The data we used came from an experimental farm, which provides some advantages over field data. For instance, weight recordings were performed in a standardized manner; weight at birth was measured within 12 h after lambing and weight at day 45 was measured very close to the actual 45th day of life. This avoided approximations by interpolation in the calculation of the ADG. However, the use of such experimental data has the disadvantage of including relatively few records and special attention must be paid to make sure that the data can disentangle direct and maternal effects. In this particular dataset, we are confident that this is the case for single trait analyses (mod(1)) because of the strong genetic relationships between individuals, especially cousin relationships. The mean number of records per dam, sire and maternal granddam for single reared-lambs was low (1.5, 6.1 and 8.6, respectively). However, these animals were also parents of twin reared-lambs. Consequently, records from twins provided the necessary information to estimate random parameters for single reared-lambs (if correlated) and helped to disentangle the direct and maternal effects for single reared-lambs when estimated in the case of multiple-trait assumptions. This was confirmed by the consistency of the estimates of heritabilities and correlations between models.
We decided to analyze the hypothetical differences between single- and twin-reared lambs by testing for differences between singles and twins for all random components of the model. At present, the results reported in the literature are in favour of a difference between the effects associated with singles and twins. Concerning direct effects, it has been reported that the behaviour of single-reared lambs is different from that of twin-reared lambs. On pasture, single-reared lambs were usually further from their dams than were multiple-reared lambs . It has also been shown that single lambs suckled less frequently but longer than twins [7, 14]. In other species, it has been reported that the behavioural mechanisms of sibling competition range from very aggressive interactions, through various milder agonistic interactions, to scramble competition . Although, to our knowledge, such mechanisms have not been reported in sheep, we can assume that competitive behaviour also exists in this species. With regards to maternal effects, the lactation curve differs between ewes nursing single and twin lambs. Ewes suckling twins have been shown to produce more milk than those suckling single lambs; their peak yield is reached during the 3rd week of lactation, compared with the 4th week for ewes with single lambs, and they show higher persistency [3, 5]. Furthermore, ewes with twins have higher milk fat levels and produce more milk energy than those with single lambs . From a genetic point of view, these differences could be interpreted as differences in both the ewe's and lamb's environmental conditions depending on the number of lambs reared. However, the results we obtained did not support the hypothesis of a genetic by environment interaction between single and twin lambs, which we evaluated with a multiple-trait model; the genetic correlation between the direct (maternal) effects for single or twin lambs was not significantly different from 1 and their variances did not differ. These results are not consistent with those obtained by Buvanendran et al. , who reported that genetic variance and heritability were greater for twins, although heritabilities were not significantly different.
Our results demonstrate that the maternal permanent effect was not the same when ewes reared single versus twin lambs. The permanent effect of dam accounts for all environmental factors related to the dam that are not explicitly incorporated in the model but which modify the non-genetic component of the maternal environment and therefore influence the growth of the lambs. A difference in permanent effects of dams for single versus twin lambs indicates that some of those unaccounted factors exert different effects depending on the number of lambs reared. One of these factors could be impairment of one quarter due to mastitis, which would have a negative influence on the ability of the ewe to rear two lambs but not on her ability to suckle a single lamb.
Our results for the relative importance of the litter effect (7 to 12%) are in the range of those reported in previous studies (0.11 ) or slightly lower (0.26 to 0.31 ). The litter effect is a combination of everything that affects members of a litter in the same way, including environmental conditions that are not accounted for by the other effects included in the model, and maternal temporary environmental effects (ewe*year effect in our case).
The results obtained here are in favour of different residual variance for single- versus twin-reared lambs. The raw data showed that single lambs have a higher ADG and a higher standard deviation than twins. The difference in variance was not due to a mean and variance relationship. In fact, the data were normally distributed and the slope of the regression linking the standard deviation of the raw data to the mean (with 10 g steps) was null (3.2.10-4).
Variances of dam permanent and residual effects were higher for single- than twin-reared lambs. One possible explanation for these differences is that, in the case of single-reared lambs, the observed ADG represents the "optimal" growth that can be obtained for the corresponding lamb-ewe-environment combination, while the competition between twin-reared lambs results in only part (α%) of this optimal growth to be expressed. In other words, if we only consider random factors:
refer to the observed ADG for the single or twin lamb i of ewe j, respectively, and other notations are the same as for the general model. Under this assumption, the variances of all random factors for single lambs are higher than for twins and this is consistent with the results obtained in this study. In fact, although not significantly different from 1 for the genetic effects, the ratio between the variances of random factors for single and twin lambs varied from 0.7 to 0.9 for the different factors in mod(7). Although convenient, this hypothesis oversimplifies the problem because the correlation between the permanent effects of the dam is not equal to 1 between single- and twin-reared lambs.
Our estimates of heritability are consistent with most of the heritabilities reported in the literature for pre-weaning ADG in sheep. Bromley et al.  reported heritabilities varying from 0.07 to 0.20 for direct effects and from 0.04 to 0.05 for maternal effects, depending on the breed. In a review, Safari et al.  reported an average heritability of 0.15 for the direct effect and 0.05 for the maternal effect. Heritability was also higher for the direct effect (0.21) than for the maternal effect (0.01) in Mousa et al. . Hagger , when comparing models in two breeds, obtained heritabilities varying from 0.08 to 0.16 for direct effects and from 0.02 to 0.10 for maternal effects. On the contrary, Snowder and Van Vleck  reported a low heritability for direct effects (0.03) and a higher heritability for maternal effects (0.28). Estimates of the genetic correlation between direct and maternal effects obtained in previous studies vary to a much greater extent, from -0.52  to 0.52 . Our close to 0 estimate of the genetic correlation is consistent with the review by Safari et al. (-0.02 (0.08)) . It is a well-known fact that estimates of this correlation are particularly sensitive to data structure [22–24] but, as previously mentioned, working with experimental data from a single herd probably overcomes this bias. The genetic parameters used in the French genetic evaluation model are heritabilities of 0.20 for the direct effect and 0.30 for the maternal effect, and -0.4 for the genetic correlation, (J.P. Poivey, personal communication). The discrepancy between these parameters and those estimated in the present study indicates that it may be of interest to update the parameters for field data.
We did not find any spurious changes in the maternal EBV of ewes rearing twin lambs for the first time after having reared single lambs the previous years, as had been reported from the field. One explanation for this result is that problems reported from field data are due to the quality of the data recorded, especially absence of recording lamb deaths which introduces bias in the type of rearing factor. This problem does not exist for the experimental data used for this study.
In this study, we focused on the possible heterogeneity of variance components for pre-weaning growth in sheep due to the number of lambs reared in order to check if the multiplicative coefficient assumptions made in the French genetic evaluation system are valid. Several other factors have been reported in the literature to affect early growth but are not included at present in the French genetic evaluation model and can introduce biases. A non-exhaustive list of these factors is the following: an environmental covariance between dam and offspring [25, 26], sire*year, sire*herd*year [23, 27], sire*dam, dam*number born  combinations, etc. The importance of these factors should be tested on field data when updating the French genetic evaluation model.