J. Anim. Sci. 2006. 84:538-545
© 2006 American Society of Animal Science
Genetic parameters for various random regression models to describe total sperm cells per ejaculate over the reproductive lifetime of boars
S. H. Oh*,
M. T. See*,1,
T. E. Long
and
J. M. Galvin,2
* North Carolina State University, Raleigh, NC 27695 and
and
Smithfield Premium Genetics, Roanoke Rapids, NC 27870
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Abstract
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The objective of this study was to model the variances and covariances of total sperm cells per ejaculate (TSC) over the reproductive lifetime of AI boars. Data from boars (n = 834) selected for AI were provided by Smithfield Premium Genetics. The total numbers of records and animals were 19,629 and 1,736, respectively. Parameters were estimated for TSC by age of boar classification with a random regression model using the Simplex method and DxMRR procedures. The model included breed, collector, and year-season as fixed effects. Random effects were additive genetic, permanent environmental effect of boar, and residual. Observations were removed when the number of data at a given age of boar classification was <10 records. Preliminary evaluations showed the best fit with fifth-order polynomials, indicating that the best model would have fifth-order fixed regression and fifth-order random regressions for animal and permanent environmental effects. Random regression models were fitted to evaluate all combinations of first- through seventh-order polynomial covariance functions. Goodness of fit for the models was tested using Akaike’s Information Criterion and the Schwarz Criterion. The maximum log likelihood value was observed for sixth-, fifth-, and seventh-order polynomials for fixed, additive genetic, and permanent environmental effects, respectively. However, the best fit as determined by Akaike’s Information Criterion and the Schwarz Criterion was by fitting sixth-, fourth-, and seventh-order polynomials; and fourth-, second-, and seventh-order polynomials for fixed, additive genetic, and permanent environmental effects, respectively. Heritability estimates for TSC ranged from 0.27 to 0.48 across age of boar classifications. In addition, heritability for TSC tended to increase with age of boar classification.
Key Words: genetic parameter • pig • random regression • semen
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INTRODUCTION
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Artificial insemination plays an important role in animal breeding by allowing greater use of genetically superior sires. It has been shown that there is an opportunity for genetic improvement of male fertility traits (Brandt and Grandjot, 1998
; Oh et al., 2006). However, the genetic control of semen traits in pigs has not been extensively studied. Moreover, total sperm cells per ejaculate (TSC) are longitudinal data, i.e., total sperm cells change over age. In previous studies, this type of data was analyzed by multiple trait methods, choosing the most important time points as separate traits. Because of the number of potential observations over a boar’s lifetime, it would be difficult to analyze this type of data thoroughly because of computational limits. Semen data have also been analyzed similarly to growth curves, ignoring genetic effects (Morant and Gnanasakthy, 1989), or were considered simple repeated measurements, ignoring time dependency.
In many cases, the assumption of a univariate repeated model is not appropriate, and a full multivariate model with the number of traits equal to the number of ages would result in a highly overparameterized analysis. Therefore, a model using the minimum number of traits is required (Meyer and Hill, 1997). Random regression models have been extensively applied to the test-day model analysis of milk yield of dairy cattle (Jamrozik and Schaeffer, 1997; Olori et al., 1999; Strabel and Misztal, 1999). Random regression models have also been fitted to weight data of pigs (Huisman et al., 2002). Random regression models provide a method for analyzing independent components of variation that reveal specific patterns of change over time. The objective of this study was to model the variances and covariances for TSC over the reproductive lifetime of AI boars.
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MATERIALS AND METHODS
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Data
Total sperm cell records (n = 19,629) for 834 boars were provided by Smithfield Premium Genetics (Table 1). One thousand seven hundred thirty-six individuals were included in the pedigree file. Boars represented 3 breeds and were housed on 2 farms. Each farm was similar in the numbers of boars of each breed. These data were collected by thirty-four collectors over 5 yr with 4 seasons/yr. Data were collected on farm 1 from 1998 through 2002, and data were recorded on farm 2 only in 2000 and 2001.
The TSC were determined by multiplying the total semen volume, measured as the weight of the ejaculate, by the sperm concentration, measured using a self-calibrating photometer.
Observations were removed when the number of data at a given age of boar classification time point was <10 records or when the TSC were missing, 0, or <0 (Figure 1). Weights of ejaculates were measured from 1998 to 2002 with approximately one-half recorded in 2000. Data were distributed evenly across seasons (Table 1
). Differences between boar collection date and birth date were used to provide each record with a fixed age of boar classification in weeks. When a boar had 2 observations during 1 wk of age, the record closest to the whole week was utilized.
The average collection interval of boars was 9 d, and 75% of the records in this study had a collection interval between 6 to 8 d. Collections from boars with <6 d of rest accounted for 5% of the records. The coefficient of determination of regression analysis between the collection interval and the TSC was 0.0034; thus, no relationship between the collection interval and the TSC was assumed here.
Statistical Analysis
Parameters were estimated for TSC by age of boar classification under a random regression model using DxMRR (Meyer, 1998). The model included breed, collector, and year-season as fixed effects and additive genetic effects, permanent environmental effect of boar, and residual as random effects. Random regression models were fitted to evaluate all combinations of first-through seventh-order polynomial covariance functions for the fixed effects of age of boar classification, additive genetic, and permanent environmental effects. This resulted in the evaluation of 343 models. Methods to reduce the order of orthogonal polynomials were studied using eigenvalues (Meyer and Hill, 1997; Schaeffer, 2000). However, the absolute standard is ambiguous, and the number of effective eigenvalues was different for every fitted model. Here, all combinations from first to seventh orders for fixed, additive genetic, and permanent environmental effects were analyzed. Goodness of fit for models was tested using Akaike’s Information Criterion (AIC) and the Schwarz Criterion (SC):
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where p is the number of parameters estimated, N is the sample size, and r(X) is the rank of the coefficient matrix of fixed effects (Meyer, 2001a).
The general random regression model was:
where yij is record j from animal i, Fij is a set of fixed effects, wij is the standardized (– 1 to 1) age at recording, 
n(wij is Legendre polynomial of age n, ß n represents the fixed regression coefficients that model the population mean, 
in represents the random regression coefficients for additive genetic effects, 
in represents the random regression coefficients for permanent environmental effects, and 
ij represents random residuals. The corresponding orders of fit are denoted as kF, kA, and kP.
In matrix notation,
where
- y
- vector of N observations measured on ND animals,
- b
- vector of fixed effects (including Fij and ß n),
- a
- vector of kA x NA additive genetic random regression coefficients,
- p
- vector of kP x ND permanent environmental random regression coefficients,
- e
- vector of N residuals,
- X, Z, and C
- design matrices relating elements of y to elements of b, a, and p, respectively, and
- kA and kP
- the order of fit for a and p and corresponding genetic and permanent environmental covariance function GA and EP.
The variance-covariance matrix of random effects was:
where KA and KP are matrices of coefficients of covariance functions for additive genetic and permanent environmental effects. Matrix A is the numerator relationship matrix, and is an identity matrix. It is assumed that the residuals have a mean of zero and a common variance (
2e).
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RESULTS AND DISCUSSION
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The mean and standard deviation of TSC were 111.69 x 109 and 42.40, respectively. The average TSC increased linearly with age. However, fluctuations were observed after approximately 140 wk of age because of decreasing numbers of records. Standard deviations of the average TSC were consistent over time (Figure 2).
The random regression model that fitted kF = 6, kA = 5, and kP = 7 coefficients for fixed, additive genetic, and permanent environmental effects showed the largest log likelihood value. This model was the fourth best-fitting model based on AIC and the 52nd best-fitting model based on SC. Generally, log likelihood value will increase as the number of parameters in the model increases. Therefore, log likelihood values are less conservative than AIC and SC values, which are weighted by the number of parameters estimated in the model (Table 2). The SC is stricter than AIC. The AIC showed the best fit when kF = 6, kA = 4, and kP = 7, and this was the third best-fitting model based on log likelihood and 20th best-fitting model based on SC. The SC showed the best fit when kF = 4, kA = 2, and kP = 7, and this model was ranked the 10th best-fitting model by log likelihood and second best-fitting model by AIC. Considering the conservative nature of SC and the relative ranking by AIC, this model may be the best based on overall fit.
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Table 2. Order of fit for fixed (kF), additive genetic (kA), and permanent environmental (kP) effects, number of parameters (p), log likelihood (logL; – 77,000), Akaike’s Information Criterion (AIC; +154,300), Schwarz Criterion (SC; +154,000), and ranks of log likelihood, AIC, and SC
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Log likelihood, AIC, and SC values (Figure 3) for all combinations of kA and kP showed a similar pattern in that their values tended to decrease as the order of the orthogonal polynomials increased. However, second-and third-order orthogonal polynomials for additive genetic had the lowest AIC and SC values and, hence, the best fit.
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Figure 3. Log likelihood, Akaike’s Information Criterion (AIC), and the Schwarz Criterion (SC) values by polynomial order of additive genetic effects and polynomial order of permanent environmental (PE) effects.
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Additive genetic and permanent environmental effects and eigenvalues for the 3 best-fitting models are shown in Table 3. Based on the number of nonzero eigenvalues (
) or eigenvalues relatively closer to zero, 1) the model with kF = 6, kA = 5, and kP = 7 could be reduced to 1 with kA = 3 and kP = 5; 2) the model with kF = 6, kA = 4, and kP = 7 could be reduced to kA = 2 and kP = 4; and 3) the model with kF = 4, kA = 2, and kP = 7 could be reduced to kA = 2 and kP = 6. The methods to reduce the orders of orthogonal polynomials were studied using eigenvalues (Meyer and Hill, 1997; Schaeffer, 2000). However, the absolute standard is ambiguous, and the number of effective eigenvalues was different in every result for each model fitted. This makes it difficult to determine the optimum order of orthogonal polynomials for the models studied here.
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Table 3. Estimates of variances (diagonal), covariances (below diagonal), and correlations (above diagonal) between random regression coefficients and eigenvalues ( ) of the coefficient matrix for models with order of fit of 6, 5, 7; 6, 4, 7; and 4, 2, 7 for fixed, additive genetic, and permanent environmental effects, respectively
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Heritability estimates over week of age are presented in Figure 4. These values are the means and standard deviations of heritability at each week from all 343 model combinations of first- to seventh-order orthogonal polynomials. Heritability estimates for TSC ranged from 0.27 to 0.48. These values strongly agreed with the estimate of repeatability for this trait (0.37) reported previously by Oh et al. (2006). Standard deviations tended to decrease from 33 wk of age to about 45 wk, maintained consistent intervals until 100 wk of age, and then increased rapidly. Heritability of TSC tended to increase as boars grew older (Figure 4). Huisman et al. (2002) reported a similar increase in heritability estimates in an evaluation of pig BW.
Heritability estimates for TSC here were similar to those reported in the literature. Masek et al. (1977) estimated a repeatability of 0.24 using a 2-factorial hierarchical analysis of variance. Du Mesnil du Buisson et al. (1978) reported that the heritability for the number of spermatozoa produced per ejaculate under comparable collection rate conditions was 0.35. Huang and Johnson (1996) estimated repeatability of total number of sperm as 0.26 for 3 collections/wk and 0.16 for daily collections. Brandt and Grandjot (1998) reported a heritability of 0.24 and a repeatability of 0.46 for number of sperm cells.
Three-dimensional graphs (Figure 5) showed similar trends for covariance components for models with 1) kF = 6, kA = 5, and kP = 7; 2) kF = 6, kA = 4, and kP = 7; and 3) kF = 4, kA = 2, and kP = 7. Genetic variances tended to increase with age for each model. Additive genetic covariance estimates between ages decreased as the interval between ages increased. This was in contrast to permanent environmental and phenotypic variances that were relatively consistent over age of boar, and it was linked to the increase in heritability estimates. Graphs of permanent environmental effects for each model were similar, although values differed. Sudden increases in permanent environmental variances and covariances after 140 wk of age may be due to the limited amount of data for those ages. Graphs of phenotypic covariances were similar to those observed for permanent environmental covariances. Residuals were assumed to be homogeneous across ages of boar. However, Meyer (2001b) and Lidauer and Mäntysaari (2001) suggested adjustments be made for heterogeneous residual variances.
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Figure 5. (Co)variance components between ages with different polynomial order of fits for fixed, additive genetic, and permanent environmental effects, respectively.
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Genetic correlations, such as genetic covariances, were high between adjacent ages and decreased as the interval between ages increased. The polynomial of order kF = 6, kA = 4, and kP = 7 showed the highest genetic correlations even between distant ages. Genetic correlations ranging from 0.4 to 0.5 from the model with polynomials of order kF = 4, kA = 2, and kP = 7 decreased as the intervals of ages increased. These results indicate that later performance may be harder to predict accurately from records at an early age.
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IMPLICATIONS
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Genetic variance of total sperm cells per ejaculate increased during the productive life of the boar, resulting in heritability estimates increasing from 0.27 to 0.48. Genetic correlations between total sperm cells per ejaculate at different ages were larger for adjacent ages. Random regression models with comparatively high-order polynomials for fixed, additive genetic, and permanent environmental effects provided the best fit.
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Footnotes
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2 Present address: Halifax Community College, 100 College Drive, Weldon, NC 27890. 
1 Corresponding author: todd_see{at}ncsu.edu
Received for publication April 29, 2005.
Accepted for publication November 4, 2005.
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LITERATURE CITED
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Brandt, H., and G. Grandjot. 1998. Genetic and environmental effects on male fertility of AI boars. Proc. 6th World Cong. Genet. Appl. Livest. Prod. 23:527–530.
Du Mesnil du Buisson, F., M. Paquignon, and M. Courot. 1978. Boar sperm production: Use in artificial insemination—A review. Livest. Prod. Sci. 5:293–302.
Huang, Y. T., and R. K. Johnson. 1996. Effect of selection for size of testes in boars on semen and testis traits. J. Anim. Sci. 74:750–760.[Abstract]
Huisman, A. E., R. F. Veerkamp, and J. A. M. van Arendonk. 2002. Genetic parameters for various random regression models to describe the weight data of pigs. J. Anim. Sci. 80:575–582.[Abstract/Free Full Text]
Jamrozik, J., and L. R. Schaeffer. 1997. Estimates of genetic parameters for a test day model with random regressions for yield traits of first lactation Holsteins. J. Dairy Sci. 80:762–770.[Abstract/Free Full Text]
Lidauer, M., and E. Mäntysaari. 2001. A multiplicative random regression model for test-day data with heterogeneous variance. Proc. 2001 Interbull Meeting, Budapest, Hungary. Bulletin 27:167–171.
Masek, N., J. Kuciel, J. Masek, and L. Maca. 1977. Genetical analysis of indicators for evaluating boar ejaculates. Acta Universitatis Agriculturae. Facultas Agronomica, Brno. 25:133–139.
Meyer, K. 1998. "DXMRR" – A program to estimate covariance functions for longitudinal data by restricted maximum likelihood. Proc. 6th World Cong. Genet. Appl. Livest. Prod. 27:465–466.
Meyer, K. 2001a. Estimates of direct and maternal covariance functions for growth of Australian beef calves from birth to weaning. Genet. Sel. Evol. 33:487–514.[Medline]
Meyer, K. 2001b. Estimating genetic covariance functions assuming a parametric correlation structure for environmental effects. Genet. Sel. Evol. 33:557–585.[Medline]
Meyer, K., and W. G. Hill. 1997. Estimation of genetic and phenotypic covariance functions for longitudinal or ‘repeated’ records by restricted maximum likelihood. Livest. Prod. Sci. 47:185–200.
Morant, S. V., and A. Gnanasakthy. 1989. A new approach to the mathematical formulation of lactation curves. Anim. Prod. 49:151–162.
Oh, S.-H., M. T. See, T. E. Long, and J. M. Galvin. 2006. Estimates of genetic correlations between production and semen traits in boar. Asian-Australasian J. Anim. Sci.
Olori, V. E., W. G. Hill, B. J. McGuirk, and S. Brotherstone. 1999. Estimating variance components for test day milk records by restricted maximum likelihood with a random regression animal model. Livest. Prod. Sci. 61:53–63.
Schaeffer, L. R. 2000. Random regression models. Lecture notes. Available: http://www.aps.uoguelph.ca/~lrs/ANSC637/LRS14/ Accessed Oct. 31, 2005.
Strabel, T., and I. Misztal. 1999. Genetic parameters for first and second lactation milk yields of Polish Black and White Cattle with random regression test-day models. J. Dairy Sci. 82:2805–2810.[Abstract]
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Abstract
INTRODUCTION
