Equal Block Sizes Research Articles

Robustness of General Efficiency Balanced Block Designs against Missing Observations

In the present communication, some results on robustness of generalized efficiency balanced (GEB) block designs with equal block sizes (binary and non-binary) are presented. The necessary & sufficient conditions are obtained for robustness as per criteria 1 (Ghosh, 1982) when the missing observations appear in the following patterns: (i) t (≥1) observations of a design pertaining to the same treatment are missing and (ii) all observations in a block are missing but the residual design still becomes connected. Sufficient conditions for robustness according to Criteria 1 are examined in each type of GEB designs. Another criterion (Criteria 2) for robustness of a block design is the small amount loss of efficiency values of the residual design after deletion of t (≥1) observations. The declining trend of efficiency values after deletion of treatments one by one are presented graphically for measuring the robustness of designs under study.

Tri-hierarchical incomplete block designs

A tri-hierarchical incomplete block (TriHIB) design is defined as an arrangement of v equi-replicated treatments in three levels of blocking such that each block of the first level contains m1 blocks of second level and that of second level contains m2 blocks of the third level. Every level of a TriHIB design, treated independently, is an incomplete block design (IBD) and ignoring any one level of blocking, results into a bi-hierarchical incomplete block (BiHIB) design. Here, two general methods of construction of TriHIB designs are proposed. In the first method, popular association schemes (viz., triangular, Latin square and rectangular) of partially balanced IBDs have been used for developing three series of designs with equal/unequal block sizes at higher levels of blocking. Second method yields another series of designs, through cyclic development of two initial sequences, with equal block sizes at first two levels and equal/unequal block sizes at the third level of blocking. A list of parameters of TriHIB designs for number of treatments up to 30 is presented, along with efficiency factors. Further, for easy accessibility and wider adaptability of these designs, an R package ‘Tri.Hierarchical.IBDs’ has been developed for the generation of the proposed designs.

A Method of Constructing p-rep Designs

Background: Early generation breeding trials (EGBTs) are very important in plant breeding programmes. In most cases, a large number of breeding lines are to be tested, often with very few available resources and it is also required to repeat these trials in a number of environments. For such trials, an alternative is to use partially replicated designs, where a proportion of the test lines are replicated at each environment. Methods: Here, a general method of constructing a series of efficient partially replicated designs for EGBTs in equal block sizes, through initial blocks is developed. Result: Taking all environments together, the designs obtained are equi-replicate and are partially balanced. They are cost effective in terms of resources as they require lesser replications.

Variable block-size compressed sensing for depth map coding

Compressed sensing (CS) is the theory and practice of sub-Nyquist sampling of sparse signals of interest. Perfect reconstruction is possible with much fewer than the Nyquist required number of data samples. In this work, we consider a variable block-size CS architecture for fast compression of depth maps for three-dimensional video (3DV) applications. While existing CS-based depth map coding methods encode depth maps with equal block size, the proposed algorithm partitions a depth map into smooth and edge blocks of variable sizes via rate-distortion optimized quad-tree decomposition. CS is then performed on edge blocks, and eight-bit encoding is performed on smooth blocks. At the decoder, high quality depth map reconstruction is achieved by minimizing the spatial total-variation. Experimental results show that at a small extra expense of encoder complexity, the proposed variable block-size compressed sensing has enhanced significantly the rate-distortion performance over existing low-complexity CS-based depth map coding algorithms.

Optimality of block designs under the model with the first-order circular autoregression

In this paper optimal properties of some circular balanced block designs under the model with circular autoregression of order one are studied. Universal optimality of some balanced block designs with equal block sizes is proven and E-optimality of complete balanced block designs with the number of blocks equal to the number of treatments or the number of treatments reduced by two is shown.

Least mean square error-based image compression using block truncation coding

Block truncation coding is one of the image compression techniques which has low computational complexity. This lossy compression technique divides the image into equal block sizes and the pixels in each block are replaced with a high mean and low mean value based on the mean as the threshold. The bit plane thus obtained after quantisation is transmitted along with the mean and variance. This gives a bit rate of 2 bits per pixel. Nevertheless, when the block size is increased for obtaining higher compression ratio, the image is dominated by severe blocking artefacts and blurred edges thereby reducing the visual quality of the image. In the proposed method the mean square error is calculated iteratively for the blocks. The pixels are grouped into higher order and lower order blocks and the group which produces the least mean square error is transmitted along with the mean values of the higher order and lower order block. This implies that the image which is retrieved in the receiver gives the optimised visual quality in the least mean square sense.

Least mean square error-based image compression using block truncation coding

Block truncation coding is one of the image compression techniques which has low computational complexity. This lossy compression technique divides the image into equal block sizes and the pixels in each block are replaced with a high mean and low mean value based on the mean as the threshold. The bit plane thus obtained after quantisation is transmitted along with the mean and variance. This gives a bit rate of 2 bits per pixel. Nevertheless, when the block size is increased for obtaining higher compression ratio, the image is dominated by severe blocking artefacts and blurred edges thereby reducing the visual quality of the image. In the proposed method the mean square error is calculated iteratively for the blocks. The pixels are grouped into higher order and lower order blocks and the group which produces the least mean square error is transmitted along with the mean values of the higher order and lower order block. This implies that the image which is retrieved in the receiver gives the optimised visual quality in the least mean square sense.

Novel sampling design for respondent-driven sampling

Respondent-driven sampling (RDS) is a method of chain referral sampling popular for sampling hidden and/or marginalized populations. As such, even under the ideal sampling assumptions, the performance of RDS is restricted by the underlying social network: if the network is divided into communities that are weakly connected to each other, then RDS is likely to oversample one of these communities. In order to diminish the "referral bottlenecks" between communities, we propose anti-cluster RDS (AC-RDS), an adjustment to the standard RDS implementation. Using a standard model in the RDS literature, namely, a Markov process on the social network that is indexed by a tree, we construct and study the Markov transition matrix for AC-RDS. We show that if the underlying network is generated from the Stochastic Blockmodel with equal block sizes, then the transition matrix for AC-RDS has a larger spectral gap and consequently faster mixing properties than the standard random walk model for RDS. In addition, we show that AC-RDS reduces the covariance of the samples in the referral tree compared to the standard RDS and consequently leads to a smaller variance and design effect. We confirm the effectiveness of the new design using both the Add-Health networks and simulated networks.

Diallel cross designs for test versus control comparisons

Block designs for diallel crosses for comparing a group of test lines with a control line are suitable when the experimenters are interested to estimate test vs. control comparisons with respect to their general combining ability effects with a smaller variance. In this paper, some families of small block designs for diallel cross experiments for test vs. control comparisons have been obtained. Out of these, some designs are suitable when the experimenter cannot have blocks of equal size while the others yield designs with equal block size. As the number of lines increases, the variance of estimated contrasts pertaining to test vs. test lines as well as test vs. control lines decreases. Moreover, test vs. control comparisons are made with less variance as compared to test vs. test comparisons, in all the cases.

Families of proper generalized neighbor designs

A generalized neighbor design relaxes the equality condition on the number of times two treatments occur as neighbors in the design. In this article we have constructed a new series of generalized neighbor designs with equal block sizes, a series of neighbor designs of Rees [1967. Some designs of use in serology. Biometrics 23, 779–791] and a series of neighbor designs with two distinct block sizes. Two more new series of GN 2 designs are also constructed for even number of treatments. It has been shown that quasi neighbor designs introduced by Preece [1994. Balanced Ouchterlony neighbor designs. J. Combin. Math. Combin. Comput. 15, 197–219] are special cases of generalized neighbor designs with t = 2 . All the designs given here are binary. A new definition—partially balanced circuit design is introduced which is a special case of generalized neighbor designs with binary blocks.

Construction of central composite designs for balanced orthogonal blocks

Box & Hunter (1957) recommended a set of orthogonally blocked central composite designs (CCD) when the region of interest is spherical. In order to achieve rotatability along with orthogonal blocking, the block size for those designs becomes unequal and it may not be attractive or practical to use such unequally blocked designs in many practical situations. In this paper, a construction method of orthogonally blocked CCD under the assumption of equal block size is proposed and an index of block orthogonality is introduced.

A REVIEW OF BOUNDS FOR THE EFFICIENCY FACTOR OF BLOCK DESIGNS

summaryThis paper draws together bounds for the efficiency factor of block designs, starting with the papers of Conniffe & Stone (1974) and Williams & Patterson (1977). By extending the methods of Jarrett (1983), firstly to cover supercomplete block designs and then to cover resolvable designs, a set of bounds is obtained which provides the best current bounds for any block design with equal replication and equal block size, including resolvable designs and two‐replicate resolvable designs as special cases. The bounds given for non‐resolvable designs apply strictly only to designs which are either regular‐graph (John & Mitchell, 1977) or whose duals are regular‐graph. It is conjectured (John & Williams, 1982) that they are in fact global bounds. Similar qualifications apply to the bounds for resolvable designs.

External quicksort

An external sorting algorithm based on quicksort is presented. The file to be sorted is kept on a disk and only those blocks are fetched into the main memory which are currently needed. At each time, a block is kept in the main memory, if the expected space-time cost of holding it until its next use is smaller than the expected space-time cost of removing it and fetching it again. The efficiency of the algorithm is tested through simulation experiments and the results are compared to those achieved with mergesort in a corresponding environment. The total execution time and the main memory space-time integral are used for measuring the performance. When equal block sizes are used, external quicksort results in a much smaller average space requirement than mergesort. On the other hand, mergesort is somewhat faster than external quicksort. The main memory space-time integral of quicksort is always considerably smaller than that of mergesort. External quicksort is less sensitive to the block size and to the file size. With faster disks, the performance of external quicksort improves faster than that of mergesort. The relative difference of the algorithms is independent of the file size. The external quicksort is also analytically compared to some previous external versions of quicksort. It is shown to be satisfied with less space and fewer block fetches than the others.

On the Classification of Translation Group-divisible Designs

We determine the possible types of translation divisible designs having equal block sizes; we also describe the partitions of their translation groups.

A Comparison of Upper Bounds for Efficiency Factors of Block Designs

SUMMARY A comparison is made of four competing upper bounds for the efficiency factor of an incomplete block design, in the case of connected designs with equal treatment replication and equal block size. It is shown that one of the four bounds is generally the tightest over a wide range of parameter values.

A note on the construction of phase-confounded designs

In experiments comparing the effects of different crop rotations it sometimes happens that blocks of reasonably small size can only be achieved by not letting all sequences of each rotation occur in each block. If one of the rotations carries the test crop more than once such a restriction usually leads to the confounding of some phase contrasts. Stevens (1956) and Patterson (1964) provided two classes of phase-confounded designs. The two designs, however, lack the desirable property that if, in a particular block, some of the rotations carry the test crop in a given year, other rotations also do. Furthermore, the designs do not involve equal block sizes in the sense that, in any year, in any block, the number of plots carrying the test crop is the same.

A Class of Phase‐Confounded Designs

AbstractA new class of phase‐confounded designs is suggested. The class possesses the property that if in a year one of the rotations carries the test crop all other rotations also do. It also leads to equal block sizes when the rotations are structured in specified ways. Conditions under which designs of this class exist as well as method of constructing them are discussed.

Two-Factor Experiments in Non-Orthogonal Designs

Summary Some of the problems of asymmetrical factorial experiments using incomplete block designs of equal block size are considered. A set of conditions is established under which a non-orthogonal design for an m × n factorial experiment will give an orthogonal analysis of the main effects and interaction. These conditions are based upon the variances and covariances of the estimated treatment effects in the basic design, and cover all known classes of designs that give an orthogonal analysis. Simple expressions for the adjusted main effects and interaction sums of squares are obtained.