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.

Aphids, because of their short generation time and low developmental threshold temperatures, are an insect group expected to respond particularly strongly to environmental changes. Forty years of standardized, daily data on the abundance of flying aphids have been brought together from countries throughout Europe, through the EU Thematic Network 'EXAMINE'. Relationships between phenology, represented by date of first appearance in a year in a suction trap, of 29 aphid species and environmental data have been quantified using the residual maximum likelihood (REML) methodology. These relationships have been used with climate change scenario data to suggest plausible changes in aphid phenology. In general, the date of first record of aphid species in suction traps is expected to advance, the rate of advance varying with location and species, but averaging 8 days over the next 50 years. Strong relationships between aphid phenology and environmental variables have been found for many species, but they are notably weaker in species living all year on trees. Canonical variate analysis and principal coordinate analysis were used to determine ordinations of the 29 species on the basis of the presence/absence of explanatory variables in the REML models. There was strong discrimination between species with different life cycle strategies and between species feeding on herbs and trees, suggesting the possible value of trait-based groupings in predicting responses to environmental changes.

Summary This study reviews 52 field experiments, mostly from the UK, studying the effects of cultivation techniques, sowing date, crop density and cultivar choice on Alopecurus myosuroides infestations in cereal crops. Where possible, a statistical meta‐analysis has been used to calculate average responses to the various cultural practices and to estimate their variability. In 25 experiments, mouldboard ploughing prior to sowing winter cereals reduced A. myosuroides populations by an average of 69%, compared with non‐inversion tillage. Delaying drilling from September to the end of October decreased weed plant densities by approximately 50%. Sowing wheat in spring achieved an 88% reduction in A. myosuroides plant densities compared with autumn sowing. Increasing winter wheat crop density above 100 plants m−2 had no effect on weed plant numbers, but reduced the number of heads m−2 by 15% for every additional increase in 100 crop plants, up to the highest density tested (350 wheat plants m−2). Choosing more competitive cultivars could decrease A. myosuroides heads m−2 by 22%. With all cultural practices, outcomes were highly variable and effects inconsistent. Farmers are more likely to adopt cultural measures and so reduce their reliance on herbicides, if there were better predictions of likely outcomes at the individual field level.

The statistical analysis of late-stage variety evaluation trials using a mixed model is described, with one-or two-stage approaches to the analysis. Two sets of trials, from Australia and the UK, were used to provide realistic scenarios for a simulation study to evaluate the different methods of analysis. This study showed that a one-stage approach gave the most accurate predictions of variety performance overall or within each environment, across a range of models, as measured by mean squared error of prediction or realized genetic gain. A weighted two-stage approach performed adequately for variety predictions both overall and within environments, but a two-stage unweighted approach performed poorly in both cases. A generalized heritability measure was developed to compare methods.

Likelihood ratio tests for ®xed model terms are proposed for the analysis of linear mixed models when using residual maximum likelihood estimation. Bartlett-type adjustments, using an approximate decomposition of the data, are developed for the test statistics. A simulation study is used to compare properties of the test statistics proposed, with or without adjustment, with a Wald test. A proposed test statistic constructed by dropping ®xed terms from the full ®xed model is shown to give a better approximation to the asymptotic 1 2 -distribution than the Wald test for small data sets. Bartlett adjustment is shown to improve the 1 2 -approximation for the proposed tests substantially.