A Sparse Implementation of the Average Information Algorithm for Factor Analytic and Reduced Rank Variance Models

Factor analytic variance models have been widely considered for the analysis of multivariate data particularly in the psychometrics area. Recently Smith, Cullis & Thompson (2001) have considered their use in the analysis of multi‐environment data arising from plant improvement programs. For these data, the size of the problem and the complexity of the variance models chosen to account for spatial heterogeneity within trials implies that standard algorithms for fitting factor analytic models can be computationally expensive. This paper presents a sparse implementation of the average information algorithm (Gilmour, Thompson & Cullis, 1995) for fitting factor analytic and reduced rank variance models.     

NEAREST NEIGHBOUR ANALYSIS OF UNEQUALLY REPLICATED TRIALS

There has been much work on the use of neighbouring plots to control environmental variation in the analysis of agricultural field experiments. In particular, the Residual Maximum Likelihood Neighbour (REMLN) analysis of Gleeson&Cullis (1987) appears very promising. The application of the REMLN analysis to an unequally replicated field trial augmented with an additional variety planted every six plots in a grid system is here compared with a covariance (COV) analysis using the neighbouring grid or check plot values as the covariate. The results indicate that the REMLN analysis gives more accurate estimates of treatment contrasts than the COV analyses, but that the estimate of treatment means can be biased. The bias depends on the mean of the check plot. This bias can be removed by adjusting the estimates of the treatment means such that their average equals the average of the raw means rather than that of the raw data.     

The Analysis of Crop Variety Evaluation Data in Australia

The major aim of crop variety evaluation is to predict the future performance of varieties. This paper presents the routine statistical analysis of data from late-stage testing of crop varieties in Australia. It uses a two-stage approach for analysis. The data from individual trials from the current year are analysed using spatial techniques. The resultant table of variety-by-trial means is combined with tables from previous years to form the data for an overall mixed model analysis. Weights allow for the data being estimates with varying accuracy. In view of the predictive aim of the analysis, variety effects and interactions are regarded as random effects. Appropriate inferential tools have been developed to assist with interpretation of the results. Analyses must be conducted in a timely manner so that variety predictions can be published and disseminated to growers immediately after harvest each year. Factors which facilitate this include easy access to historic data and the use of specialist mixed model software.     

‘Spite effects’ in tax evasion experiments

This paper sets out to consider individuals’ motivations to evade taxation. Experimental results indicate that individuals do not simply maximize pecuniary welfare. Their behaviour is consistent with the presence of a ‘spite effect’ when their perceptions are that enforcement variables are ‘excessive’.     

The Analysis of Designed Experiments and Longitudinal Data by Using Smoothing Splines

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 first 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 final example is a multienvironment agricultural field 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 fixed and random effects, random coefficients and variance modelling to provide simultaneous modelling of trends and covariance structure. The key result that allows coherent and flexible 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 non-smooth components. Estimation and inference, the analysis of the three examples and a discussion of extensions and unresolved issues are also presented.     

PROPERTIES OF THE PATTERNED VECH OPERATOR

The vec operator arranges the columns of a matrix one below the other. When the matrix is symmetric such elements are not distinct but an extraction of only the distinct elements on or below the diagonal forms the operation denoted by vech. For other types of patterned matrices a ‘patterned vech’ operator is defined. The transformations from vech to vec are not uniquely defined. Here we examine properties of linear transformations which overcome the lack of uniqueness and develop properties of such linear transformations.     

Prediction in linear mixed models

Following estimation of effects from a linear mixed model, it is often useful to form predicted values for certain factor/variate combinations. The process has been well defined for linear models, but the introduction of random effects into the model means that a decision has to be made about the inclusion or exclusion of random model terms from the predictions. This paper discusses the interpretation of predictions formed including or excluding random terms. Four datasets are used to illustrate circumstances where different prediction strategies may be appropriate: in an orthogonal design, an unbalanced nested structure, a model with cubic smoothing spline terms and for kriging after spatial analysis. The examples also show the need for different weighting schemes that recognize nesting and aliasing during prediction, and the necessity of being able to detect inestimable predictions.     

ON AN APPROXIMATE OPTIMALITY CRITERION FOR THE DESIGN OF FIELD EXPERIMENTS UNDER SPATIAL DEPENDENCE

The design of large‐scale field trials where the residuals are correlated has been of recent interest, in large part because of advances in statistical and computational methods of analysis. The construction of designs for correlated data has typically used A‐optimality and is computationally intensive. This involves calculating the inverse of the information matrix for treatments under the supervision of an optimization strategy that explores the design space. We propose an approximation to A‐optimality, using nearest‐neighbour balance, that is less computationally demanding and can achieve at least 95% efficiency relative to A‐optimality in many practical situations.     

Some hypotheses and evidence on tax knowledge and preferences

The article comments on a national quota sample of 900 adults in the UK. Questions were asked concerning beliefs as to the sources of government revenue to pay for services; whether taxes should he increased to improve services; and whether any extra tax levied should go on incomes or the goods people buy. The overall results showed widespread ignorance of government sources of revenue. The majority preferred to keep the level of taxation and expenditure at present levels, but if taxes had to he increased these should he levied on goods rather than incomes. Predictions of the overall and disaggregated results from economic theory and past survey research met with varying success, the implications of which are discussed.