INTRODUCTION:

The choice of an optimal number of components is still a relevant issue for partial least squares (PLS) and principal component regression (PCR) statistical methods when applied to genomic selection (GS). Methods such as degrees-of-freedom (DoF) technique, which is based on statistical information theory (^{KRÄMER & SUGIYAMA, 2011}), or cross-validation (CV) approach, which is based on empirical independent data set samples, have been proposed; however, no comparisons between the two methods, with a focus on genomic prediction, have yet been reported in the literature.

Pork pH traits are directly related to water retention, tenderness, juiciness, and meat appearance. Furthermore, these traits are evaluated after slaughter; thus, the identification of genetically superior individuals for selection can be accomplished through GS methods, such as PLS or PCR. Therefore, we aimed to apply PLS (and also PCR and traditional multiple regression) to GS for pork pH traits (in this case, pH measured at 45min and 24h after slaughter), as well as to identify the optimal number of PLS components based on the DoF and CV methods.

METHODOLOGY:

The data was obtained from an F2 population containing 345 animals whose pork pH was measured at 45min (pH_{45}) and 24h (pH_{u}) after slaughter. These measurements were performed by inserting a glass electrode (DIGIMED, DME-CV1) connected to a previously calibrated pH meter (DIGIMED DM-20) into a *Longissimus dorsi* muscle taken from the region immediately posterior to a pig's last rib.

DNA was extracted from these animals and subjected to genotyping using a low-density customized SNP (single nucleotide polymorphism) chip with 384 markers based on the Illumina Porcine SNP60 BeadChip. These SNPs were selected according to QTL positions previously identified for this population using meta-analyses and fine mapping (^{HIDALGO et al., 2013}; ^{VERARDO et al., 2015}). Thus, although a small number of markers were used, this customized SNPchip ensured an appropriate coverage of the relevant genomic regions in this population. After SNP quality-control analysis, a total of 237 SNPs were considered and distributed as follows on **
Sus scrofa
** chromosomes: SSC1 (56), SSC4 (54), SSC7 (59), SSC8 (30), SSC17 (25), and SSCX (13), with the average distance within each chromosome equal to 5.17, 2.37, 2.25, 3.93, 2.68, and 11.0 Mb, respectively.

The form components t_{i} (i = 1,...,m) of the PLS method are designed to capture the largest possible amount of information arranged in explanatory variables X_{1},...,X_{p} (SNP genotypes, assuming 0, 1, and 2 correspond, respectively, to aa, aA, and AA) in order to predict the dependent variable, Y (pH_{45} and pH_{u}). According to this approach, the regression equation is expressed as

where t_{i} are the column vectors that make up the matrix, T and are estimates of the regression coefficients between Y and T, . The correlation between any pair of components is equal to 0, that is, , . Thus, the PLS method reduces the number of terms in the regression equation (1), since this number is usually less than the number of variables, X. The main difference between the PCR and PLS methods is that the first takes account of only the explanatory variables in the component construction; whereas the second also takes dependent variables into account (^{GARTHWAITE, 1994}).

DoF are used to quantify the intrinsic complexity of a regression method. According to ^{KRAMER & SUGIYAMA (2011}), the complexity of PLS analysis depends upon the collinearity of the predictor variables. The higher the collinearity value, the lower is the complexity; therefore, the lower is the number of DoFs (i.e., the number of components is smaller). ^{KRÄMER et al. (2011)} presented an unbiased estimate of DoFs of PLS with m latent components,, which is given by . To calculate the trace of the derivative in (2), it is necessary to use an algorithm based on the orthogonal decomposition of X, which is unique for fixed y and m (^{KRAMER & SUGIYAMA, 2011}). A more practical way of determining the number of components is CV, by which the original data set is divided into N subsets (folds) and N analyses are performed, such that in each one, one of the subsets is used for validation. Thus, the values predicted by the equations estimated in each analysis can be directly compared with the removed (observed) values. All analyses were performed using R software (R DEVELOPMENT CORE TEAM, 2015) by means of the pcr, pls.model, and pls.cv functions of the plsdof package.

The original population of 345 individuals was fractionated into two different populations: training and validation. In order to separate the groups while taking lower parentage into account, the PEDIGREE VIEWER software was used to provide the two groups with more within-group relationships and fewer relationships between groups. The predicted values from the validation population (**Ŷ _{p}**) were obtained by , with

**X**being the genotypes of the SNP markers of individuals in the validation population and being the vector of estimates of marker effects from training population. Correlation between vectors of the predicted and observed phenotypes in the validation population is called predictive ability and provides the efficiency of a GS method. After the best method (PLS, PCR, and traditional multiple regression) in terms of predictive ability was decided, the absolute values of the estimated marker effects were used to identify possible chromosomal regions (QTLs) directly associated with the studied traits.

_{p}RESULTS:

The curves in figure 1 (with DoF and MSE standing, respectively, for "degrees of freedom" and "cross-validation") indicated that stabilization occurred with 30 components under both methods. Above this value, there was an increase in the complexity of the model, but its performance remained unchanged. Although both methods pointed to the same optimal number of components, the CV method was more expensive in terms of computational demand and; therefore, in practical terms, the DoF method is recommended. However, from a statistical viewpoint, DoF may be preferred because of its theoretical statistical background because CV can be characterized as an empirical method based on computational effort.

Because the optimal number of PLS components was consistently reported to be 30, this number was also used in predictive ability analysis to provide a situation in which the DoF and CV methods were directly comparable. Predictive abilities of the PLS method were 0.84 and 0.81, respectively, for the pH45 and pHu traits; whereas, those for PCR method were, 0.78 and 0.76, respectively. Finally, the traditional ordinary least squares (OLS) multiple regression method provided lower values, 0.39 and 0.33, respectively, for pH45 and pHu.

DISCUSSION:

According to ^{AZEVEDO et al. (2014}), the advantage of PLS over PCR is due to the fact that the former method takes account of variable response (phenotypic observations) when processing components' composition, whereas the PCR method considers only the covariates themselves. Poor performance of the OLS method may be due to the lack statistical features to control the problems of multicollinearity, thereby leading to overfitting problems (^{SILVEIRA et al., 2014}) when compared to the PLS and PCR methods.

Besides the predictive analysis inherent to genomic selection, it is also interesting to identify the SNP markers with higher estimated effects (^{CAMPOS et al., 2015}) because the PLS method can directly estimate these for each marker. These markers may reveal relevant chromosomal regions affecting each trait. For both phenotypes (pH_{45} and pH_{u}), the SNP markers ALGA0026103, ALGA0026237, ALGA0026100, ALGA0026241, and ALGA0026109 located on chromosome 4 (between 80 and 90Mbp) presented stronger effects as compared with other markers. Other studies have reported the relevance of this same chromosomal region. Among these, ^{STRATZ et al. (2012)} and ^{MA et al. (2013}) stands out, who also used F2 populations involving commercial lines and identified QTL for pH_{45} at positions of 79.5 and 88.26Mbp, respectively, and ^{PONSUKSILI et al. (2010}), which reported QTL for pH_{u} at 82Mbps using different populations derived from the Duroc breed.

CONCLUSION:

Both methods (DoF and CV) pointed to the same optimal number of components. From a statistical viewpoint, DoF may be preferable because of its theoretical statistical background because CV is an empirical method based on computational effort. The PLS method is efficient for genomic selection (predictive ability is around 80%) for pork pH traits using only genotypic information based on SNP markers. This method revealed a relevant region (around 80 and 90Mbp) on chromosome 4 that may affect the pork pH values measured at 45min and 24h after slaughter