## Abstract

^{0.75}, and change in BW (ΔBW), along with parity, a fifth-order polynomial on days in milk (DIM), and the interaction between this polynomial and parity in a first-stage model. The residuals from this analysis were considered to be a phenotypic measure of RFI. Estimated partial regression coefficients of DMI on MilkE and on BW

^{0.75}ranged from 0.29 to 0.47 kg/Mcal for MilkE across research stations, whereas estimated partial regression coefficients on BW

^{0.75}ranged from 0.06 to 0.16kg/kg

^{0.75}. Estimated partial regression coefficients on ΔBW ranged from 0.06 to 0.39 across stations. Heritabilities for country-specific RFI were based on fitting second-stage random regression models and ranged from 0.06 to 0.24 depending on DIM. The overall heritability estimate across all research stations and all DIM was 0.15 ± 0.02, whereas an alternative analysis based on combining the first- and second-stage model as 1 model led to an overall heritability estimate of 0.18 ± 0.02. Hence future genomic selection programs on feed efficiency appear to be promising; nevertheless, care should be taken to allow for potentially heterogeneous variance components and partial relationships between DMI and other energy sink traits across environments when determining RFI.

## Key words

## Introduction

**RFI**) has been proposed as a measure of dairy feed efficiency as an alternative to ratio-based (i.e., input/output) measures, partly because of its ability to take into account changes in energy dynamics over the course of a lactation due to BW changes, for example (

**MilkE**), metabolic BW (

**MBW**) defined as BW

^{0.75}, BW changes (

**ΔBW**), and cohort effects. The resulting estimated residuals from this first stage model are deemed to be RFI phenotypes. These RFI phenotypes are then used as the response variables in a second stage quantitative genetics model to estimate heritabilities and, subsequently, breeding values for feed efficiency.

- Pryce J.E.
- Arias J.
- Bowman P.J.
- Davis S.R.
- Macdonald K.A.
- Waghorn G.C.
- Wales W.J.
- Williams Y.J.
- Spelman R.J.
- Hayes B.J.

*J. Dairy Sci.*2012; 95: 2108-2119

- Vallimont J.E.
- Dechow C.D.
- Daubert J.M.
- Dekleva M.W.
- Blum J.W.
- Barlieb C.M.
- Liu W.
- Varga G.A.
- Heinrichs A.J.
- Baumrucker C.R.

*J. Dairy Sci.*2011; 94: 2108-2113

- Vallimont J.E.
- Dechow C.D.
- Daubert J.M.
- Dekleva M.W.
- Blum J.W.
- Barlieb C.M.
- Liu W.
- Varga G.A.
- Heinrichs A.J.
- Baumrucker C.R.

*J. Dairy Sci.*2011; 94: 2108-2113

## Materials and Methods

### Data

**MY**), DMI, BW, fat (

**FAT%**), protein (

**PROT%**), and lactose (

**LACT%**) components, were collected from Holstein cows on research stations within 3 countries: the United States (

**US**), the Netherlands (

**NL**), and the United Kingdom (

**UK**). Data from US were derived from 6 research stations: Iowa State University (

**ISU**; Ames), Michigan State University (

**MSU**; East Lansing), the University of Florida (

**UF**; Gainesville), the University of Wisconsin-Madison (

**UW**), the United States Dairy Forages Research Center (

**FRC**; Madison, WI), and the USDA Animal Genomics and Improvement Laboratory (

**AGIL**; Beltsville, MD). Some of the data provided by ISU has been analyzed previously and described in more detail by

**TGEN**) in Lelystad previously described by

**NBZ**) herd located near Leeuwarden and also previously described by

**ZOM**) based on the work by

**NLN**based on data collected from various nutritional experiments. Data on all variables (i.e., DMI, MY, BW, FAT%, PROT%, and LACT%) from each of these 4 NL studies were summarized on a weekly basis before further analysis.

**LAN**) farm near Edinburgh from 1992 to 2001 and from the Scottish Agricultural College (

**SAC**) Dairy Research Centre based at Crichton Royal Farm near Dumfries with data collection from 2003 to 2011. These data are described in more detail by

### Phenotypes

Country | Station/study^{1}UF=University of Florida, ISU=Iowa State University, MSU=Michigan State University, FRC=USDA Forage Research Center, UW=University of Wisconsin-Madison, AGIL=USDA Animal Genomics Improvement Laboratory, TGEN=‘t Gen, NBZ=Nij Bosma Zaathe, ZOM=data from Zom et al. (2012), NLN=miscellaneous compilation of studies, LAN=Langhill, SAC=Scottish Agricultural College. | No. of cows | No. of lactations | No. of weekly records | Average number of weekly records per lactation | Years of data collection |
---|---|---|---|---|---|---|

United States | UF | 205 | 220 | 1,663 | 7.6 | 2009–2013 |

ISU | 692 | 729 | 8,306 | 11.4 | 2008–2013 | |

MSU | 163 | 192 | 1,807 | 9.4 | 2011–2013 | |

FRC | 115 | 117 | 1,280 | 10.9 | 2009–2011 | |

UW | 412 | 463 | 5,372 | 11.6 | 2007–2013 | |

AGIL | 288 | 448 | 3,617 | 8.1 | 2007–2011 | |

Netherlands | TGEN | 671 | 671 | 11,760 | 17.5 | 1991–1998 |

NBZ | 100 | 100 | 525 | 5.3 | 2003–2004 | |

ZOM | 783 | 1,232 | 10,401 | 8.4 | 1991–2001 | |

NLN | 412 | 531 | 5,795 | 10.8 | 2003–2012 | |

United Kingdom | LAN | 590 | 1,217 | 24,242 | 19.9 | 1990–2001 |

SAC | 462 | 904 | 15,786 | 17.4 | 2003–2011 | |

TOTAL | 4,893 | 6,824 | 90,554 | 13.3 |

### Statistical Analyses

where

*y*is any of the 6 variables; $\text{parity}\times \sum _{k=0}^{5}\text{DI}{\text{M}}^{k}$ are fixed effects of parity (primiparous vs. multiparous) and specific fifth-order Legendre polynomial regressions of

*y*on DIM, whereas ration(expt) are the fixed effects of ration within experiment; test week is the random effect of the midweek calendar date, whereas $\text{animal}\times \sum _{k=0}^{{n}_{k}}\text{DI}{\text{M}}^{k}$ is the random effects Legendre polynomial of order

*nk*≤ 3 of

*y*on DIM for the animal effect. Finally,

*e*is a normally distributed residual. Parity was not fitted for TGEN data, as all TGEN cows were primiparous. The variance of each individual $\text{animal}\times \sum _{k=0}^{{n}_{k}}\text{DI}{\text{M}}^{k}$ term was an

*nk*×

*nk*unstructured covariance matrix with independence assumed across animals. For 18 of the 69 (12 stations × 6 variables − 3 stations missing LACT%) analyses, it was required to specify

*nk*< 3 in Equation [1], as convergence was not otherwise attained; for the other 51 analyses,

*nk*= 3. Upon all such analyses, records of each of the 6 key variables were designated to be residual outliers if the absolute values of the corresponding externally studentized residuals were greater than 5, with 1 exception: if the absolute studentized residual exceeded 5 for DMI and the absolute studentized residual for MY exceeded 3 with the same sign and vice versa (i.e., exceeding 5 for MY and exceeding 3 for DMI, both with the same sign), then both DMI and MY records were kept as provided. This exception was intended to keep biological outliers in the data so as to best reflect energy balance between DMI and its energy sinks (i.e., as based on the other 5 variables), whereas the critical intent of data editing was to correct residual outliers (<0.5% of the data) that were most likely to be recording errors. Outlier date effects for each trait were also targeted if the corresponding date estimate exceeded 3.5 date effect standard deviations (based on the square root of the estimated variance component for date). Any records on the corresponding trait for an outlier date were corrected relative to the estimated effect of the most previous nonoutlier date to preserve any potential temporal trends typically associated with date effects.

^{0.75}. The weekly ΔBW was determined based on a simple linear regression of actual or projected BW on days from the beginning to the end of a week. The ration assignment for any particular week was based on the ration that was fed at the midpoint of the week.

where $\text{parity}\times \sum _{k=0}^{5}DI{M}^{k}$ are the fixed effects of parity (primiparous vs. multiparous) and specific fifth-order Legendre polynomial regressions of DMI on DIM,

*b*

_{1}is the partial regression coefficient of DMI on MilkE,

*b*

_{2}is the partial regression coefficient of DMI on MBW,

*b*

_{3}is the partial regression coefficient of DMI on ΔBW, ration(expt) is the random effect of experiment-specific rations, whereas test week is the random effect of test week with ε being the residual. The estimated residuals $\left(\stackrel{\u02c6}{\u03f5}\right)$ from the corresponding analysis were determined to be the weekly RFI for use in a subsequent second stage quantitative genetic analysis.

where RFI is fitted as a function of overall mean μ with $\text{cow}\times \sum _{k=0}^{{n}_{a}}\text{DI}{\text{M}}^{k}$ being the order

*na*Legendre polynomial on DMI for random additive genetic effects (with correlation based on the numerator relationship matrix with genetic groups), $\text{cow}\times \text{parity}\times \sum _{k=0}^{{n}_{p}}\text{DI}{\text{M}}^{k}$ being the order

*np*Legendre polynomial on DMI for cow-parity-specific lactations (based on the correlation matrix being the identity matrix) to model within random lactation permanent environmental effects, cow being the random intercept of cow (based on the correlation matrix being the identity matrix), and

*e*being the residual with variance ${\sigma}_{e}^{2}.$ This last term (cow) was required to separate between-lactation permanent environmental effects from within-lactation permanent environmental effects when multiple lactations per cow occurred. The Legendre polynomial orders were constrained up to

*na*=

*np*= 3 until the next polynomial was no longer statistically significant based on simple Wald tests on the variance components, or until REML convergence was no longer attainable.

*na*=

*np*= 0) were fitted to animal-specific effects to infer an overall heritability estimate in each case. We also considered additional terms in this joint model:

where Equation [2]

^{RHS}and Equation [3]

^{RHS}pertain to terms on the right hand side of Equations [2] and [3], respectively. Furthermore, ration(expt) × (MilkE MBW ΔBW) are the effects of ration and its interaction effects with MilkE, MBW, and ΔBW. These interaction effects, along with the random intercept term ration(expt) from Equation [2], were modeled as correlated random effects to more reliably infer the potential (co)variability in the energy sink partial regression coefficients across rations while jointly accounting for animal sources of variability (i.e., genetic and permanent environment). This joint model additionally specified station × (intercept MilkE MBW ΔBW) as the random effects of research stations and their interactions with MilkE, MBW, and ΔBW to allow inference on station-specific partial regression coefficients just as we implicitly conducted with the station-specific first-stage energy sink model analyses in Equation [2]. We also specified station-specific residual variances. These 1-step Equation [4] analyses were conducted separately for each country and jointly. Variance components were estimated by REML using the ASREML package (

## Results

### Summary of Key Variables

### Estimated Partial Regression Relationships from First Stage Energy Sink Model

*P*< 0.05) from 0. Furthermore, some of these standard errors were small enough to suggest that some of the partial regression coefficients were different from each other using a simple

*z*-test. For example, to compare any 2 stations

*j*and

*j*’ for

*b*

_{1}(partial regression on MilkE), one could readily determine the

*z*-test statistic as

and declare a significant difference (

*P*< 0.05) if

*z*> 1.96. Nevertheless, these standard errors are badly understated, as discussed herein, and hence formal separation of these coefficients based on

*P*-values is not provided in Table 2.

Country | Station/study^{1}UF=University of Florida, ISU=Iowa State University, MSU=Michigan State University, FRC=USDA Forage Research Center, UW=University of Wisconsin-Madison, AGIL=USDA Animal Genomics Improvement Laboratory, TGEN=‘t Gen, NBZ=Nij Bosma Zaathe, ZOM=data from Zom et al. (2012), NLN=miscellaneous compilation of studies, LAN=Langhill, SAC=Scottish Agricultural College. | MilkE | MBW | ΔBW | Intercept |
---|---|---|---|---|---|

United States | UF | 0.35 ± 0.016 | 0.16 ± 0.007 | 0.13 ± 0.03 | N/A |

ISU | 0.36 ± 0.005 | 0.10 ± 0.003 | 0.30 ± 0.03 | 22.39 ± 0.15 | |

MSU | 0.40 ± 0.009 | 0.10 ± 0.004 | 0.36 ± 0.04 | 22.26 ± 0.21 | |

FRC | 0.34 ± 0.013 | 0.12 ± 0.007 | 0.18 ± 0.06 | 22.74 ± 0.25 | |

UW | 0.44 ± 0.008 | 0.11 ± 0.004 | 0.27 ± 0.04 | 23.39 ± 0.24 | |

AGIL | 0.37 ± 0.008 | 0.08 ± 0.004 | 0.39 ± 0.08 | 22.43 ± 0.16 | |

Netherlands | TGEN | 0.40 ± 0.005 | 0.10 ± 0.002 | 0.08 ± 0.01 | 23.29 ± 0.11 |

NBZ | 0.47 ± 0.010 | 0.07 ± 0.010 | 0.27 ± 0.06 | N/A | |

ZOM | 0.34 ± 0.005 | 0.06 ± 0.002 | 0.07 ± 0.01 | 20.34 ± 0.12 | |

NLN | 0.29 ± 0.005 | 0.07 ± 0.003 | 0.18 ± 0.03 | 20.40 ± 0.42 | |

United Kingdom | LAN | 0.35 ± 0.003 | 0.06 ± 0.001 | 0.35 ± 0.02 | 17.83 ± 0.06 |

SAC | 0.36 ± 0.003 | 0.07 ± 0.001 | 0.06 ± 0.01 | 19.18 ± 0.14 |

^{0.75}, and ΔBW = 0. N/A refers to estimates with unusually large SE or predictions beyond allowable parameter space (i.e., negative intercepts) because DIM range for that station did not include 125 d.

*P*< 0.05) partial regression coefficients for DMI on MBW, ranging from 0.06 to 0.16 kg/kg

^{0.75}. The partial regression coefficients of DMI on ΔBW were also all deemed to be statistically significant (

*P*< 0.05) for all research stations. Furthermore, all such partial regression coefficients were positive, intuitively suggesting that as ΔBW increases, DMI increases and vice versa; nevertheless, there was substantial range (0.07 to 0.36 kg of DMI/kg of ΔBW) in these estimates. These coefficients were particularly heterogeneous within the 2 UK stations (0.06 to 0.35). One could use the same

*z*-test to suggest that some of these partial regression coefficients appeared to be different from each other. Again, this heterogeneity could be a reflection of differences in energy or statistical considerations, as addressed herein.

### Quantitative Genetic Analysis of RFI from Two-Stage Analyses

*na*= 1 and

*np*= 2 on UK,

*na*= 0 (i.e., just intercept) and

*np*= 1 on US,

*na*= 2 and

*np*= 3 on NL data, and

*na*= 2 and

*np*= 3 on the overall data. Based on these fits, DIM-specific genetic variances, within-lactation permanent environmental variances, and heritabilities were estimated. In addition, a between-lactation permanent environmental variance component $\left({\sigma}_{b}^{2}\right)$ was also estimated because some cows had multiple lactations. This term was important for the joint analysis $\left({\stackrel{\u02c6}{\sigma}}_{b}^{2}=0.33\pm 0.05\text{\hspace{0.17em}}\text{k}{\text{g}}^{2}\right)$ and for each country-specific analysis, but appeared to be substantially larger in the US $\left({\stackrel{\u02c6}{\sigma}}_{b}^{2}=0.51\pm 0.17\text{\hspace{0.17em}}\text{k}{\text{g}}^{2}\right)$ relative to NL $\left({\stackrel{\u02c6}{\sigma}}_{b}^{2}=0.22\pm 0.06\text{\hspace{0.17em}}\text{k}{\text{g}}^{2}\right)$ or UK $\left({\stackrel{\u02c6}{\sigma}}_{b}^{2}=0.32\pm 0.07\text{k}{\text{g}}^{2}\right).$

### One-Step Model Analysis

Country | Station/study^{1}UF=University of Florida, ISU=Iowa State University, MSU=Michigan State University, FRC=USDA Forage Research Center, UW=University of Wisconsin-Madison, AGIL=USDA Animal Genomics Improvement Laboratory, TGEN=‘t Gen, NBZ=Nij Bosma Zaathe, ZOM=data from Zom et al. (2012), NLN=miscellaneous compilation of studies, LAN=Langhill, SAC=Scottish Agricultural College. | MilkE | MBW | ΔBW | Intercept |
---|---|---|---|---|---|

United States | UF | 0.34 ± 0.03^{3} | 0.13 ± 0.01 | 0.14 ± 0.04 | 23.64 ± 0.39 |

ISU | 0.34 ± 0.04 | 0.11 ± 0.02 | 0.21 ± 0.06 | 23.82 ± 0.70 | |

MSU | 0.36 ± 0.02 | 0.14 ± 0.01 | 0.25 ± 0.04 | 23.82 ± 0.38 | |

FRC | 0.37 ± 0.03 | 0.12 ± 0.02 | 0.10 ± 0.06 | 23.74 ± 0.39 | |

UW | 0.38 ± 0.02 | 0.10 ± 0.01 | 0.08 ± 0.04 | 24.18 ± 0.29 | |

AGIL | 0.30 ± 0.08 | 0.11 ± 0.04 | 0.06 ± 0.14 | 23.08 ± 1.36 | |

Netherlands | TGEN | 0.26 ± 0.08 | 0.16 ± 0.04 | 0.32 ± 0.12 | 24.35 ± 1.36 |

NBZ | 0.46 ± 0.08 | 0.02 ± 0.04 | 0.29 ± 0.11 | 22.27 ± 1.17 | |

ZOM | 0.28 ± 0.02 | 0.07 ± 0.01 | 0.08 ± 0.02 | 21.80 ± 0.26 | |

NLN | 0.18 ± 0.02 | 0.10 ± 0.01 | 0.14 ± 0.02 | 20.92 ± 0.27 | |

United Kingdom | LAN | 0.20 ± 0.06 | 0.07 ± 0.03 | 0.27 ± 0.08 | 17.83 ± 0.97 |

SAC | 0.18 ± 0.06 | 0.05 ± 0.03 | 0.04 ± 0.08 | 19.11 ± 0.97 |

*P*< 0.05) from each other.

^{0.75}, and ΔBW = 0.

^{2}, 0.0012 ± 0.00022 (kg/kg

^{0.75})

^{2}, 0.0060 ± 0.00095 (kg/Mcal)

^{2}, and 0.013 ± 0.0026, respectively. The estimated standard deviations of these effects can then be determined as the square roots of these variance component estimates; for example, $\sqrt{1.69}=1.3$ kg for intercept, $\sqrt{0.0012}=0.035$ kg/kg

^{0.75}for the partial regression on MBW, $\sqrt{0.0060}=0.077$ kg/Mcal for partial regression on MilkE, and $\sqrt{0.013}=0.11$ for the partial regression on ΔBW. Thus, assuming that ration-specific partial regression coefficients on MilkE are normally distributed, at a mean of around 0.40 kg/Mcal (i.e., based on roughly averaging the second columns of either Tables 2 and 3), one should then roughly anticipate a range of ±2 SD about the mean between various rations [i.e., 0.40 ± 2 (0.077) = (0.25, 0.55) kg/Mcal for the partial regression of DMI on MilkE].

## Discussion

### Potential Heterogeneity in the Partial Regression Coefficients

*b*

_{1}) of DMI on MilkE for some research stations might imply that these cows may be more efficient at converting DMI to MilkE. Nevertheless, partial regression coefficients for DMI on MilkE for most other stations or studies were in reasonable agreement with those presented by the National Research Council (NRC, 2001). On average, the partial regression coefficients for MilkE from the first-stage model (Table 2) are less than those reported on page 4 of

^{0.75}at midlactation. Our results imply that great care may be needed to create the RFI variable for subsequent quantitative genetic analysis. This recognition was made by

Lu, Y. F., M. J. Vandehaar, K. A. Weigel, L. E. Armentano, D. M. Spurlock, C. R. Staples, E. E. Connor, and R. J. Tempelman. 2014. An alternative approach to modeling genetic merit of feed efficiency in dairy cattle. Proc. 10th World Congr. Genet. Appl. Livest. Prod., Vancouver, Canada. Am. Soc. Anim. Sci., Champaign, IL.

### Genetic and Permanent Environmental Variability in RFI

## Conclusions

## Acknowledgments

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