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.     

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.     

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.     

A COMPARISON OF MIXED MODEL SPLINES FOR CURVE FITTING

Three types of polynomial mixed model splines have been proposed: smoothing splines, P‐splines and penalized splines using a truncated power function basis. The close connections between these models are demonstrated, showing that the default cubic form of the splines differs only in the penalty used. A general definition of the mixed model spline is given that includes general constraints and can be used to produce natural or periodic splines. The impact of different penalties is demonstrated by evaluation across a set of functions with specific features, and shows that the best penalty in terms of mean squared error of prediction depends on both the form of the underlying function and the signal:noise ratio.     

A COMPARISON OF ANALYSIS METHODS FOR LATE‐STAGE VARIETY EVALUATION TRIALS

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.