G. J. Ross

Minimum spanning trees (MST) and single linkage cluster analysis (SLCA) are explained and it is shown that all the information required for the SLCA of a set of points is contained in their MST. Known algorithms for finding the MST are discussed. They are efficient even when there are very many points; this makes a SLCA practicable when other methods of cluster analysis are not. The relevant computing procedures are published in the Algorithm section of the same issue of Applied Statistics. The use of the MST in the interpretation of vector diagrams arising in multivariate analysis is illustrated by an example.

Abstract Sodium pyrophosphate (0.1 M ) extracts of some soil samples from Canada and East Malaysia were subjected to low‐speed centrifugation both without and with Superfloc, high‐speed centrifugation, and ultrafiltration of the centrifugates prior to determination of their Fe and Al contents. Relative to ultrafiltration, none of the centrifugation methods cleared the extracts completely and thus part of the extracted Fe p and Al p measured in the supernatants was present in suspended solids. Suspended solids, with mineralogy similar to that of the clay of corresponding samples, were particularly abundant in the supernatants of the tropical soil samples after low‐speed centrifugation without Superfloc. Adding Superfloc to these extracts and centrifuging them at low speed sedimented practically all the aluminum silicates but concentrated the goethite in the material that remained suspended. High‐speed centrifugation gave similar results but with less concentration of goethite. Smaller amounts of suspended solids, also with mineralogy similar to that of the clay of corresponding samples, were present in the supernatants of samples from Canadian soils, except for that from a spodic horizon sample in which suspended minerals were absent. Ultrafiltration of the centrifugates through a 0.025‐µm millipore filter cleared the extracts effectively and is recommended for tropical soils and for soils giving doubtful results with centrifugation methods only. High‐speed centrifugation is probably adequate for most temperate‐region soils, as suspended Fe and Al in the extracts tested accounted for 0.1 dag/kg or less of sample weight.

Maximum‐likelihood estimation problems can be solved numerically using function minimization algorithms, but the amount of computing required and the accuracy of the results depend on the way the algorithms are used. Attention to the analytical properties of the model, to the relationship between the model and the data, and to descriptive properties of the data can greatly simplify the problem, sometimes providing a method of solution on a desk calculator. This paper describes how parameter transformation, sequential minimization and nested minimization can be used to solve particular problems. Applications to well‐known problems of distribution fitting, quantal responses and least‐squares curve fitting are described. The implications for computer programming are discussed.