The Use of Multidimensional-Scaling in Analyzing Multi-Attribute Genotype Response Across Environments

Three-way Multidimensional Scaling is presented as a single analysis of genotype × environment × attribute data. The concept and underlying model of this technique are discussed, and its usefulness is investigated by applying the analysis to a well-known soybean data set. The measured attributes were considered to represent the line response in each environment. It was assumed the lines could be considered in an underlying space of r dimensions. The environments were able to perceive different response patterns in that different importances could be placed on these underlying dimensions. By applying this technique a spatial representation of the lines in a low-dimensional space was obtained. Thus a single multi-attribute analysis is achieved rather than attempting to combine separate analyses for each attribute. The resulting line response pattern was examined in relation to previous reports on this data set. The inclusion of all six attributes revealed information not previously obtained by separate analysis of two of these attributes. If general inference about genotype response is to be made rather than inference on one particular attribute, then the use of a technique such as this is recommended

Intercomparing Residuals to Find Outliers in Randomized Blocks

The results of experiments laid out in a randomized complete block often appear to contain one or more unusual values. Techniques for detecting such exotic values, understanding of alternative descriptions of exoticity, one or the other of which applies rather often, leading to what might be done when exotic values are detected, can all be of practical use. This study was designed to investigate such a two-way data set: (a) by using for detection the combination of a median polish to highlight unusual values and an analysis on its residuals that is guided by the order statistic medians from a unit half-Gaussian distribution, (b) by exploring an alternate analysis on a logarithmic scale, and (c) by either Winsorizing or applying missing-value techniques to those values found to be exotic. It details the processes and the ways in which the results may be interpreted by application to data from an experiment conducted by the Rubber Research Institute in Malaysia. This article provides a concise practical tool for the investigation of such data, leading into more appropriate statistical analysis

Cluster-Analysis in a Randomized Complete Block Design

The mixture maximum likelihood approach to clustering is used to allocate treatments from a randomized complete block design into relatively homogeneous groups. The implementation of this approach is straightforward for fixed but not random block effects. The density function in each underlying group is assumed to be normal and clustering is performed on the basis of the estimated posterior probabilities of group membership. A test based on the log likelihood under the mixture model can be used to assess the actual number of groups present. The technique is demonstrated by using it to cluster data from a randomized complete block experiment

Likelihood Estimation with Normal Mixture-Models

We consider some of the problems associated with likelihood estimation in the context of a mixture of multivariate normal distributions. Unfortunately with mixture models, the likelihood equation usually has multiple roots and so there is the question of which root to choose. In the case of equal covariance matrices the choice of root is straightforward in the sense that the maximum likelihood estimator exists and is consistent. However, an example is presented to demonstrate that the adoption of a homoscedastic normal model in the presence of some heteroscedasticity can considerably influence the likelihood estimates, in particular of the mixing proportions, and hence the consequent clustering of the sample at hand