S.J. Welham

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.

Transcriptome analysis of grain development in hexaploid wheat

BackgroundHexaploid 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.ResultsThe transcriptome of developing caryopses from hexaploid wheat (Triticum aestivum, cv. Hereward) was determined using Affymetrix wheat GeneChip® oligonucleotide arrays which have probes for 55,052 transcripts. Of these, 14,550 showed significant differential regulation in the period between 6 and 42 days after anthesis (daa). Large changes in transcript abundance were observed which were categorised into distinct phases of differentiation (6–10 daa), grain fill (12–21 daa) and desiccation/maturation (28–42 daa) and were associated with specific tissues and processes. A similar experiment on developing caryopses grown with dry and/or hot environmental treatments was also analysed, using the profiles established in the first experiment to show that most environmental treatment effects on transcription were due to acceleration of development, but that a few transcripts were specifically affected. Transcript abundance profiles in both experiments for nine selected known and putative wheat transcription factors were independently confirmed by real time RT-PCR. These expression profiles confirm or extend our knowledge of the roles of the known transcription factors and suggest roles for the unknown ones.ConclusionThis transcriptome data will provide a valuable resource for molecular studies on wheat grain. It has been demonstrated how it can be used to distinguish general developmental shifts from specific effects of treatments on gene expression and to diagnose the probable tissue specificity and role of transcription factors.

Likelihood Ratio Tests for Fixed Model Terms using Residual Maximum Likelihood

Likelihood ratio tests for fixed 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 fixed terms from the full fixed model is shown to give a better approximation to the asymptotic χ2‐distribution than the Wald test for small data sets. Bartlett adjustment is shown to improve the χ2‐approximation for the proposed tests substantially.

An efficient computing strategy for prediction in mixed linear models

After estimation of effects from a linear mixed model, it is often useful to form predicted values for certain factor/variate combinations. This process has been well-defined for linear models, but the introduction of random effects 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 fitted model. In many cases, the size of the matrices required to calculate predictions and their covariance matrix directly can be prohibitive. An efficient 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.

Regulatory Hotspots Are Associated with Plant Gene Expression under Varying Soil Phosphorus Supply in Brassica rapa

Gene expression is a quantitative trait that can be mapped genetically in structured populations to identify expression quantitative trait loci (eQTL). Genes and regulatory networks underlying complex traits can subsequently be inferred. Using a recently released genome sequence, we have defined cis- and trans-eQTL and their environmental response to low phosphorus (P) availability within a complex plant genome and found hotspots of trans-eQTL within the genome. Interval mapping, using P supply as a covariate, revealed 18,876 eQTL. trans-eQTL hotspots occurred on chromosomes A06 and A01 within Brassica rapa; these were enriched with P metabolism-related Gene Ontology terms (A06) as well as chloroplast- and photosynthesis-related terms (A01). We have also attributed heritability components to measures of gene expression across environments, allowing the identification of novel gene expression markers and gene expression changes associated with low P availability. Informative gene expression markers were used to map eQTL and P use efficiency-related QTL. Genes responsive to P supply had large environmental and heritable variance components. Regulatory loci and genes associated with P use efficiency identified through eQTL analysis are potential targets for further characterization and may have potential for crop improvement.

Estimating the spatial scales of regionalized variables by nested sampling, hierarchical analysis of variance and residual maximum likelihood

The variogram is essential for local estimation and mapping of any variable by kriging. The variogram itself must usually be estimated from sample data. The sampling density is a compromise between precision and cost, but it must be sufficiently dense to encompass the principal spatial sources of variance. A nested, multi-stage, sampling with separating distances increasing in geometric progression from stage to stage will do that. The data may then be analyzed by a hierarchical analysis of variance to estimate the components of variance for every stage, and hence lag. By accumulating the components starting from the shortest lag one obtains a rough variogram for modest effort. For balanced designs the analysis of variance is optimal; for unbalanced ones, however, these estimators are not necessarily the best, and the analysis by residual maximum likelihood (reml) will usually be preferable. The paper summarizes the underlying theory and illustrates its application with data from three surveys, one in which the design had four stages and was balanced and two implemented with unbalanced designs to economize when there were more stages. A Fortran program is available for the analysis of variance, and code for the reml analysis is listed in the paper.

A Hypomethylated population of Brassica rapa for forward and reverse Epi-genetics

BackgroundEpigenetic marks superimposed on the DNA sequence of eukaryote chromosomes provide agility and plasticity in terms of modulating gene expression, ontology, and response to the environment. Modulating the methylation status of cytosine can generate epialleles, which have been detected and characterised at specific loci in several plant systems, and have the potential to generate novel and relatively stable phenotypes. There have been no systematic attempts to explore and utilise epiallelic variation, and so extend the range of phenotypes available for selection in crop improvement. We developed an approach for generating novel epialleles by perturbation of the DNA methylation status. 5- Azacytidine (5-AzaC) provides selective targeting of 5mCG, which in plants is associated with exonic DNA. Targeted chemical intervention using 5-AzaC has advantages over transgenic or mutant modulation of methyltransferases, allowing stochastic generation of epialleles across the genome.ResultsWe demonstrate the potential of stochastic chemically-induced hypomethylation to generate novel and valuable variation for crop improvement. Systematic analysis of dose–response to 5-AzaC in B. rapa guided generation of a selfed stochastically hypomethylated population, used for forward screening of several agronomic traits. Dose–response was sigmoidal for several traits, similar to that observed for chemical mutagens such as EMS. We demonstrated transgenerational inheritance of some phenotypes. BraRoAZ is a unique hypomethylated population of 1000 E2 sib lines. When compared to untreated controls, 5-Aza C-treated lines exhibited reduced immuno-staining of 5mC on pachytene chromosomes, and Methylation Sensitive Amplified Polymorphism (MSAP) profiles that were both divergent and more variable. There was coincident phenotypic variation among these lines for a range of seed yield and composition traits, including increased seed protein content and decreased oil content, as well as decreased erucic acid and corresponding increases in linoleic and/or palmitic acid. Each 5-AzaC-treated line represents a unique combination of hypomethylated epialleles.ConclusionsThe approach and populations developed are available for forward and reverse screening of epiallelic variation and subsequent functional and inheritance studies. The generation of stochastically hypomethylated populations has utility in epiallele discovery for a wide range of crop plants, and has considerable potential as an intervention strategy for crop improvement.

A variance shift model for detection of outliers in the linear mixed model

A variance shift outlier model (VSOM), previously used for detecting outliers in the linear model, is extended to the variance components model. This VSOM accommodates outliers as observations with inflated variance, with the status of the ith observation as an outlier indicated by the size of the associated shift in the variance. Likelihood ratio and score test statistics are assessed as objective measures for determining whether the ith observation has inflated variance and is therefore an outlier. It is shown that standard asymptotic distributions do not apply to these tests for a VSOM, and a modified distribution is proposed. A parametric bootstrap procedure is proposed to account for multiple testing. The VSOM framework is extended to account for outliers in random effects and is shown to have an advantage over case-deletion approaches. A simulation study is presented to verify the performance of the proposed tests. Challenges associated with computation and extensions of the VSOM to the general linear mixed model with correlated errors are discussed.

Pasture and sheep responses to lime application in a grazing experiment in a high-rainfall area, south-eastern Australia. I. Pasture production

‘Managing Acid Soils Through Efficient Rotations (MASTER)’ is a long-term pasture–crop rotation experiment commenced in 1992. One of the objectives was to demonstrate the extent of crop, pasture, and animal responses to lime on a typical acidic soil in the 500–800 mm rainfall zone in south-eastern Australia. Two types of pastures (perennial v. annual pastures) with or without lime application were established in 1992. This paper presents the results of the pasture dry matter (DM) responses to lime application over 6 years from 1992 to 1997. Results showed that both perennial and annual pastures responded positively to lime on a highly acidic soil on the south-west slopes of New South Wales. Averaged across pasture types and 5 growing seasons, the limed pastures produced 18% more pasture DM (520 kg/ha, P < 0.05) than the unlimed pastures. Significant responses to lime were detected on perennial pastures (610 kg DM/ha, P < 0.05), but not on annual pastures, although the limed annual pastures produced more DM (420 kg/ha, P = 0.20) than the unlimed annual pastures. There was a large seasonal variation in pasture growth rate with the significant lime responses in winter and spring on both perennial pastures (P < 0.05) and annual pastures (P < 0.10 in winter and P < 0.05 in spring), but no responses in autumn and summer on either perennial or annual pastures. The extra growth in winter is of importance as winter is the period when feed is normally inadequate and limits stocking rates. It is recommended that perennial-based pastures should be promoted for the purposes of productivity, in terms of increasing pasture production and improving feed quality, and for the environmental benefits in terms of alleviating the soil acidity problem and reducing the risk of dryland salinity in the high-rainfall zone in south-eastern Australia.

A novel transcriptomic approach to identify candidate genes for grain quality traits in wheat.

A novel methodology is described in which transcriptomics is combined with the measurement of bread-making quality and other agronomic traits for wheat genotypes grown in different environments (wet and cool or hot and dry conditions) to identify transcripts associated with these traits. Seven doubled haploid lines from the Spark x Rialto mapping population were selected to be matched for development and known alleles affecting quality. These were grown in polytunnels with different environments applied 14 days post-anthesis, and the whole experiment was repeated over 2 years. Transcriptomics using the wheat Affymetrix chip was carried out on whole caryopsis samples at two stages during grain filling. Transcript abundance was correlated with the traits for approximately 400 transcripts. About 30 of these were selected as being of most interest, and markers were derived from them and mapped using the population. Expression was identified as being under cis control for 11 of these and under trans control for 18. These transcripts are candidates for involvement in the biological processes which underlie genotypic variation in these traits.