REML Estimates Research Articles

Pivotal estimation with applications for the intraclass correlation coefficient in the balanced one-way random effects model

In this paper pivotal quantities are used to obtain estimators of parameters. As an application of this technique, a family of estimators is presented for the intraclass correlation coefficient in the balanced one-way random effects model. The family is derived by equating a pivotal quantity to different values of the pivoting distribution, and includes the familiar REML and ML estimators. It is shown that under certain conditions, in terms of mean-squared error, most members of the family of estimators are admissible within the family. The authors provide guidance concerning the choice of an individual member of the family for estimation purposes and indicate how the method can be extended to unbalanced designs.

EM-REML estimation of covariance parameters in Gaussian mixed models for longitudinal data analysis.

This paper presents procedures for implementing the EM algorithm to compute REML estimates of variance covariance components in Gaussian mixed models for longitudinal data analysis. The class of models considered includes random coefficient factors, stationary time processes and measurement errors. The EM algorithm allows separation of the computations pertaining to parameters involved in the random coefficient factors from those pertaining to the time processes and errors. The procedures are illustrated with Pothoff and Roy's data example on growth measurements taken on 11 girls and 16 boys at four ages. Several variants and extensions are discussed.

Reproducibility of the six-minute walking test in chronic heart failure patients.

The six-minute walking test (WT) is used in trials and clinical practice as an easy tool to evaluate the functional capacity of chronic heart failure (CHF) patients. As WT measurements are highly variable both between and within individuals, this study aims at assessing the contribution of the different sources of variation and estimating the reproducibility of the test. A statistical model describing WT measurements as a function of fixed and random effects is proposed and its parameters estimated. We considered 202 stable CHF patients who performed two baseline WTs separated by a 30 minute rest; 49 of them repeated the two tests 3 months later (follow-up control). They had no changes in therapy or major clinical events. Another 31 subjects performed two baseline tests separated by 24 hours. Collected data were analysed using a mixed model methodology. There was no significant difference between measurements taken 30 minutes and 24 hours apart (p = 0.99). A trend effect of 17 (1.4) m (mean (SE)) was consistently found between duplicate tests (p < 0.001). REML estimates of variance components were: 5189 (674) for subject differences in the error-free value; 1280 (304) for subject differences in spontaneous clinical evolution between baseline and follow-up control, and 266 (23) for the within-subject error. Hence, the standard error of measurement was 16.3 m, namely 4 per cent of the average WT performance (403 m) in this sample. The intraclass correlation coefficient was 0.96. We conclude that WT measurements are characterized by good intrasubject reproducibility and excellent reliability. When follow-up studies > or = 3 months are performed, unpredictable changes in individual walking performance due to spontaneous clinical evolution are to be expected. Their clinical significance, however, is not known.

The PX-EM algorithm for fast stable fitting of Henderson's mixed model.

This paper presents procedures for implementing the PX-EM algorithm of Liu, Rubin and Wu to compute REML estimates of variance covariance components in Henderson's linear mixed models. The class of models considered encompasses several correlated random factors having the same vector length e.g., as in random regression models for longitudinal data analysis and in sire-maternal grandsire models for genetic evaluation. Numerical examples are presented to illustrate the procedures. Much better results in terms of convergence characteristics (number of iterations and time required for convergence) are obtained for PX-EM relative to the basic EM algorithm in the random regression.

Parameter estimates for direct and maternal genetic effects on yearling, eighteen-month, and slaughter weights of Korean native cattle.

Data collected by the National Livestock Research Institute of the Rural Development Administration of Korea were used to estimate genetic parameters for yearling (YWT, n = 5,848), 18-mo (W18, n = 4,585), and slaughter (SWT, n = 2,279) weights for Korean Native cattle. Nine animal models were used to obtain REML estimates of genetic parameters: DP-2 included genetic, uncorrelated dam, and residual random effects; DQ-2 included genetic, sire x region x year-season interaction, and residual random effects; DPQ-2 was based on DQ-2 but included both interaction and dam effects; DMP-2 was based on DP-2 but with dam effect partitioned to include maternal genetic and permanent environmental effects; and DMPQ-2 was based on DMP-2 but also included sire interaction effects. Those five models included two fixed factors: region x year-season and age of dam x sex effects. Models DP-3, DQ-3, DPQ-3, and DMPQ-3 were based on DP-2, DQ-2, DPQ-2, and DMPQ-2 but included as a third fixed factor whether or not identification of the sire was known. Estimates of heritability with DMPQ-3 for YWT, with DPQ-3 for W18 and SWT when analyzed with single-trait analyses were .14, .11, and .17, respectively, and were nearly the same with bivariate analyses. Estimate of maternal heritability for YWT from single-trait analysis was .04, with estimates for other traits near zero. For bivariate analyses, the estimate for YWT was .01. With single trait analysis, estimate of the direct-maternal genetic correlation for YWT was negative (-.81). Estimates of direct genetic correlations between YWT and W18, YWT and SWT, and W18 and SWT were .99, 1.00, and .97, respectively. Estimates of environmental correlations varied from .60 to .81; the largest was between W18 and SWT. Including a fixed factor for whether sire identification was missing or not missing reduced the estimate of heritability for slaughter weight. The results suggest that the sire x region x year-season interaction is important for yearling weight and may be needed in a model for slaughter weight. Maternal effects may be of slight importance for yearling weight but of no importance for W18 and SWT. Models for national cattle evaluations for Korean Native cattle for YWT should be considered that include maternal genetic and permanent environmental as well as sire x region x year-season interaction effects, but those effects seem not to be needed for models for W18 and SWT. Not much reranking of sires occurred when ranked was based on the different models for W18 and SWT.

Response to 13 generations of selection for increased 8-week body weight in lines of mice carrying a sheep growth hormone-based transgene.

The purpose of this study was to evaluate selection in lines of transgenic mice. Two replicates of lines that either carried or did not carry the sheep metallothionein-1a sheep growth hormone transgene (oMt1a-oGH) were established. The host lines had been previously selected for rapid growth or selected randomly. Within-litter selection for increased 8-wk body weight was carried out for 13 generations. The frequency of oMt1a-oGH was monitored in all generations in the transgenic lines, but no genotypic information regarding the transgene was used as an aid to selection. The oMt1a-oGH was activated from weaning, at 3 wk, until 8 wk of age by adding ZnSO4 to the drinking water. Zinc stimulation of the transgene was not done during mating, gestation, or lactation. Data on body weights and weight gains were analyzed with a conventional mixed model and with an animal model. Genetic progress was achieved in all lines subjected to directional selection. In the control background, response to selection for 8-wk body weight was larger in the nontransgenic lines than in the transgenic lines, whereas no difference was found in the selected background. The frequency of the transgene was increased from the initial .5 to .62 in the randomly selected background but decreased to .04 in lines from a selected background. The REML estimates of variance components and genetic gain estimates varied greatly between the two methods. In general, there was better agreement between the realized heritability estimates and the heritability estimates obtained from the conventional mixed model analysis than between realized heritability estimates and results obtained using the animal model. Favorable correlated responses were obtained for 3- and 6-wk body weights and on 3- to 6- and 6- to 8-wk weight gains. Correlated responses to selection were larger in the selected than in the nonselected background but were not affected by the presence of the transgene. Results suggest that constructs similar to the oMt1a-oGH, which allow tight regulation, may be successfully incorporated into commercial livestock and should have larger effects in populations that have not been subject to selection.

Estimation of genetic parameters from pedigreed populations: lessons from analysis of alevin weight in Coho salmon ( Oncorhynchus kisutch)

The purpose of the study was to assess the impact of various model structures on REML estimates of variance components using data on alevin weight from two replicate populations from the Genetic Improvement Program for Coho salmon (Chile). Data consisted of 130 d alevin weight from a dams-nested-within-sires mating design over two consecutive generations. Relationship information included direct and collateral relatives but parental individuals lacked records. The construction of a range of animal models considered random effects of direct additive genetic, maternal additive genetic and full-sib family effects as well as the covariance of direct and maternal genetic effects. Fixed effects of year (generation) and spawn date of dams within year were considered and also evaluated. The relative effectiveness of various models in describing the data set were assessed using likelihood ratio tests. The results demonstrated the importance of the correct interpretation of effects in the data set, particularly those effects that can influence the resemblance between relatives. The data structure, as well as the animal model applied, markedly influenced the magnitude of variance component estimates. Models based on year as the only fixed effect did not describe the data nearly as effectively as models containing both year and spawn data of dams within year. Simple models based on animal as the only random effect gave upward biased estimates of additive genetic variance. The most appropriate model for the data set was one based on both year and spawn date as fixed effects, and animal and full-sib family as random effects. The results from models combining maternal genetic and full-sib family effects to exploit the full covariance structure of the data showed that there was confounding between these variance component estimates. The most consistent interpretation of this result was that common environmental effects and non-additive genetic effects were more important sources of variability than maternal genetic effects. The study also demonstrated high variability in parameter estimates for replicate populations.

ON USING PROC MIXED FOR LONGITUDINAL DATA

PROC MIXED has become a standard tool for analyzing repeated measures data. Its popularity results from a wide choice of correlated error models compared to other software, e.g. PROC GLM. However, PROC MIXED's versatility comes at a price. Users must take care. Problems may result from MIXED defaults. These include: questionable criteria for selecting correlated error models; starting values that may impede REML estimation of covariance components; and biased standard errors and test statistics. Problems may be induced by inadequate design. This paper is a survey of current knowledge about mixed model methods for repeated measures. Examples are presented using PROC MIXED to demonstrate these problems and ways to address them.

STARTING VALUES FOR PROC MIXED WITH REPEATED MEASURES DATA

A major advantage of PROC MIXED for repeated measures data is that one could choose from many different correlated error models. However, MIXED uses default starting values that may cause difficulty obtaining REML estimates of the covariance parameters for several of the models available. This can take the form of excessively long run times or even failure to converge. We have written a program to obtain initial covariance parameter estimates that result in greatly improved performance of the REML algorithm. We will use two covariance models frequently of interest in animal health experiments, the first-order ante-dependence model [ANTE(l)] and the Toeplitz model with heterogeneous variances [TOEPH], to illustrate the use of our procedure.

Bias Reduction of Approximate Maximum Likelihood Estimates for Heritability in Threshold Models

Approximate maximum likelihood estimates, such as penalized quasi-likelihood and iterated reweighted REML estimates, for heritability or intraclass correlation in threshold models for binary data can be seriously biased. Two approaches to correct for this bias are studied, with emphasis on animal breeding models for binary data. Scope for reduction of bias of heritability estimates is found to be slim because of the commonly large number of fixed effects in animal breeding models. Minimal dimensions for the data are identified such that bias and root-mean-squared error are of modest size and useful inference on heritability (or intraclass correlation) is feasible.

USING CONFIDENCE INTERVALS TO OBTAIN A FAMILY OF ESTIMATORS OF THE INTRACLASS CORRELATION COEFFICIENT (OR HERITABILITY)

A family of point estimators is presented for the intraclass correlation coefficient (or heritability) in the balanced one-way random effects model. The family is obtained by equating a pivotal quantity to different values of the pivoting distribution, and includes the familiar ML and REML estimators. In terms of mean-squared error, most members of the family of estimators are admissible within the family. A sire model is used to illustrate the estimation of heritability. The authors provide guidance concerning the choice of an individual member of the family for estimation purposes and indicate how the method can be extended to unbalanced designs.

AN ALTERNATIVE FOR MIXED MODEL ANALYSES OF LARGE, MESSY DATA SETS (MTDFREML)

Portable Fortran based programs (MTDFREML) were developed using a derivative-free algorithm to obtain REML estimates of (co)variance components. Computations are based on Henderson's mixed model equations for multiple-trait models with missing observations on some traits and incorporation of relationships among relatives. Many fixed and random factors are allowed with number of levels dependent on computer memory. Data sets with more than 40,000 genetic effects have been analyzed. Options allow solving MME at convergence. Constraints are automatically imposed. Expectations, standard errors of contrasts of solutions for fixed effects and prediction error variances of solutions for random effects can be obtained. Dimensions can be changed to match data with computer capability. A Fortran compiler is necessary. No fee is charged but the University of Waterloo must certify a license has been obtained for sparse matrix subroutines (SPARSPAK) used in the program. As an example, birth weights of 4891 progeny of 389 sires nested within 12 breeds and of 2893 dams nested within 3 breeds of dam were analyzed to estimate components of variance due to sires and dams and to estimate differences among breeds of sires. For MTDFREML the analysis was trivial but for PROC MIXED the analysis was impossible unless dams were dropped from the model.

The genetic evaluation of beef sires used for crossing with dairy cows in the UK 2. Genetic parameters and sire merit predictions for calving survey traits

AbstractGenetic parameters were estimated for traits recorded in a calving survey for beef sires used in commercial dairy herds in England and Wales. The five traits included in the survey were calving difficulty score, calf mortality to 48 h after birth, gestation length, and subjectively assessed calf size and conformation. The data file examined included over 88 000 records, on 323 sires of eight beef sire breeds, namely Aberdeen Angus, Belgian Blue, Blonde d'Aquitaine, Charolais, Hereford, Limousin, Piedmontese and Simmental.Estimates of heritability and genetic and phenotypic correlations were obtained for all traits by REML procedures using a sire model. The effects of cow age, calf sex, year and season of the calving and sire breed were included in the model, together with interactions.Heritability estimates were 0·09 (s.e. 0·01) for calving difficulty score, 0·32 (s.e. 0·03) for gestation length, 0·02 (s.e. 0·003) for mortality, 0·09 (s.e. 0·01) for calf size and 0·06 (s.e. 0·01) for calf conformation.Heritability estimates for the scored categorical and binomial (mortality) traits were also obtained using threshold model analysis. As expected, these estimates were higher than the REML estimates; 0·16 (s.e. 0·01) for calving difficulty score, 0·08 (s.e. 0·01) for mortality, 0·14 (s.e. 0·01) for calf size and 0·09 (s.e. 0·01) for calf conformation. Calving difficulty score was strongly correlated genetically with calf size (-0·84), mortality (0·74) and calf conformation (-0·72), and moderately correlated with gestation length (0·26). Calf size and conformation were very highly correlated genetically (0·86), and both were also strongly correlated with mortality (-0·63 and -0·52 respectively). In all cases, the genetic correlations were stronger than the phenotypic correlations.Sire genetic merit was predicted for all traits and the distribution of these predictions is described. Sire predictions of genetic merit for calving difficulty score on the underlying scale were backtransformed to predict the expected incidence of serious difficulties in future calvings and these predictions are illustrated for the major sire breeds.

Growth of Holstein dairy heifers in the United States.

Data were collected on the heart girth (n = 8,565; a measure of body weight) and height at withers (n = 8,568) of Holstein dairy heifers from 659 dairy farms as part of the National Dairy Heifer Evaluation Project during 1991 and 1992. Means and standard deviations for weight and height were determined for ages .5 to 23.5 mo. Third-order polynomial regression equations were derived to study the relationship of weight and height to age. Stepwise mixed-model regression, using REML estimation, was used to identify factors associated with Holstein heifer growth and to build a multivariate model describing Holstein heifer growth across the United States. Holstein heifers on the sample of dairy farms in this study were heavier and taller at the withers than standards published 30 to 50 yr ago. Increased rolling herd average milk production was associated with a greater rate of growth in Holstein heifers. Holstein heifers in the West and Midwest were larger at a given age than those in the Northeast and Southeast.

Wald consistency and the method of sieves in REML estimation

We prove that for all unconfounded balanced mixed models of the analysis of variance, estimates of variance components parameters that maximize the (restricted) Gaussian likelihood are consistent and asymptotically normal--and this is true whether normality is assumed or not. For a general (nonnormal) mixed model, we show estimates of the variance components parameters that maximize the (restricted) Gaussian likelihood over a sequence of approximating parameter spaces (i.e., a sieve) constitute a consistent sequence of roots of the REML equations and the sequence is also asymptotically normal. The results do not require the rank p of the design matrix of fixed effects to be bounded. An example shows that, in some unbalanced cases, estimates that maximize the Gaussian likelihood over the full parameter space can be inconsistent, given the condition that ensures consistency of the sieve estimates.

Asymptotics for REML estimation of spatial covariance parameters

In agricultural field trials, restricted maximum likelihood estimation (REML) of the spatial covariance parameters is often preferred to maximum likelihood. Although it has either been conjectured or assumed that REML estimators are asymptotically Gaussian, conditions under which such asymptotic results hold are clearly needed. This article gives checkable conditions for spatial regression when sampling locations are either on a rectangular grid or are irregularly spaced but satisfy certain growth conditions.

ML, REML and MINQUE Estimates of Variance Components and Heritabilites for Milk Yield Characteristics of Brown Swiss

Variance components and heritabilites were estimated using by ML, REML, MINQUE methods for milk yield, fat percentage, fat yield, protein percentage. ML estimates were smaller for variance components and higher for heritabilites, than estimates obtained by REML and MINQUE. REML and MINQUE estimates were similar for variance components and heritabilites.

Estimation and interpretation of direct and maternal genetic parameters for weights of Australian Angus cattle

Abstract REML estimates of genetic parameters in field data from Australian Angus cattle were similar to estimates for other British breeds in Australia. The main difference was a smaller permanent environmental effect of dam than in breeds such as the Hereford. Heritabilities of birth, weaning, yearling and 600 day weight were estimated to be 0.35, 0.20, 0.24 and 0.25, respectively, for direct effects and 0.08, 0.09, 0.06 and 0.04 for maternal effects. Dam environmental effects were 0.05, 0.14, 0.07 and 0.04 and phenotypic variances 14, 501, 754 and 1152 kg2. Correlations between birth and later weights were 0.52–0.59 (direct genetic), 0.45–0.76 (maternal genetic), 0.37–0.54 (maternal environmental) and 0.12–0.19 (residual). Correlations among post birth weights were 0.84–0.90 (direct genetic), 0.86–0.87 (maternal genetic), 0.84–0.94 (maternal environmental) and 0.49–0.62 (residual). When fitted, estimates of the correlations between direct and maternal genetic effects were large and negative. Examination of the correlations between estimated breeding values and the fitting of alternative models suggested, however, that such correlations were a consequence of other effects in the data, rather than evidence of a true negative genetic relationship.

Mass selection for 70-day body weight in rabbits.

Five generations of phenotypic mass selection for increased 70-d body weight (market age, MW) were conducted in an experimental population of rabbits. Market-age weight was recorded for 1,616 rabbits from 336 litters (sired by 121 bucks and reared by 321 does). Additional growth and carcass traits measured in control (C) and select (S) lines were 28-d weaning weight (WW), 28- to 70-d average daily gain (ADG), carcass yield percentage (CY), loin primal cut percentage (LP), and lean-to-bone ratio in the loin primal cut (LBR). Average inbreeding coefficients in litters by the fifth generation were 10.3 and 11.2% for C and S lines, respectively. After five generations the difference in weighted cumulative selection differential between lines for MW was 1,015 g. Using an animal model with REML and regression procedures, observed and realized estimated heritabilities for MW were .12 and .11, respectively. Estimates of heritability by REML were .04, .17, .37, .25, and .35 for WW, ADG, CY, LP, and LBR, respectively. Bivariate REML estimates of direct genetic, common litter, and residual correlations between MW and a second growth trait (WW or ADG) were all positive and moderate to high in magnitude (range of .56 to .98). At generation five, mean direct breeding values of S and C line rabbits for MW were 133.2 and 12.0 g. Genetic trends were 29.1 and 29.4 g per generation for MW, estimated according to regressions of estimated mean breeding value (obtained from mixed-model analyses) or of observed S minus C line mean differences on generation number, respectively. Correlated response for WW, ADG and LBR was consistently favorable, whereas results for other carcass traits studied were inconclusive. Selection was effective for increasing MW performance.

Multiple-trait Gibbs sampler for animal models: flexible programs for Bayesian and likelihood-based (co)variance component inference.

A set of FORTRAN programs to implement a multiple-trait Gibbs sampling algorithm for (co)variance component inference in animal models (MTGSAM) was developed. The MTGSAM programs are available to the public. The programs support models with correlated genetic effects and arbitrary numbers of covariates, fixed effects, and independent random effects for each trait. Any combination of missing traits is allowed. The programs were used to estimate variance components for 50 replicates of simulated data. Each replicate consisted of 50 animals of each sex in each of four generations, for 400 animals in each replicate for two traits. For MTGSAM, informative prior distributions for variance components were inverted Wishart random variables with 10 df and means equal to the simulation parameters. A total of 15,000 Gibbs sampling rounds were completed for each replicate, with 2,000 rounds discarded for burn-in. For multiple-trait derivative free restricted maximum likelihood (MTDFREML), starting values for the variance components were the simulation parameters. Averages of posterior mean of variance components estimated using MTGSAM with informative and flat prior distributions for variance components and REML estimates obtained using MTDFREML indicated that all three methods were empirically unbiased. Correlations between estimates from MTGSAM using flat priors and MTDFREML all exceeded .99.