Brian R. Cullis

Average information REML: An efficient algorithm for variance parameter estimation in linear mixed models

A strategy of using an average information matrix is shown to be computationally convenient and efficient for estimating variance components by restricted maximum likelihood (REML) in the mixed linear model. Three applications are described. The motivation for the algorithm was the estimation of variance components in the analysis of wheat variety means from 1,071 experiments representing 10 years and 60 locations in New South Wales. We also apply the algorithm to the analysis of designed experiments by incomplete block analysis and spatial analysis of field experiments.

Accounting for natural and extraneous variation in the analysis of field experiments

We identify three major components of spatial variation in plot errors from field experiments and extend the two-dimensional spatial procedures of Cullis and Gleeson (1991) to account for them. The components are nonstationary, large-scale (global) variation across the field, stationary variation within the trial (natural variation or local trend), and extraneous variation that is often induced by experimental procedures and is predominantly aligned with rows and columns. We present a strategy for identifying a model for the plot errors that uses a trellis plot of residuals, a perspective plot of the sample variogram and, where possible, likelihood ratio tests to identify which components are present. We demonstrate the strategy using two illustrative examples. We conclude that although there is no one model that adequately fits all field experiments, the separable autoregressive model is dominant. However, there is often additional identifiable variation present.

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.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 nonsmooth components. Estimation and inference, the analysis of the three examples and a discussion of extensions and unresolved issues are also presented.

The analysis of crop cultivar breeding and evaluation trials: An overview of current mixed model approaches

The analysis of series of crop variety trials has a long history with the earliest approaches being based on ANOVA methods. Kempton (1984) discussed the inadequacies of this approach, summarized the alternatives available at that time and noted that all of these approaches could be classified as multiplicative models. Recently, mixed model approaches have become popular for the analysis of series of variety trials. There are numerous reasons for their use, including the ease with which incomplete data (not all varieties in all trials) can be handled and the ability to appropriately model within-trial error variation. Currently, the most common mixed model approaches for series of variety trials are mixed model versions of the methods summarized by Kempton (1984). In the present paper a general formulation that encompasses all of these methods is described, then individual methods are considered in detail.

On the Design of Early Generation Variety Trials With Correlated Data

This article considers the design of early generation variety trials with a prespecified spatial correlation structure and introduces a new class of partially replicated designs called p-rep designs in which the plots of standard varieties are replaced by additional plots of test lines. We show how efficient p-rep designs can be readily generated using the modified Reactive TABU search algorithm. The expected and realized genetic gain of p-rep and grid plot designs is compared in a simulation study.

Residual Maximum Likelihood (REML) Estimation of a Neighbour Model for Field Experiments

A spatial analysis of field experiments is proposed which takes account of association between neighbouring plots. The residual maximum likelihood (REML) method of Patterson and Thompson (1971, Biometrika 58, 545-554) is used to estimate parameters of a general neighbour model, which can be expressed as an autoregressive moving average (ARMA) model. Three data sets are analysed to (i) highlight the need for a model selection procedure, (ii) illustrate the differing results between incomplete block and neighbour analysis and the effect of including treated border plots in the design, and (iii) illustrate the environmental variation within an experiment using prediction of trend.

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.

SPATIAL ANALYSIS OF MULTI-ENVIRONMENT EARLY GENERATION VARIETY TRIALS

A fully efficient approach for the analysis of multi-environment early stage variety trials is considered that accommodates a general spatial covariance structure for the errors of each trial. The analysis simultaneously produces best linear unbiased predictors of the genotype and genotype by environment interaction effects and residual maximum likelihood estimates of the spatial parameters and variance components. Two motivating examples are presented and analyzed, and the results suggest that the previous approximate analyses can seriously affect estimation of the genetic merit of breeding lines, particularly for models with more complex variance structures.

Joint modeling of additive and non-additive genetic line effects in single field trials

A statistical approach is presented for selection of best performing lines for commercial release and best parents for future breeding programs from standard agronomic trials. The method involves the partitioning of the genetic effect of a line into additive and non-additive effects using pedigree based inter-line relationships, in a similar manner to that used in animal breeding. A difference is the ability to estimate non-additive effects. Line performance can be assessed by an overall genetic line effect with greater accuracy than when ignoring pedigree information and the additive effects are predicted breeding values. A generalized definition of heritability is developed to account for the complex models presented.

Reliability of higher seeding rates of wheat for increased competitiveness with weeds in low rainfall environments

SUMMARY Increasing crop competitiveness using higher seeding rates is a possible technique for weed manage- ment in low input and organic farming systems or when herbicide resistance develops in weeds. A range of wheat seeding rates were sown and resulted in crop densities between 50-400 plants/m 2 (current recommendations are 100-150 plants/m 2 ) in the presence and absence of annual ryegrass (Lolium rigidum Gaud.) in three wheat cultivars at nine experiments in southern Australia. Wheat densities of at least 200 plants/m 2