Mixed Model Methodology Research Articles

Semiparametric regression during 2003-2007.

Semiparametric regression is a fusion between parametric regression and nonparametric regression that integrates low-rank penalized splines, mixed model and hierarchical Bayesian methodology - thus allowing more streamlined handling of longitudinal and spatial correlation. We review progress in the field over the five-year period between 2003 and 2007. We find semiparametric regression to be a vibrant field with substantial involvement and activity, continual enhancement and widespread application.

An overview on quantitative and genomic tools for utilising dominance genetic variation in improving animal production

In addition to genetic progress made by selection on additive genetic values, short-term gains can be produced by recovering possible inbreeding depression or utilising putative overdominance. These are both caused mainly by dominance genetic variation which can be quantified using mixed model methodology. Inbreeding brings along a requirement for extra parameterisation in expressing and estimating dominance variance. The extra parameters specify how dominance is affecting the mean and (co)variances among inbred animals. The full description for breed crosses contains a very large set of parameters. The benefits from crossbreeding are highest with widely deviating allele frequencies between the breeds. Maximisation of heterosis can be done only on a temporary basis as a continued exploitation leads to stagnation in the overall genetic progress. Therefore efficient methods with immediate returns are needed to find the most promising breeds jointly with the most potential mating pairs. One possibility is the use of genomic tools in assessing populations for crossbreeding and in searching for major genes mediating dominance variation. The analyses are providing markers that can be used in choosing mating pairs that produce desirable dominance deviations in analysed marker brackets. Genome-wide marker sets can be used for discovering genome segments with maximum heterosis effect. The phenotypic records are available for such analyses, soon are also the large marker sets and their genotypes: the analytical tools need developers.;

Effects of Breed and Feeding System on Milk Production, Body Weight, Body Condition Score, Reproductive Performance, and Postpartum Ovarian Function

The objective of this study was to investigate the potential differences among Holstein-Friesian (HF), Montbéliarde (MB), Normande (NM), Norwegian Red (NRF), Montbéliarde × Holstein-Friesian (MBX), and Normande × Holstein-Friesian (NMX) across 2 seasonal grass-based systems of milk production. The effects of breed and feeding system on milk production, body weight, body condition score, fertility performance, hormone parameters, ovarian function, and survival were determined by using mixed model methodology, generalized linear models, and survival analysis. The 5-yr study comprised up to 749 lactations on 309 cows in one research herd. The HF produced the greatest yield of solids-corrected milk, the MB and NM produced the least yields, and NRF, MBX, and NMX were intermediate. The NRF had the lowest body weight throughout lactation, the NM had the highest, and the other breeds were intermediate. Body condition score was greatest for MB and NM, least for HF, and intermediate for NRF, MBX, and NMX. The HF had a lower submission rate and overall pregnancy rate compared with the NRF. The NRF survived the longest in the herd, the HF survived the shortest, and the NM, MB, MBX, and NMX were intermediate. Breed of dairy cow had no effect on selected milk progesterone parameters from 5 d postpartum until 26 d after first artificial insemination. Breed of dairy cow did not influence insulin and insulin-like growth factor-1 around parturition or at the start of the breeding season. Animals offered a high-concentrate diet had greater milk yield, but they did not have improved reproductive performance. Differences observed between the different breeds in this study are a likely consequence of the past selection criteria for the respective breeds.

Genetic and Environmental Trends for Milk Production Traits in Sheep Estimated with Test-day Model

Data from milk performance testing were used to analyze genetic and environmental trends for purebred Tsigai, Improved Valachian and Lacaune sheep. 103,715 (Tsigai), 212,962 (Improved Valachian) and 2,196 (Lacaune) test-day records gathered by the State Breeding Institute of the Slovak Republic entered the analyses. The respective pedigree data comprised 23,724 (Tsigai), 51,401 (Improved Valachian) and 438 (Lacaune) records. The multiple-trait, mixed model methodology was used to predict the breeding values for daily milk yield, fat and protein content and to estimate the fixed and remaining random effects assumed to affect the above mentioned traits, separately for each breed. The breeding values for daily milk yield were adjusted for 150-day standardized lactation length by multiplying with the constant 150, as the breeding goal of the selection scheme in Slovakian sheep is to increase 150-day milk production and constant heritability throughout the whole lactation is assumed. The genetic trends were expressed as changes in averages of breeding values across birth years of animals. For Tsigai and Lacaune breeds, cumulative genetic changes over the analyzed period were 3.8 and 5.1 kg for 150-day milk, 0 and -0.16% for fat content and 0 and -0.12% for protein content. For Improved Valachian breed, either a low (1.6 kg for 150-day milk yield) or zero (fat and protein content) cumulative genetic change was found. The environmental trends were calculated as averages of solutions for flock-test day effect across years and months in which measurements were taken. A distinctive cyclical pattern which reflected short-time variation in milk production traits was found. Possible explanations for this phenomenon are given and discussed.

Bayesian Analysis of Mass Spectrometry Proteomic Data Using Wavelet‐Based Functional Mixed Models

In this article, we apply the recently developed Bayesian wavelet-based functional mixed model methodology to analyze MALDI-TOF mass spectrometry proteomic data. By modeling mass spectra as functions, this approach avoids reliance on peak detection methods. The flexibility of this framework in modeling nonparametric fixed and random effect functions enables it to model the effects of multiple factors simultaneously, allowing one to perform inference on multiple factors of interest using the same model fit, while adjusting for clinical or experimental covariates that may affect both the intensities and locations of peaks in the spectra. For example, this provides a straightforward way to account for systematic block and batch effects that characterize these data. From the model output, we identify spectral regions that are differentially expressed across experimental conditions, in a way that takes both statistical and clinical significance into account and controls the Bayesian false discovery rate to a prespecified level. We apply this method to two cancer studies.

Fitting Conditional Survival Models to Meta‐Analytic Data by Using a Transformation Toward Mixed‐Effects Models

Frailty models are widely used to model clustered survival data. Classical ways to fit frailty models are likelihood-based. We propose an alternative approach in which the original problem of "fitting a frailty model" is reformulated into the problem of "fitting a linear mixed model" using model transformation. We show that the transformation idea also works for multivariate proportional odds models and for multivariate additive risks models. It therefore bridges segregated methodologies as it provides a general way to fit conditional models for multivariate survival data by using mixed models methodology. To study the specific features of the proposed method we focus on frailty models. Based on a simulation study, we show that the proposed method provides a good and simple alternative for fitting frailty models for data sets with a sufficiently large number of clusters and moderate to large sample sizes within covariate-level subgroups in the clusters. The proposed method is applied to data from 27 randomized trials in advanced colorectal cancer, which are available through the Meta-Analysis Group in Cancer.

Variabilidade genética e seleção para caracteres de crescimento da seringueira

O presente estudo objetivou estimar parâmetros e valores genéticos para os caracteres altura e diâmetro de progênies de seringueira, pelo método de modelos mistos (procedimento REML/BLUP). Progênies de meio-irmãos foram estabelecidas em área do município de Dois Irmãos do Buriti (MS), sob delineamento de blocos ao acaso, com 28 tratamentos (progênies), cinco repetições e dez plantas por parcela. A coleta dos dados foi realizada em março de 2004. Aos 12 meses de idade foram avaliados os caracteres altura e diâmetro. Nas herdabilidades individuais no sentido restrito para altura (0,1877) e diâmetro (0,1809), bem como nas herdabilidades médias no sentido restrito de progênies para altura (0,3385) e diâmetro (0,4124) observaram-se boas magnitudes. Ganhos genéticos preditos na ordem de 14,04% a 16,54% foram obtidos para o caráter altura e 25,76% a 91,78% para o caráter diâmetro. Os resultados estimulam a continuidade do programa de melhoramento genético, com a possibilidade de maximização dos ganhos nas gerações seguintes.

Defining the experimental unit for the design and analysis of site-specific experiments in commercial cotton fields

Designing experiments involves several processes. The first process identifies the experimental units generated by conducting the experiment. The second process is the application of planned treatments to the experimental units. Other processes are the analysis of the experiment and interpretation of results. While traditional experimental small plot designs are useful for investigating many aspects of cotton production, on a large scale, like an entire commercial field, they are difficult to implement because of routine farming operations and topographical variability. As an alternative, by using the capabilities of a variable-rate controller, it is possible to intersect the swath width of the largest farm implement’s boom (or tool bar) along its geo-referenced paths of travel with one or more geo-registered field zones to create experimental units having different shapes and sizes. Defining the experimental units in this manner establishes a site-specific experiment throughout the entire field. Spatial information recovered by Geographic Information System (GIS) processing from these asymmetrical experimental units, when coupled with general linear mixed model methodology, permits the assessment of effects on geo-referenced yield points due to topography, the site-specific and/or traditional farm management practices, and various interactions among them. In this paper, a general methodological approach for analyzing field-sized site-specific experiments is developed and described. An application is demonstrated by analyzing an unreplicated cotton variety trial that included a single site-specific application of a plant growth regulator.

The use of linear mixed model theory for the genetic analysis of repeated measures from clonal tests of forest trees. I. A focus on spatially repeated data

The paper reviews the linear mixed models (LMM) methodology that is suitable for the statistical and genetic analyses of spatially repeated measures collected from clonal progeny tests. For example, we consider a poplar clonal trial where progenies of different families are propagated by cuttings, and only one ramet per clone is planted on each block. Modeling covariance structures following the LMM theory allows improving genetic parameter estimation based on clonal testing. Besides variance components, we also obtained an estimate of the covariance between residuals (within clonal effects in two different blocks). This covariance is due to planting more than one ramet from the same genotype in the same trial, which generates correlated residual effects from different blocks. Its estimation can significantly improve the comparison among clones within a progeny test or between tests in a clonal testing network. Results indicate that the covariance is also a component of the genetic variance estimator and plays a significant role in assessing the variance of specific (micro) environmental effects. A positive covariance implies that ramets show a similar performance in more than one block. Thus, a larger and more positive covariance implies a stronger genetic effect controlling the expression of the trait in the local environment and a smaller variance of specific environmental effects. On the contrary, a negative covariance implies that the performance of individual ramets is affected by strong microenvironmental effects, specific to one or more blocks, which can also directly increase the within-clone variability.

Correlated responses of pre- and postweaning growth and backfat thickness to six generations of selection for ovulation rate or prenatal survival in French Large White pigs

Correlated effects of selection for components of litter size on growth and backfat thickness were estimated using data from 3 pig lines derived from the same base population of Large White. Two lines were selected for 6 generations on either high ovulation rate at puberty (OR) or high prenatal survival corrected for ovulation rate in the first 2 parities (PS). The third line was an unselected control (C). Genetic parameters for individual piglet BW at birth (IWB); at 3 wk of age (IW3W); and at weaning (IWW); ADG from birth to weaning (ADGBW), from weaning to 10 wk of age (ADGPW), and from 25 to 90 kg of BW (ADGT); and age (AGET) and average backfat thickness (ABT) at 90 kg of BW were estimated using REML methodology applied to a multivariate animal model. In addition to fixed effects, the model included the common environment of birth litter, as well as direct and maternal additive genetic effects as random effects. Genetic trends were estimated by computing differences between OR or PS and C lines at each generation using both least squares (LS) and mixed model (MM) methodology. Average genetic trends for direct and maternal effects were computed by regressing line differences on generation number. Estimates of direct and maternal heritabilities were, respectively, 0.10, 0.12, 0.20, 0.24, and 0.41, and 0.17, 0.33, 0.32, 0.41, and 0.21 (SE = 0.03 to 0.04) for IWB, IW3W, IWW, ADGBW, and ADGPW. Genetic correlations between direct and maternal effects were moderately negative for IWB (-0.21 +/- 0.18), but larger for the 4 other traits (-0.59 to -0.74). Maternal effects were nonsignificant and were removed from the final analyses of ADGT, AGET, and ABT. Direct heritability estimates were 0.34, 0.46, and 0.21 (SE = 0.03 to 0.05) for ADGT, AGET, and ABT, respectively. Direct and maternal genetic correlations of OR with performance traits were nonsignificant, with the exception of maternal correlations with IWB (-0.28 +/- 0.13) and ADGPW (0.23 +/- 0.11) and direct correlation with AGET (-0.23 +/- 0.09). Prenatal survival also had low direct but moderate to strong maternal genetic correlations (-0.34 to -0.65) with performance traits. The only significant genetic trends were a negative maternal trend for IBW in the OR line and favorable direct trends for postweaning growth (ADGT and AGET) in both lines. Selection for components of litter size has limited effects on growth and backfat thickness, although it slightly reduces birth weight and improves postweaning growth.

Correlated responses for litter traits to six generations of selection for ovulation rate or prenatal survival in French Large White pigs

Effects of selection for reproductive traits were estimated using data from 3 pig lines derived from the same Large White population base. Two lines were selected for 6 generations on high ovulation rate at puberty (OR line) or high prenatal survival corrected for ovulation rate in the first 2 parities (PS line). The third line was an unselected control line. Genetic parameters for age and BW at puberty (AP and WP); number of piglets born alive, weaned, and nurtured (NBA, NW, and NN, respectively); proportions of stillbirth (PSB) and survival from birth to weaning (PSW); litter and average piglet BW at birth (LWB and AWB), at 21 d (LW21 and AW21), and at weaning (LWW and AWW) were estimated using REML methodology. Heritability estimates were 0.38 +/- 0.03, 0.46 +/- 0.03, 0.16 +/- 0.01, 0.08 +/- 0.01, 0.09 +/- 0.01, 0.04 +/- 0.01, 0.04 +/- 0.02, 0.19 +/- 0.02, 0.10 +/- 0.02, 0.10 +/- 0.02, 0.36 +/- 0.02, 0.27 +/- 0.01, and 0.24 +/- 0.01 for AP, WP, NBA, PSB, NW, NN, PSW, LWB, LW21, LWW, AWB, AW21, and AWW, respectively. The measures of litter size showed strong genetic correlations (r(a) >/= 0.95) and had antagonistic relations with PSB (r(a) = -0.59 to -0.75) and average piglet BW (r(a) = -0.19 to -0.46). They also had strong positive genetic correlations with prenatal survival (r(a) = 0.67 to 0.78) and moderate ones with ovulation rate (r(a) = 0.36 to 0.42). Correlations of litter size with PSW were negative at birth but positive at weaning. The OR and PS lines were negatively related to PSW and average piglet BW. Puberty traits had positive genetic correlations with OR and negative ones with PS. Genetic trends were estimated by computing differences between OR or PS and control lines at each generation using least squares and mixed model methodologies. Average genetic trends were computed by regressing line differences on generation number. Significant (P < 0.05) average genetic trends were obtained in OR and PS lines for AP (respectively, 2.1 +/- 0.9 and 3.2 +/- 1.0 d/generation) and WP (respectively, 2.0 +/- 0.5 and 1.8 +/- 0.5 d/generation) and in the PS line for NBA (0.22 +/- 0.10 piglet/generation). Tendencies (P < 0.10) were also observed for LWB (0.21 +/- 0.12 kg/generation) and AWW (-0.25 +/- 0.14 kg/generation) in the PS line. Selection on components of litter size can be used to improve litter size at birth, but result in undesirable trends for preweaning survival.

Direct responses to six generations of selection for ovulation rate or prenatal survival in Large White pigs

Effects of selection for ovulation rate or prenatal survival were examined using data from 3 pigs lines derived from the same base Large White population. Two lines were selected for 6 generations on high ovulation rate at puberty (OR line) or high prenatal survival corrected for ovulation rate in the first 2 parities (PS line). The third line was an unselected control line. Genetic parameters for ovulation rate on the left, right, and both ovaries at puberty (ORPL, ORPR, and ORP, respectively) and at fertilization (ORFL, ORFR, and ORF, respectively), total number of piglets born (TNB) per litter, prenatal survival (PS = TNB/ORF), and PS corrected for ovulation rate (CPS = PS + 0.018ORF) were estimated using REML methodology. Responses to selection were estimated by computing differences between OR or PS and control lines at each generation using least squares and mixed models methodology. Average genetic trends were computed by regressing line differences on generation number. Realized heritabilities were estimated using standard procedures. Heritability estimates were 0.17, 0.11, 0.34, 0.13, 0.09, 0.33, 0.14, 0.11, and 0.17 (SE = 0.01 to 0.03) for ORPL, ORPR, ORP, ORFL, ORFR, ORF, PS, CPS, and TNB, respectively. Realized heritabilities were 0.37 +/- 0.08 and 0.10 +/- 0.09 for ORP and CPS, respectively. The different measures of ovulation rate had strong genetic correlations (r(g) > 0.7). The ORF had midrange negative genetic correlations with PS and CPS (-0.45 +/- 0.07 and -0.42 +/- 0.08, respectively). The ORP also had an antagonistic genetic relationship with PS (-0.26 +/- 0.07) but was almost independent from CPS (-0.02 +/- 0.11). The TNB was moderately correlated with ORP and ORF (r(g) = 0.41 +/- 0.09 for both traits). Average genetic trends in OR and PS lines were, respectively, 0.49 +/- 0.10 and 0.11 +/- 0.10 for ORP, and 0.43 +/- 0.11 and 0.11 +/- 0.11 for ORF. Responses to selection were slightly superior in the left than in the right ovary. No significant difference was found for PS or CPS in any of the lines. The TNB did not change in the OR line but significantly improved in the PS line (0.24 +/- 0.11 piglets/generation).

Abordagens freqüentista e bayesiana para avaliação genética de bovinos da raça Canchim para características de crescimento

Este trabalho foi realizado com os objetivos de: a) comparar os componentes de (co)variância obtidos por meio dos métodos da Máxima Verossimilhança Restrita (REML) e da inferência bayesiana (IB); b) realizar a avaliação genética do peso à desmama (P240) e aos 18 meses de idade (P550) de bovinos da raça Canchim, padronizados ou não para 240 e 550 dias de idade, respectivamente, utilizando-se a metodologia dos modelos mistos e a obtenção dos componentes de (co)variância por REML ou IB; e c) verificar a semelhança entre os animais selecionados considerando-se a avaliação genética realizada com os pesos reais ou padronizados e por meio de abordagens freqüentista ou bayesiana. Foram obtidos os componentes de (co)variância, herdabilidade e correlação genética para P240 e P550. Os valores genéticos obtidos foram utilizados para simular um processo de seleção em que 10% dos touros e 50% das vacas com os maiores valores genéticos aditivos diretos teriam chance de reproduzir. Os componentes de (co)variância e os parâmetros genéticos estimados por REML, na maioria dos casos, foram inferiores às médias a posteriori obtidas por IB. Ocorreram diferenças quanto aos animais selecionados, provavelmente em decorrência das diferenças entre os componentes de (co)variância e dos parâmetros genéticos obtidos. Adotando-se a IB, a inclusão da idade do animal no momento da pesagem como covariável no modelo estatístico não provocou grande alteração dos touros e vacas selecionados.

Modelling Concurrency of Events in On-Line Auctions via Spatiotemporal Semiparametric Models

Summary We introduce a semiparametric approach for modelling the effect of concurrent events on an outcome of interest. Concurrency manifests itself as temporal and spatial dependences. By temporal dependence we mean the effect of an event in the past. Modelling this effect is challenging since events arrive at irregularly spaced time intervals. For the spatial part we use an abstract notion of ‘feature space’ to conceptualize distances among a set of item features. We motivate our model in the context of on-line auctions by modelling the effect of concurrent auctions on an auction's price. Our concurrency model consists of three components: a transaction-related component that accounts for auction design and bidding competition, a spatial component that takes into account similarity between item features and a temporal component that accounts for recently closed auctions. To construct each of these we borrow ideas from spatial and mixed model methodology. The power of this model is illustrated on a large and diverse set of laptop auctions on eBay.com. We show that our model results in superior predictive performance compared with a set of competitor models. The model also allows for new insight into the factors that drive price in on-line auctions and their relationship to bidding competition, auction design, product variety and temporal learning effects.

Esperanças matemáticas dos quadrados médios: uma análise essencial

Este trabalho teve como objetivo avaliar e identificar qual o tipo de soma de quadrados mais apropriada para testar hipóteses de interesse, assim como discutir alternativas mais adequadas para a solução de inconvenientes expressos por meio da análise da esperança matemática dos quadrados médios utilizados em modelos lineares mistos. A análise das esperanças matemáticas dos quadrados médios pode ser uma ferramenta de grande importância nas inferências a partir de dados experimentais, tanto incompletos (casela vazia) quanto não-balanceados. Desta forma, foram utilizados quatro exemplos, cada qual com sua peculiaridade em função do experimento ser completo ou incompleto com dados balanceados ou não-balanceados e na presença de casela vazia. O pacote estatístico SAS, versão Learning Edition, foi empregado para analisar os experimentos. O resultado da análise das esperanças matemáticas dos quadrados médios indicou que a soma de quadrados do tipo I somente apresentou condições de ser utilizada em presença de dados completamente balanceados. De modo contrário, os resultados apontam que a soma de quadrados tipo III é a soma de quadrados mais apropriada no caso de dados não-balanceados. As somas de quadrados tipo II e IV são as mais importantes no caso de caselas vazias, fato que corrobora a necessidade de avaliar sempre as esperanças matemáticas dos quadrados médios.

A Mixed Model Approach for Geoadditive Hazard Regression

Abstract. Mixed model based approaches for semiparametric regression have gained much interest in recent years, both in theory and application. They provide a unified and modular framework for penalized likelihood and closely related empirical Bayes inference. In this article, we develop mixed model methodology for a broad class of Cox‐type hazard regression models where the usual linear predictor is generalized to a geoadditive predictor incorporating non‐parametric terms for the (log‐)baseline hazard rate, time‐varying coefficients and non‐linear effects of continuous covariates, a spatial component, and additional cluster‐specific frailties. Non‐linear and time‐varying effects are modelled through penalized splines, while spatial components are treated as correlated random effects following either a Markov random field or a stationary Gaussian random field prior. Generalizing existing mixed model methodology, inference is derived using penalized likelihood for regression coefficients and (approximate) marginal likelihood for smoothing parameters. In a simulation we study the performance of the proposed method, in particular comparing it with its fully Bayesian counterpart using Markov chain Monte Carlo methodology, and complement the results by some asymptotic considerations. As an application, we analyse leukaemia survival data from northwest England.

Mapping Quantitative Trait Loci for Longitudinal Traits in Line Crosses

Quantitative traits whose phenotypic values change over time are called longitudinal traits. Genetic analyses of longitudinal traits can be conducted using any of the following approaches: (1) treating the phenotypic values at different time points as repeated measurements of the same trait and analyzing the trait under the repeated measurements framework, (2) treating the phenotypes measured from different time points as different traits and analyzing the traits jointly on the basis of the theory of multivariate analysis, and (3) fitting a growth curve to the phenotypic values across time points and analyzing the fitted parameters of the growth trajectory under the theory of multivariate analysis. The third approach has been used in QTL mapping for longitudinal traits by fitting the data to a logistic growth trajectory. This approach applies only to the particular S-shaped growth process. In practice, a longitudinal trait may show a trajectory of any shape. We demonstrate that one can describe a longitudinal trait with orthogonal polynomials, which are sufficiently general for fitting any shaped curve. We develop a mixed-model methodology for QTL mapping of longitudinal traits and a maximum-likelihood method for parameter estimation and statistical tests. The expectation-maximization (EM) algorithm is applied to search for the maximum-likelihood estimates of parameters. The method is verified with simulated data and demonstrated with experimental data from a pseudobackcross family of Populus (poplar) trees.

Ventricular Diastolic Stiffness Predicts Perioperative Morbidity and Duration of Pleural Effusions After the Fontan Operation

We validated the clinical relevance of ventricular stiffness by examining surgical morbidity in children with univentricular hearts undergoing Fontan operation. We hypothesized that ventricular stiffness affects Fontan morbidity, particularly duration of pleural effusions. Sixteen children with right ventricular (RV) (n =11) or left ventricular (LV) (n =5) dominance were studied intraoperatively at a median age of 3.3 years (1.8 to 5.1). Transesophageal long-axis echocardiograms and ventricular pressure by micromanometer provided end-diastolic pressure (P) area (A) relations during initiation and conclusion of cardiopulmonary bypass. Curve fitting to the equation P=alphae(betaA) defined the ventricular stiffness constant, beta. Changes in beta and clinical correlations were examined. Ventricular stiffness increased after bypass in patients with complete pre-bypass and post-bypass data (n =11, P=0.023, mixed models methodology). Pre-bypass beta correlated well with duration of chest tube (CT) drainage (r=0.90, n =16), net perioperative fluid balance (r=0.71, n=14), and length of stay (LOS) (r=0.81, n =16). CT duration and LOS also correlated significantly with post-bypass beta (r=0.77 for both, n=11), but insignificantly with preoperative catheterization pressures. Intraoperative beta predicts duration of CT drainage, net perioperative fluid balance, and LOS after the Fontan operation. These observations could improve risk stratification and clinical management of children at high-risk undergoing the Fontan operation.

Modeling Genotype × Environment Interaction Using Additive Genetic Covariances of Relatives for Predicting Breeding Values of Wheat Genotypes

In plant breeding, multienvironment trials (MET) may include sets of related genetic strains. In self‐pollinated species the covariance matrix of the breeding values of these genetic strains is equal to the additive genetic covariance among them. This can be expressed as an additive relationship matrix, A, multiplied by the additive genetic variance. Using Mixed Model Methodology, the genetic covariance matrix can be estimated and Best Linear Unbiased Predictors (BLUPs) of the breeding values obtained. The effectiveness of exploiting relationships among strains tested in METs and usefulness of these BLUPs of breeding values for simultaneously modeling the main effects of genotypes and genotype × environment interaction (GE) have not been thoroughly studied. In this study, we obtained BLUPs of breeding values using genetic variance–covariance structures constructed as the Kroneker product (direct product) of a structured matrix of genetic variances and covariances for sites and a matrix of genetic relationships between strains, A. Results are compared with those from traditional fixed effects and random effects models for studying GE ignoring genetic relationships. A CIMMYT international wheat trial was used for illustration. Results showed that direct products of factor analytic structures with matrix A efficiently model the main effects of genotypes and GE. These models showed the lowest standard error of the BLUPs [SE(BLUP)] of breeding values. Genotypes that were related to other genotypes had small SE(BLUP). Related genotypes can clearly be visualized in biplots.

Seleção para peso pós-desmama em um rebanho Gir. 1. Resposta direta e correlacionada em 21 anos de seleção

A seleção para peso pós-desmama tem sido aplicada em um rebanho Gir (80 vacas, seis touros) desde 1976, e um rebanho Nelore Controle, NeC (60 vacas, quatro touros), tem sido mantido para avaliação da mudança ambiental. O critério de seleção dos machos foi o peso aos 378 dias (P378), obtido em prova de ganho de peso, e o das fêmeas, o peso aos 550 dias (P550) em recria a pasto. A média dos diferenciais de seleção efetivos dos pais foi, respectivamente, 19,5 e 1,8 kg para Gir e NeC, correspondendo a 0,72 e 0,07 unidades de desvio-padrão (u.d.p.). A média dos diferenciais de seleção efetivos acumulados dos pais, na progênie de 2001, foi 83,7 e 7,2 kg para Gir e NeC, correspondendo a 3,17 e 0,29 u.d.p. O intervalo de geração médio no rebanho Gir foi 5,65 anos e no NeC 5,0 anos e o coeficiente de geração médio individual na progênie de 2001 atingiu 4,25 (Gir) e 4,65 (NeC) gerações de seleção. As estimativas de tendência genética no rebanho Gir, obtidas por quadrados mínimos, foram PN = 0,16; P210 = 0,81 kg/ano; GPRE = 3,02 g/dia/ano; P378 = 2,88; P550 = 2,80 kg/ano; G112 = 11,2 g/dia/ano, AM = 0,003 e AF = 0,004 m/ano. Com base no modelo misto, as estimativas para essas características foram menores (0,03 e 0,28 kg/ano; 1,15 g/dia/ano; 1,45 e 1,07 kg/ano; 3,73 g/dia/ano; 0,003 e 0,002 m/ano). As estimativas de herdabilidade realizadas para P378 e P550 foram 0,69 e 0,64, respectivamente. Os resultados indicaram que a seleção para peso pós-desmama, com base na informação de desempenho individual, promoveu respostas genéticas significativas tanto nas características sob seleção direta quanto nas respostas correlacionadas em características de crescimento da fase pré e pós-desmama no rebanho Gir.