Mixed Model Methodology Research Articles

Estimates of genetic and phenotypic correlations for ewe productivity traits of three breeds under 8-month breeding cycles and artificial rearing of lambs

Paternal half-sib estimates for genetic and phenotypic correlations for ewe productivity of three synthetic sheep breeds (Canadian, Outaouais and Rideau Arcotts) were based on 11 106 ewe records from 3528 ewes and their 814 sires collected over 8 years (1983–1990) and 24 lambings in 8-month breeding cycles. Two separate flocks of ewes, housed indoors year-round in a controlled environment, were mated to lamb alternately at 4-month intervals, at the end of a period of short days, in February, June and October. Artificial lighting regimens and exogenous hormones were used to synchronize estrus. All lambs were raised on milk replacer diets from birth to weaning at 21 days of age. Genetic and phenotypic correlations based on mixed model methodology were estimated by breed and involved ewe weight at breeding, reproduction (prolificacy, multiple birth and fecundity) and total lamb weights (at birth, 21 and 91 days of age). Ewe weight at breeding was found to have neither genetic nor phenotypic relationships with the reproductive traits. Prolificacy was associated, both genetically (0.50–0.80) and phenotypically (0.77–0.79), with total lamb weight at birth for the three breeds. However, genetic correlations between prolificacy and total lamb weights at 21 days (0.85) and 91 days (0.94), important ( P<0.05) in the Outaouais breed, failed to demonstrate significance in the Canadian and Rideau breeds. Estimates of genetic correlations between fecundity and total lamb weights (0.88–0.96 at birth, 0.67–1.06 at 21 days, and 0.72–1.06 at 91 days) were always significant for the three breeds, suggesting that selection for fecundity would result in correlated genetic improvement of ewe productivity. Within total lamb weight, the genetic and phenotypic relationships between weights at birth and either 21 days or 91 days were similar. Corresponding estimates between 21 and 91 days of age tended to be larger. The magnitude of the phenotypic and environmental correlations were alike. In general, genetic and phenotypic correlations based on least squares analysis, maximum likelihood and restricted maximum likelihood procedures were similar, however, there may be some cause for concern in the absence of associated standard errors. The results provide estimates of genetic and phenotypic correlations for ewe productivity traits in the three Arcott sheep breeds developed in Canada under conditions of an 8-month breeding cycle when lambs were fed with a milk replacer diet and raised artificially in a controlled environment.

Stocking density and tank size in the design of breed improvement programs for body size of tilapia

Various aspects of cultural conditions were investigated in an effort to design an efficient selective breeding scheme for tilapia. High density culture has the potential of reducing confounding environmental effects associated with fish behavior. Body size of fry reared through 56 days was not affected by any of the stocking densities used (10–200 fry/l) with uniform water inflow. However, when water inflow was proportional to fish density, high density culture of fingerlings (18 fish/l) did appear to promote growth and result in improved normality of the distribution of body size relative to fish stocked at lower (2 fish/l) or higher (200 fish/l) densities. Although this could be interpreted as a consequence of diminished social interactions, a definitive conclusion could not be made in the absence of direct observational data on individual fish. Mixed-model methodology was used to estimate genetic parameters for body size. Estimated heritability for body weight was about 0.25 for all ages from 56 to 126 days. Phenotypic and genetic correlations among body weights at various ages were positive and near unity.

Heritability of scrotal circumference adjusted and unadjusted for body weight in nellore bulls, using univariate and bivariate animal models

Mixed model methodology under univariate and bivariate animal models was used to estimate genetic parameters of scrotal circumference (SC) unadjusted (h U 2) and adjusted (h A 2) for body weight, from field data on 875 Nellore males sired by 80 bulls. The models included random effect of animal, the fixed effects of age of dam, year of birth, and month and year of measurement, and the covariate age of the animal, in addition, one of the models also included the covariate body weight. Using univariate models, SC estimates of h U 2 were 0.18, 0.65, 0.77 and 0.60, and of h A 2 they were 0.15, 0.60, 0.71 and 0.50, at 9, 12, 18 and 24 mo of age, respectively. Bivariate estimates of heritability ranged from 0.18 to 0.77 for h U 2, and from 0.13 to 0.70 for h A 2. The results suggest that it would be better to consider unadjusted SC rather than adjusted SC in selection programs. Bivariate body weight estimates of heritability ranged from 0.15 to 0.45. Genetic correlations between SC and body weight ranged from 0.58 to 0.71 when body weight was not included in the model, and from 0.33 to 0.64 when body weight was included. These results suggest that simultaneous selection for body weight and SC is possible in Nellore cattle.

Application of mixed model methodology in breeding strategies for laying fowl

The main objective of this review is to present the advantages and disadvantages of applying the best linear unbiased prediction (BLUP) procedure with an animal model to the evaluation of laying fowl within a breeding programme. Following an outline of the traditional selection index-based procedure there are descriptions of mixed model methodology including modelling and data transformation, genetic parameter estimation and breeding value prediction. Some computer programs for carrying out BLUP animal model algorithms are described and the results of empirical and simulation comparisons involving different livestock species are given. New possibilities arising from the application of the animal model to breeding strategy are discussed. Poultry breeding evaluations using the traditional selection index do not always result in the same conclusions as arise when using BLUP. However it seems likely that, as has happened with other livestock species, the BLUP animal model method will be increasingly applied in breeding programmes to improve egg production.

Sparse matrix tools for Gaussian models on lattices

Sparse matrix methods are considered for Gaussian models on lattices. The approach depends on the applicability of mixed model methodology, which is feasible when observational errors are present. Moreover, lattice models with nearest-neighbor interactions lead to a spatial variance-covariance matrix having a sparse inverse, and this lends itself to sparse-matrix manipulation. Modern technology involving the Cholesky decomposition and backward differentiation are described for prediction, parameter estimation, cross validation, and conditional simulation of spatial processes.

Heritability and repeatability estimates for ewe productivity traits of three breeds under 8-month breeding cycles and artificial rearing of lambs

Heritability and repeatability were estimated for ewe productivity of three synthetic sheep breeds (Canadian, Outaouais and Rideau Arcotts) using 11 106 ewe records from 3528 ewes and their 814 sires collected over 8 years (1983–1990) and 24 lambings. Two flocks of ewes were housed indoors year-round in a controlled environment. Ewes were mated to lamb in February, June and October at the end of a period of short days, using artificial lighting regimens and exogenous hormones to synchronize estrus to lamb alternately at 4-month intervals in an 8 month breeding cycle. All lambs were raised on milk replacer diets from birth to weaning at 21 days of age. Traits examined included fertility, prolificacy, multiple birth, fecundity, ewe weight at breeding, lamb mortality at 21 and 91 days of age, and total lamb weight per ewe lambing (and per 100 kg ewe weight) at birth, 21 and 91 days. Paternal half-sib estimates of heritability based on least squares mixed model methodology ranged from −0.04 to 0.20 for the reproductive traits, 0.05 to 0.17 for lamb mortality, 0.03 to 0.21 for ewe weight at breeding and 0.01 to 0.22 for total lamb weight. Corresponding values for repeatability estimates ranged from 0.05 to 0.21, 0.21 to 0.28, 0.06 to 0.13 and 0.03 to 0.19, respectively. The heritability and repeatability estimates did not vary among breeds ( P > 0.05) and were significantly different from zero for all traits except multiple birth. There was a tendency for estimates based on restricted maximum likelihood and maximum likelihood procedures to be lower than those based on the least squares procedure for fertility, multiple birth and lamb mortality ( P > 0.05). The study demonstrates that there is significant potential to genetically increase reproductive rate and lamb weight and for reducing lamb mortality in Canadian, Outaouais and Rideau sheep under conditions of an 8 month breeding cycle when lambs are fed a milk replacer diet and raised artificially in a controlled environment.

Teacher and classroom context effects on student achievement: Implications for teacher evaluation.

The Tennessee Value-Added Assessment System (TVAAS) has been designed to use statistical mixed-model methodologies to conduct multivariate, longitudinal analyses of student achievement to make estimates of school, class size, teacher, and other effects. This study examined the relative magnitude of teacher effects on student achievement while simultaneously considering the influences of intraclassroom heterogeneity, student achievement level, and class size on academic growth. The results show that teacher effects are dominant factors affecting student academic gain and that the classroom context variables of heterogeneity among students and class sizes have relatively little influence on academic gain. Thus, a major conclusion is that teachers make a difference. Implications of the findings for teacher evaluation and future research are discussed.

A generalized Michaelis-Menten response surface.

We describe an application of recently developed generalized Michaelis-Menten response surface and non-linear mixed model methodologies to model glucose utilization in foetal sheep. More specifically, we model the response surface of glucose utilization rate in the foetal sheep as a function of glucose and insulin concentrations using a three-dimensional analogue of the Michaelis-Menten pharmacokinetic model. To account for multiple measurements per sheep, we apply the non-linear mixed effects model proposed by Lindstrom and Bates using the EM algorithm computational scheme presented by Hirst et al.

Simulation study of the glmm method applied to the analysis of clustered survival data

A simulation study is carried out for the analysis of clustered survival data based on the generalised linear mixed model (GLMM) methodology. Data are simulated for a proportional hazards model containing random components corresponding to a cluster effect and a subject effect. The performance of approximate maximum likelihood (ML) and residual maximum likelihood (REML) estimation of the regression parameter and variance components are evaluated. For a range of true parameter values,average biases of estimators, standard error of average bias and standard error of estimates over simulations are reported. A discussion of the use of random effects predictions is also given.

Genetic trends over 20 years of selection in the three synthetic Arcotts, Suffolk and Finnish Landrace sheep breeds. 1. Early growth traits

Mixed model methodology based on a multi-trait animal model was used to estimate genetic trends for lamb weights in three synthetic Arcott (Canadian Outaouais and Rideau) breeds and two random bred control (Suffolk and Finnish Landrace) breeds. Selection in the Canadian Arcott was primarily on lamb weight at 91 d with a lesser emphasis on litter size. Outaouais and Rideau Arcotts were selected primarily for litter size with a lesser emphasis on lamb weight at 91 d. All lambs were housed indoors year-round in a controlled environment under conditions of artificial rearing and 8-mo breeding cycles. Genetic improvement over 20 yr based on predicted breeding values for lamb weights at birth, 21, 70 and 91 d of age were 0.3, 0.9, 2.1 and 2.8 kg, respectively in the Canadian, 0.2, 0.6, 1.2 and 1.5 kg in the Outaouais, and 0.3, 0.5, 0.9 and 1.4 kg in the Rideau breeds. Corresponding values for the unselected Suffolk were 0.3, 0.4, 0.4 and 0.5 kg, respectively, and for the Finnsheep were 0.2, 0.2, 0.3 and 0.5 kg. Alternative methods of estimating genetic trends using control populations were based on deviation of the predicted mean year-season phenotype in the selected breeds from that in the corresponding random bred control breeds (Suffolk for Canadian, and Finnsheep for Outaouais and Rideau) and also as a deviation from the year-season estimate in the corresponding random bred control breeds. Genetic trends estimated from the alternative methods, which took into account the contribution of drift, were larger than those based on predicted breeding values. These values were reflected in the realized heritabilities estimated from regression of cumulative selection differential on the genetic trends. In the Arcott breeds, annual genetic trend for lamb weights varied according to breed and the estimation procedure which amounted to 0.25–0.81% of the mean. It is evident selection offers one of the most practical means for rapid and permanent improvement of early lamb growth in multi-breed synthetic populations under conditions of artificial rearing, 8-mo breeding cycle and a controlled environment. Key words: Sheep breeds, genetic trends, multi-trait animal model and growth traits

Mixed Model Analysis of DNA Sequence Evolution

Nucleotides in a DNA sequence may be changing at different rates, because they are located in different structural and functional regions of the gene, and are thus subject to different mutational pressures or selective restrictions. Knowledge of substitution rates at specific sites is important for understanding the forces and mechanisms that have shaped the evolution of the DNA sequences. The gamma distribution has previously been proposed to model such rate variation among nucleotide sites. Based on mixed model methodology we present in this paper a method for predicting substitution rates at nucleotide sites by using homologous DNA sequences. The predictor is unbiased and "best" in the sense that it minimizes the mean squared error and maximizes the correlation between the predictor and the true value. It is also quite robust to errors in estimates of parameters in the model. A numerical example is given, with guidelines for the practical use of the approach. The most influential factor affecting the accuracy of prediction is the number of sequences; to get a correlation of over .7 between the predictor and the true value, about six to seven sequences are needed, depending on the overall similarity of the sequences.

Performance of hair breeds and prolific wool breeds of sheep in southern Illinois: lamb production of F1 adult ewes

Two- and three-year-old F1 crossbred ewes produced from Suffolk and Targhee dams and sires of the prolific wool breeds of Finnsheep, Combo-6, and Booroola Merino and the hair breeds of St. Croix and Barbados were pasture-mated to Dorset rams over a 4-yr period at the Dixon Springs Agricultural Center in Southern Illinois. Booroola Merino sires were homozygous for the FecB allele for high number of ovulations. The effects of age of ewe, sex of lamb (where appropriate), breed of dam, and breed of sire on condition score, breeding weight, ovulation rate, breeding to lambing interval, fertility, prolificacy, lamb survival, weaning weight, and ewe productivity were estimated. All traits were analyzed using mixed-model methodology. Crossbred ewes from Suffolk dams had a higher (P < .01) condition score, heavier (P < .01) breeding weight, higher (P < .01) ovulation rate, shorter (P < .01) breeding to lambing interval, greater (P < .01) prolificacy, heavier (P < .01) lambs at weaning, and greater (P < .05) ewe productivity than crossbred ewes from Targhee dams. Ewes from hair-breed sires had lighter (P < .01) breeding weight, lower (P < .05) ovulation rate, shorter (P < .05) breeding to lambing interval, higher (P < .05) fertility, higher (P < .05) lamb survival, lower (P < .10) lamb weaning weight, and higher (P < .05) ewe productivity than ewes sired by the prolific wool-breed rams. The breed of sire x breed of dam interaction was a significant source of variation for breeding weight, fertility, lamb survival, and ewe productivity.(ABSTRACT TRUNCATED AT 250 WORDS)

The tennessee value-added assessment system (TVAAS): Mixed-model methodology in educational assessment

The tennessee value-added assessment system (TVAAS): Mixed-model methodology in educational assessment

Additive and non-additive genetic effects of growth-curve parameters of Holstein, Ayrshire and crossbred females

Crossbreeding parameters (line additive, dominance and additive × additive epistatic effects for individual and for maternal performance) on growth traits of females from a crossbreeding experiment between Holstein-based (HS) and Ayrshire-based (AS) lines were estimated by individual animal models, incorporating all known additive genetic relationships among animals, through restricted maximum likelihood and mixed-model methodologies. The growth traits [asymptotic weight (A), rate parameter (k), inflection parameter (m), average lifetime absolute growth rate (AGR), average lifetime absolute maturing rate (AMR) and average lifetime relative growth rate (RGR)] were estimated by fitting the Richards function to the observed growth curve of 3076 individuals. The statistical model included the random effect of the animal breeding value and the fixed effects of genetic group and station–year–season of birth. Results indicated that the HS exceeded (P &lt; 0.001) the AS in additive effects for individual performance for both A and AGR. The HS exceeded (P &lt; 0.05) the AS in additive effects for maternal performance for A. Individual heterosis was positive for A (P &lt; 0.001) and for AGR (P &lt; 0.01). Maternal heterosis was negative for A (P &lt; 0.05) and positive for AMR (P &lt; 0.05). Total heterosis (TH) had positive effects on both AGR and AMR (P &lt; 0.05). Heterosis retained in advanced crossbred generations was not significant (P &gt; 0.05) for any of the studied traits. The results suggest that crossbreeding designed to exploit TH can alter the shape of the growth curve of dairy cattle. Key words: Crossbreeding, dairy cattle, growth curve

Solutions to a System of Equations Involving a First-Order Autoregressive Process

A first-order autoregressive process is proposed for modeling certain random effects in instances for which a more general model would involve too many (co)variance components. This process requires only two parameters, and its inverse—needed for its inclusion in mixed model methodology—is tridiagonal and easy to obtain. The (co)variance matrix for effects that follow a first-order autoregressive structure is introduced, and the rules for obtaining its inverse are derived and outlined in an algorithm (an example is also given). Incorporation of such a structure into the mixed model equations is also discussed, and an iterative procedure for obtaining solutions to this potentially large system of equations is outlined.

Estimation of genetic trends for weaning weight and teat number in Iberian pigs using mixed model methodology

Mixed model methodology was used to estimate genetic and environmental trends over 23 years for the traits weaning weight and number of teats in three populations of Iberian pigs. Full genealogy was available. REML estimates of variance components were used to obtain the predicted breeding values. Annual rate of genetic trend was estimated as the regression of the yearly mean predicted breeding value on year. Environmental trend was directly obtained from the estimate of the effect of batch. Results obtained for weaning weight were consistent with positive selection differentials and the low selection intensity applied. Estimated environmental trends reflected a change to worse management conditions that occurred in the herd. However, the methodology seems to have failed to separate genetic from environmental effects correctly for teat number. The reason may be in the special characteristics of the trait, which is not normally distributed, and for which the theoretical genetic model may differ from that assumed in mixed model analysis.

Precision and information in linear models of genetic evaluation

Some criteria for measuring the overall precision of a genetic evaluation using linear mixed-model methodology are presented. They are derived via an extension of the coefficient of determination to linear combinations of estimates and via the use of the Kullback information. A parallel is drawn between inestimability of fixed-effects contrasts and the zero coefficient of determination for contrasts of random effects. The procedure is illustrated with 2 minor hypothetical examples of genetic evaluation based on an animal model and on a sire model.

Marginal likelihood and Bayesian approaches to the analysis of heterogeneous residual variances in mixed linear Gaussian models

A statistical procedure for assessing the extend and sources of heterogeneity of residual variances in Gaussian mixed linear models is presented. The method assumes a structural linear model for logvariances. Hypothesis testing about the effects of factors potentially involved in the variation of dispersion parameters is based on a likelihood ratio test using marginal likelihood functions where location parameters are integrated out. This method allows us to identify meaningful sources of variation in heterogeneous variances and to assess how these factors act (additivity, interactions). Dispersion parameters are estimated either by maximum marginal likelihood or by the mode of their posterior distribution using a log inverted gamma distribution as prior. Algorithms so obtained have remarkable similarities with the mixed model methodology of Henderson for location parameters. The analysis of a small example is used to illustrate the theory

Selection for postweaning growth in inbred Hereford cattle: the Fort Keogh, Montana line 1 example.

Demographic characteristics and genetic trends in birth weight and pre- and postweaning ADG were examined in a population of Hereford cattle (Line 1). Line 1 was founded largely from two paternal half-sib sires and has been selected for postweaning growth. There were pedigree records on 951 members of the base population that predated 1935, when data collection began. Numbers of records analyzed using mixed-model methodology were 4,716 birth weight, 4,427 preweaning ADG, and 3,579 postweaning ADG. Birth weight and preweaning ADG were considered to have direct and maternal genetic components. Inbreeding accumulated rapidly from 1935 to 1960 and more slowly (.22%/yr) thereafter. Any reduction in additive genetic variance due to inbreeding and selection may have been offset by a concurrent reduction in generation interval that was observed as time progressed. Expected selection differential for 365-d weight, averaged over sexes, was 31.2 kg per generation. For birth weight, annual genetic trends in direct and maternal effects were 42 +/- 3 g and 15 +/- 3 g, respectively. Annual direct and maternal genetic trends for preweaning ADG were .70 +/- .06 g/d and .63 +/- .06 g/d, respectively. Direct response in postweaning ADG was linear and equal to 5.3 +/- .6 g.d-1.yr-1. As a result, estimated breeding values of birth weight, 200-d weight, and 365-d weight increased by 3.2 kg, 14.5 kg, and 62.4 kg, respectively, from 1935 to 1989. Selection within Line 1 was effective in increasing genetic potential for growth over 13 generations. No selection plateau was observed in any of the traits examined.

Selection response of growth rate in rabbits for meat production

Genetic and environmental trends in 2 lines of rabbit (B and R) selected on individual weight gain (WG) from weaning (4 wk) to slaughter (11 wk) were estimated using mixed model methodology. Line B was derived from the California breed and line R was a synthetic of stock of different origin. The data were collected from a single herd and comprised 7 718 individuals in line B and 9 391 in line R, the lines having 12 and 9 generations of selection respectively. Realized responses in the 2 lines were 2.7% and 2.2% of the initial mean per year respectively and showed that selection on WG was effective but was less than expected. Selection on slaughter weight (SW) and effects of selection on other economic traits are discussed. It is concluded that selection on either WG or SW is a simple method for improving growth rate in rabbit sire line stocks.