# Comparison of Estimated Genetic Parameters Between Random Regression Model and Multi-Trait Animal Model

North Carolina State University Swine Genetics Research from 2004-2005.

By S.H. Oh and M.T. See

Evidence has been reported that changes in animal performance with increasing age are influenced by genetic factors. Animal breeders are interested in genetic parameters that describe the change of traits over time. Analysis of these changes can be undertaken using repeatability (Henderson, 1984), multiple trait (Reents et al., 1995) or random regression models (RRM). Random regression (Meyer, 1998) allows for the calculation of (co) variances at every age. Multiple trait animal models have traditionally been used for traits measured over time by defining observations at distinct ages as different traits. However, computational requirements need to explain the number of traits equal to the number of ages (Meyer and Hill, 1997). Therefore, records collected over ages are often analyzed as repeated measurements or as different traits that are separated by specific intervals. The objective was to compare the (co) variance of total sperm cells (TSC; 10^{9}) over the active lifetime of AI boars between random regression model and multi-traits animal model.

**Materials and Methods**

*Data source.*Data were provided by Smithfield Premium Genetics. 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 years. 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. Total number of records and animals for random regression model were 19,629 and 1,736, respectively. Observations were removed when the number of data at a given age of boar classification time point was less than 10 records, or TSC were missing, zero or less than zero (Figure 1). Weights of ejaculates were measured from 1998 to 2002 with approximately one-half recorded in 2000. Records for multi-traits animal model analysis were edited to include only records produced at 9, 12, 15, 18, 21, 24, and 27 months of age, and were used as separate traits. Number of observations at 9, 12, 15, 18, 21, 24, and 27 months of age were 305, 413, 370, 306, 248, 200 and 109, respectively (Table 1). Number of animals with valid records was 750. Frequency of records was highest at 12 months of age, decreasing gradually over time.

*Statistical analysis of multi-trait animal model analysis.* Variance components for the multiple trait analysis were estimated by derivative free REML using MTDFREML (Boldman et al., 1995). Fixed effects for the model were year-season, breed, collector, and farm. A permanent environmental effect for boar was included to account for repeated measures. The (co)variance structure was estimated from twenty-one separate five-trait analyses. Convergence was considered to have been reached when the variance of the -2 log likelihood in the simplex was less than 1×10^{-9}. Twenty-one combinations of five trait analyses were conducted resulting in means for each parameter estimate, and standard deviations were considered as standard errors.

*Statistical analysis of random regression model analysis.* Parameters were estimated for TSC by age of boar classification under a random regression model using DxMRR (Meyer, 1998). 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. Goodness of fit for models was tested using Akaike’s Information Criterion (AIC) and Schwarz Criterion (SC).

**Results and Discussion**

*Multi-traits animal model analysis.* Heritability estimates were 0.28, 0.29, 0.26, 0.27, 0.30, 0.79, and 0.41 for 9, 12, 15, 18, 21, 24, and 27 months of age, respectively (Table 2). Heritability of TSC at 24 months of age was high because of high genetic variance at the age and may be due in part to selection of records for specific age points. Genetic correlations between measures of TSC at different ages averaged 0.64. Genetic correlations between adjacent ages were higher than those between more distant ages. Decreasing genetic correlations with increasing age may also be due to the limited amount of data and the selection of records to defined age ranges.

*Random regression model analysis.*The random regression model, fitting 6th, 5th, and 7th order for fixed, additive genetic and permanent environmental effects showed the largest log likelihood value. This model was the 4th best fitting model based on AIC and the 52nd best fitting model based on SC. AIC showed best fit when, respectively, 6th, 4th, and 7th order fixed, additive genetic and permanent environmental effects were fit. This was the 3rd best fitting model based on log likelihood and 20th best fitting model based on SC. Schwarz Criterion showed the best fit when 4th, 2th, and 7th order polynomials were fit for fixed, additive genetic and permanent environmental effects, respectively. This model was ranked with the 10th best fit by log likelihood and 2nd 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. Heritability estimates over week of age are presented in Figure 2. These values are the means and standard deviations of heritability at each week from all 343 models combinations of first to seventh order orthogonal polynomials. Heritability estimate for TSC ranged from 0.27 to 0.48. Standard deviations tended to decrease from 33 weeks of age to about 45 weeks, maintained consistent intervals by 100 weeks of age, and then increased rapidly. This increase in variance closely follows the numbers of TSC records over age as shown in Figure 3. Comparing Figures 2 and 3 it would also appear that heritability tends to increase when there is less information. Huissman et al. (2002) reported a similar observation in an evaluation of pig body weights. Heritability estimates in this study were similar to those reported in the literature. Masek et al. (1977) estimated 0.24 as repeatability using two-factorial hierarchical analysis of variance. Du Mesnil du Buisson et al. (1978) reported that the heritability for the number of spermatozoa produced per ejaculate in comparable collection rate conditions was 0.35 even though the standard deviations were too high to affirm interpreting the results. Huang and Johnson (1996) estimated repeatability of total number of sperm (billion) as 0.26 for three collections per week, and 0.16 for daily collections. On the other hand, Brandt and Grandjot (1998) reported that heritability and repeatability of number of sperm cells was 0.24 and 0.46 on average, respectively.

*Comparison between models.*Estimates of genetic parameters from both multiple trait and random regression methods would indicate that measures of TSC at different ages are genetically different traits. Figure 2 shows the comparison of heritability estimates between the three best fit models determined from random regression model analysis and the evaluation of seven ages by multi-trait animal model analysis. The results are very similar except for the heritability at 24 months of age from MTDFREML that had very high genetic variance. Other than 24 months the results are consistent but it appears that the multiple trait method resulted in an overestimation of heritability of TSC. This overestimation may be due to the reduced amount of data or due to age classifications. The ability to accurately estimate genetic correlations between different ages is reduced by limiting records to specific ages. Therefore, multiple traits methods may not be appropriate for analyzing longitudinal data. However, this method may be improved with sufficient numbers of records at each age and availability of computer resources. Random regression analysis provides much more detail with regard to the changes of the variance components with age. Genetic correlations between TSC 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.

**Conclusions**

These studies conclusively show there is an opportunity for genetic selection on semen traits. Estimates of genetic parameters would indicate that measures of TSC at different ages are genetically different traits. However, the ability to accurately estimate genetic correlations between different ages is reduced by limiting records to specific ages. Therefore, multiple traits methods may not be appropriate for analyzing longitudinal data. This method may be appropriate with sufficient numbers of records at each age and availability of computer resources. Random regression methods are the most appropriate to analyze semen traits as they are longitudinal data measured over the boars lifetime. Additional work is needed to understand the relative economic importance of semen traits in the development of breeding objectives.

**References**

Boldman, K. G., L. A. Kriese, L. D. Van Vleck, C. P. Van Tassell and S. D. Kachman. (1995). A Manual for Use of MTDFREML. A Set of Programs to Obtain Estimates of Variances and Covariances [Draft]. U.S. Department of Agriculture, Agricultural Research Service.

Brandt, H. and G. Grandjot. 1998. Genetic and environmental effects on male fertility of AI boars. Proc. 6th World Congr. 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.

Henderson, C. R. (1984). Applications of linear models in animal breeding. Univ. of Guelph, Guelph, Canada.

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.

Huisman, A. E., Veerkamp, R. F., and van Arendonk, J. A. M. (2002). Genetic parameters for various random regression models to describe the weight data of pigs. J. Anim. Sci. 80:575-582.

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). Modeling ‘repeated’ records: covariance functions and random regression models to analyse animal breeding data. 6th World Congr. Genet. Appl. Livest. Prod. 25:517-520.

Meyer, K., and Hill, W. G. (1997). Estimation of genetic and phenotypic covariance functions for longitudinal or ‘repeated’ records by restricted maximum likelihood. Livest. Prod. Sci. 47:185-200.

Reents, R., Dekkers, J. C. M., and Schaeffer, L. R. (1995). Genetic evaluation for somatic cell score with a test day model for multiple lactations. J. Dairy Sci. 78:2858-2870.

#### Table 1. Statistic of data analyzed

Age (month) | 9 | 12 | 15 | 18 | 21 | 24 | 27 |
---|---|---|---|---|---|---|---|

Number of records | 305 | 413 | 370 | 306 | 248 | 200 | 109 |

Mean | 85.16 | 106.3 | 114.3 | 117.9 | 115.5 | 116.4 | 125.8 |

SD | 36.42 | 40.21 | 43.89 | 44.81 | 44.83 | 44.19 | 42.06 |

CV | 42.77 | 37.85 | 38.41 | 38.02 | 38.83 | 37.98 | 33.43 |

#### Table 2. Estimates of heritability and genetic correlation for TSC over the active lifetime of AI boars

Age (month) | 9 | 12 | 15 | 18 | 21 | 24 | 27 |
---|---|---|---|---|---|---|---|

9 | 0.28±0.05 | ||||||

12 | 0.73±0.07 | 0.29±0.04 | |||||

15 | 0.77±0.07 | 0.87±0.13 | 0.26±0.03 | ||||

18 | 0.50±0.49 | 0.77±0.18 | 0.71±0.23 | 0.27±0.05 | |||

21 | 0.16±0.53 | 0.52±0.42 | 0.62±0.22 | 0.62±0.29 | 0.30±0.05 | ||

24 | -0.05±0.14 | 0.18±0.26 | 0.29±0.16 | 0.32±0.09 | 0.37±0.11 | 0.79±0.09 | |

27 | 0.21±0.05 | 0.44±0.22 | 0.50±0.14 | 0.49±0.12 | 0.52±0.12 | 0.54±0.05 | 0.41±0.09 |