Selection indices for residual feed intake derived from milk spectra

Improving production efficiency and minimizing the environmental impact of dairy farming are 2 important goals of the dairy industry. Achieving these objectives requires improving the feed-to-milk conversion efficiency. One way to achieve this goal is through genetic selection. However, measuring feed efficiency in commercial herds is currently not feasible. As such, we conducted a study to evaluate the genetic accuracy of various selection indices derived from Fourier transform mid-infrared (FTIR)- spectra or milk composition. We use 7,793 weekly records on 537 genotyped cows (78,964 SNPs), with information on residual feed intake (RFI), and FTIR-spectra. We fitted various types of selection indexes using the complete FTIR-spectra of milk samples. The estimated heritability of RFI was 0.12 ± 0.02. The accuracy of indirect selection using the FTIR-spectra was maximized using a principal components selection index (0.16 ± 0.07), followed by a Lasso-type penalized selection index (0.14 ± 0.06). We determined that an index based on milk spectral data recorded on ~25 daughters produced a progeny average with an accuracy comparable to direct phenotypic selection for RFI. We conclude that indirect selection for RFI using FTIR-spectra data can be effective for sires with progeny; however, future studies with a larger sample size are needed to validate these results.


INTRODUCTION
Dairy cow feed efficiency is a major factor affecting the economic and environmental sustainability of the dairy sector.Several studies have shown that feed efficiency is heritable (Veerkamp et al., 1995;Parker Gaddis et al., 2021;Cavani et al., 2022) and hence can be improved through genetic selection.However, measuring all component traits of feed efficiency is currently not economically feasible on most commercial farms because estimating residual feed intake (RFI) using an energy sink model requires measuring live weight and dry matter intake.One way to generate genetic evaluations for feed efficiency is to use genomic models trained with data collected from feeding trials (Pryce et al., 2014).This strategy uses SNP marker estimates from training data to predict the breeding values of individuals who do not have phenotypic data.Recent studies show that genome-enabled prediction of RFI can offer moderately accurate predictions; however, the prediction accuracy quickly decays for animals distantly related to those that have been phenotyped (Li et al., 2020).Indirect selection based on traits genetically correlated with feed efficiency could offer a complementary approach to genomic prediction.
The RFI of cows measured in experimental conditions has been used, together with Fourier transformed midinfrared spectra (FTIR-spectra) data, to calibrate equations to predict RFI from milk-spectra data, e.g., Berry andCrowley, (2013), McParland et al., (2014), McParland and Berry, (2016).One advantage of this approach is that FTIR-spectra data is routinely collected (and used to predict milk composition) in many countries, including the US, Europe, Australia, and many more countries (ICAR, 0.2020) Most of the studies concerned with predicting feed efficiency using milk FTIR-spectra data have been focused Selection indices for residual feed intake derived from milk spectra H. O. Toledo-Alvarado, 1 * R. J. Tempelman, 2 M. Lopez-Cruz, 3 M. J. VandeHaar, 2 J. E. P. Santos, 4 F. Peñagaricano, 5 P. Khanal, 2,6 and G. de los Campos 3,7,8 * on calibrating equations to predict feed efficiency phenotypes.Furthermore, when multiple models are evaluated, comparisons are often based on the phenotypic correlation of predictions with measured RFI in testing data sets (McParland and Berry, 2016).This approach is appropriate when the objective is to predict feed efficiency at the phenotypic level.However, from a selection perspective, this approach has 2 potential drawbacks.First, phenotypic prediction equations may be sub-optimal from a breeding perspective because the phenotypic prediction of RFI from FTIR-spectra leverages both environmental and genetic covariances between RFI and individual wavelengths.For that reason, Lopez-Cruz et al. (2020) argued that, from a breeding perspective, a better approach might be to develop selection indices (Falconer and Mackay, 1996) that directly predict the genetic component of a selection goal (e.g., RFI) based on indirect measurements (e.g., FTIR-spectra).The distinction between phenotypic and genetic prediction is particularly important in this context because previous studies have shown that estimated heritabilities of individual wavelengths vary substantially across the FTIR spectrum (e.g., Bittante and Cecchinato, 2013;Rovere et al., 2019).Second, model comparisons based on phenotypic correlations may be inadequate because phenotypic correlations may be differentially influenced by environmental and genetic covariances between the measured traits and the selection objective.Thus, from a selection perspective, it is better to compare models based on the accuracy of indirect selection achieved by their genetic rather than phenotypic predictions (e.g., Lopez-Cruz et al., 2020).
Therefore, the overarching goal of our study was to develop and compare alternative strategies to derive selection indices for RFI from FTIR-spectra.We evaluated regularized (Lasso-penalized and principal-componentsbased) selection indices recently proposed by Lopez-Cruz et al. (2020).Our results show that selection indices derived from FTIR spectra can achieve moderate accuracy of indirect selection for dairy cow feed efficiency.

Data
Data from trials conducted at the University of Florida and Michigan State University (7,793 weekly records from 537 Holstein cows) were used in this study.The daily dry matter intake (DMI) of each cow was calculated as the amount of feed offered minus the amount of feed refused multiplied by the dry matter content of the feed in a particular week.Additional data collected included daily milk yield (MY), body weight records, FTIR-derived milk composition 4 times a week, and the FTIR-spectra data associated with milk samples.All milk samples were analyzed using Bentley FTS instruments with 899 spectral points from wavenumber (wn) 649 to 3,998 cm −1 .Animals were genotyped on various available chips and imputed to the standard SNP set included in the Council on Dairy Cattle Breeding (USA) evaluations.After standard filters, we had 78,964 SNPs available for subsequent analyses.We refer the reader to Parker Gaddis et al. (2021), Nehme Marinho et al. (2021), and Khanal et al. (2022) for a more detailed description of the phenotypic and genomic data and the filters applied.
RFI Calculation.An energy sink model was used to estimate the weekly RFI using the net energy content in milk (NE L ) as determined in NRC ( 2001): NE L = (0.0929 × fat% + 0.0563 × protein% + 0.0395 × lactose%) × MY.The weekly metabolic body weight (MBW) was calculated as the weekly average BW 0.75 , and weekly body weight change ΔBW was calculated using a linear regression of (actual or projected) BW on days from the beginning to the end of a week (Berry and Crowley, 2013).To obtain RFI we first fitted the following model where DMI is the weekly dry matter intake, are the fixed effects of parity (primiparous or multiparous) and specific fifth-order Legendre polynomial regressions of DMI on DIM and their interaction, b 1 is the partial regression coefficient of DMI on NE L , b 2 is the partial regression coefficient of DMI on MBW, b 3 is the partial regression coefficient of DMI on change on ΔBW, ration(expt) is the random effect of experiment-specific rations (there were 16 nutritional experiments in Michigan State University (MSU) and 5 in University of Florida (UF), each having more than one ration), whereas testwk is the random effect of test week.The fitted residuals ε ( ) from the above model were used as the RFI phenotypes in subsequent genetic analyses.

Estimation of heritabilities, genetic correlations, and the accuracy of indirect selection
The accuracy of indirect selection (AIS) of an indicator trait (I i , e.g. an individual wavenumber) is the correlation between the indicator trait and the genetic merit for the selection goal, that is Cor g I y i i , .
( )  (Falconer and Mackay, 1996).Therefore, to estimate the AIS of individual wavenumbers, we fitted a sequence of 899 bivariate GBLUP models using RFI as one trait and one wavelength at a time as the other trait.Likewise, we fitted bivariate GB-LUP models using RFI as one of the traits and a milkrelated phenotype (MBW, NE L , fat yield [FY], protein yield [PY], lactose yield [LY], and MY) as the other trait.
The bivariate models used for estimating genetic parameters were as follows: where Y is a (n×2) matrix of phenotypes (observations in rows, traits in columns), µ = (µ 1 , µ 2 )′ is a vector of traitspecific intercepts, X is an (n×p) incidence matrix for the fixed effects of lactation (primiparous vs. multiparous), herd (MSU vs. UF), experiment-ration (a factor with 23-levels), and DIM, B (p×2) is a matrix of coefficients for the fixed effects, Z (n×q) is an incidence matrix connecting the observations and the q = 537 cows, C (q×2) and U (q×2) are matrices containing the permanent and additive-genetic (random) effects of the cows on each of the traits, respectively, and E is a n×2 matrix with error terms.
The random effects were assumed to be normally distributed, specifically: (i) each row of E was assumed to follow a multivariate normal distribution ) where R 0 is a 2×2 environmental co- variance matrix, (ii) each row of C was assumed ( ) where P 0 is a 2×2 covariance matrix for the permanent environmental effects, and, (iii) the vector of additive effects u G G ~, .MVN 0 0 ⊗ ( ) Here, u is a 2q×1 vector containing the two columns of U, G 0 is a 2×2 genetic covariance matrix between traits, and G is a q×q genomic additive relationship matrix (for the 537 cows with genotypes) derived from centered SNPs and scaled to have an average diagonal value of 1 (VanRaden, 2008).The (co)variance matrices R 0 , P 0 , and G 0 were unstructured, and we assumed them to be homogeneous across all records.
For each analysis, we generated a total of 50,000 MCMC samples, the first 10,000 were discarded as burnin and the remaining ones were thinned at an interval of 1/5.Convergence was assessed by visual inspection of trace plots of the MCMC samples of variances and covariances.We used the remaining samples after burnin and thinning to estimate posterior means for heritabilities, genetic correlations and the accuracy of indirect selection of each indicator trait.Since RFI was present in all bi-variate models, for heritability estimation for this trait, we fitted a single-trait model (with the same effects as in the bi-variate model above-described) using the BGLR-R package using the BGLR function.The genetic parameters for all the other traits were derived from bivariate models fitted using the BGLR-R package using the Multitrait function (Pérez and de los Campos, 2014;  Pérez-Rodríguez and de los Campos, 2022).
Alternative selection indices.We evaluated different approaches to predict the genetic merit of RFI based on FTIR milk spectra: the first approach was a standard Selection Index (SI) using the complete FTIR-spectra (x i ) as input, that is where β is a vector containing the weights assigned to each of the measured phenotype in the index.These weights are derived by minimizing the expected squared deviation between the genetic merit for the selection goal ( , u yi e.g. the genetic merit for RFI of the ith genotype) and the index, as follows: The solution to the above-optimization problem can be shown to be ˆ, , β = − P g x xy where P x = Cov(x i ) is a phenotypic covariance matrix of the indicator traits and g x,y is a vector containing the genetic covariances between the indicator traits (899 wn) and RFI.However, since the number of measured phenotypes (and therefore the number of coefficients that need to be estimated) is large and the wavelengths are highly colinear, estimation errors in P x and g x,y could lead to an index having a low heritability and low accuracy of indirect selection (Lopez-Cruz et al., 2020).Therefore, our second approach was a Lasso 1-penalized selection index (L-SI) derived from FTIR spectra.Here, the coefficients of the SI were derived using an Lasso 1-penalized objective function: where λ is a penalty parameter (λ = 0 yields the coefficients from the standard SI) and ∑ β is an L-1 norm which induces both shrinkage of estimates and variable selection (Lopez-Cruz et al., 2020).The last approach was a SI using FT-IR-derived principal components (PC-SI) as the indicator phenotypes.To derive PC, we used the singular value decomposition of the FTIR-matrix X = UDV′ where X is a matrix containing the (centered) FTIR-spectra data (observations in rows, wavlengths in columns), U and V are the left-and right-singular vectors, respectively, and D is a diagonal matrix containing the singular values of X.The top-q PCs are the first q columns of U. We computed the singular value decomposition of the FTIR-matrix using the svd() function of the base R package (R Core Team, 2023).The PC-SI model has the form of the SI with x i being a vector containing the first q principal components; we fitted models and evaluated the AIS for models using q=1,2,…,899.

Estimation of the accuracy of indirect selection in cross-validation.
To evaluate the accuracy of indirect selection achieved by each of the SIs we: (i) estimated the genetic parameters needed to derive the coefficients of the indices using a training data set which included 70% of the available data, (ii) and used those parameter estimates to derive the coefficients of each of the indices, (iii) applied the index to the left-out testing set (30%) to derive RFI predictions, and (iv) estimated the accuracy of indirect selection for each of the indices by fitting using testing data bi-variate G-BLUP models with RFI as one of the traits and the index as the other trait.We conducted 50 training-testing partitions.For each partition we sampled at random experiments roughly composing ~70% of the data, those experiments constituted the training data (16 experiments), and the data from the other experiments was used for testing (7 experiments).This procedure guarantees all data from a particular experiment was kept together either within training or testing group.The estimated accuracies that we report correspond to the averages across the 50 training-testing partitions.
Accuracy of sire selection as a function of the number of daughters per sire.Milk spectra are measured in cows' milk and the methods presented in this study can be used to predict genetic merit for the RFI of an individual cow from the milk spectra of the same cow.To evaluate the potential use of FTIR-derived selection indices for sire selection, we estimated the accuracy of progeny means of those indices.This was obtained using: , 1948) where r XG is the accuracy of the progeny average, n is the number of daughters selected, g is the accuracy of indirect selection for RFI estimated for each of the methods above-described.
For this conceptual exercise, we varied the number of daughters from 5 to 100 (Table 3).All the analyses were conducted using the R-software (R Core Team, 2023).

RESULTS AND DISCUSSION
The estimated heritabilities (h 2 ), and genetic correlations (r g ) for RFI, NE L , MY, and milk components are presented in Table 2 (trace plots of genetic and environmental parameters, are presented in the Appendix Figures A2 through A7.)The heritability estimate for RFI (0.12 ± 0.02) was within the range of previously reported estimates for similar populations.For instance, using random regressions, Tempelman et al. (2015) reported heritability estimates for RFI of ~0.10, ~0.14, and ~0.05, at 50 DIM, 100 DIM, and 200 DIM, respectively.Khanal et al. ( 2023) estimated a heritability of 0.24 ± 0.02 in US Holstein heifers across 182 trials conducted between 2014 and 2022 at the STgenetics Ohio Heifer Center.Other estimates of the heritability of RFI have ranged from 0.03 to 0.38, depending on the breed, number of animals, the model used, and other factors (Berry and Crowley, 2013).Among the traits reported in Table 2, MBW was the trait with the highest estimated heritability (0.439 ± 0.068) in agreement with previous studies (Hardie et al., 2017, Lu et al., 2018).Although the estimated genetic parameters were similar to other publications, we emphazie that the relatively small sample size available leads to considerable uncertainty about estimates, as highlighted by the posterior standard deviations reported.(Figure A1) The genetic correlations between the RFI and MBW, NE L , FY, PY, LY, and MY, were not statistically different from zero; this was expected given that these phenotypes are used in the model to derive the RFI; therefore, at the phenotypic level, RFI should be not correlated with those traits.We estimated high genetic correlations between NE L and milk components (FY 0.896 ± 0.031; PY 0.868 ± 0.036; LY 0.829 ± 0.053), and also with milk yield (0.813 ± 0.049).The estimated genetic correlations between FY, with PY, and LY were high (FY-PY 0.562 ± 0.105; FY-LY 0.530 ± 0.128); other studies have estimated high genetic correlations between these traits, for FY-PY values from 0.82 to 0.85 and between FY-LY values from 0.40 to 0.79 in Holstein cows and Holstein-Jersey crosses (Welper and Freeman, 1992;Sneddon et al., 2015).We estimated a very high genetic correlation between PY and LY

Genetic correlations between RFI and individual wavelengths
The heritability estimates varied substantially between wavelengths as did the estimated genetic correlations between wavelengths and RFI (Figure 1).Note, however, that there is a smooth variation in genetic correlations across adjacent wavelengths.This is a direct consequence of the high phenotypic correlation in absorbances between nearby wavenumbers.The highest values of heritabilities were observed in the region from 1,406 to 1,290 cm −1 , with heritability estimates around ~0.20.This region has been associated with C-H groups of aliphatic hydrocarbons and amide III stretching vibrations of proteins (Wang et al., 2016).Bittante and Cecchinato (2013) estimated a heritability of 0.182 ± 0.09 at wavenumber 1,380 cm −1 similar to our results.This peak is related to alkyl bonds in specific methyl bonds.In contrast, Rovere et al. (2019), estimated a heritability of 0.25 ± 0.01 at band 1,372 cm −1 , and for this band an association with acetone has been inferred (Hansen, 1998).In this region, our estimates of genetic correlations and of AIS were not statistically different than zero.
The highest positive genetic correlations were estimated in the region from band 3,506 to 3,260 cm −1 and the highest values of AIS were observed around the wavenumber 3,476 cm −1 , given an estimated genetic correlation with RFI of 0.29 ± 0.13, an h 2 = 0.17 ± 0.03 and an AIS of 0.11 ± 0.05.This region has been associated with alcohols, phenols (O-H bonds), and primary amines (N-H bonds).In contrast with our results, very   low heritabilities have been previously estimated in these regions in other studies.For example, Bittante and Cecchinato ( 2013), estimated heritabilities from 0.009 ± 0.02 to 0.02 ± 0.02 whereas Rovere et al. ( 2019) estimated heritabilities around 0.03 ± 0.005.It should be noted that their populations are rather different, and our study is based on 2 herds, which may not be as representative as the sources of variation in the studies by Bittante and Cechinato (2013) or Rovere et al. (2019).
On the other hand, we estimated the highest negative genetic correlations with RFI for bands between 3,107 to 3,099 cm −1 , averaging −0.25 ± 0.14, with an estimated h 2 = 0.16 ± 0.03, and an AIS = 0.10 ± 0.06.Bittante and Cecchinato (2013) estimated an h 2 = 0.10 within this region which includes wavelengths whose absorbances are characteristic of methine bonds C = CH 2 .
For the wavelength region 1,786 to 1,725 cm −1 (fat A) known to be associated with fat (Kaylegian et al., 2006;Lynch et al., 2006), we estimated a relatively low AIS of ~0.05 ± 0.05.For fat B, spanning 2,870 to 2,778 cm −1 , we observed a similar AIS around ~0.04 ± 0.05.A similar situation was observed for regions usually associated with protein (1,547 to 1,478 cm −1 ) and lactose (1,049 to 1,042 cm −1 ) which detect nitro-compounds (Bittante and Cecchinato, 2013).Even though the heritabilities for the bands in these regions were moderate (h 2 = 0.16 ± 0.3 and 0.18 ± 0.03), the estimates for the genetic correlation with RFI were low with high standard errors (r g ~-0.05 ± 0.16 and −0.05 ± 0.15 respectively) leading to a low AIS of 0.02 ± 0.06 for both regions.In agreement with our results, several wavelengths in this area have been reported to be moderately heritable (~0.20 Bittante and Cecchinato, 2013) with differences in the variance com-ponents and heritabilities across parities and different stages of lactation (Rovere et al., 2019).Considering that several milk components are associated with this region (Bittante and Cecchinato, 2013) it was not surprising to estimate genetic correlations with RFI close to zero and consequently low AIS of RFI.This happens because, as noted earlier, RFI is defined to be phenotypically independent of milk energy which is a function of milk components.

Selection indices derived from FTIR-spectra data
We used the estimated phenotypic and genetic correlations to derive various SIs for RFI obtained from the spectra.The results are presented in Figure 2. The standard selection index using the complete spectrum (SI 899 wn) had a very low accuracy (AIS = 0.023 ± 0.01).This is due to the index having a near-zero heritability, which is a sign of over-fitting.The standard selection index was designed for prediction using a few number of predictors, but is suboptimal when using high dimensional multivariate predictors.The principal-components selection index (PC-SI) achieved the maximum AIS with 201 principal components (Figure 2); for that number of components, the AIS was 0.159 ± 0.07.With a smaller number of predictors, 47 active wavelengths, the penalized selection index (L-SI) achieved an AIS of 0.137 ± 0.06 (Figure 2).After reaching its maximum value, the AIS tends to gradually decrease thus providing evidence that overfitting begins being a problem.The relative efficiency of the optimal PC-SI (L-SI), i.e., the ratio of the accuracy to the square-root of the heritability of RFI was 0.44 (0.35).Conceptually, our results are similar to those reported by Lopez-Cruz et al. (2020) for crop imaging data in that PC-SI and L-SI achieve higher AIS than any individual wavelength and much higher than a traditional (non-penalized) SI using the complete spectrum.
For US Holsteins, Li et al. (2020) reported a that genomic predictions for RFI had a reliability of 0.34 for animals with phenotypes and 0.13 for all genotyped animals.These reliabilities cannot be directly compared with the AIS that we report because the reliabilities reported by Li et al. are model-based and our estimates of AIS were derived using testing data.The greatest limitation for genomic prediction of RFI was the size of the data set used to estimate marker effects.The same limitation applies for the SIs considered in this studies because those were derived from a relatively small training data set for which RFI and milk spectra were measured in the same cows.

Accuracy of progeny averages
The AIS for RFI that a single milk-spectra record can deliver is limited because the SIs considered in this study do not borrow information between animals in the way that traditional or genomic prediction does.However, the spectra-derived for RFI could be used as traits in genetic evaluations, thus getting the benefit of leveraging the spectra data that is available for large number of animals and the power offered by genetic evaluations that enable borrowing of informations between realtives.To illustrate this, we estimated the accuracy of progeny averages for SIs with AIS ranging from 0.10 to 0.25 (e.g., PC-SI or L-SI).The results, which are presented in Table 3, show that progeny averages of PC-SI or L-SI penalized selection indexes derived from 25 daughters can offer an accuracy close to 0.35.Therefore, even though for a single record the AIS of penalized SI for RFI is considerably smaller than the square-root of the heritability of RFI, progeny averages of spectra-derived predictions of RFI can be moderately accurate.This is relevant because, as noted, FTIR-spectra is routinely recorded in commercial farms; thus, making the derivation of FTIR-derived SIs for RFI at large scale feasible.
Progeny averages are not available for young bulls, and predictions based on a small number of daughters (e.g., < 10) have low accuracy (Table 3).To address this, one possibility would be to use genomic prediction for such animals.Another approach could be using spectraderived predictions of RFI derived using PC-SI or L1-SI as traits in genetic evaluations thus enabling borrowing of information between relatives.

Final considerations
In this manuscript we investigated ways to predict genetic merit for RFI using milk-spectra data.Addressing whether RFI should be a selection goal or what may be the indirect response to selection for RFI is outside of the scope of this manuscript.However, our study provides information that can inform those questions.At the phenotypic level, RFI is orthogonal to energy requirements and (approximately) uncorrelated with the production traits that are used to estimate those energy requirements.Although RFI is not guaranteed to be genetically uncorrelated with production traits, we do not have evidence supporting nonzero correlations between RFI and production traits (see Table 1).Therefore, we think it is reasonable to expect that selection for RFI would not result in a sizable indirect response to selection on standard production traits such as milk yield, fat% or protein%.However, we reported many nonzero genetic correlations between RFI and several wavelengths (Figure 1) which have been associated with different fatty acids and proteins.Therefore, one could expect that selection for RFI may lead to changes in milk composition for phenotypes (e.g., in the fatty acid or protein proviles) that are not considered in the derivation of RFI and may relevant from an industrial or nutritional point of view.

CONCLUSIONS
Genetic selection for feed efficiency in dairy cattle is important.However, measuring RFI in commercial farms remains infeasible.Genomic prediction has been proposed as an approach to develop genetic evaluations for RFI.However, developing large reference data sets to derive genomic predictions remains challenging.A complementary approach is to predict genetic merit for RFI using correlated traits.In this study, we show that milk-spectra, a phenotype that is routinely collected in dairy farms, can be used to predict the genetic component of RFI of cows with moderate genetic accuracy (relative efficiency ~0.4).These predictions can be used as 'daughter phenotypes' in the evaluation of sires.Our results suggest that such approach can achieve moderately high accuracy for sires with more than 25 daughters.However, future studies, with larger sample size are needed to validate our results, and to investigate possible ways of combining genomic prediction with the indices discussed in this study.
Toledo-Alvarado et al.: INDIRECT SELECTION FOR RFI USING SPECTRA Figure 1.Heritability, genetic correlation, and the accuracy of indirect selection (AIS) for RFI by wavelength (±posterior sd).
Figure 2. Accuracy of indirect selection for RFI using a principal components selection index (PP-SI) or a penalized Lasso selection index (L-SI).The horizontal axis indicate either the number of principal components used (for the PC-SI) or the number of wavelengths active in an L1-penalized selection index (for the L-SI).

Figure A3 .
Figure A3.Trace plots from MCMC sampling over iterations for genetic variance (VarG), Permanent environmental variance (VarPe) and genetic covariance (CovPe), in bivariate models for RFI and milk energy (NE L ).

Table 1 .
Toledo-Alvarado et al.: INDIRECT SELECTION FOR RFI USING SPECTRA Descriptive

Table 3 .
Accuracy of progeny testing using either direct or indirect selection for RFI different values of the accuracy of indirect selection as a function of number of progeny * The columns provide the accuracy of progeny testing assuming that the genetic merit of RFI of the cows is predicted from spectra data using indices with an accuracy of inderct selection of 0.10, 0.15, and 0.20.** Accuracy of direct selection, assuming that RFI is measured in cows.

Table 2 .
Heritability (diagonal, ± posterior sd), and genetic correlation (above the diagonal, ± posterior sd) estimates for RFI, metabolic body weight (MBW, kg), milk energy (NE L ), milk components and milk production : Estimated accuracy of either direct (RFI) or indirect (for the other 6 traits) phenotypic selection for RFI.The results in italics correspond to estimates for which a 95% Bayesian credibility region included zero.Phenotypic and environmental correlations and plots of the (estimated) posterior distribution of the correlations are presented in Appendix 1. *