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«Heritability and repeatability of the number of lambs born and reared estimated using linear and threshold models Dariusz Piwczyński, Bogna ...»

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Archiv Tierzucht 54 (2011) 3, 271-279, ISSN 0003-9438

© Leibniz Institute for Farm Animal Biology, Dummerstorf, Germany

Heritability and repeatability of the number of lambs born

and reared estimated using linear and threshold models

Dariusz Piwczyński, Bogna Kowaliszyn and Sławomir Mroczkowski

Department of Genetics and General Animal Breeding, Faculty of Biology and Animal Breeding, University of

Agriculture and Life Sciences in Bydgoszcz, Poland


The research was conducted on 3 844 Polish Merino lamb dams born in 1991-2001, used in 15 flocks from the Pomerania and Kujawy region in Poland. The assessed parameters were the number of lambs born from a dam after lambing (LSB) (1, 2, 3) and the number of lambs reared (LSW) (0, 1, 2, 3). The genetic parameters LSB and LSW were estimated with the use of two methods: Average Information – REML (AI-REML) and Gibbs sampling (GS). For estimation of components by means of the AI-REML method the animal’s linear model was used, and in the case of the GS method a threshold model was also used alongside the linear one. The LSB heritability estimated using the AI-REML and GS methods in combination with a linear model were similar and their values were respectively 0.025 and 0.029, with similar standard errors for variance components. Applying the GS method combined with a threshold model resulted in a two times higher heritability (0.054) compared to when linear models were used. A similar tendency was found to exist in  respect of estimated repeatability. When using linear models, the obtained values were closely matched: 0.064 (AI-REML) and 0.065 (GS). The highest repeatability occurred when a threshold model was used (0.118). The LSW heritability was low and, depending on the model and method (0.016-0.020). Similar values LSW repeatability were obtained with the use of linear models (0.048 – REML and 0.049 – GS), and when a threshold model was used the result was higher – 0.070.

Keywords: sheep, linear model, threshold model, genetic parameters, litter size, Merino Zusammenfassung Bewertung der Heritabilität und Wiederholbarkeit der Anzahl von geborenen und aufgezogenen Lämmern mittels Linearen Modells und Schwellenmodells Die Untersuchungen wurden an 3 844 Mutterschafen der Rasse Polnische Merino durchgeführt. Die Tiere stammten aus 15 Herden und wurden zwischen 1991 und 2001 in der Region Pommern und Kujawien in Polen geboren. Bewertet wurden die Anzahl der durch das Mutterschaf (LSB) geworfenen Lämmer (1, 2, 3) und die Anzahl der aufgezogenen Lämmer (LSW) (0, 1, 2, 3). Die genetischen Parameter von LSB und LSW wurden mit Hilfe von zwei Verfahren bewertet: Average Information-REML (AI-REML) sowie Gibbs-Sampling-Verfahren (GS). Bei der Komponentenbewertung mit AI-REML wurde das lineare Modell des Tieres angewendet, bei GS zusätzlich das Schwellenmodell. Die Werte der Heritabilitätskennziffern von LSB, die unter Einsatz des Verfahrens AI-REML und GS in Verbindung mit dem 272 Piwczyński et al.: Genetic parameters for lambs born and reared estimated using linear and threshold models linearen Modell bewertet wurden, waren angenähert und betrugen: 0,025 und 0,029 - bei vergleichbaren Fehlerwerten in Bezug auf die standardmäßige Varianzkomponente. Die Anwendung des GS-Verfahrens mit dem Schwellenmodell ergab einen doppelt höheren Wert der Heritabilität (0,054) als beim linearen Modell.

Eine ähnliche Tendenz wurde im Bereich der Wiederholbarkeitskennziffern festgestellt.

Angenäherte Werte wurden beim Einsatz des linearen Modells erreicht: 0,064 (AI-REML) und 0,065 (GS). Eine deutlich höhere Wiederholbarkeitskennziffer ergab sich bei der Anwendung des Schwellenmodells: 0,118. Die Heritabilität der Anzahl aufgezogener Lämmer war niedrig und schwankte abhängig von dem eingesetzten Modell und der angewendeten Methode zwischen 0,016 und 0,020.

Vergleichbare Werte der LSW-Wiederholbarkeit wurden bei Nutzung vom linearen Modell (0,048 – REML und 0,049 – GS) erreicht, ein deutlich höheres Ergebnis (0,07) wurde dagegen bei Verwendung des Schwellenmodells errreicht.

Schlüsselwörter: Schaf, Lineares Modell, Schwellenmodell, genetische Parameter, Wurfgröβe, Merino Introduction The Polish Merino is the most commonly used sheep breed in Poland. It comes from the French Merino Precoz which was improved after World War II with the breeds of the German, Caucasian Merino. The breeding of the Polish Merino is centred in the regions of Wielkopolska, and Pomerania and Kujawy. Ewes of this breed in 2008 constituted approximately 14 % of female sheep being assessed in terms of their performance in Poland (Sheep and Goats Breeding in Poland in 2008, 2009).

Merino Sheep are much more common in countries to the west of Poland, for instance, in Germany. Based on the research conducted by Süß et al. (2004) it was established that the proportion of German Mutton Merino and Merinoland Sheep in the Saxon-Anhalt region is as high as 57.7 %. The used merinos are most frequently of the wool-meat type. The mean prolificacy of the breed in 2008 was relatively low and equalled 128.3 %. It is a clearly lower level than that presented by Strittmatter (2004) with respect to HB-German Mutton Merino – 146-168.8 %.

Variables like number of lambs born and reared take discrete values. These traits are considered threshold variables, that is traits which are discontinuous in their expression, usually representing a number of categorized phenotypic values, however, they are conditioned by polygenes (Dempster & Lerner 1950, Gianola 1982). Their characteristic property is the fact that exceeding the so-called threshold referring to the genes possessed results in a fundamental change of the phenotype.

The distribution of these variables is incompatible with a normal distribution. Therefore prior to estimation of the (co)variance components the said traits must undergo the Snell transformation (Naazie et al. 1991), or a probit transformation (Piwczyński 2004), then linear model calculations may be made using the REML method. Matos et al. (1997a), De Vries et al.

(2005) and Olesen et al. (1994), estimating genetic parameters, applied, as one of the models which it is possible to use for estimation of prolificacy variance components, the Poisson Arch Tierz 54 (2011) 3, 271-279 model. Results of numerous research studies (Altarriba et al. 1998, Matos et al. 1997a, 1997b, Olesen et al. 1994, Yazdi et al. 1999) show on the other hand that the genetic parameters of discreet reproduction traits should be estimated with the use of threshold models. Genetic parameters of reproduction traits may be estimated with the use of multitrait models (Hagger 2002, Vries et al. 1998) as well as repeatability models; results for these models are usually published in subject-related scientific literature (Dobek et al. 2004, Matos et al.

1997a, Noor et al. 2001, Olesen et al. 1994, Saboulard et al. 1995, Schmalwasser et al. 1991, Piwczyński 2009). It must be emphasised though that multitrait models have advantage over repeatability models as they take into account covariance between values of a trait obtained in subsequent measurements (Szyda 2001). Apart from establishing correct model selection, the researcher must choose between estimation methods. Usually, genetic parameters of the aforementioned reproduction traits are determined with the use of the Restricted Maximum Likelihood (REMLS), and Gibbs sampling.

Estimations of genetic parameters conducted so far indicate low influence of genetic assumptions on reproduction traits in Merino sheep (h2: fertility – 0.0720-0.203; number of lambs born from a mother that had lambed before – 0.060-0.260; number of lambs reared by a mated mother – 0.039-0.183 (Duguma et al. 2002, Lee et al. 2009, Mroczkowski et al. 1981, Olivier et al. 1998, 2001, Piwczyński 2009). Such significant differences in presented values of heritability indices may be caused, among other things, by the method or model used in statistical analysis.

The objective of the study was to compare the effects of heritability and repeatability estimates for lambs born and reared, obtained by means of the Average Information REML and Gibbs sampling using a linear and a threshold model.

Material and methods The research was conducted on 3 844 dams of the Polish Merino breed, born in the years 1991-2001 and used in 15 flocks from Pomerania and Kujawy, Poland. Data on descent and performance of sheep came from the breeding documentation from the years 1990-2004, made available by the Local Sheep and Goat Breeders Association in Bydgoszcz. The animals which underwent reproduction performance assessments between 1993 and 2003 were at the age of 2 to 12. The assessed parameters were the number of lambs born from a dam after lambing (LSB) (1, 2, 3) and the number of lambs reared by a mated dam (LSW) (0, 1, 2, 3).

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The pedigree information of the studied animal population was, if possible, completed up to 3rd generation. In total the pedigree database comprised 9 297 animals. In order to calculate the inbreeding index in the studied population, the INBREED procedure from the SAS package (SAS 2008) was used. Twenty inbred animals were found with the mean inbreeding of 14.76 % (SD=10.04 %).

274 Piwczyński et al.: Genetic parameters for lambs born and reared estimated using linear and threshold models

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As part of the statistical determination, the basic measures of location and variability of controlled traits were calculated. An explorative analysis of LSB and LSW was carried out using multiple logistic regression (SAS 2008). In the course of the analysis, using the selection method of the forward type regression model, the following variables, associated with the above traits, were selected: flock, year of birth, ewe’s age and birth type, and flock × year of birth interaction. The significance of parameters, i.e. the selected variables, was evaluated by means of the Wald statistics (SAS 2008). The statistical analysis was conducted with the use of the SAS computer package, applying the LOGISTIC procedure (SAS 2008).

The LSB and LSW genetic parameters were estimated by means of two methods: Average Information – Restricted Maximum Likelihood (AI-REML) and Gibbs sampling (GS). Estimating components by means of the AI-REML method the animal’s linear model (LM) was applied, and in the case of the GS method, a threshold model was used alongside a linear one (TM).

The GIBBS1F90 software (Misztal 2007) was used to estimate the LSB and LSW genetic parameters according to a linear model, and the THRGIBBS1F90 (Tsuruta & Misztal 2006) according to a threshold model.

Estimating variance components by means of the Gibbs sampling method, 100 000 samples were generated, 40 000 of which were considered as so-called »burn-in« samples.

Due to occurrence of autocorrelation of results obtained from adjacent samples, genetic parameters were determined based on values obtained from every 100th sample. Variance components as well as heritability and repeatability indices were therefore determined based on results of 600 samples. The POSTGIBBS1F90 computer software (Tsuruta & Misztal

2006) was used to determine the number of samples initially rejected.

Below is the linear model applied to estimate variance components.

y = Xfy βfy + Xw βw + Xt βt + Zaa + Zpe pe + e (1) where y is the 15 938 × 1 observation vector, βfy, βw, βt are the fixed effects vectors: flock-year of birth (146 × 1), dam’s age (6 × 1); dam’s birth type (2 × 1), a is the 9 297 × 1 random genetic additive effects vector, pe is the 9 297 × 1 random permanent environment effects vector, Xfy, Xw, Xt are the incidence matrices for permanent effects: flock-year of birth (15 938 × 146), dam’s age (15 938 × 6); dam’s birth type (15 938 × 2), Za is the 15 938 × 9 297 incidence matrix for random direct additive genetic effects, Zpe is the 15 938 × 9 297 incidence matrix for random permanent environment effects and e is the 15 938 × 1 random errors vector.

Arch Tierz 54 (2011) 3, 271-279

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where A is the 9 297 × 9 297 dimensional additive relationship matrix, In and Iq are the identity matrices, σa2 is the direct additive genetic variance, σpe is the random permanent environment effects variance, σe2 is the error variance and σp is the phenotypic variance (σp=σa2+ σpe+ σe2).

While estimating variance components with the use of a threshold model, the same random and fixed effects were taken into account, the difference, however, was that with this model the modelled parameter was the so-called »unobserved tendency«.

Estimating genetic parameters with the AI-REML method, the same convergence index, equal to 10−10, was adopted for all models. Errors of estimated (co)variances were approximated according to the method described by Klei & Tsuruta (2008). In the case of the GS method, standard variance components errors and genetic parameters, determined based thereon, were calculated as standard deviations for the values of these components, and indices obtained from the abovementioned 600 samples.

Heritability (h2) and repeatability (r’) were obtained applying the following formulas,


(2) σa2 (σa2 + σpe ) h2 = r’= σ σ p p Results Table 1 shows basic measures of location and variability of the number of lambs born and reared. The share of multiple litters in the studied population was approximately 30 (Table 2).

Using the multiple regression method the following independent variables were selected, which were significantly associated with the number of lambs born and reared: flock, year of birth, age, and ewe’s birth type, flock × year of birth interaction. The selected variables were taken into account while estimating (co)variance components.

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