Balanced Block Designs Research Articles

A Difference Theorem for the Construction of Balanced Block Designs with Variable Replications

A Difference Theorem for the Construction of Balanced Block Designs with Variable Replications

An Extension of Block Design Methods and an Application in the Construction of Redundant Fault Reducing Circuits for Computers

In a balanced block design one's aim is to arrange a number of objects in a number of blocks according to certain rules. There is a certain class of design in which the blocks can be conceived as operating upon the objects to produce new objects which, in turn, may enter a further design. In this case not only does the block operator have to be taken into account, but arrangements of objects and blocks, normally regarded as indistinct in a block design, can have an important effect on the properties of the design. These ideas are applied to the design of simple detecting and correcting computer redundancy circuits. Here the criterion of goodness for design is not just the reducing properties obtained by a single pass of information through the circuit, but the further effect of information encountering a sequence of such circuits. To detect, and more ambitiously to correct, faults which occur in the generation and transmission of information, it is necessary to generate and transmit extra redundant information. This may be achieved either by carrying redundant information bits with the information word (parity bits), or by parallel transmission of the information words in a bundle of channels, and comparison at their destination. The latter method would appear wasteful in the amount of redundancy which it demands but has been studied by Pierce (Pierce, 1962) and others, particularly as it is used in biological mechanisms. However, the redundant equipment required is very simple, and the striking success which can be obtained with simple devices suggests that their use in computer circuitry would be effective and economic. This paper uses the block design method for the study of redundancy circuits based on majority vote takers. A computer has been used in a systematic study of these extended block design conditions. A very simple redundancy circuit with good reducing properties is described. Though the terms detection and correction are commonly used, we prefer to use the word fault in this paper and to reserve error for the numerical errors studied in Numerical Analysis.

On the Properties and Construction of Balanced Block Designs with Variable Replications

On the Properties and Construction of Balanced Block Designs with Variable Replications

On Balanced Block Designs with Three Different Numbers of Replica Lions

On Balanced Block Designs with Three Different Numbers of Replica Lions

Independency and operationism in Q-sorting

Misgivings about the independency of items in Q-sorting are shown to be due to misunderstandings about what is operational and what is not, and to the fallacy of thinking of items in a Q-sample in logical instead of in psychological-situational terms. Q-samples are composed of balanced block designs for systematic, logical reasons. But Q-sorting has no such logical involvement; it concerns, instead, the subject’s understanding or apperception of a situation. What is independent, and what not, is therefore a matter of fact, not of definition. And because operations are inductive, Q-sortings are analyzed by inductive factor analysis rather than by variance analysis, though for some purposes the latter is of course in order.

Balanced Block Designs with Two Different Numbers of Replicates

In most standard types of incomplete block designs (e.g. balanced incomplete block designs, and partially balanced designs, including lattice designs) all treatments occur the same number of times. This requirement may be inconvenlient in practice. For example, only small amounts of seed might be available for some new varieties, while seed of other varieties was abundant. To avoid such difficulties by taking only as many replicates of all treatments as the limitations on some treatments allowed would be an inefficient approach, involving a sacrifice of information on those treatments for which limitations did not exist, as well as on comparisons between the two kinds of treatment. For the last reason, also, the alternative of investigating the two groups in separate experiments is unacceptable. In situations of this sort there is an obvious practical need for incomplete block designs in which the number of replicates is not the same for all treatments. Such designs would also be of value when some treatments are of more interest than others: though interest in these other treatments might be enough to justify their inclusion in the experiment, the experimenter wou]d wish to spend less on investigating them than on the treatments of major interest. Several discussions on designs with varying numbers of replicates exist in statistical literature, but most of these are concerned with the analysis of designs that happen to have unequal replications of treatments, e.g. Anderson and Manning [1948], Corsten [1958], Federer [1961], Gomes and Guimaraes [1958], Graybill and Pruitt [1958], Justesen and Keuls [1958], Pearce [1948], Rao [1947], Youden and Connor [1953].

On Balanced Unequal Block Designs

On Balanced Unequal Block Designs

On balanced unequal block designs

On balanced unequal block designs

Automorphisms of Steiner Triple Systems

This paper treats the following problem in combinatorial analysis: Find an incomplete balanced block design D with parameters b, v, r, k, and λ = 1. possessing an automorphism group G which is doubly transitive on the elements of D and such that the subgroup H of G fixing all the elements of a block is transitive on the remaining elements. Also find transitive extensions of such groups G. If the block design is a finite projective plane, the plane is necessarily Desarguesian. Thus, these properties of the automorphism group may be considered as a “Desarguesian” property of the designs. This paper considers the case in which D is a Steiner triple system. The main result is that a “Desarguesian” Steiner triple system is either 1) a projective geometry over GF(2) or 2) an affine geometry over GF(3). Two intermediate results are of interest: 1) A Steiner triple system has for each element an involution fixing only this element, if and only if every triangle generates an S(9), a Steiner triple system with 9 elements; 2) if a Steiner triple system has for each triple an involution fixing only the elements of this triple, then every triangle generates an S(7) or an S(9).