Partially Balanced Incomplete Block Designs Research Articles

Higher associate class partially balanced incomplete block designs using graphs for agricultural experiments

Some new association schemes and construction methods of 3- and 4-associate class partially balanced incomplete block (PBIB) designs based on these schemes using chosen graphs have been given. A catalogue is included of efficient PBIB designs with number of treatments (v) ≤ 100.

Constant block-sum Youden-m square and PBIB designs using Galois field

In the present paper, we have discussed the construction of constant block-sum designs using Galois field. In this direction, the construction of constant block-sum Youden-m square designs and the constant block-sum Partially Balanced Incomplete Block (PBIB) designs has been discussed. In addition to this, a generalized m-associate class association scheme, on which the PBIB designs constructed in this paper are based, is also discussed.

A Construction Technique for Group Divisible (v-1,k,0,1) Partially Balanced Incomplete Block Designs (PBIBDs)

Group Divisible PBIBDs are important combinatorial structures with diverse applications. In this paper, we provided a construction technique for Group Divisible (v-1,k,0,1) PBIBDs. This was achieved by using techniques described in literature to construct Nim addition tables of order 2n, 2≤n≤5 and (k2,b,r,k,1)Resolvable BIBDs respectively. A “block cutting” procedure was thereafter used to generate corresponding Group Divisible (v-1,k,0,1) PBIBDs from the (k2,b,r,k,1)Resolvable BIBDs. These procedures were streamlined and implemented in MATLAB. The generated designs are regular with parameters(15,15,4,4,5,3,0,1);(63,63,8,8,9,7,0,1);(255,255,16,16,17,15,0,1) and (1023,1023,32,32,33,31,0,1). The MATLAB codes written are useful for generating the blocks of the designs which can be easily adapted and utilized in other relevant studies. Also, we have been able to establish a link between the game of Nim and Group Divisible (v-1,k,0,1) PBIBDs.

Combinatorial Construction of Second Order Rotatable Designs

In this paper, we constructed a new series for the construction of Second Order Rotatable Design using Partially Balanced Incomplete Block Designs (PBIBD).

Construction of three associate PBIB designs using K - maps

A new series of s (s ≥ 3) associate class Partially Balanced Incomplete Block (PBIB) designs have been constructed by applying hamming distance on Karnaugh maps. In this series, every character is expressed in binary set and calculate the weight, then hamming distance for each literal of k – map. We considersꞌ different factors (s≥3) simultaneously and each factor has two levels, it means each factor is introduced in primal as well as dual form.

Three-associate class partially balanced incomplete block designs through kronecker product

Kronecker product of matrices can be advantageously used to obtain three-associate class partially balanced incomplete block (PBIB) designs from two-associate class PBIB designs, for a larger number of treatments. This method has been described here and two such series of PBIB designs are obtained. If the experimenter is constrained of resources, theses designs can be used as an alternative to balanced incomplete block designs or 2-associate class partially balanced incomplete block designs.

A Necessary Condition for the Existence of a Certain Resolvable Pairwise Balanced Designa

A mathematical topic using the property of resolvability and affine resolvability was introduced in 1850 and the designs having such concept have been statistically discussed since 1939. Their combinatorial structure on existence has been discussed richly since 1942. This concept was generalized to<i>α</i>-resolvability and affine <i>α</i>-resolvability in 1963. When <i>α</i> = 1, they are simply called a resolvable or an affine resolvable design, respectively. In literature these combinatorial arguments are mostly done for a class of block designs with property of balanced incomplete block (BIB) designs and <i>α</i>-resolvability. Due to these backgrounds, Kadowaki and Kageyama have tried to clarify the existence of affine <i>α</i>-resolvable partially balanced incomplete block (PBIB) designs having association schemes of two associate classes. The known 2-associate PBIB designs have been mainly classified into the following types depending on association schemes, i.e., group divisible (GD), triangular, Latin-square (L<SUB>2</SUB>), cyclic. First, it could be proved that an affine <i>α</i>-resolvable cyclic 2-associate PBIB design does not exist for any <i>α</i> ≥ 1. Also, Kageyama proved the non-existence of an affine <i>α</i>-resolvable triangular design for 1 ≤ <i>α</i> ≤ 10 in 2008. Furthermore, the existence of affine resolvable GD designs and affine resolvable L<SUB>2</SUB> designs with parameters <i>υ</i> ≤ 100 and <i>r</i>, <i>k</i> ≤ 20 was mostly clarified by Kadowaki and Kageyama in 2009 and 2012. As a result, only three designs (i.e., two semi-regular GD designs, only one L<SUB>2</SUB> design) are left unknown on existence within the practical range of parameters. In the present paper, a necessary condition for the existence of a certain resolvable pairwise balanced (PB) design (i.e., some block sizes are not equal) is newly provided. Existence problems on PB designs are far from the complete solution. By use of the necessary condition derived here, we can also show a non-existence result of the affine resolvable L<SUB>2</SUB> design which is left as only one unknown among L<SUB>2</SUB> designs.

A new technique for comparing PBIBDs

We introduce here a quick and easy new technique for comparing efficiencies of Partially Balanced Incomplete Block Designs (PBIBDs) with the same values of the parameters v, b, r and k. We demonstrate the use of the newly proposed technique by first constructing some PBIBDs based on some association scheme and then comparing some of the newly constructed PBIBDs with designs already existing in literature.

A Key Pre-Distribution Scheme Based on µ-PBIBD for Enhancing Resilience in Wireless Sensor Networks.

Many key pre-distribution (KPD) schemes based on combinatorial design were proposed for secure communication of wireless sensor networks (WSNs). Due to complexity of constructing the combinatorial design, it is infeasible to generate key rings using the corresponding combinatorial design in large scale deployment of WSNs. In this paper, we present a definition of new combinatorial design, termed “µ-partially balanced incomplete block design (µ-PBIBD)”, which is a refinement of partially balanced incomplete block design (PBIBD), and then describe a 2-D construction of µ-PBIBD which is mapped to KPD in WSNs. Our approach is of simple construction which provides a strong key connectivity and a poor network resilience. To improve the network resilience of KPD based on 2-D µ-PBIBD, we propose a KPD scheme based on 3-D Ex-µ-PBIBD which is a construction of µ-PBIBD from 2-D space to 3-D space. Ex-µ-PBIBD KPD scheme improves network scalability and resilience while has better key connectivity. Theoretical analysis and comparison with the related schemes show that key pre-distribution scheme based on Ex-µ-PBIBD provides high network resilience and better key scalability, while it achieves a trade-off between network resilience and network connectivity.

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.

Climate smart agriculture adaptation strategies for rain-fed agriculture in drought-prone areas of Central Kenya

The drier parts of Central Kenya are characterised by poor crop harvest due to unpredictable, unreliable and poor rainfall distribution patterns. The field experiment was laid out in Partially Balanced Incomplete Block Design (PBIBD) with 36 treatments replicated three times. The treatments of tied ridges and contour furrows under sorghum alone and intercrop plus external soil amendment of 40 kg P/ha + 20 kg N/ha + manure 2.5 t/ha had the highest grain yield of 3.1 t/ha. The soil fertility levels differed significantly from one another (p = 0.0001) in terms of sorghum grain yield. Generally, all experiment controls had the lowest grain yields as low as 0.3 t/ha to 0.5 t/ha. Therefore, integration of minimal organic and inorganic inputs under various water harvesting technologies could be considered as an alternative food security initiative towards climate smart agriculture for climate change mitigation in drought-prone areas of Central Kenya.

English

In this paper, we have constructed partially balanced incomplete block (PBIB) designs with two and three associate classes by establishing a link between PBIB designs and graphs through the chosen lines and number of triangles. We have considered six configurations for this purpose and constructed four two associate class PBIB designs and two three class PBIB designs.

Analysis of Covariance of Reinforced Balanced Incomplete Block Designs With a Single Explanatory Variable

Reinforced balanced incomplete block designs (BIBDs) are very useful in statistical planning of experiments as they can be constructed for any number of treatments for given numbers of replications. Das (1958) was first to introduce the statistical analysis of variance (ANOVA) of these designs, and in the same year Giri also developed the same statistical analysis of variance for reinforced partially balanced incomplete block designs (PBIBDs). In this article, we focus on the method of statistical analysis of covariance (ANCOVA) of reinforced balanced incomplete block design (BIBD) when a single explanatory variable is available in the experiment.

English

Soil fertility degradation remains the major biophysical cause of declining per capita crop production on smallholder farms in Central Kenya highlands. A study was conducted to compare farmers&rsquo; perception and biophysical data on selected water harvesting and integrated soil fertility management technologies on sorghum (Sorghum bicolor (L.) Moench) and cowpea (Vigna unguiculata L.) production in Central highlands of Kenya. Three hundred and seventy one smallholder farmers were invited to evaluate thirty six plots laid out in Partially Balanced Incomplete Block Design (PBIBD) replicated three times. The treatment which was ranked best overall rated as &lsquo;good&rsquo; by the farmers was farmers practice with a mean score of (2.78) and yielding (3.5 t/ha) under sorghum alone plus external soil amendment of 40 &nbsp;kg P /ha+20 kg N /ha. This was closely followed by tied ridges and contour furrows overall rated as &lsquo;good&rsquo; by the farmers under sorghum alone plus external soil amendment of 40 kg P /ha+20 kg N /ha+manure 2.5 t/ha and 40 &nbsp;kg P /ha+40 &nbsp;kg N /ha+manure 5 t/ha both with a mean score of (2.7) and yielding (3.0 t/ha) and (2.9 t/ha) respectively. &nbsp;Generally, all experiment controls were overall scored as &lsquo;poor&rsquo; yielding as low as 0.3 t/ha to 0.6 t/ha. Therefore, integration minimal addition of organic and inorganic inputs on highly valued traditional crops with adequate rainfall under normal farmers practice in semi arid lands could be considered as an alternative option contribution to food security in central highland of Kenya.&nbsp; &nbsp; Key words: Food security, water harvesting, integrated soil fertility management, Central Kenya. &nbsp;

A web solution for Partially Balanced Incomplete Block experimental designs

Heterogeneity in the experimental material is an important problem to deal with the designing of scientific experiments. Block designs are useful in controlling heterogeneity arising due to one source. A Randomized Complete Block (RCB) design is the simplest and commonly used block design. When the number of treatments in an experiment increases, incomplete block designs with smaller block sizes can be adopted. Balanced Incomplete Block (BIB) and Partially Balanced Incomplete Block (PBIB) designs are two important types of such designs. PBIB designs are extremely wide spread in literature. For ready referencing and potential use of these designs, online software for generation and analysis of these designs is highly desirable. This paper describes the development of a complete web solution for generation and analysis of PBIB designs using client–server architecture. An e-learning material on these designs is also prepared that can be used as reference material by researchers and students working in this area. WS-PBIBD is accessible any time from arbitrary platforms through internet. This software would help researchers in planning, designing and analyzing their experiments through web.

Construction of Three Associate Class PBIB Designs with Three Replicates Using Pairing in Triplets System

A Series of three associate class Partially Balanced Incomplete Block (PBIB) designs with three replicates is introduced in this paper. The method of construction of new designs is based on method of pairing in symbols of triplets system. In the construction of three associate class PBIB designs, we take nine blocks with three replicates. We then paired the iterative values of triplets with its initial values under different conditions to get blocks of new designs. Association scheme of new series of designs is also based on occurrence of treatment pairs. Efficiency factors of three kinds of comparisons E1, E2, E3 and overall efficiency factor E along with illustration have also been given in the paper.

Perceptual Evaluation of Inter-song Similarity in Western Popular Music

We describe and test the methodological set up for a web-based listening experiment that assesses the perception of inter-song similarity optimizing the trade-off between stimulus coverage and experimental time. The experiment used a relatively large set of stimuli of Western popular music: 78 song excerpts selected from 13 genres, involving 78 participants. The experiment used triadic comparisons of song excerpts to present the participants with a low-complexity task, and a partially balanced incomplete block design (PBIBD) to reduce the number of stimulus comparisons with the consequent possibility of extending the stimulus set. The three control variables used in the excerpt selection, genre, tempo and timbre, showed statistically significant saliency and a hierarchical degree of impact on participants' pair rankings (genre > tempo > timbre). We investigated the participants' perceptual space using a combination of numerical and analytical methods that help to reduce and represent the dimensionality of the data. We used a combination of scaling and discriminant functions to gain insight into the important factors underlying the organization of the participants' perceptual space. In the perceptual space calculated through multidimensional scaling, we used quadratic discriminant analysis to search for axes that maximized the separation of the excerpt classes. We identified three axes that were a posteriori labelled as ‘slow–fast’, ‘vocal–non-vocal’, and ‘synthetic–acoustic’. We found a high correlation between the excerpt tempo in beats per minute and the excerpt projections on the slow–fast axis. A final analysis showed that the relevance of the factors responsible for the grouping of excerpt subsets is context dependent.

A Complete Generalization of Clatworthy Group Divisible Designs

Partially balanced incomplete block designs (PBIBDs) have a long history and have been extensively used in agriculture and industrial experiments. Since the book of Clatworthy on two-associate-class partially balanced designs was published in 1973, little progress has been made on the construction of these designs. Group divisible designs (GDDs) are an important type of PBIBD with two associate classes. The existence of a GDD with block size $k=3$ was completely settled by Fu, Rodger, and Sarvate. In their works, the most difficult case to solve was when the number of groups, m, is less than the block size k. The existence of GDDs when $m<k$ is, in general, a difficult case to solve. Indeed, when $k=4$, very little is known about the existence of such GDDs. In this paper, we present two general construction methods for GDDs. The first one is a generalization of Wilson's fundamental construction in combinatorial design theory. The second is an extension of the traditional construction using double group divisible designs. As an application of our new construction methods, a complete solution is provided for GDDs that generalize all eleven designs in the old table of Clatworthy which have block size four, three groups, and replication number at most $10$. For these GDDs, no progress has been made until very recently.

Key pre-distribution using partially balanced designs in wireless sensor networks

We propose two deterministic key predistribution schemes in a wireless sensor network (WSN), in which sensor nodes are deployed randomly. Both the schemes are based on combinatorial designs, called partially balanced incomplete block designs (PBIBD). An important feature of our scheme is that every pair of nodes within communication range can communicate directly, making communication faster and efficient. The number of keys per node is of the order of $\sqrt N$, where $N$ is the number of nodes in the network. Our second design has the added advantage that we can introduce new nodes in the network keeping the key pool fixed. This makes the network scalable. We study the resiliency of the network under node compromise and show that our designs give better results than the existing ones.

Optimum covariate designs in partially balanced incomplete block (PBIB) design set-ups

The use of covariates in block designs is necessary when the covariates cannot be controlled like the blocking factor in the experiment. In this paper, we consider the situation where there is some flexibility for selection in the values of the covariates. The choice of values of the covariates for a given block design attaining minimum variance for estimation of each of the parameters has attracted attention in recent times. Optimum covariate designs in simple set-ups such as completely randomised design (CRD), randomised block design (RBD) and some series of balanced incomplete block design (BIBD) have already been considered. In this paper, optimum covariate designs have been considered for the more complex set-ups of different partially balanced incomplete block (PBIB) designs, which are popular among practitioners. The optimum covariate designs depend much on the methods of construction of the basic PBIB designs. Different combinatorial arrangements and tools such as orthogonal arrays, Hadamard matrices and different kinds of products of matrices viz. Khatri–Rao product, Kronecker product have been conveniently used to construct optimum covariate designs with as many covariates as possible.