Linear and generalized linear models for the detection of QTL effects on within-subject variability

Quantitative trait loci (QTLs) may affect not only the mean of a trait but also its variability. A special aspect is the variability between multiple measured traits of genotyped animals, such as the within-litter variance of piglet birth weights. The sample variance of repeated measurements is assigned as an observation for every genotyped individual. It is shown that the conditional distribution of the non-normally distributed trait can be approximated by a gamma distribution. To detect QTL effects in the daughter design, a generalized linear model with the identity link function is applied. Suitable test statistics are constructed to test the null hypothesis H(0): No QTL with effect on the within-litter variance is segregating versus H(A): There is a QTL with effect on the variability of birth weight within litter. Furthermore, estimates of the QTL effect and the QTL position are introduced and discussed. The efficiency of the presented tests is compared with a test based on weighted regression. The error probability of the first type as well as the power of QTL detection are discussed and compared for the different tests.     

Estimation of genetic trends from 1977 to 1998 for farrowing characteristics in the French Large White breed using frozen semen

The objective of the study was to estimate genetic trends from 1977 to 1998 in the French Large White (LW) breed for stillbirth and associated traits measured at farrowing using frozen semen. Two groups of pigs (G77 and G98) were obtained by inseminating LW sows with semen from LW boars born either in 1977 or in 1998. A second generation was produced by inter se mating in each group. Farrowing was thoroughly supervised through both direct observations and video recording all long farrowing on a total of 137 first- and second-parity litters produced by sows from this second generation (68 G77 and 69 G98 litters, respectively). Measurements included birth time, weight and birth characteristics (including orientation, presence of cyanosis or oedema, membrane obstruction, umbilical cord length/content) of each piglet, as well as sow traits (weight and backfat thickness, farrowing duration, litter size and within-litter variation of weights at birth). The data were analysed using linear or generalised linear mixed models, according to the definition of the trait (continuous or binary data). The importance of several effects to piglet probability of stillbirth was then quantified by computing the reduction of variance associated with the addition of each effect in the model. Litter size did not significantly differ in first parity, but was higher in G98 second-parity sows: the differences for global (including pre partum dead piglets) and total numbers of piglets born per litter were +2.3 ± 1.1 and +1.3 ± 0.6, respectively. G98 sows also had a higher number of stillbirths in both parities (+0.7 ± 0.3 stillborn per litter). Piglets from G98 litters were heavier at birth (+130 ± 40 g for birth weight adjusted for litter size), without any increase in within-litter heterogeneity of birth weight. No significant difference was detected between G77 and G88 groups for farrowing length and the distribution of time interval between piglet births. G98 stillborn piglets had longer and more often empty umbilical cords at birth. G98 piglets born alive also had more often umbilical nodes than G77 piglets. These characteristics were considered as indicators of increased farrowing difficulties and risk of hypoxia at birth in G98 pigs. Time of birth of each piglet, sow fatness at farrowing and time of first placenta expulsion were the main factors of variation of the piglet's probability of stillbirth.     

A general approach to mixed effects modeling of residual variances in generalized linear mixed models

We propose a general Bayesian approach to heteroskedastic error modeling for generalized linear mixed models (GLMM) in which linked functions of conditional means and residual variances are specified as separate linear combinations of fixed and random effects. We focus on the linear mixed model (LMM) analysis of birth weight (BW) and the cumulative probit mixed model (CPMM) analysis of calving ease (CE). The deviance information criterion (DIC) was demonstrated to be useful in correctly choosing between homoskedastic and heteroskedastic error GLMM for both traits when data was generated according to a mixed model specification for both location parameters and residual variances. Heteroskedastic error LMM and CPMM were fitted, respectively, to BW and CE data on 8847 Italian Piemontese first parity dams in which residual variances were modeled as functions of fixed calf sex and random herd effects. The posterior mean residual variance for male calves was over 40% greater than that for female calves for both traits. Also, the posterior means of the standard deviation of the herd-specific variance ratios (relative to a unitary baseline) were estimated to be 0.60 ± 0.09 for BW and 0.74 ± 0.14 for CE. For both traits, the heteroskedastic error LMM and CPMM were chosen over their homoskedastic error counterparts based on DIC values.     

Immunoneutralization of inhibin suppresses reproduction in female mink.

This study determined whether immunoneutralization of inhibin affected gonadotropin secretion, embryo development, and ovarian function in mink. Adult female mink (n = 10) were immunized with bovine inhibin alpha 1-26 gly-tyr (bINH, 100 micrograms) conjugated to human alpha globulins (HAG), or with HAG alone (n = 10, controls), mixed with Freund's complete adjuvant. A series of five boosters containing bINH or HAG were then administered during a 2-yr period. Titers of bINH antibodies and serum concentrations of gonadotropins were determined for each breeding season in 1990 and 1991. Each year after whelping, we determined gestation length; sex, number, and weight of live and dead kits per litter at birth; and number and weight of kits per litter 3 wk after whelping. Results were pooled for statistical analysis. Bovine INH antibody titers (percent 125I-bINH bound to serum diluted 1:8000) were 53 +/- 3% vs. 2 +/- 0.6%, and serum concentrations of FSH were higher (p < 0.05) in bINH-immunized mink compared with controls (144 +/- 23 vs. 67 +/- 12 ng/ml). However, number (3.8 +/- 0.2 vs. 5 +/- 0.4) and weight (8 +/- 0.3 vs. 9.7 +/- 0.4 g) of kits per litter at birth and number of kits per litter alive 3 wk after birth (2.9 +/- 0.5 vs. 4.7 +/- 0.4) were lower (p < 0.05) in bINH-immunized mothers compared with controls. During the nonbreeding season in 1991, a single injection of hCG (100 IU) was administered to bINH-immunized and control mink; 24 h later blood was sampled, and organ weights were determined.(ABSTRACT TRUNCATED AT 250 WORDS)     

Economic weights of production and functional traits for Merinolandschaf, Romney, Romanov and Sumavska sheep in the Czech Republic

Economic weights of production and functional traits were estimated for the dual-purpose sheep breeds Romney (RY), Merinolandschaf (ML), Romanov (RV) and Sumavska (SA). A bio-economic computer model simulating the profit function for production systems with one lambing per year was used for the calculations. First, marginal economic values were calculated which were defined as partial derivatives of the profit function with respect to the average level for each trait. Applying gene-flow methodology, absolute and relative economic weights for direct and maternal components of the traits were then estimated for pure-bred production systems and systems which included partial terminal cross-breeding. The following traits were evaluated: birth weight, daily gain till weaning and during rearing, mature weight, dressing percentage, fleece weight, conception rate of ewe lambs and ewes, litter size at lambing (total number of lambs born per ewe lambing), survival rate of lambs at birth and till weaning and productive lifetime of ewes. Standardized economic weights were calculated as the product of the economic weight and the genetic standard deviation of each trait or trait component. The total economic importance was defined as the sum of the absolute (not taking into account the sign) standardized economic weights over all traits and trait components, and relative economic weights were then computed as the standardized economic weights expressed as percentages of total economic importance. The direct components of survival rate at birth and till weaning, the direct component of daily gain till weaning and litter size had the highest relative economic weights (greater than 10%) in pure-bred systems in all four breeds. In ML and RV, the maternal component of lamb survival rate till weaning also had a relative economic weight exceeding 10%. In systems with partial terminal cross-breeding for all four breeds, relative economic weights of maternal traits and trait components were slightly larger than in pure-bred production systems, particularly for litter size. Based upon comparison of relative economic weights of traits among breeds, separate breeding goals for RY and RV are recommended, whereas a common breeding goal is deemed appropriate for ML and SA.     

Biometrical modeling of twin and family data using standard mixed model software

Summary .   Biometrical genetic modeling of twin or other family data can be used to decompose the variance of an observed response or 'phenotype' into genetic and environmental components. Convenient parameterizations requiring few random effects are proposed, which allow such models to be estimated using widely available software for linear mixed models (continuous phenotypes) or generalized linear mixed models (categorical phenotypes). We illustrate the proposed approach by modeling family data on the continuous phenotype birth weight and twin data on the dichotomous phenotype depression. The example data sets and commands for Stata and R/S-PLUS are available at the Biometrics website.     

Joint analysis of beef growth and carcass quality traits through calculation of co-variance components and correlations

A joint growth-carcass model using random regression was used to estimate the (co)variance components of beef cattle body weights and carcass quality traits and correlations between them. During a four-year period (1994-1997) of the Australian "southern crossbreeding project", mature Hereford cows (N = 581) were mated to 97 sires of Jersey, Wagyu, Angus, Hereford, South Devon, Limousin, and Belgian Blue breeds, resulting in 1141 calves. Data included 13 (for steers) and 8 (for heifers) body weight measurements approximately every 50 days from birth until slaughter and four carcass quality traits: hot standard carcass weight, rump fat depth, rib eye muscle area, and intramuscular fat content. The mixed model included fixed effects of sex, sire breed, age (linear, quadratic and cubic), and their interactions between sex and sire breed with age. Random effects were sire, dam, management (birth location, year, post-weaning groups), and permanent environmental effects, and their interactions with linear, quadratic and cubic growth, when possible. Phenotypic, sire and dam correlations between body weights and hot standard carcass weight and rib eye muscle area were positive and moderate to high from birth to feedlot period. Management variation accounted for the largest proportion of total variation in both growth and carcass traits. Management correlations between carcass traits were high, except between rump fat depth and intramuscular fat (r = 0.26). Management correlations between body weight and carcass traits during the pre-weaning period were positive except for intramuscular fat. The correlations were low from birth to weaning, then increased dramatically and were high during the feedlot period.     

Supplementation of dextrose to the diet during the weaning to estrus interval affects subsequent variation in within-litter piglet birth weight

Effects of supplementation of dextrose to the diet of sows during the weaning-to-estrus interval (WEI) on subsequent litter size and within-litter variation were investigated. After weaning, 223 sows (first to fifth parity) were fed 3.5 kg/d. Half of the sows additionally received 150 g of dextrose per day as topdressing on the feed. WEI and estrus duration were determined as well as subsequent pregnancy rate and litter size. Piglets were weighed individually at birth and at weaning (day 26.4; S.D.: 2.5). Supplementation of dextrose to the diet during the WEI did not affect WEI (106 h), pregnancy rate (88.2%), farrowing rate (84.2%), subsequent litter size (total born: 13.70), or birth weight (1599 g). The within-litter variation in birth weight was lower in sows on the dextrose treatment (CV: 17.5% versus 21.2% for the dextrose and control group, respectively, P=0.03). From this experiment, we concluded that addition of dextrose during the weaning to estrus interval did not increase litter size, but seems to affect the uniformity in birth weight of the litter.     

广义岭回归在家禽育种值估计中的应用

讨论了岭回归方法应用于混合线性模型方程组中估计家禽育种值的方法,其实质是将传统的混合线性模型方程组理解为一种广义岭回归估计,为确定遗传参数的估计提供了一种途径;同时,以番鸭为例,考虑了一个性状和两个固定效应,采用广义岭回归法对公番鸭育种值进行了估计,并与最佳线性无偏预测法(BLUP 法)进行了比较,结果表明,广义岭回归方法和BLUP 法估计的育种值及其排序非常接近,其相关系数和秩相关系数分别达到了0.998~(**)和0.986~(**),且采用广义岭回归法预测的误差率低(在±10%以内);表明在混合线性模型方程组中使用广义岭回归估计动物育种值的方法具有可行性,并可省去估计遗传参数的过程,使BLUP 法在动物选育中的应用更具实用性.     

Overdispersed fungus germination data: statistical analysis using R

ABSTRACT

Proportion data from dose-response experiments are often overdispersed, characterised by a larger variance than assumed by the standard binomial model. Here, we present different models proposed in the literature that incorporate overdispersion. We also discuss how to select the best model to describe the data and present, using R software, specific code used to fit and interpret binomial, quasi-binomial, beta-binomial, and binomial-normal models, as well as to assess goodness-of-fit. We illustrate applications of these generalized linear models and generalized linear mixed models with a case study from a biological control experiment, where different isolates of Isaria fumosorosea (Hypocreales: Cordycipitaceae) were used to assess which ones presented higher resistance to UV-B radiation. We show how to test for differences between isolates and also how to statistically group isolates presenting a similar behaviour.     

Isolating the cow-specific part of residual energy intake in lactating dairy cows using random regressions

The ability to properly assess and accurately phenotype true differences in feed efficiency among dairy cows is key to the development of breeding programs for improving feed efficiency. The variability among individuals in feed efficiency is commonly characterised by the residual intake approach. Residual feed intake is represented by the residuals of a linear regression of intake on the corresponding quantities of the biological functions that consume (or release) energy. However, the residuals include both, model fitting and measurement errors as well as any variability in cow efficiency. The objective of this study was to isolate the individual animal variability in feed efficiency from the residual component. Two separate models were fitted, in one the standard residual energy intake (REI) was calculated as the residual of a multiple linear regression of lactation average net energy intake (NEI) on lactation average milk energy output, average metabolic BW, as well as lactation loss and gain of body condition score. In the other, a linear mixed model was used to simultaneously fit fixed linear regressions and random cow levels on the biological traits and intercept using fortnight repeated measures for the variables. This method split the predicted NEI in two parts: one quantifying the population mean intercept and coefficients, and one quantifying cow-specific deviations in the intercept and coefficients. The cow-specific part of predicted NEI was assumed to isolate true differences in feed efficiency among cows. NEI and associated energy expenditure phenotypes were available for the first 17 fortnights of lactation from 119 Holstein cows; all fed a constant energy-rich diet. Mixed models fitting cow-specific intercept and coefficients to different combinations of the aforementioned energy expenditure traits, calculated on a fortnightly basis, were compared. The variance of REI estimated with the lactation average model represented only 8% of the variance of measured NEI. Among all compared mixed models, the variance of the cow-specific part of predicted NEI represented between 53% and 59% of the variance of REI estimated from the lactation average model or between 4% and 5% of the variance of measured NEI. The remaining 41% to 47% of the variance of REI estimated with the lactation average model may therefore reflect model fitting errors or measurement errors. In conclusion, the use of a mixed model framework with cow-specific random regressions seems to be a promising method to isolate the cow-specific component of REI in dairy cows.     

不同繁育饲养设施对Beagle犬生产性能的影响

目的探索不同繁育饲养设施对Beagle犬主要生产性能的影响。方法使用两种不同Beagle犬繁育饲养设施（设施Ⅰ、设施Ⅱ）对母犬产仔状况、仔犬初生重、60 d离乳犬只存活率、60 d离乳犬只体重、1～6月龄犬只体重月增长等生产性能进行了比较研究。结果两种设施比较母犬产仔状况,仔犬初生重无差异（P〉0.05）;设施Ⅱ比设施Ⅰ的60 d离乳犬只存活率高18.4%,差异极显著（P〈0.01）,60日龄离乳犬只体重平均多（0.62±0.13）kg,差异极显著（P〈0.01）;1～6月龄犬只体重月增长平均多0.29±0.14 kg,差异近显著（P=0.08〉0.05）。结论设施Ⅱ优于设施Ⅰ。     

Genetic analysis of ewe productivity traits in Moghani sheep

Genetic and environmental (co)variance components for productivity traits in Moghani sheep were estimated using data from 1344 ewes. The data were collected in the Jafarabad breeding station, north-east of Iran, during a 13-year period (1995-2008). The studied traits were litter size at birth (LSB), litter size at weaning (LSW), litter mean weight per lamb born (LMWLB), litter mean weight per lamb weaned (LMWLW), total litter weight at birth (TLWB) and total litter weight at weaning (TLWW). A model including direct additive genetic effects as well as permanent environmental effects related to repeated records of ewe was the most appropriate model for all the studied traits. Genetic parameters were estimated applying restricted maximum likelihood (REML) procedure. Direct heritability estimate for LSB, LSW, LMWLB, LMWLW, TLWB and TLWW were 0.11, 0.02, 0.15, 0.07, 0.07 and 0.06, respectively. Corresponding values for repeatability estimates were 0.16, 0.19, 0.18, 0.11, 0.13 and 0.09. Genetic correlations between the studied traits ranged from −0.99 for LSB-LMWLB and LSW-LMWLB to 0.99 for LSB-TLWB. Phenotypic and environmental correlation estimates were generally lower than those of genetic ones. Estimates of permanent environmental correlation among traits were positive and medium to high. Although low direct heritabilities were estimated for the reproductive traits, as these traits are of interest then they should be included in a breeding program.     

Genetic heterogeneity of residual variance - estimation of variance components using double hierarchical generalized linear models

Background

The sensitivity to microenvironmental changes varies among animals and may be under genetic control. It is essential to take this element into account when aiming at breeding robust farm animals. Here, linear mixed models with genetic effects in the residual variance part of the model can be used. Such models have previously been fitted using EM and MCMC algorithms.

Results

We propose the use of double hierarchical generalized linear models (DHGLM), where the squared residuals are assumed to be gamma distributed and the residual variance is fitted using a generalized linear model. The algorithm iterates between two sets of mixed model equations, one on the level of observations and one on the level of variances. The method was validated using simulations and also by re-analyzing a data set on pig litter size that was previously analyzed using a Bayesian approach. The pig litter size data contained 10,060 records from 4,149 sows. The DHGLM was implemented using the ASReml software and the algorithm converged within three minutes on a Linux server. The estimates were similar to those previously obtained using Bayesian methodology, especially the variance components in the residual variance part of the model.

Conclusions

We have shown that variance components in the residual variance part of a linear mixed model can be estimated using a DHGLM approach. The method enables analyses of animal models with large numbers of observations. An important future development of the DHGLM methodology is to include the genetic correlation between the random effects in the mean and residual variance parts of the model as a parameter of the DHGLM.     

Testing for Genetic Association in the Presence of Linkage and Gene–Covariate Interactions

In order to study family‐based association in the presence of linkage, we extend a generalized linear mixed model proposed for genetic linkage analysis (Lebrec and van Houwelingen (2007), Human Heredity 64 , 5–15) by adding a genotypic effect to the mean. The corresponding score test is a weighted family‐based association tests statistic, where the weight depends on the linkage effect and on other genetic and shared environmental effects. For testing of genetic association in the presence of gene–covariate interaction, we propose a linear regression method where the family‐specific score statistic is regressed on family‐specific covariates. Both statistics are straightforward to compute. Simulation results show that adjusting the weight for the within‐family variance structure may be a powerful approach in the presence of environmental effects. The test statistic for genetic association in the presence of gene–covariate interaction improved the power for detecting association. For illustration, we analyze the rheumatoid arthritis data from GAW15. Adjusting for smoking and anti‐cyclic citrullinated peptide increased the significance of the association with the DR locus.     

Growth and development of adipose tissue and gut and related endocrine status during early growth in the pig: impact of low birth weight

With genetic selection, the increase in litter size has led to higher variation in within-litter birth weights in pigs. This has been associated with a reduction in mean birth weights and a rise in the proportion of piglets weighing less than 1 kg at birth. Low birth weight pigs exhibit lower postnatal growth rates and feed efficiency, which may be explained by an inadequate digestion and/or nutrient use as a consequence of prenatal undernutrition. It is now documented that there is a relationship between birth weight and subsequent pattern of growth and development of tissues and organs. During the neonatal period, the rapid somatic growth is accompanied by tremendous anatomical, physiological and chemical composition changes. The present review focuses primarily on the influence of low birth weight on adipose tissue and the gastrointestinal tract growth and development during the suckling period. The importance of the somatotropic axis, insulin, thyroid hormones, glucocorticoids, epidermal growth factor and leptin in the regulation of these developmental processes is also considered.     

Unbiasedness of the Estimator of the Function of Expected Value in the Mixed Linear Model

The traditional method for estimating the linear function of fixed parameters in mixed linear model is a two-stage procedure. In the first stage of this procedure the variance components estimators are calculated and next in the second stage these estimators are taken as true values of variance components to estimating the linear function of fixed parameters according to generalized least squares method. In this paper the general mixed linear model is considered in which a matrix related to fixed parameters and or/a dispersion matrix of observation vector may be deficient in rank. It is shown that the estimators of a set of functions of fixed parameters obtained in second stage are unbiased if only the observation vector is symmetrically distributed about its expected value and the estimators of variance components from first stage are translation-invariant and are even functions of the observation vector.     

A reproductive study of Weiser-Maples guinea pigs]

The Weiser-Maples (WE) guinea pig strain was introduced by Backshire Co., Ltd. (USA) in 1977. We have been breeding WE strain guinea pigs for skin melanization research. The WE guinea pig colony produced 1271 pups in 417 litters from May 1978 through December 1983. Breeding date are shown below. The mean litter size was 3.05, the stillborn rate was 15.2%, the weaning rate for live-born pups was 93.5% and the sex ratio was 1.01. The average age at first vaginal membrane rupture was 31.4 days at which time body weight was 290.5g. The mean length of the first 7 estrous cycles was about 17 days, with no cyclical variation in length. The mean duration of gestation was 67.9 days. Duration of pregnancy varied with litter size. There was an inverse relationship between litter size and duration of pregnancy. Most of the pups were delivered alive in mid-pregnancy with a parturition range of 56 to 76 days. The probability of pup death depends on gestational length: the lowest incidence of mortality was seen in litters born at 70 days. The mortalities were related to litter size but not to parity. There was an inverse relationship between birth weight and litter size. In WE guinea pigs, the mean weight for a litter of 1 was 120 g; for a litter of 5, the mean body weight was 58g. Male body weights were slightly heavier than female at birth and at weaning age. The mean body weights are shown below, date of birth: female 88.3g, male 93.3g, weaning age (2 weeks): female 181.1g, male 198.8g and 30 weeks: female 758.7g, male 1018.0g. These date for WE guinea pigs are comparable to those of other strains.     

Selection for environmental variation: a statistical analysis and power calculations to detect response

Data from uterine capacity in rabbits (litter size) were analyzed to determine whether the environmental variance was partly genetically determined. The fit of a classical homogeneous variance mixed linear (HOM) model and that of a genetically structured heterogeneous variance mixed linear (HET) model were compared. Various methods to assess the quality of fit favor the HET model. The posterior mean (95% posterior interval) of the additive genetic variance affecting the environmental variance was 0.16 (0.10; 0.25) and the corresponding number for the coefficient of correlation between genes affecting mean and variance was -0.74 (-0.90;-0.52). It is argued that stronger support for the HET model than that derived from statistical analysis of data would be provided by a successful selection experiment designed to modify the environmental variance. A simple selection criterion is suggested (average squared deviation from the mean of repeated records within individuals) and its predicted response and variance under the HET model are derived. This is used to determine the appropriate size and length of a selection experiment designed to change the environmental variance. Results from the analytical expressions are compared with those obtained using simulation. There is good agreement provided selection intensity is not intense.     

Variation in maternal investment in a small cervid; the effects of cohort,sex, litter size and time of birth in roe deer (Capreolus capreolus) fawns

 We investigated the effects of cohort, sex, litter size and time of birth on birth weights and postnatal growth rates of roe deer fawns in a highly reproductive Norwegian population. By repeatedly recapturing radio-collared individuals, a total of 950 weights were obtained from 231 fawns of known age. In accordance with earlier studies, there was a period of linear growth during the first month following birth. Mean postnatal growth rates of 155 g/day are the highest yet recorded for roe deer; however, the mean birth weights of fawns were lower than those reported from populations in continental Europe. During the period of linear growth, we found no sex differences. However, growth rates were affected both by time of birth and litter size; fawns born early had lower growth rates than fawns born during or after the peak calving period, and fawns in triplet – groups had lower growth rates than either fawns in twin – groups or single fawns. Despite a fourfold increase in population density during the study, this factor was not able to explain variation in postnatal growth rates, although cohort effects on birth weight were evident. Received: 13 May 1996 / Accepted: 26 June 1996