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Acta Agriculturae Scandinavica, Section A — AnimalScience

ISSN: 0906-4702 (Print) 1651-1972 (Online) Journal homepage: http://www.tandfonline.com/loi/saga20

Simultaneous genetic evaluation of simulatedmastitis susceptibility and recovery ability using abivariate threshold sire model

B. G. Welderufael, D. J. de Koning, L. L. G. Janss, J. Franzén & W. F. Fikse

To cite this article: B. G. Welderufael, D. J. de Koning, L. L. G. Janss, J. Franzén & W. F. Fikse(2016) Simultaneous genetic evaluation of simulated mastitis susceptibility and recovery abilityusing a bivariate threshold sire model, Acta Agriculturae Scandinavica, Section A — AnimalScience, 66:3, 125-134, DOI: 10.1080/09064702.2016.1275761

To link to this article: http://dx.doi.org/10.1080/09064702.2016.1275761

© 2017 The Author(s). Published by InformaUK Limited, trading as Taylor & FrancisGroup

Published online: 17 Jan 2017.

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Simultaneous genetic evaluation of simulated mastitis susceptibility and recoveryability using a bivariate threshold sire modelB. G. Welderufaela,b*, D. J. de Koninga, L. L. G. Janssb, J. Franzénc and W. F. Fiksea

aDepartment of Animal Breeding and Genetics, Swedish University of Agricultural Sciences, Uppsala, Sweden; bCenter for Quantitative Geneticsand Genomics, Department of Molecular Biology and Genetics, Aarhus University, Tjele, Denmark; cDepartment of Statistics, StockholmUniversity, Stockholm, Sweden

ABSTRACTThe aim of this study was to develop a new approach for joint genetic evaluation of mastitis andrecovery. Two mastitis incidences (0.28 and 0.95) measured via somatic cell count and threebetween traits genetic correlations (0.0, 0.2, and −0.2) were simulated for daughter group sizesof 60 and 240. A transition model was applied to model transitions between healthy anddisease state. The RJMC package in DMU was used to estimate (co)variances. Heritabilitieswere consistent with the simulated value (0.039) for susceptibility and a bit upward biased forrecovery. Estimates of genetic correlations were −0.055, 0.205, and −0.192 for the simulatedvalues of 0.0, 0.2, and −0.2, respectively. For daughter group size of 60, accuracies of sire EBVranged from 0.56 to 0.69 for mastitis and from 0.26 to 0.48 for recovery. The studydemonstrated that both traits can be modeled jointly and simulated correlations could becorrectly reproduced.

ARTICLE HISTORYReceived 10 June 2016Revised 12 December 2016Accepted 13 December 2016

KEYWORDSDairy cow; health state;transition probability; udderhealth; recovery

1. Introduction

Mastitis, an inflammation of the mammary gland, isusually caused by bacterial infection. Occasionally, itmight arise from chemical, mechanical, or thermal inju-ries. It is a common and costly disease in modern dairyfarms (Halasa et al., 2007; Geary et al., 2012). In additionto its high incidence and economic importance (due todiscarded milk, reduced milk production, culling cows,and treatments cost), dairy cows welfare and consu-mers’ demand for antibiotics-free milk and milk pro-ducts makes mastitis a concern both to the dairyfarms and the community. The well-documented(Carlén et al., 2004; Koivula et al., 2005) genetic antag-onism between milk production and clinical mastitis isan additional reason to include mastitis in a dairycattle breeding goal. Developing better models forgenetic evaluation of mastitis is among the top priori-ties for breeding dairy cows with improved mastitisresistance.

Somatic cell counts (SCCs) and recorded cases of clini-cal mastitis (CM) are the variables that are mostly usedfor genetic evaluations of mastitis (Franzén et al., 2012).The lack of routine records (Carlén et al., 2006) of mastitis

has led to the use of indirect but related traits, such as:SCC, udder conformation (type traits), milking speed,and electrical conductivity (Schukken et al., 1997).Among these related traits, SCC is widely accepted as aproxy for mastitis and considered as the best measure(Emanuelson et al., 1988; Uhler, 2009) due to its ease ofrecording and high genotypic and phenotypic corre-lation with mastitis (Emanuelson et al., 1988; Gernandand Konig, 2014).

Franzén et al. (2012) developed a model using tran-sition probabilities to analyze SCC changes during lac-tation to assess genetic merit for mastitis susceptibilityin dairy cows. The model was built on the idea thatmeasurements are taken at regular intervals (say,weekly) and that for each measurement an individualcow is classified into one of two possible states(healthy or diseased), whereupon transition probabil-ities between these states are analyzed. This methodmodels transitions to and from states of infection, i.e.both the disease susceptibility and the recoveryprocess are considered, enhancing the genetic evalu-ation of mastitis. The SCC-based analysis by Franzénet al. (2012) ignored possible genetic correlationbetween susceptibility and recovery ability from

© 2017 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis GroupThis is an Open Access article distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives License (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited, and is not altered, transformed, or built upon inany way.

CONTACT B. G. Welderufael berihu.welderufael@slu.se

ACTA AGRICULTURAE SCANDINAVICA, SECTION A — ANIMAL SCIENCE, 2016VOL. 66, NO. 3, 125–134http://dx.doi.org/10.1080/09064702.2016.1275761

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mastitis. Though both mastitis susceptibility and recov-ery ability were considered, the traits were analyzedseparately with a single trait model. A simpleproduct-moment correlation between estimatedbreeding values (EBVs) failed to reproduce differentvalues of simulated genetic correlations between mas-titis susceptibility and recovery ability. Literature onestimation of genetic correlation between mastitis sus-ceptibility and recovery ability is not available. To beable to investigate whether a genetic correlationexists between mastitis susceptibility and recoveryability a bivariate model had to be developed.

The objectives of the current study were, therefore, (1)to develop a new approach for a joint genetic evaluationof mastitis susceptibility and recovery ability and evalu-ate its identifiability by estimating genetic correlationbetween mastitis susceptibility and recovery abilityusing a bivariate threshold sire model, and (2) to evaluatethe effect of daughter group sizes and level of mastitisincidence on breeding value accuracies for given simu-lated values of genetic correlations between mastitis sus-ceptibility and recovery ability.

2. Material and methods

2.1. Data simulation

Simulation software by Carlén et al. (2006), further devel-oped by Franzén et al. (2012), was used to generate SCCobservations for individual cows on the basis of simu-lated mastitis cases. Mastitis and SCC records were gen-erated on a weekly basis for first lactation cows. A cow’smastitis history for the whole lactation was simulated onthe basis of weekly values for susceptibility to becomeinfected with and to recover from mastitis. A cow thatwas healthy in week t−1 was assumed to have mastitisin week t if the mastitis susceptibility for that week wasabove a certain threshold (that depended on thedesired incidence; see below). A cow with mastitis inweek t−1 was assumed to have recovered from mastitisin week t if the recovery ability for that week was above acertain threshold (that depended on the average recov-ery rate). The mastitis susceptibility and recovery abilityfor an individual cow were generated as the sum of aherd effect, the animal’s breeding values, permanentenvironmental cow (lactation) effect, and a (weekly)

temporary environmental effect. The effect of herd onmastitis susceptibility and recovery ability was assumedto be uncorrelated, likewise for permanent and temporalenvironmental effects. Breeding values for mastitis sus-ceptibility and recovery ability could be correlated (seebelow). All effects were assumed to follow normal distri-butions; variances for the random effects are in Table I.

Weekly SCC observations were generated on the basisof the mastitis history for each cow separately. The levelof SCC on non-mastitic test-days followed a baseline SCClactation curve (denoted as L(t)) to which a randomnoise, generated as the exponential of a randomnormal deviate (SD = 0.56), was added (Figure 1). Onmastitic test-days, the SCC level was elevated followedby a successive decline towards the baseline level. Thepattern (Figure 2) is a generalization of the effects of Sta-phylococcus aureus, E. coli, Streptococcus dysgalactiae,and Streptococcus uberis on SCC, reported by de Haaset al. (2002).

Two levels of mastitis incidence or cases per lactationwere considered: scenario one (0.28 cases/lactation)and scenario tow (0.95 cases/lactation). Scenario onewas assumed to reflect previous estimates of incidenceof mastitis in field data of Swedish first-parity cows(Carlén et al., 2004; Franzén et al., 2012). The secondscenario was not based on previous studies but waschosen to take into consideration for possibilities ofhigher frequencies of mastitis in multiparous cowsand for merely theoretical interest to investigate the

Table I. Phenotypic records and selected genetic parameters used in simulations.Mastitis cases/lactation Trait

Geneticvariance

Variance of permanentenvironment (lactation)

Variance of temporaryenvironment (week)

Herdvariance

Number ofobservations

0.28 Mastitis susceptibility 0.036 0.252 0.64 0.072 1,170,000Recovery ability 0.036 0.252 0.64 0.072 10,300

0.95 Mastitis susceptibility 0.036 0.252 0.64 0.072 1,120,000Recovery ability 0.036 0.252 0.64 0.072 35,000

Figure 1. Baseline SCC (solid line) and baseline ± 1 SD (brokenlines) for healthy cows.

126 B. G. WELDERUFAEL ET AL.

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difference in performance of the method for differentlevels of mastitis incidence (Franzén et al., 2012).Three different genetic correlations between mastitissusceptibility and recovery ability were considered:rg = 0.0, rg = 0.2, and rg = −0.2. We chose these geneticcorrelations to make our simulations comparable toFranzén et al. (2012). The cow permanent environ-mental correlation between mastitis and recovery wassimulated as zero for all scenarios. As described byFranzén et al. (2012), the simulated heritability forboth mastitis susceptibility and recovery ability was0.036 (within-herd heritability 0.039) on a weeklybasis. This is similar to the heritability of mastitis of esti-mated with survival analysis by Carlén et al. (2005). Thecows were daughters of either 400 or 100 unrelatedsires distributed over 1200 herds with a fixed herdsize of 20, which resulted into daughter group sizes of60 and 240 respectively. All combinations of mastitisincidences, daughter group sizes, and genetic corre-lations were produced in 20 replicates. More detailson the parameters used to simulate the data can befound in Franzén et al. (2012).

2.2. Transition probabilities

For a healthy, lactating cow, there is a risk of becominginfected and for a mastitic cow there is a possibility ofrecovering from mastitis, which can be conceptualizedinto probabilities of mastitis and recovery. For eachcow, several transitions from healthy state (H) to dis-eased state (D), denoted as HD, and from D to H,denoted as DH, can occur within a lactation (Franzénet al., 2012). The probabilities of changing states orremaining in the current state can be summarized in atransition probability matrix, Ti , for cow i, like inFranzén et al. (2012):

Ti = 1− pHDi pHD

ipDHi 1− pDH

i

[ ], (1)

where 1− pHDi is the probability of remaining in the H

state for cow i, pHDi the probability of moving from

H to D state for cow i, pDHi the probability of moving

from D to H state for cow i, and 1− pDHi the probability

of remaining in the D state for cow i.In the transition probability matrix in Equation (1), the

first row contains the probabilities of being in either ofboth states at time t + 1 for cow i in healthy state (H) attime t, and the second row contains probabilities ofbeing in either of both states at t + 1 for cow i in diseased(D) at time t. The transition matrix is desired to have highvalues of 1− pHD

i (probability of remaining in the H statefor cow i) andpDH

i (probability ofmoving from the D stateto H state, i.e. fast recovery if cow i hadmoved fromH to Dstate), and consequently low values ofpHD

i (probability ofmoving from H to D state) and 1− pDH

i (probability ofremaining in the D state) (Franzén et al., 2012).

Thus, a sequence of H’s and D’s, indicating whether ornot a cow had mastitis on subsequent test weeks, can beconverted into a new sequence of state changes: 0 if acow remains in the same state and 1 if the cowchanges state. These transitions result into a sequenceof longitudinal records (repeated measurements) of mas-titis susceptibility or recovery for a cow along the lacta-tion, which can be used to make inferences about pHD

i

and pDHi .

2.3. Trait definition

Weekly SCC observations (generated from the simulatedmastitis status) were used to assess whether a cow was ina diseased (D) or healthy (H) state. If a cow’s observedweekly SCC exceeded a predefined boundary (denotedas B(t)), 2,000,000 cells/mL or 10 × L(t), where L(t) is theSCC level along the lactation curve), the cow was con-sidered mastitic and in the D state; otherwise the cowwas in the H state. As defined and explained inFranzén et al. (2012) a multiplier factor of 10 was usedto lower the risk of misclassifications. Let SCCijkt be theobserved SCC value for cow i, daughter of sire j, fromherd k at time t, for t = 1, 2, 3,… ti where ti is thenumber of observations for cow i within the same lacta-tion. A binary response hijkt stated whether the tth obser-vation of SCC for cow i was below (H) or above (D) theboundary as indicated in Figure 3. This binary responseis formally expressed as follows:

hijkt = 1 if SCCijkt . B(t),0 if SCCijkt ≤ B(t).

{(2)

Following the methods developed by Franzén et al.(2012) and described in the previous section “tran-sition probabilities”, two new datasets were createdfrom the binary response in Equation (2). The first

Figure 2. Increase of SCC due to mastitis as a function of thenumber of days post-infection.

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dataset contained transitions from healthy to dis-eased states (HD) and was used to make inferenceabout mastitis susceptibility. The second dataset con-tained transitions from diseased to healthy states(DH) and was used to make inference about recoveryability. The HD dataset was much larger than theDH dataset because it included all cows, while theDH dataset contained only cows with at leastone case of mastitis that thus had an opportunityto express recovery ability. All the transitionsbetween states were recorded as a binary variablewhich indicates whether or not a transition tookplace between two consecutive observations, attime t and t + 1. Thus, for each cow and lactation,two series of binary transition indicators werecreated according to:

yHDijkt =1 if hijkt = 0 and hijk(t+1) = 1,

0 if hijkt = 0 and hijk(t+1) = 0,

missing if hijkt = 1 and hijk(t+1) = 0 or 1,

⎧⎪⎨⎪⎩

(3)

for the H to D transitions and

yDHijkt =1 if hijkt = 1 and hijk(t+1) = 0,

0 if hijkt = 1 and hijk(t+1) = 1,

missing if hijkt = 0 and hijk(t+1) = 0 or 1,

⎧⎪⎨⎪⎩

(4)

for D to H transitions for t = 1,2.3… , ti − 1.A cow without a case of classified mastitis did there-

fore not have any data for transitions in the D to H

direction. At the same time, the duration of a diseasedstate was usually much shorter than that of a healthystate. This resulted in a much smaller dataset for D toH transitions in comparison to the data in the H to Ddirection. Cow, sire, and herd indicators were repeatedas many times as there were binary responses in the lac-tation in both transitions. Each sequence was just a repli-cate of the same indicator duplicated as many times asthe length of yHDijkt and yDHijkt for susceptibility and recovery,respectively.

2.4. Statistical model

A bivariate threshold sire model was fitted to the longi-tudinal series of binary transitions indicators. When atrait is defined as a binary variable, the sire model is pre-ferred over the animal model for convergence reasons(Carlén et al., 2005). The threshold model assumes thebinary data are the outcome of a continuous variableon an unobserved scale called liability (Gianola andFoulley, 1983). The observed binary transitions take thevalue 1 if the liability is larger or smaller than thedefined threshold and 0 otherwise as explained above.The variable for the observed transitions (yHDijkt and yDHijkt )was modeled with a bivariate probit threshold siremodel describing susceptibility to and recovery frommastitis:

Pr(yHDijkt = 1) = pHDijkt = Pr(lHDijkt , 0), and

lHDijkt = bHD + HHDk + SHDj + CHD

ij + eHDijkt , (5)

Pr(yDHijkt = 1) = pDHijkt = Pr(lDHijkt , 0), and

lDHijkt = bDH + HDHk + SDHj + CDH

ij + eDHijkt , (6)

whereyHDijkt = 1 for H to D transition, 0 for H to H transition,

and missing otherwise for observation t.YDHijkt = 1 for D to H transition, 0 for D to D transition,

and missing otherwise for observation t.bHD, bDH =mean mastitis susceptibility and recovery

ability respectively for an average cow.HHDk , HDH

k = random effect of herd k for mastitis sus-ceptibility and recovery ability respectively.

CHDij , CDH

ij = random permanent effect of cow i of sire jfor mastitis susceptibility and recovery ability.

SHDj , SDHj = random sire effect of sire j for mastitis sus-ceptibility and recovery ability respectively.

eHDijkt , eDHijkt = random residual effect of cow i of sire j from

herd k for the tth observationThe mean mastitis susceptibility and recovery ability

were fitted as fixed effects. The random effects wereassumed to be normally distributed with zero means

Figure 3. SCC level based boundary (B(t), solid line) andobserved SCC (dashed line) as a function of time in lactation (t,weeks after calving) for a given primiparous cow. According tothe observed SCC, the cow in this figure has made a transitionfrom healthy (H) to disease (D) state at week 15 and stayedbeing diseased for 2 weeks and recovered at week 18 to stayin H state until week 35 and moved to D state in the next twoweeks and finally recovered to remain in H state throughoutthe remaining lactation period.

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and (co)variances in a matrix notation:

Var

HHDk

HDHk

CHDij

CDHij

SHDjSDHjeHDijkteDHijkt

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

Is2HHD

0 0 0 0 0 0 0

0 Is2HDH

0 0 0 0 0 0

0 0 Is2CHD

IsCHD,DH 0 0 0 0

0 0 IsCDH,HD Is2CDH

0 0 0 0

0 0 0 0 Is2SHD IsSHD,DH 0 0

0 0 0 0 IsSDH,HD Is2SDH 0 0

0 0 0 0 0 0 Is2eHD 0

0 0 0 0 0 0 0 Is2eDH

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

where I are identity matrices of appropriate size. Thes2HHD

, s2CHD

, s2SHD , and s2

eHD are variances for herd, cow per-manent environmental effect, sire additive genetic, andresidual effects, respectively, for mastitits susceptibility.The s2

HDH, s2

CDH, s2

SDH , and s2eDH are variances for herd,

cow permanent environmental effect, sire additivegenetic, and residual effects, respectively, for recoveryability. The sCHD,DH and sSHD,DH stand for the permanentenvironmental and sire covariances, respectively,between mastitis susceptibility and recovery ability.However, as the residual variances are not identifiablefor binary traits for both traits, it was fixed to one. Thetwo traits were never observed simultaneously and theresidual covariance between them was therfore con-strained to zero.

2.5. Analysis, sampling, and Bayesian inference

Bayesian analysis with the RJMC package in DMU(Madsen and Jensen, 2013), which implements MCMCmethods and Gibbs sampling, was performed to obtainthe posterior distribution of all model parameters. Flatprior distributions were employed for all (co)variancecomponents in the model. We ran single chains of50,000 iterations with the first 10,000 iterations discardedas burn-in and a sampling interval of 25 to produce aposterior distribution of sample size of 2000 fromwhich point estimates of parameters were derived.Trace and density plots as well as autocorrelationswere (visually) inspected using the BOA softwarepackage (Smith, 2007) in R (R Core Team, 2015), to evalu-ate mixing and convergence of MCMC chains to the pos-terior distribution.

The bivariate model provides an option to estimate agenetic (rg) and permanent environmental (rc) corre-lations between the two traits. The correlationsbetween mastitis susceptibility and recovery abilitywere calculated as follows:

rg = sSHD,DH�������������s2SHD

× s2SDH

√ for genetic correlation (7)

and

rc = s cHD,DH�������������s2cHD × s2

cDH

√

for permanent environmental correlation

(8)

wheresSHD,DH and scHD,DH are the posterior mean estimate of

genetic and permanent environmental covariance,respectively, between the traits.

s2SHD and s2

cHD are the posterior mean estimates ofgenetic and permanent environmental variances,respectively, for mastitis susceptibility.

s2SDHand s2

cDH are the posterior mean estimates ofgenetic and permanent environmental variances,respectively, for recovery ability.

Heritability for each trait was calculated taking intoaccount the sire model. Under this model heritabilityfor mastitis susceptibility (h2HD) was estimated using pos-terior means of variances as follows:

h2HD = 4× s2SHD

s2SHD

+ s2CHD

+ s2eHD

, (9)

whereh2HD is the posterior mean heritability of mastitis sus-

ceptibility, s2SHD is the posterior mean sire additive

genetic variance for mastitis susceptibility, s2CHD is the

posterior mean permanent environmental variance ofcow for mastitis susceptibility and s2

eHD the residual var-iance for mastitis susceptibility (fixed to 1).

The posterior mean heritability for recovery ability(h2DH) was calculated as follows:

h2DH = 4× s2SDH

s2SDH

+ s2CDH

+ s2eDH

, (10)

whereh2DH is the posterior mean heritability of recovery

ability, s2SDH is the posterior mean sire additive genetic

variance for recovery ability, s2CDH is the posterior mean

permanent environmental variance of cow for recovery

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ability and s2eDH the residual variance for recovery ability

(fixed to one).Correlations between true breeding values (TBVs), gen-

erated during the simulation process, and the estimatedbreeding values (EBVs, posterior means) from the MCMCanalysis were calculated to quantify the accuracy forboth directions of the disease, that is susceptibility to mas-titis and recovery ability. The estimation accuracy for thehealth to disease direction (rHD) was defined as thePearson correlation between sire TBV and EBV as follows:

rHD = sTBVHD, EBVHD�����������������s2TBVHD

× s2EBVHD

√ , (11)

where sTBVHD, EBVHD is the covariance between TBV andEBV, s2

TBVHDands2EBVHD are the TBV and EBV variances for

mastitis susceptibility.The estimation accuracy for the disease to health

direction (rDH) of the trait was defined as the Pearson cor-relation between sire TBV and EBV and calculated asfollows:

rDH = sTBVDH, EBVDH�����������������s2TBVDH

× s2EBVDH

√ , (12)

where sTBVDH, EBVDH is the covariance between TBV andEBV, s2

TBVDHands2EBVDH are the TBV and EBV variances for

recovery ability.

3. Results

3.1. Genetic correlation and heritability

In scenario one (low mastitis incidence), the mean (of 20replicates) of estimated heritabilities for mastitis

susceptibility ranged from 0.035 to 0.037 and from0.041 to 0.044 for the daughter group size of 60 and240, respectively (Table II). In scenario two (high mastitisincidence), estimated heritabilities for mastitis suscepti-bility ranged from 0.036 to 0.042 over all daughtergroups. The mean of the estimated heritabilities forrecovery ability ranged from 0.036 to 0.066 over allcases of mastitis incidence and daughter groups(Table II). While the heritabilities for recoveryability were upward biased, the posterior modes ofheritabilities were closer to the simulated (true) value(Table II).

Estimated genetic correlations between mastitis sus-ceptibility and recovery ability (rg) were mostly in agree-ment with the simulated correlations (rg), except inscenario one for the daughter group size of 60(Table II). The mean (of 20 replicates) estimate ofgenetic correlation deviated by −0.055, 0.005, and0.008 for the alternative with simulated values of rg =0.0, rg = 0.2, and rg =−0.2, respectively, over all daughtergroup sizes and mastitis incidences (Table II). For thedaughter group size of 60, however, the genetic corre-lations were imprecise and deviated considerably, butnot significantly from the simulated values. The widthof the 95% HPD for rg was larger than one for the alterna-tives with low mastitis incidence for the daughter groupsize of 60, and even for those (high mastitis incidence)alternatives with more information about recovery the95% HPD for rg was large with a mean of 0.69(Table III). For 92.5%, 95%, and 96.25% of the replicatesthe true value for the genetic correlations between thetraits was within the 95% HPD interval for rg = 0.0, rg =0.2, and rg =−0.2, respectively, over all daughtergroups and mastitis incidences (Table IV).

Table II. Means and modes (SE in subscript) of genetic (rg) and permanent environmental (rc) correlations between mastitissusceptibility and recovery ability, heritability for mastitis susceptibility (h2HD) and recovery ability (h2DH).

60 daughters/sire 240 daughters/sire

0.28 mastitis cases/lactation 0.95 mastitis cases/lactation 0.28 mastitis cases/lactation 0.95 mastitis cases/lactation

Mean Mode Mean Mode Mean Mode Mean Mode

rgrg = 0.0 −0.1130.060 −0.1310.069 −0.0100.031 −0.0090.032 −0.0450.040 −0.0600.048 −0.0520.035 −0.0590.038rg = 0.2 0.1220.058 0.1600.067 0.2100.046 0.2170.048 0.2310.039 0.2620.044 0.2560.030 0.2640.031rg =−0.2 −0.0730.054 −0.0990.068 −0.2440.026 −0.2480.026 −0.2160.062 −0.2520.069 −0.2350.038 −0.2490.040rcrg = 0.0 −0.0470.017 −0.0210.013 0.0080.006 0.0080.006 −0.1440.061 −0.1230.068 0.0020.007 0.0020.007rg = 0.2 −0.0590.029 −0.0640.051 0.0230.006 0.0220.006 −0.0110.011 −0.0060.011 0.0180.006 0.0180.006rg =−0.2 −0.0850.014 −0.0500.009 −0.0140.006 −0.0150.006 −0.0560.013 −0.0350.011 −0.0100.007 −0.0100.007h2HDrg = 0.0 0.0370.001 0.0360.001 0.0390.001 0.0380.001 0.0440.002 0.0410.002 0.0420.002 0.0400.002rg = 0.2 0.0370.002 0.0360.002 0.0380.001 0.0370.001 0.0410.002 0.0390.002 0.0410.002 0.0390.002rg =−0.2 0.0350.002 0.0340.002 0.0360.001 0.0360.001 0.0410.002 0.0380.002 0.0410.001 0.0390.001h2DHrg = 0.0 0.0570.006 0.0420.006 0.0370.003 0.0340.003 0.0540.006 0.0460.006 0.0420.003 0.0390.003rg = 0.2 0.0570.006 0.0340.005 0.0360.003 0.0340.003 0.0500.004 0.0430.005 0.0410.002 0.0380.002rg =−0.2 0.0660.008 0.0480.008 0.0420.002 0.0400.003 0.0540.004 0.0440.003 0.0460.003 0.0440.003

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Estimates of permanent environmental effect corre-lations (rc) between mastitis susceptibility and recoveryability were mostly negative. The permanent environ-mental correlations were very weak (close to 0) withthe exception of scenario one for the daughter groupsize of 240 (Table II). A negative rc (−0.144) wasobserved in scenario one for the daughter group sizeof 240.

3.2. Accuracies of breeding values

The mean of accuracies of breeding values for mastitissusceptibility, in scenario one, was 0.57 for the daughter

group size of 60 (Table V). For the daughter group size of240, the accuracies ranged from 0.81 to 0.83. Similarly, inscenario two, the accuracy of breeding values for mastitissusceptibility markedly improved from 0.69 for thedaughter group size of 60 to 0.89 for the daughtergroup size of 240 (Table V).

The accuracies of breeding values for recovery abilitywere, in all cases, low compared to those for mastitis sus-ceptibility. Accuracies in scenario one ranged from 0.26to 0.27 for the daughter group size of 60 and from 0.49to 0.50 for the daughter group size of 240. In scenariotwo, the accuracy for recovery ability improved from aminimum value of 0.44 for the daughter group size of

Table III. Length of 95% highest posterior density interval (SE in subscript) for genetic correlations between mastitis susceptibility andrecovery ability for each simulated genetic correlation (rg).

rg

60 daughters/sire 240 daughters/sire

0.28 mastitis cases/lactation 0.95 mastitis cases/lactation 0.28 mastitis cases/lactation 0.95 mastitis cases/lactation

Width Lower Upper Width Lower Upper Width Lower Upper Width Lower Upper

0.0 1.1300.056 −0.6900.070 0.4400.075 0.6970.028 −0.3600.036 0.3370.038 0.9860.055 −0.5450.052 0.4410.048 0.6280.019 −0.3660.035 0.2620.0360.2 1.1860.047 −0.4720.080 0.7140.062 0.7030.034 −0.1320.054 0.5700.050 0.9450.047 −0.2230.054 0.7220.036 0.5930.012 −0.0410.032 0.5510.027−0.2 1.1230.061 −0.6490.058 0.4740.074 0.6630.023 −0.5900.030 0.0740.027 0.8570.044 −0.6500.059 0.2070.074 0.5740.015 −0.5210.033 0.0530.042

Table IV. Proportion of replicates for which the 95% highest posterior density interval includedthe simulated value for genetic correlations between mastitis susceptibility and recovery ability(rg), heritability for mastitis susceptibility (h2HD) and recovery ability (h2DH) for each simulatedgenetic correlation (rg).

60 daughters/sire 240 daughters/sire

0.28 mastitiscases/lactation

0.95 mastitiscases/lactation

0.28 mastitiscases/lactation

0.95 mastitiscases/lactation

rgrg = 0 0.80 1.00 0.95 0.95rg = 0.2 0.90 0.95 0.95 1.00rg =−0.2 0.90 1.00 0.85 0.90

h2HDrg = 0 0.90 1.00 0.95 1.00rg = 0.2 0.90 1.00 1.00 0.95rg =−0.2 0.80 0.90 1.00 1.00

h2DHrg = 0 0.95 0.95 0.90 0.85rg = 0.2 1.00 0.90 1.00 0.95rg =−0.2 0.95 1.00 1.00 1.00

Table V. Mean (SE in subscript) correlations between true and estimated breeding values ofsires for mastitis susceptibility (rHD) and recovery ability (rDH) for each simulated geneticcorrelation (rg).

60 daughters/sire 240 daughters/sire

0.28 mastitiscases/lactation

0.95 mastitiscases/lactation

0.28 mastitiscases/lactation

0.95 mastitiscases/lactation

rHDrg = 0.0 0.570.01 0.690.01 0.810.01 0.890.00rg = 0.2 0.560.01 0.690.01 0.830.01 0.890.01rg =−0.2 0.570.01 0.690.01 0.810.01 0.890.00rDHrg = 0.0 0.260.01 0.460.01 0.500.02 0.710.01rg = 0.2 0.270.01 0.440.01 0.500.01 0.710.01rg =−0.2 0.270.01 0.480.01 0.490.01 0.720.01

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60 to a minimum value of 0.71 for the daughter groupsize of 240 (Table V).

4. Discussion

4.1. Genetic correlation and heritability

Our main objective was to propose a bivariate model forgenetic evaluation of both; mastitis susceptibility andrecovery ability in a joint analysis. We put emphasis onthe correlations (rg) between the traits to evaluatewhether the model is identifiable. Franzén et al. (2012)studied whether or not the different values of simulatedgenetic correlations were reproducible from single traitanalyses. Their results showed that the correlationsbetween EBVs were very scattered. Though they con-firmed the simulated pattern, the positive and zerovalues of rg (0.2 and 0.0) were very scattered comparedto the simulated values. However, for the simulatednegative values of rg (−0.2) their approach failed toreproduce the negative pattern of correlations. Thecurrent results match in pattern to the simulatedvalues, with the exception of the alternative with smalldaughter group size and low mastitis incidence; disre-garding this alternative, the mean estimated correlationdeviated at most 0.05 from the simulated value.However, estimates were characterized by a relativelyhigh degree of uncertainty (e.g. the length of the 95%HPD width for rg in scenario one for daughter groupsize of 60 was larger than one). Nevertheless, we foundthat for a large proportion of the replicates, the true par-ameter value was within the 95% HPD interval. To thebest of our knowledge the new approach for analyzingSCC data in the current study is a first approach tomodel mastitis susceptibility and recovery ability in asimultaneous analysis using a bivariate threshold siremodel. Therefore, even though we are not able tofurther compare the results with previous studies, byreproducing the simulated correlations, we have demon-strated that the new approach is valid and capable toestimate genetic correlations between mastitis suscepti-bility and recovery ability, in a joint analysis. In situationswhere there is little information about recovery (e.g. dueto low mastitis incidence) the genetic correlation canonly be estimated with moderate precision.

The low estimated permanent environmentalcorrelation (rc) between the traits is not unexpected asthe effect of cow permanent environment on mastitissusceptibility and recovery ability was assumed uncorre-lated during the data simulation. But, the expectedvalue for rc from the analysis is not zero, because thecow permanent environmental effect absorbs three quar-ters of the genetic variance in a sire model .The average of

rc =−0.023 (for rg = 0.0) and rc = 0.020 (for rg = 0.2) are,therefore, approximately consistent to the expectedvalue of 0.019. For rg =−0.2, the mean rc =−0.047 showsslight deviation from the expected value of −0.019. Forthe small daughter group size combined with low mastitisincidence deviations from expected value are higher. Asobserved for the genetic correlations, the lower precision,especially observed for the daughter group size of 60 inscenario one could be due to little information aboutrecovery because of low mastitis incidence.

The heritability estimates for mastitis susceptibilityinferred from the weekly SCC data were approximatelyin agreement with the simulated (within-herd) heritabil-ity of 0.039. In a study with the same simulation design,different models for genetic evaluation of mastitis (linear,threshold, and survival analysis) were compared and arange of heritability estimates were reported (Carlénet al., 2006). For survival analysis of time to first mastitisthey reported a value of 0.037, which is very close tothe simulated value of 0.039. This supports our estimates,because their survival analysis model is conceptuallyequivalent to analyzing state transitions with a probitmodel with repeated observations as in the currentstudy. One difference, and advantage, is that ourmodel allows for repeated mastitis cases, whereas theirsurvival analysis did not.

Estimated mean of heritabilities for recovery abilitywere slightly upward biased and higher than for mastitissusceptibility. However, the posterior modes of heritabil-ities for both mastitis susceptibility and recovery abilitywere closer to the true value. It is clearly shown that pos-terior mode predicts better than posterior mean. A simu-lation study by Lipschutz-Powell et al. (2012) showedfailure of conventional models (such as sire models) tocapture the genetic variation in susceptibility even if itis present in a disease data, our new bivariate siremodel did capture all the variations and resulted inslightly overestimated heritabilities. The over estimationcan be due to the nature of the trait and the low numberof observations (Table I) for the DH dataset. Recoveryability can only be observed if there has been mastitis,but with low incidence level the number of observationsis low for the DH dataset.

Sire models with the MCMC method and Gibbssampling (the model used in the current study) are per-tinent in the genetic analysis of disease traits or binarydata like mastitis (Kadarmideen et al., 2001). Suchthreshold models have been favored over the restrictedmaximum likelihood (REML) applied to generalized linearmixed models (GLMM) in several studies (e.g. Sørensenet al., 2009; Franzén et al., 2012; Gernand et al., 2012).This is because in threshold models where the trait ofinterest is defined on an underlying susceptibility scale,

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there are theoretical advantages as they are more appro-priate to depict the physiological background of adisease data compared to linear models and the associ-ated model assumptions such as normality (Yin et al.,2014).

4.2. Accuracies of breeding values

The model generated more accurate EBVs for the highermastitis frequency, where the proportion of false classi-fied cases is expected to be lower (Franzén et al.,2012). Likewise, the increase in accuracy with theincrease in the number of daughters (from 60 to 240) isexpected as accuracy is highly dependent on theamount of performance information of an animal andinformation from its relatives. Especially in dairy cattlethe large number of progeny evaluated is very usefulto achieve a high level of accuracies. The accuracy (corre-lations between EBV and TBV) for the daughter groupsize of 60 was higher than the values reported byFranzén et al. (2012), despite that the simulation designwas exactly the same. This could be traced back to a pro-gramming mistake in the conversion of health states tostate transitions that affected the results in Franzénet al. (2012) negatively (Fikse and Franzén, pers.comm.), which was corrected in the current study. Inan earlier analysis of single trait threshold sire modelusing the same simulation design, the accuracy of breed-ing values ranged from 0.53 to 0.60 (Carlén et al., 2006)for the daughter group of 60 for the lower mastitis fre-quency per lactation. The corresponding accuracy inthe current study (ranging from 0.56 to 0.57) was consist-ent with that study.

The accuracies of breeding values for recovery abilitycompared to mastitis susceptibility were low in all cases.Accuracies of breeding values for recovery ability thatranged from 0.24–0.48 were reported by Franzén et al.(2012). The corresponding accuracy in the currentstudy (ranging from 0.26 to 0.48) was a bit higher. Webelieve the improvement in accuracy is due to the cor-rection of the previously mentioned programmingmistake in Franzén et al. (2012). The other reason forthe lower accuracies for recovery ability (compared tomastitis susceptibility) is the much reduced informationin the DH data; because not all cows did get mastitis,which means small number of transitions from D to Hstate. The number of records for mastitis susceptibility(HD) was over one million per replicate, whereas it wasaround 10,000 and 35,000 for recovery ability (DH) forlow and high mastitis frequency, respectively (Table I).Nevertheless, we have achieved much higher accuracies(as high as 0.72) by increasing the daughter group sizeto 240.

In general, as expected the accuracy of breedingvalues was highly dependent on daughter group sizesand the frequency of mastitis, like the findings byFranzén et al. (2012) where higher mastitis frequencycombined with more daughters per sire resulted inhigher accuracies. The different simulated values ofgenetic correlations (rg = 0.0, rg = 0.2, and rg =−0.2)between mastitis susceptibility and recovery abilitieshad effect in neither the current nor in the previousstudy (Franzén et al., 2012). Accuracies of breedingvalues were not affected even with higher genetic corre-lations (e.g. rg =−0.5 (unreported results)). The observedlow heritabilities are one reason for this.

The method was tested on simulation data. In real lifethe data are expected to have different quality and quan-tity. Applying the method and investigating the analysiswith real-life data will be the next step. One aspect is toexpand models (5) and (6) with additional systematiceffects, for example to allow for difference in mastitis sus-ceptibility and recovery ability over the course of lacta-tion. In addition, classification of cows as healthy ormastitic based on SCC data is critical, and algorithmslike those proposed by Sørensen et al. (2016) offer poten-tial for this purpose. It is also important to investigate theperformance of the model to utilize direct information onCM and recovery instead of only SCC which is indirectmeasure of mastitis.

In conclusion, we have developed a new approachwith a bivariate model for joint estimation of breedingvalues for susceptibility and recovery from mastitis onthe basis of state transitions according to changes inSCCs. The developed bivariate model could be used toestimate possible genetic correlations between mastitissusceptibility and recovery. The high accuracies of breed-ing values (0.89 in scenario two) demonstrate the impor-tance and potential of SCC in the genetic evaluation ofmastitis susceptibility. Effect of level of mastitis incidenceon accuracy of breeding value for both mastitis suscepti-bility and recovery ability was as important as thenumber of daughters per sire. The more daughters persire combined with more mastitis incidence, the moreaccurate the breeding values were observed. The newapproach with bivariate model gives an option for jointanalysis of mastitis susceptibility and recovery ability.The model identifiability was shown by reproducingthe simulated genetic correlations from the modelanalysis.

Acknowledgements

We are grateful to Yvette de Haas from the Animal Breed-ing and Genomics Centre of Wageningen UR LivestockResearch for her input and comments on earlier

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version of this manuscript. We thank Emma Carlén,Animal geneticist at Växa Sverige for her insightfulcomments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Funding

BGW benefited from a joint grant from the European Commis-sion and Swedish University of Agricultural Sciences (SLU),within the framework of the Erasmus-Mundus joint doctorateEGS-ABG.

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