A total of 71,518 days open (DO) and number of services per conception (NSC) records of 28,523 Iranian Holstein cows were analysed by random regression model. Akaike’s information criterion and likelihood ratio test suggested that a model with quadratic Legendre polynomials for additive genetic and permanent environmental was best. Heritability in different parities ranged from 0.103 to 0.045 in first parity and 0.054 to 0.030 in sixth parity for DO and NSC, respectively. Estimated genetic correlations between parities decreased continuously with increasing distance between parities for both DO and NSC. The first eigenfunction explained 89.99% and 97.22 % of the total genetic variance of DO and NSC, while the second eigenfunction accounted 6.24% and 3.18%, respectively. Different selection indices were constructed for DO and NSC. Genetic response to improve overall fertility was greater when the index included the first eigenvector than the response obtained from indices excluding it. Similar genetic gains were obtained from the first eigenvector, which had a nearly flat associated eigenfunction along lactations and from selection by the intercept of the random regression. The first eigenvector indices were responsible of changes in the level of DO and NSC in a similar manner for all parities, without altering the shape of the response curve across parities. The second and third eigenvector indices modified the shape of this curve but the improvement in genetic gains by including them in the selection index were small (DO) or negligible (NSC) due to the small amount of variability associated with these components.

Fertility is a trait with a high economic value in most selection schemes in dairy cattle because of the negative economic consequences of low fertility (

On the other hand, a wide number of statistical models have been used in the genetic evaluations of fertility traits with no clear consensus as to what approach is best for each trait. One of the features shared by most fertility traits is that several measures are available along the cow’s productive life. Treatment of longitudinal traits in animal breeding has followed several approaches from the simplest repeatability model (

In this study, the use of RRM in the analyses of DO and number of services per conception (NSC), as representative traits for two types of information commonly available for genetic improvement of fertility, has been explored to compare the use of alternative regression orders of the RRM and to develop selection indices for level and persistency of fertility along the productive life of the cows.

A total of 71,518 records of DO and NSC collected from parity 1 to 6 during the period 1981 to 2007 were used in this study. These records came from 28,523 female Holstein cows sired by 925 bulls in 15 large Holstein herds of Iran. These herds of Holstein cows located in 10 different provinces of Iran. If NSC was greater than 10, then NSC was assigned to 10, and DO was required to be between 30 and 330 days. Single trait RRM were used to estimate the genetic parameters of DO and NSC. For analyzing traits with RRM, Legendre polynomials for the regression on parity number were used. Two models with first (linear) and second (quadratic) order of fit were compared. The logarithm of the likelihood (logL) and the Akaike’s Information Criterion (AIC) were used for model comparison. Estimations were done using the Wombat package (

The RRM model in matrix notation was:

with

t*= 2 * ((t– min(T))/(max(T)-min(T)) – 1,

and covariates of the Legendre polynomials up to the second order were

[1, t*, 0.5(3t*
^{2}-1)]

The model assumptions were:

where
_{a} and
_{pe}= matrices of (co)variance between random regression coefficients for additive genetic and permanent environmental effects, respectively;
_{nr} = identity matrix; nr = number of animals with records; = Kronecker product, and

The eigendecomposition of
_{a} was obtained from,

_{a}

where
_{a} as columns (
_{i}, i=1,3) of
_{a}.

Corresponding eigenfunctions were obtained as:

_{i}(t)=

where i(t) = value of the i
^{th}eigenfunction at parity t,
^{th} eigenvector of
_{a} and

Selection to improve the overall fertility performance for the first six lactations was considered for both DO and NSC by defining as the aggregate genotype,

H =

where

Alternative selection indices were:

i) The index based on the k eigenvectors of
_{a}, I
_{k}.

I
_{k}=

where

Index weights in
_{k} and H are:

^{-1}

The genetic gain in each fertility trait from parity 1 to 6 was then estimated as:

where i is the selection intensity that was set to one in this study and
_{(6x6)} =
_{a}

ii) Partial selection indices based on one eigenvector,
_{i},

I
_{ei}=
_{i}
^{'}

or in two eigenvectors,
_{i}, and

I
_{eiej} =
_{ij}’
_{ij}’

with
_{ij}=
_{ij}'
_{ij}= [
_{i}
_{j}]

with
_{ij}= diagonal matrix with the i
^{th} and j
^{th} eigenvalues,
_{i},
_{j}

iii) Selection by the intercept, a
_{o},

where
_{o}= vector of covariances between the intercept and the other random regression coefficients and
_{oo} = variance of the intercept

iv) Selection by breeding value for first lactation,
_{L1}

_{}

_{}

_{}

_{}

_{}

_{}

with
_{1}=

Descriptive statistics of DO and NSC in parities 1 to 6 are given in

Model comparison statistics, logL, AIC and likelihood ratio tests (LRT) are presented in

The estimated covariances and correlations between the additive genetic random regression coefficients for DO and NSC are shown in ^{th }parity decreased abruptly, except for the first parity.

The three eigenfunctions associated with the additive genetic covariance matrices are shown in

The pattern of the second eigenfunction of DO is similar to the third eigenfunction of NSC. The second eigenfunction of DO almost linearly decreased from parity 1 to 4 and then stabilized. Estimated genetic response to alternative selection indices for DO and NSC are shown in _{e1},I
_{e1e2},I
_{e1e3},I
_{k}) or selection based on the intercept (I
_{ao}) was much larger than selection from indices based on the second and third eigenvalues, which account for changes in the trait level across lactations (I
_{e2},I
_{e3},I
_{e2e3}). The lowest overall genetic gain was observed for I
_{e2} for DO and for I
_{e3 }for NSC. The improvement of adding variables related to changes in the trajectory of the trait across lactations was small for DO and negligible for NSC. For the index showing the largest overall response (I
_{e1e2}), responses for parity 1 to 6 were 18, 17.2, 16.2, 15,13.4 and 11.6 days, and 0.28, 0.27,0.27, 0.25, 0.24 and 0.23 units, for DO and NSC, respectively.

Using I
_{e2} (including only the second eigenvector) resulted in positive genetic gains for DO in parity 1 and 2 and negative genetic gains in parity 3, 4, 5 and 6. Selection by I
_{e2} for NSC caused negative genetic gains in parity 2, 3 and 4 and positive genetic gains in parity 1, 5 and 6. Including third and second eigenfunction together in selection indices (I
_{e2e3}) resulted in positive genetic gains in parities 1, 2, 5 and 6 and negative genetic gains in other parities for DO, and negative genetic gains in parities 2, 3 and 4 and positive genetic gains in other parities for NSC.

Both, the LRT and AIC criteria indicated that the model fitting a quadratic Legendre polynomial for additive genetic and permanent environmental effects was better than a linear model for analyzing DO and NSC in terms of goodness of fit. Menéndez-Buxadera

The negative correlation between the intercept and the linear coefficient indicate that animals with small genetic values for the level of the trait are expected to show an increase in the trait along parities. Thus, cows with good genetic potential for fertility (small values for DO or NSC) will tend to show a decline in fertility (increased DO and NSC) performance along lactations and vice versa.

In our study for DO and NSC, the estimated additive genetic variances decreased along parities while permanent environmental variance increased. Patterns of response in other studies differ. ^{th} parity and ^{th }parity. For the same data, Ghiasi

In this study, high genetic correlations (ranging from 0.66 to 0.99) were estimated for each trait (DO or NSC). The high estimates of genetic correlations between different parities indicate that selection to decrease DO or NSC at any parity will decrease DO or NSC throughout all parities. However, because the estimated genetic correlations are not unity, differences in the pattern of response of individual animals across parities are expected. In other words, persistency of DO and NSC genetic merit across parities may differ among animals.

The first eigenfunction of DO and NSC in this study is positive and nearly constant throughout all parities. This means that selecting for the genetic value associated to the first eigenfunction will result in a similar improvement in all parities, deriving from the high and positive correlation found between different parities. The first eigenfunction is therefore responsible for scaling up and down the curve of DO and NSC without changing its shape and explains most of the observed variability in this data. But, selection for the second eigenfunction could be used to decrease DO from first parity to subsequent lactations. Selection based on the third eigenfunction of NSC would reduce the level of NSC from parity 1 to 4 but the amount of genetic response will be low because the third eigenvalue explains a low portion of the overall additive genetic variance of NSC. The pattern of the third eigenfunction of DO and the second eigenfunction of NSC showed a more complex response. The proportion of variance explained by the first eigenfunction in other studies dealing with quadratic RRM for fertility traits varied from 66% in

The overall estimated genetic response to selection (
_{e1},I
_{e1e2},I
_{e1e3},I
_{k}) or selection based on the intercept (I
_{ao}) was much larger than selection from indices based on the second and third eigenvalues, which account for changes in the trait level across lactations (I
_{e2},I
_{e3},I
_{e2e3}). The lowest overall genetic gain was observed for I
_{e2} for DO and for I
_{e3} for NSC. Including the second or third eigenvector to the selection index along with the first eigenvector index had a small impact on the direction and amount of genetic gains. This negligible impact of including the second and/or third eigenvariables in the selection index for NSC might be associated with the fact that the amount of variability explained by those variables was very small, 3%, while the same variables explained around 10% for DO. This derives from the larger genetic correlations between more distant lactations for NSC, especially for first parity, which showed a 0.9 correlation with parity 6 for this trait compared with DO, for which that correlation was 0.69. Both the very small of variability explained by the second and third eigenvariables and the large correlation between lactations imply little re-ranking of animals for different lactations and small changes in the trait across lactations. In the same way, the amount of overall genetic gain obtained from selection by first lactation performance, I
_{L1}, was around 12% lower than genetic gains obtained from selection by I
_{e1},I
_{e1e2},I
_{k }and I
_{ao} for DO and the same for NSC. Under all indices involving e
_{1} or a
_{o}, the amount of genetic gain was largest for first parity and decreased along parities. Selection by an index combining the three eigenvectors provided more even responses across lactations but also an overall smaller response. These results are in agreement with results obtained by _{e3} were near zero in all parities therefore this index could not change the shape of NSC across parities.

Eigenfunction and eigenvector indices obtained for DO and NSC indicate that it is possible to increase or decrease the amount of these traits along the different parities by making use of either the first eigenvector or the intercept of the random regression. However, the results of this study revealed that if the breeding goal were based on the second and third eigenvectors fertility performance would be improved in one parities and deteriorated in other parities. Thus, no clear benefit from using indices that include both level (first eigenvector) and shape of the trajectory along lactations (second and/or third eigenvectors) over using only variables associated with the level of the traits (first eigenvector or intercept of the random regression) was found.