Genetic parameters for various random regression

models to describe total sperm cells per

ejaculate over the active lifetime of boars

 

S.H. Oh, M.T. See,

North Carolina State University, Department of Animal Science

T.E. Long, and J.M. Galvin

Smithfield Premium Genetics, Roanoke Rapids, NC

 

Summary

The objective of this study was to model the variances and covariances of total sperm cells (× 10⁹) over the active lifetime of AI boars. Data from boars (n = 834) selected for AI were provided by NPD USA. Total number of records and animals were 19,629 and 1,736, respectively. Parameters were estimated for total sperm cells by age of boar classification under a random regression model using the Simplex method and DxMRR procedures. The analysis model included breed, collector and year-season as fixed effects. Random effects included additive genetic effect, permanent environmental effect of boar, and measurement error. All measurement errors were assumed to be equal. Observations were removed when the number of data at a given age of boar classification was less than 10. 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 environment effects. In this study, random regression models were fitted to evaluate all combinations offirst through seventh order polynomial covariance functions.Goodness of fit for models was tested using Akaike's Information Criterion and 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 Schwarz Criterion was by fitting sixth, fourth, and seventh, and fourth, second, and seventh order polynomials for fixed, additive genetic and permanent environmental effects, respectively. Heritability estimates for total sperm cells ranged from .27 to .48 across age of boar classifications. In addition, heritability for total sperm cells tended to increase with age of boar classification. Observed heritability for total sperm cells was cyclic over the active lifetime of boars and may be due in part to number of observations across seasons limiting our ability to correct for seasonal effects on sperm production.

 

Introduction

Artificial insemination plays an important role in animal breeding by allowing greater utilization of genetically superior sires. It has been shown that there is an opportunity for genetic improvement of male fertility traits (Oh et al., 2003; Brandt and Grandjot, 1998). However, the genetic control of semen traits in pigs has not been extensively studied. Moreover, total sperm cells per ejaculate (TSC) are longitudinal data where volume changes 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 thoroughly analyze this type of data due to computational limits. Semen data have also been analyzed similar 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 while 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 (RRM) developed by Meyer (1998) have been extensivelyapplied to the test-day model analysis of milk yield of dairy cattle (Jamrozik and Schaeffer, 1997; Van der Werf et al.,1998;Olori et al., 1999; Strabel and Misztal, 1999;Meyer, 2000). 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 (co)variances of total sperm cells (x 10⁹) over the active lifetime of AI boars.

 

Materials and Methods

 

data

Total sperm cell records (n=19,629) for 834 boars were provided by NPD USA (Table 1). One thousand seven hundred and thirty six individuals were included in the pedigree file. Boars represented three breeds and were housed in two farms. Each farm was similar in numbers of boars of each breed. Thirty-four collectors collected these data over 5 year 4 seasons. Total sperm cells were determined by multiplying semen volume, measured as the weight of the ejaculate volume by total 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 less than 10, or total sperm cells were missing, zero or less than zero. Total sperm cells were measured from 1998 to 2002 with approximately one-half recorded in 2000. Data were distributed evenly across seasons. 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 two observations during one week of age the record closest to the whole week was utilized.

 

 

 

statistical analysis

Parameters were estimated for total sperm cells (TSC) by age of boar classification under a random regression model using DxMRR (Meyer, 1998). In evaluation of boar semen, fixed effects previously considered were boar line, AI station, season, and boar age (Grandjot et al. 1997). The analysis model included breed, collector and year-season as fixed effects; additive genetic effects, permanent environmental effect of boar, and measurement error as random effects.Random regression models were fitted to evaluate all combinations of first through seventh order polynomial covariance functions for fixed effect of age of boar classification, additive genetic, and permanent environmental effects. This resulted in the evaluation of 343 models. Methods to reduce 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 for every fitted model. Therefore, it is not easy to determine the optimum orders of polynomials. In our study, 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 Schwarz Criterion (SC).

 

Where p is the number of parameters estimated and r(X) is the rank of the coefficient matrix of fixed effects (Meyer, 2001a).

 

The general model is:

Where  is the j-th record from the i-th animal,  is the standardized (-1 to 1) age at recording,  is the n-th Legendre polynomial of age,  is a set of fixed effects,  are the fixed regression coefficients to model the population mean,  are the random regression coefficients for additive genetic effects, and  are the random regression coefficients for permanent environmental effects, respectively. , , and  denote the corresponding orders of fit.

 

In matrix notation:

y  =   Xb   +   Za   +   Cp   +   e

Where:

 

y         : vector of N observations measured on N_D animals

b        : vector of fixed effects (including  and )

a         : vector of k_A × N_A additive genetic random regression coefficients

p        : vector of k_R × N_D permanent environmental random regression coefficients

e        : vector of N measurement errors

X, Z and C       : corresponding design matrices

k_A and k_R : the order of fit for a and p and corresponding genetic and permanent environmental covariance function A and R.

 and  are the matrices of coefficients of the covariance function for additive genetic and permanent environmental effects.  is the numerator relationship matrix, and  is an identity matrix. It is assumed that all measurement errors are equal.

 

Results and Discussion

Mean and standard deviation of total sperm cells were 111.69 (×) and 42.40, respectively. Average total sperm cells increased almost linearly with age (Figure 2). However, fluctuations were observed after approximately 140 week of age due to decreasing numbers of records. Standard deviations of average total sperm cells maintained consistent intervals over time.

 

The random regression model, fitting = 6, = 5, and = 7 for fixed, additive genetic and permanent environmental effects showed the largest log likelihood value. This model was the 4^th best fitting model based on AIC and the 52^nd best fitting model based on SC. Generally, log likelihood value will increase as number of parameters in the model increase. Therefore, log likelihood values are less conservative than Akaike’s Information Criterion (AIC) and Schwarz Criterion (SC) values, which are weighted by number of parameters (Table 1). Schwarz Criterion is stricter than AIC. AIC showed best fit when = 6, = 4, = 7, and this was the 3^rd best fitting model based on log likelihood and 20^th best fitting model based on SC. Schwarz Criterion showed the best fit when = 4, = 2, = 7, and this model was ranked 10^th best fitting model by log likelihood and 2^nd best fitting model by AIC. Based on the conservative nature of SC and the relative ranking by the other criterion this model may be the best overall fit. Generally, higher order of fit for the orthogonal polynomials on age was found to be beneficial.

 

Table 1. Order of fit for fixed (), additive genetic (), and permanent environmental () effects, number of parameters (p), log likelihood (-77000), AIC (+154300), SC (+154000), and ranks of log likelihood, AIC and SC

 

 

Figure 1. Number of total sperm cell records by age of boar

Figure 2. Total sperm cells by age of boar

 

Log likelihood values (Figure 3), AIC values (Figure 4) and SC values (Figure 5) for all combinations of  and  were similar in pattern where values tended to be more desirable as the orders were higher. However, in AIC and SC values, 2^nd and 3^rd order fits of additive genetic effects had the lowest values, which are the best fit.

 

Figure 3. Log likelihood values by polynomial order of additive genetic effects and polynomial order of permanent environmental effects

Figure 4. Akaike's Information Criterion values by polynomial order of additive genetic effects and polynomial order of permanent environmental effects

Figure 5. Schwarz Criterion values by polynomial order of additive genetic effects and polynomial order of permanent environmental effects

 

Additive genetic and permanent environmental effects and eigenvalues for the three best fit models are shown in Table 2. Based on the number of non-zero eigenvalues() or eigenvalues relatively closer to zero the model of = 6, = 5, and = 7 could be reduced to the order of = 3, = 5, and the model of = 6, = 4, = 7 could be to = 2, = 4, and the model of = 4, = 2, = 7 could be to = 2, = 6, respectively. The methods to reduce orders of orthogonal polynomials were studied using eigenvalues (Meyer and Hill, 1997; Shaeffer, 2000). However, the absolute standard is ambiguous, and the number of effective eigenvalues was different in every result of fitted model. Therefore, it is not easy to determine the optimum orders of polynomials.

 

Table 2. Estimates of variances (diagonal), covariances (below diagonal), and correlations (above diagonal) between random regression coefficients and eigenvalues () of coefficient matrix, for models with order of fit of 6, 5, 7 and 6, 4, 7, and 4, 2, 7 for fixed, additive genetic and permanent environmental effects, respectively