra-block observations are correlated. Theoretical approach is used to find Etc -optimal designs analytically. In addition, due to the computational complexity of theoretical methods, in this article a two-phase optimization algorithm is proposed to construct ϕ -optimal or nearly ϕ -optimal designs. The effectiveness of the proposed algorithm is validated by comparing our results with optimal designs presented in several prior studies. Our algorithm has the advantages of being independent from the sizes of blocks, structure of correlation, and the optimality criteria. Moreover, it takes only a few minutes to obtain the optimal designs.

Measure of Slope Rotatability for Second Order Response Surface Designs Under Tri-Diagonal Correlation Error Structure Using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

In this paper, measure of slope rotatability for second order response surface designs using symmetrical unequal block arrangements with two unequal block sizes under tri-diagonal correlation error structure is suggested and illustrated with examples.

Evaluating Measure of Reformed Rotatability for Second Degree Polynomial Using a Pair of Incomplete Block Designs with Two Unequal Block Sizes

In this paper, a study on evaluating measure of reformed rotatability for second degree polynomial using a pair of incomplete block designs with two unequal block sizes (like symmetrical unequal block arrangements with two unequal block sizes) is suggested which enables us to assess the degree of reformed (modified) rotatability for a given response surface design.

Second Order Rotatable Designs of Second Type using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

Second Order Rotatable Designs of Second Type using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

On Second Order Slope Rotatable Designs under Intra-class Correlated Structure of Errors Using a Pair of Dissimilar Incomplete Block Designs

In this paper, a study of second order slope rotatable designs under intra-class correlation error structure using two suitably chosen dissimilar incomplete block designs like balanced incomplete block designs and symmetrical unequal block arrangements with two unequal block sizes are suggested. Further, we study the variance of the estimated slopes for different values of the intra-class correlation coefficient (ρ) and the distance from the centre (d) for v factors are suggested. Some illustrative examples are also suggested.

Second Order Slope Rotatable Designs under Tri-diagonal Correlation Structure of Errors Using a Pair of Incomplete Block Designs

Box and Hunter [1] introduced the concept of rotatability for response surface designs. The concept of slope-rotatability was introduced by Hader and Park [2] as an analogous to rotatability property, which is an important design criterion for response surface design. Slope-rotatable design is that of which the variance of partial derivative is a function of distance from the design (d). Recently, a few measures of slope-rotatability for a given response surface design was introduced. In this paper, a new method of slope rotatability for second order response surface designs under tri-diagonal correlation structure of errors using a pair of symmetrical unequal block arrangements with two unequal block sizes is studied. Further, a study on the dependence of variance function of the second order response surface at different design points for different values of tri-diagonal correlation coefficient ρ which lies between -0.9 to 0.9 and the distance from centre (d) is suggested.

Efficient block designs for incomplete factorial experiments for two factors with unequal block sizes

In this article, the problem of obtaining efficient block designs with unequal block sizes for incomplete factorial treatment structure with two factors excluding one treatment combination for estimation of dual versus single treatment contrasts is considered. A method of construction for obtaining efficient block designs with unequal block sizes for incomplete factorial treatment structure is proposed.

Partially replicated block designs for two-level factorial experiments

In a two-level factorial experiment, we consider blocking in the parallel-flats design with identical flats within or between blocks. We propose three classes of optimal designs with equal or unequal block sizes.

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

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.

A new series of resolvable PBIB (2) designs in unequal block sizes with three replicates

In this paper, a new series of incomplete block designs with three replications has been proposed. These designs are partially variance balanced following a group divisible association scheme. The designs obtained here are with unequal block sizes and are suitable for experimentation in uneven land. Furthermore, these designs are resolvable, which enhances their application potential in regions where it may be difficult to accommodate complete replications at one place.

Construction of second order slope rotatable designs under tri-diagonal correlated structure of errors using symmetrical unequal block arrangements with two unequal block sizes

In this paper, second order slope rotatable design (SOSRD) under tri-diagonal correlated structure of errors using symmetrical unequal block arrangements (SUBA) with two unequal block sizes is suggested.

Feasible blocked multi-factor designs of unequal block sizes

A block design is called feasible if all the factor effects in it are estimable. This paper studies the design structures of feasible block designs when the block sizes are unequal. Based on a necessary and sufficient condition, we find two structure features which respectively lead to unfeasible and feasible designs. According to these results, an effective way to find feasible designs is provided.

Partially A-optimal blocked multi-factor designs

In this paper, we propose a partially A-optimal criterion for block designs where multiple factors are arranged. The number of levels of each factor is assumed to be arbitrary and unequal block sizes are allowed. A sufficient condition is derived for a design to be partially A-optimal among all feasible designs. Then the properties of the selected design and its relation with orthogonal arrays are studied. Methods of constructing designs satisfying the sufficient condition are also given.

A note on circular neighbor balanced designs

ABSTRACTThis 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.

Main-effect plans with blocks of natural size

Block designs have been widely used in almost all areas of human investigations. It is commonly assumed that all blocks have the same size which may not, however, always be available because of practical restrictions. In this paper, the problem of main-effect plans with blocks of unequal sizes is addressed. The sufficient condition for each pair of factors to be orthogonal through the block factor is indicated. Useful properties on ANOVA (analysis of variance) of such designs are discussed. Construction procedures via augmenting new blocks and combining main-effect plans with unequal block sizes are also provided. The resulting designs have flexible sizes of blocks and factors.

Mixture designs with orthogonal blocks

ABSTRACTConstructions of blocked mixture designs are considered in situations where BLUEs of the block effect contrasts are orthogonal to the BLUEs of the regression coefficients. Orthogonal arrays (OA), Balanced Arrays (BAs), incidence matrices of balanced incomplete block designs (BIBDs), and partially balanced incomplete block designs (PBIBDs) are used. Designs with equal and unequal block sizes are considered. Also both cases where the constants involved in the orthogonality conditions depend and do not depend on the factors have been taken into account. Some standard (already available) designs can be obtained as particular cases of the designs proposed here.

Construction of Pair-wise Balanced Design

A new procedure for construction of a pair-wise balanced design with equal replication and un-equal block sizes based on factorial design are presented. Numerical illustration also provided. It was found that the constructed pair-wise balanced design was found to be universal optimal.

An Empirical Study of Second Order Rotatable Designs under Tri-Diagonal Correlated Structure of Errors using Incomplete Block Designs

In this paper, an empirical study of second order rotatable designs under tri-diagonal correlated structure of errors using incomplete block designs like pairwise balanced designs (PBD) and symmetrical unequal block arrangements (SUBA) with two unequal block sizes are suggested. Further we study the variance function of the estimated response for different values of tri-diagonal correlated coefficient (ρ) and distance from center (d) for 6≤v≤15 (v-number of factors)).