After estimation of e ects from a linear mixed model, it is often useful to form predicted values for certain factor/variate combinations. This process has been well-deÿned for linear models, but the introduction of random e ects means that a decision has to be made about the inclusion or exclusion of random model terms from the predictions, including the residual error. For spatially correlated data, kriging then becomes prediction from the ÿtted model. In many cases, the size of the matrices required to calculate predictions and their covariance matrix directly can be prohibitive. An e cient computational strategy for calculating predictions and their standard errors is given, which includes the ability to detect the invariance of predictions to the parameterisation used in the model.

In designed experiments and in particular longitudinal studies, the aim may be to assess the effect of a quantitative variable such as time on treatment effects. Modelling treatment effects can be complex in the presence of other sources of variation. Three examples are presented to illustrate an approach to analysis in such cases. The ®rst example is a longitudinal experiment on the growth of cows under a factorial treatment structure where serial correlation and variance heterogeneity complicate the analysis. The second example involves the calibration of optical density and the concentration of a protein DNase in the presence of sampling variation and variance heterogeneity. The ®nal example is a multienvironment agricultural ®eld experiment in which a yield±seeding rate relationship is required for several varieties of lupins. Spatial variation within environments, heterogeneity between environments and variation between varieties all need to be incorporated in the analysis. In this paper, the cubic smoothing spline is used in conjunction with ®xed and random effects, random coef®cients and variance modelling to provide simultaneous modelling of trends and covariance structure. The key result that allows coherent and¯exible empirical model building in complex situations is the linear mixed model representation of the cubic smoothing spline. An extension is proposed in which trend is partitioned into smooth and nonsmooth components. Estimation and inference, the analysis of the three examples and a discussion of extensions and unresolved issues are also presented.1999 Royal Statistical Society 0035±9254/99/48269 Appl. Statist. (1999) 48, Part 3, pp. 269^311 treatments, e.g. time in the longitudinal setting, the interaction of treatments with the quantitative variable is generally of interest. The following experiments illustrate such situations and provide the motivation for the methods discussed in this paper. Data sets for the examples can be accessed at http://www.blackwellpublishers.co.uk/rss/ 1.1. Example 1: live-weights of cows An experiment was carried out to study the eects of altered dietary iron intake in dairy cattle infected with Mycobacterium paratuberculosis (see Lepper et al. (1989)). Previous studies on mice had established that the multiplication of the organism was inhibited by low rather than high dietary iron intake. The aim of the study on cows was to examine the long-term eects of iron dosing on the progress of the organism. Several variables were measured in the experiment, with the focus on the live-weight data in this paper. At the start of the experiment, 28 female calves, at most 2 weeks old, were subdivided into two groups of eight and 20 animals and accommodated in two yards. 4 weeks later, the group of 20 was inoculated with the organism (the infected group) and this is taken as time 0. The ®rst live-weights were taken 122 days later and were recorded to the nearest 5 kg. Iron dosing began on day 155 (5 days after the second measurement of live-weight). The two groups were subdivi...

Background: Hexaploid wheat is one of the most important cereal crops for human nutrition. Molecular understanding of the biology of the developing grain will assist the improvement of yield and quality traits for different environments. High quality transcriptomics is a powerful method to increase this understanding.

Geostatistical estimates of a soil property by kriging are equivalent to the best linear unbiased predictions (BLUPs). Universal kriging is BLUP with a fixed-effect model that is some linear function of spatial coordinates, or more generally a linear function of some other secondary predictor variable when it is called kriging with external drift. A problem in universal kriging is to find a spatial variance model for the random variation, since empirical variograms estimated from the data by method-of-moments will be affected by both the random variation and that variation represented by the fixed effects.The geostatistical model of spatial variation is a special case of the linear mixed model where our data are modelled as the additive combination of fixed effects (e.g. the unknown mean, coefficients of a trend model), random effects (the spatially dependent random variation in the geostatistical context) and independent random error (nugget variation in geostatistics). Statisticians use residual maximum likelihood (REML) to estimate variance parameters, i.e. to obtain the variogram in a geostatistical context. REML estimates are consistent (they converge in probability to the parameters that are estimated) with less bias than both maximum likelihood estimates and method-of-moment estimates obtained from residuals of a fitted trend. If the estimate of the random effects variance model is inserted into the BLUP we have the empirical BLUP or E-BLUP. Despite representing the state of the art for prediction from a linear mixed model in statistics, the REML-E-BLUP has not been widely used in soil science, and in most studies reported in the soils literature the variogram is estimated with methods that are seriously biased if the fixed-effect structure is more complex than just an unknown constant mean (ordinary kriging). In this paper we describe the REML-E-BLUP and illustrate the method with some data on soil water content that exhibit a pronounced spatial trend.