These articles were sponsored by IAMG under the guidance of John Tipper in association with the European Journal of Soil Science.

**List of Titles**

**1. Webster, R., 1997, Regression and functional relations: European Journal of Soil Science 48, 557-566, 3 figures.**

**Abstract:** Regression is frequently abused in soil research. Its proper use is for statistical prediction. It may also be used to calculate equations for calibration. A regression equation may be used to express a functional relation between two soil variables that are thought to be related by some simple mathematical law but only when one of the variables is known exactly. In most other circumstances regression is inappropriate. Where departures from a functional relation are a result of errors of measurement and sampling fluctuation it should be replaced by a structural analysis to find the best equation. Where the underlying relation is truly bivariate it should be described as such.

**2. Horgan, G.W., 1998, Mathematical morphology for analysing soil structure from images: European Journal of Soil Science 49, 161-173, 10 figures.**

**Abstract: **Mathematical morphology is an approach to image analysis based on set theory. It explores structures by examining the effect of transforming them by a set of operations. Such operations can be built up and combined into powerful tools for exploring, transforming and measuring the size, shape and connectivity of components of interest in images. Greylevel images are handled by regarding them as binary images in three dimensions. This paper reviews the basic ideas and illustrates them by application to studies of the size distribution of soil pores, the lengths and geometric patterns of cracks in drying soil, and the growth of fungal hyphae. It then extends them in an introductory way to random sets. Practical issues of scale, resolution, sampling, replication and noise in the use of images for soil measurement are described briefly.

**3. Lark, R.M. and Webster, R., 1999, Analysis and elucidation of soil variation using wavelets: European Journal of Soil Science 50, 185-206.**

**Abstract**: A wavelet is a compact analysing kernel that can be moved over a sequence of data to measure variation locally. There are several families of wavelet, and within any one family wavelets of different lengths and therefore smoothness and their corresponding scaling functions can be assembled into a collection of orthogonal functions. Such an assemblage can then be applied to filter spatial data into a series of independent components at varying scales in a single coherent analysis. The application requires no assumptions other than finite variance. The methods have been developed for processing signals and remote sensing imagery in which data are abundant, and they need modification for data from field sampling.

This paper descibes the theory of wavelets. It introduces the pyramid algorithm for multiresolution analysis and shows how it can be adapted for fairly small sets of transect data such as one might obtain in a soil survey. It then illustrates the application using Daubechies's wavelets to two soil transects, one of gilgai on plain land in Australia and the other across a sedimentary sequence in England. In both examples the technique revealed strongly contrasting local features of the variation that had been lost by averaging the previous analyses and expressed them quantitatively in combinations of both scale and magnitude. Further, the results could be explained as the spatial effects of change in topography or geology underlying the variation in the soil.

**4. Webster, R. and Payne, R.W., 2002, Analysing repeated measurements in soil monitoring and experimentation: European Journal of Soil Science 53, 1-13.**

**Abstract:**Field monitoring, leaching studies, and experimentation in soil biology are often now being done non-destructively using fixed installations so that measurements are made repeatedly on the same units. The resulting data for each unit (suction cup, lysimeter, incubation chamber) constitute a time series in which there may be autocorrelation. The usual methods of statistical analysis, such as the analysis of variance, must be modified or replaced by more suitable ones to take account of the possible correlation.

This paper describes the split-plot design of such experiments, shows how to assess the variance-covariance matrix of residuals for uniformity by the Greenhouse-Geisser test of significance, and how to use this statistic to adjust the degrees of freedom in a formal test of significance. It also describes more recent methods. Ante-dependence analysis identifies the extent of the temporal correlation in the data and provides approximate significance tests for the treatments. Alternatively, the paper shows how the traditional analysis of variance may be replaced by a restricted maximum likelihood analysis which gives Wald statistics.

The techniques are illustrated with data on CO2 evolved from soil incubated for 75 days in closed chambers, during which time the gas was measured on 24 occasions to give time series for three replicates of each combination of two soils (limed and unlimed) and three types of ryegrass amendment. An antedependence structure (extending to ninth order) weakened the usual significance test within the sub-unit stratum. The Wald statistics showed that there was, nevertheless, a strong interaction between the treatments and time.

**5. Lane, P.W., 2002, Generalized linear models in soil science: European Journal of Soil Science 53, 241-251.**

**Abstract:**Classical linear models are easy to understand and fit. However, when assumptions are not met, violence should not be used on the data to force them into the linear mould. Transformation of variables may allow successful linear modelling, but it affects several aspects of the model simultaneously. In particular, it can interfere with the scientific interpretation of the model. Generalized linear models are a wider class, and they retain the concept of additive explanatory effects. They provide generalizations of the distributional assumptions of the response variable, while at the same time allowing a transformed scale on which the explanatory effects combine. These models can be fitted reliably with standard software, and the analysis is readily interpreted in an analogous way to that of linear models. Many further generalizations to the generalized linear model have been proposed, extending them to deal with smooth effects, non-linear parameters, and extra components of variation. Though the extra complexity of generalized linear models gives rise to some additional difficulties in analysis, these difficulties are outweighed by the flexibility of the models and ease of interpretation. The generalizations allow the intuitively more appealing approach to analysis of adjusting the model rather than adjusting the data.