The analysis of randomized experiments with orthogonal block structure. I. Block structure and the null analysis of variance

1965 ◽  
Vol 283 (1393) ◽  
pp. 147-162 ◽  
Keyword(s):  
Covariance Matrix ◽  
Analysis Of Variance ◽  
Block Structure ◽  
Experimental Designs ◽  
Randomized Experiments ◽  
Orthogonal Idempotent ◽  
Initial Description ◽  
Block Structures ◽  
Simple Block ◽  
Randomization Distribution

Nearly all the experimental designs so far proposed assume that the experimental units are or can be grouped together in blocks in ways that can be described in terms of nested and cross-classifications. When every nesting classification employed has equal numbers of subunits nested in each unit, then the experimental units are said to have a simple block structure . Every simple block structure has a complete randomization theory. Any permutation of the labelling of the units is permissible which preserves the block structure, and this defines a valid randomization procedure. In a null experiment all units receive the same treatment, and the population of all possible vectors of yields generated by the randomization procedure gives the null randomization distribution. The covariance matrix of this distribution, the null analysis of variance, and the expectations of the various mean squares in it are all derivable from the initial description of the block structure. All simple block structures can be characterized by a set of mutually orthogonal idempotent matrices C i . Such sets of matrices may also exist for non-simple block structures (e. g. those having unequal numbers of plots in a block), and the existence of such a set defines an orthogonal block structure. Non-simple orthogonal block structures do not have a complete randomization theory and inferences from them require further assumptions.

The analysis of randomized experiments with orthogonal block structure. II. Treatment structure and the general analysis of variance

1965 ◽  
Vol 283 (1393) ◽  
pp. 163-178 ◽  
Keyword(s):  
Block Structure ◽  
General Situation ◽  
Design Matrix ◽  
Treatment Model ◽  
Randomized Experiments ◽  
Orthogonal Idempotent ◽  
Estimation Equations ◽  
Treatment Structure ◽  
General Computer Program ◽  
General Computer

Any orthogonal block structure for a set of experimental units has been previously shown to be expressible by a set of mutually orthogonal, idempotent matrices C i . When differential treatments are applied to the units, any linear model of treatment effects is expressible by another set of mutually orthogonal idempotent matrices T j . The analysis of any experiment having any set of treatments applied in any pattern whatever to units with an orthogonal block structure, is expressible in terms of the matrices C i , T j and the design matrix N, which lists the treatments applied to each unit. A unit-treatment additivity assumption and a valid randomization are essential to the validity of the analysis. The relevant estimation equations are developed for this general situation, and the idea of balance is given a generalized definition, which is illustrated by several examples. An outline is sketched for a general computer program to deal with the analysis of all experiments with an orthogonal block structure and linear treatment model.

scholarly journals Stratigraphy of amethyst geode-bearing lavas and fault-block structures of the Entre Rios mining district, Paraná volcanic province, southern Brazil

2013 ◽  
Vol 86 (1) ◽  
pp. 187-198 ◽  
Author(s):  
LÉO A. HARTMANN ◽  
LUCAS M. ANTUNES ◽  
LEONARDO M. ROSENSTENGEL
Keyword(s):  
Block Structure ◽  
Underground Mines ◽  
Mining District ◽  
Volcanic Province ◽  
Fault Block ◽  
River Bed ◽  
World Class ◽  
Entre Rios ◽  
Block Structures

The Entre Rios mining district produces a large volume of amethyst geodes in underground mines and is part of the world class deposits in the Paraná volcanic province of South America. Two producing basalt flows are numbered 4 and 5 in the lava stratigraphy. A total of seven basalt flows and one rhyodacite flow are present in the district. At the base of the stratigraphy, beginning at the Chapecó river bed, two basalt flows are Esmeralda, low-Ti type. The third flow in the sequence is a rhyodacite, Chapecó type, Guarapuava subtype. Above the rhyodacite flow, four basalt flows are Pitanga, high-Ti type including the two mineralized flows; only the topmost basalt in the stratigraphy is a Paranapanema, intermediate-Ti type. Each individual flow is uniquely identified from its geochemical and gamma-spectrometric properties. The study of several sections in the district allowed for the identification of a fault-block structure. Blocks are elongated NW and the block on the west side of the fault was downthrown. This important structural characterization of the mining district will have significant consequences in the search for new amethyst geode deposits and in the understanding of the evolution of the Paraná volcanic province.

scholarly journals Stability Analysis and Monitoring Method for the Key Block Structure of the Basic Roof of Noncoal Pillar Mining with Automatically Formed Gob-Side Entry

2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Jianning Liu ◽  
Manchao He ◽  
Yajun Wang ◽  
Ruifeng Huang ◽  
Jun Yang ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Coal Mine ◽  
Block Structure ◽  
Monitoring Method ◽  
Roof Caving ◽  
Pillar Mining ◽  
Key Block ◽  
The Stability ◽  
Block Structures ◽  
Basic Roof

The key block of the basic roof is the main contributor to the structural stability of a roadway. Research on the stability of the key block structure is of great significance for the promotion of noncoal pillar mining with automatically formed gob-side entry (GEFANM) technology. This paper is set in the engineering context of the GEFANM experiment at the Ningtiaota Coal Mine. The study fully considered the differences in the gob roof caving on the roof-cutting-line side, and the range of rotation angles to maintain a stable key block was determined. Based on this range of rotation angles, the range of safe bulking coefficients of gangue was calculated. The bulking coefficient of the gangue on the gravel side of the roadway was used as the metric in a new monitoring method and in the calculation of the field parameters. The range of safe bulking coefficients was determined to be 1.40–1.37. Field monitoring was conducted to obtain the gangue bulking coefficient on the gravel side. Combining the roof and floor convergence data, when the bulking coefficient fell within the safe range, the convergence was 95–113 mm. In this stage, the key block structure was stable. When the gangue bulking coefficient fell outside the safe range, the convergence was larger, and cracks were observed. The key block may be vulnerable to instability. The results affirmed that the gangue bulking coefficient can be used as a monitoring metric to study the stability of key block structures.

Learning Block Structures in U-statistic Based Matrices

Biometrika ◽  
2020 ◽  
Author(s):  
Weiping Zhang ◽  
Baisuo Jin ◽  
Zhidong Bai
Keyword(s):  
Matrix Theory ◽  
Block Structure ◽  
Group Structure ◽  
Asymptotic Properties ◽  
Finite Sample ◽  
Eigenvalues And Eigenvectors ◽  
Sample Matrix ◽  
U Statistics ◽  
Large Matrix ◽  
Block Structures

Abstract We introduce a conceptually simple, efficient and easily implemented approach for learning the block structure in a large matrix. Using the properties of U-statistics and large dimensional random matrix theory, the group structure of many variables can be directly identified based on the eigenvalues and eigenvectors of the scaled sample matrix. We also established the asymptotic properties of the proposed approach under mild conditions. The finite-sample performance of the approach is examined by extensive simulations and data examples.

Some Design Problems in Crop Experimentation. II. Multiple Blocking Systems

1995 ◽  
Vol 31 (3) ◽  
pp. 279-290
Author(s):  
S. C. Pearce
Keyword(s):  
Field Experiments ◽  
Block Structure ◽  
Design Problems ◽  
Split Plot ◽  
Alternative Approaches ◽  
Simple Block

SummaryThis paper describes the construction, usefulness and randomization of several designs for field experiments in which there is more than one set of blocks, namely: (a) row-and-column designs, in which there are two crossing sets of blocks, treatments being applied to the plots formed by their intersections; (b) row-and-column designs in which factors are applied to complete rows or complete columns, that is, criss-cross (or strip-plot) designs; and (c) split-plot designs, in which the plots in a study of one factor are used as blocks in the study of another. All are examples of a wider class of designs, with many ramifications, said to have ‘simple block structure’. It is suggested here that some of the assumptions underlying row-and-column designs are questionable. Some alternative approaches are indicated.

scholarly journals Hierarchical Organization of Reading Subskills: Statistical Approaches

1975 ◽  
Vol 7 (3) ◽  
pp. 223-228 ◽  
Author(s):  
R. G. Stennett ◽  
P. C. Smythe ◽  
Madeline Hardy
Keyword(s):  
Cluster Analysis ◽  
Factor Analysis ◽  
Multiple Regression ◽  
Analysis Of Variance ◽  
Hierarchical Organization ◽  
Experimental Designs ◽  
Stepwise Multiple Regression ◽  
Scaling Methods ◽  
Statistical Approaches ◽  
Reading Subskills

This article describes a number of alternate methodological solutions to the problem of isolating reading subskills and establishing the nature of their hierarchical organization. The authors briefly review and evaluate Stepwise Multiple Regression, Factor Analysis, Cluster Analysis, Scaling Methods, Analysis of Variance and Transfer Designs, as potential tools in research on the hierarchical organization of reading subskills. With the exception of transfer-type experimental designs, none of the approaches seems satisfactory, the authors conclude.

Sign in / Sign up

Export Citation Format

Share Document