Genomic selection (GS) has been substantially developed in the last few years (Usai et al., 2009; Ogutu et al., 2012), which estimates the total genetic value for animals utilizing the genomic information of a dense marker map covering all the chromosomes (Meuwissen et al., 2001). In practice, GS has been applied in an attempt to increase the accuracy of breeding values in various fields such as crop and livestock breeding (Ibañez-Escriche and Gonzalez-Recio, 2011; Ogutu et al., 2012; Würschum et al., 2013). For pig breeding, the different implementations of GS have been conducted to increase in accuracy of the breeding values (Simianer, 2009; Cleveland et al., 2010; Lillehammer et al., 2013). As interest in GS increased, the computational cost and the prediction accuracy of GS become important issues especially when the data are high-dimensional, i.e., the number of variables is larger than the number of observations.

Previously genome-wide association studies that utilize individual genes or a few quantitative trait loci (QTL) were popular (Meuwissen et al., 2001; Dekkers, 2002). However, it has a limitation in that it cannot reflect the effects of the neighborhood variables. In fact, the single nucleotide polymorphisms (SNPs) are ordered by physical locations on the chromosomes, therefore, adjacent SNPs are correlated with similar associations (Liu, 2011).

The main objective of this study is to provide significant SNPs that affect the average litter size of Yorkshire pigs using regularized regression approaches and to predict the litter values with selected SNPs in the final model. To accommodate the above property of SNPs properly, we consider some regularized regression approaches: least absolute shrinkage and selection operator (LASSO) (Tibshirani, 1996), fused LASSO (Tibshirani et al., 2005), and elastic net (Zou and Hastie, 2005). It is well known that these methods identify significant variables efficiently, improve the accuracy of the prediction and handle high-dimensional data simultaneously.

In fact, Ogutu et al. (2012) performed gnomic selection using some regularized regression methods. However, there are some key features that distinguish this study from Ogutu et al. (2012). First, in this study, we have analyzed ultrahigh-dimensional data that contain 519 sows with 47,112 SNPs, while Ogutu et al. (2012) considered a dataset that has 3,000 observations with 9,990 SNPs. It is well known that the performance of regularized regressions varies over the degree of dimensionality. Thus, this study can be considered as an extension of high-dimensional case of Ogutu et al. (2012). Second, an important issue of regularized regressions is their implementation when the data are high-dimensional, because it heavily involves solving optimization. To handle such ultrahigh-dimensional data, we have used three regularized regression methods with efficient algorithms introduced by Liu et al. (2010); and hence, many researchers can easily implement the methods for GS with various high-dimensional data. Finally, we have focused on the fussed LASSO. It is designed for a problem with features that can be ordered in some meaningful ways as well as a dataset where the number of features is much greater than the sample size. The abovementioned issues can be considered as main contributions of this study.

The Rural Development Administration (RDA) provided the Illumina Porcine 60K SNP Beadchip on 703 sows and their litter size. Since this study is focused on the litter values of pigs, multiparous sows were used in the analyses. In this section, we explain the details of data in the analysis and the methods used for analysis.

To compare the performance of the three methods, PE and the Pearson correlation between the true litter values and the predicted values were used as measures of the accuracy. The number of non-zero coefficients was also used as an indicator of efficiency. If the structure of the training set is totally different from the arrangement of the validation set because of the limitation of data, then the evaluation of the method might not be reliable. To clarify the comparison for the results of methods, we extracted new samples from the original data using the bootstrap technique with a 35% resampling rate.

The overall results of each method with partitioned bootstrap samples are presented in Table 1 to 3. Table 1 describes PEs by LASSO, fused LASSO, and elastic net, and MFPE for each fold. The average PEs obtained by LASSO, fused LASSO, and elastic net were 0.8675, 0.8476, and 0.8628, respectively. As listed, the fused LASSO outperformed other methods when the explanatory variables were correlated. From Table 2, the fused LASSO provided the highest accuracy over all folds. Table 3 lists the number of non-zero coefficients selected by LASSO, fused LASSO, and elastic net. Overall, the number of non-zero coefficients by the fused LASSO was slightly high, compared to LASSO and elastic net. In the case of fold 4, for example, the consecutively selected variables by the fused LASSO were DRGA0005762, DRGA0005763, DRGA0005767, and DRGA0005770, while the LASSO selected only DRGA0005770. According to the results of the above comparison study, the fused LASSO performed well. Ogutu et al. (2012) analyzed a dataset with 9,900 SNP markers on 3,000 progenies of 20 sires and 200 dams by using various GS methods such as ridge regression, ridge regression best linear unbiased prediction (BLUP), LASSO, adaptive LASSO, elastic net, and adaptive elastic net. In their analysis, elastic net and LASSO worked well for GS, compared to others.

To construct the final model, we used the whole data set, and calculated PEs and correlation values between the given target variable and the predicted value to indicate accuracy. In the results of the final model obtained by the fused LASSO, the number of significant non-zero coefficients among 47,112 SNPs was 1,499 SNPs. Table 4 lists the names of the 10 selected significant SNPs with large estimated coefficients (in absolute value) of the SNP effects. Onteru et al. (2012) provided some important genes for reproductive traits in the QTL regions, where they employed the Bayes C model introduced by Kizilkaya et al. (2010). Figure 4 shows the scatter plot between the predicted values ŷ and the original values y. Note that the sample correlation coefficient between litter size and predicted litter size by the final model was 0.7041. It seems that the fused LASSO is suitable for selecting adjacent SNPs as well as for predicting the values.

In summary, we have considered three regularized regressions for GS with 47,112 SNPs of 519 sows. We compared three methods using PEs and correlation coefficient values, and obtained the final model to predict the litter size of pigs. From the data analysis, we observed that the fused LASSO seems to be a good choice for GS.

Finally, the regression methods for estimating the random effect in a mixed model have recently developed (Onteru et al., 2012; Resende et al., 2012). As a future study, it is worth confirming the effects of a random effect model compared to existing GS methods.