Estimation and prediction of parameters and breeding values in soybean using REML/BLUP and Least Squares

The aim of this study was to compare REML/BLUP and Least Square procedures in the prediction and estimation of genetic parameters and breeding values in soybean progenies. F(2:3) and F(4:5) progenies were evaluated in the 2005/06 growing season and the F(2:4) and F(4:6) generations derived thereof were evaluated in 2006/07. These progenies were originated from two semi-early, experimental lines that differ in grain yield. The experiments were conducted in a lattice design and plots consisted of a 2 m row, spaced 0.5 m apart. The trait grain yield per plot was evaluated. It was observed that early selection is more efficient for the discrimination of the best lines from the F(4) generation onwards. No practical differences were observed between the least square and REML/BLUP procedures in the case of the models and simplifications for REML/BLUP used here.

The Relationship between Intraocular Pressure Reduction and Rates of Progressive Visual Field Loss in Eyes with Optic Disc Hemorrhage

Purpose: To evaluate rates of visual field progression in eyes with optic disc hemorrhages and the effect of intraocular pressure (IOP) reduction on these rates. Design: Observational cohort study. Participants: The study included 510 eyes of 348 patients with glaucoma who were recruited from the Diagnostic Innovations in Glaucoma Study (DIGS) and followed for an average of 8.2 years. Methods: Eyes were followed annually with clinical examination, standard automated perimetry visual fields, and optic disc stereophotographs. The presence of optic disc hemorrhages was determined on the basis of masked evaluation of optic disc stereophotographs. Evaluation of rates of visual field change during follow-up was performed using the visual field index (VFI). Main Outcome Measures: The evaluation of the effect of optic disc hemorrhages on rates of visual field progression was performed using random coefficient models. Estimates of rates of change for individual eyes were obtained by best linear unbiased prediction (BLUP). Results: During follow-up, 97 (19%) of the eyes had at least 1 episode of disc hemorrhage. The overall rate of VFI change in eyes with hemorrhages was significantly faster than in eyes without hemorrhages (-0.88%/year vs. -0.38%/year, respectively, P < 0.001). The difference in rates of visual field loss pre- and post-hemorrhage was significantly related to the reduction of IOP in the post-hemorrhage period compared with the pre-hemorrhage period (r = -0.61; P < 0.001). Each 1 mmHg of IOP reduction was associated with a difference of 0.31%/year in the rate of VFI change. Conclusions: There was a beneficial effect of treatment in slowing rates of progressive visual field loss in eyes with optic disc hemorrhage. Further research should elucidate the reasons why some patients with hemorrhages respond well to IOP reduction and others seem to continue to progress despite a significant reduction in IOP levels. Financial Disclosure(s): Proprietary or commercial disclosure may be found after the references. Ophthalmology 2010; 117: 2061-2066 (C) 2010 by the American Academy of Ophthalmology.

Sampling, WLS, and Mixed Models

Mixed models may be defined with or without reference to sampling, and can be used to predict realized random effects, as when estimating the latent values of study subjects measured with response error. When the model is specified without reference to sampling, a simple mixed model includes two random variables, one stemming from an exchangeable distribution of latent values of study subjects and the other, from the study subjects` response error distributions. Positive probabilities are assigned to both potentially realizable responses and artificial responses that are not potentially realizable, resulting in artificial latent values. In contrast, finite population mixed models represent the two-stage process of sampling subjects and measuring their responses, where positive probabilities are only assigned to potentially realizable responses. A comparison of the estimators over the same potentially realizable responses indicates that the optimal linear mixed model estimator (the usual best linear unbiased predictor, BLUP) is often (but not always) more accurate than the comparable finite population mixed model estimator (the FPMM BLUP). We examine a simple example and provide the basis for a broader discussion of the role of conditioning, sampling, and model assumptions in developing inference.

Leverage analysis for linear mixed models

We consider a generalized leverage matrix useful for the identification of influential units and observations in linear mixed models and show how a decomposition of this matrix may be employed to identify high leverage points for both the marginal fitted values and the random effect component of the conditional fitted values. We illustrate the different uses of the two components of the decomposition with a simulated example as well as with a real data set.

Predicting random effects with an expanded finite population mixed model

Prediction of random effects is an important problem with expanding applications. In the simplest context, the problem corresponds to prediction of the latent value (the mean) of a realized cluster selected via two-stage sampling. Recently, Stanek and Singer [Predicting random effects from finite population clustered samples with response error. J. Amer. Statist. Assoc. 99, 119-130] developed best linear unbiased predictors (BLUP) under a finite population mixed model that outperform BLUPs from mixed models and superpopulation models. Their setup, however, does not allow for unequally sized clusters. To overcome this drawback, we consider an expanded finite population mixed model based on a larger set of random variables that span a higher dimensional space than those typically applied to such problems. We show that BLUPs for linear combinations of the realized cluster means derived under such a model have considerably smaller mean squared error (MSE) than those obtained from mixed models, superpopulation models, and finite population mixed models. We motivate our general approach by an example developed for two-stage cluster sampling and show that it faithfully captures the stochastic aspects of sampling in the problem. We also consider simulation studies to illustrate the increased accuracy of the BLUP obtained under the expanded finite population mixed model. (C) 2007 Elsevier B.V. All rights reserved.