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ANIMAL GENETICS |
* Station de Génétique Quantitative et Appliquée, Institut National de la Recherche Agronomique, 78352 Jouy-en-Josas Cedex, France;
and
SUISAG (AG für Dienstleistungen in der Schweineproduktion), Allmend, CH-6204 Sempach, Switzerland
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Abstract |
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 Top Abstract INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONIMPLICATIONSAPPENDIXLITERATURE CITED |
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Key Words: exterior trait • feet and leg score • longevity • sow • teat
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INTRODUCTION |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONIMPLICATIONSAPPENDIXLITERATURE CITED |
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). Their major impact comes from the fact that domestic animals must be alive and healthy to reproduce normally. A longer length of productive life substantially decreases replacement costs and enables achievement of maximum performance in the herd by having more mature sows.
The number of longevity studies in pig breeding has increased in the recent past (e.g., Yazdi et al., 2000a,b; Serenius and Stalder, 2004; Tarrés et al., 2005). Nowadays, keeping a sow in a herd depends on a combination of morphological and production characteristics. Leg weakness is a serious problem and is a common culling reason in sows (Dial and Koketsu, 1996; Friendship et al., 1996). The improvement of production traits (e.g., growth rate) and the acceleration of the reproductive rhythm of sows for the past 20 yr have led to an increase in the incidence of leg weakness. It is a problem that affects production, and it has been studied in different species (Jørgensen and Vestergaard, 1990). Furthermore, leg problems are a welfare concern because of the pain many sows must endure before they are slaughtered (Jørgensen and Sørensen, 1998). In addition, leg problems of selection candidates decrease selection opportunities and thus selection intensity.
The aim of this study was to evaluate the influence of several exterior traits (number of teats and feet and leg scores) on the longevity of Swiss sows using survival analysis techniques. The response to indirect selection for longevity was also examined.
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MATERIALS AND METHODS |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONIMPLICATIONSAPPENDIXLITERATURE CITED |
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Data
The data in this study were from Large White, purebred sows from Swiss nucleus herds. The data were provided by SUISAG, a company that is owned by the Swiss pig producers and that is responsible for the Swiss pig breeding program (herdbook, performance testing, genetic evaluation). At first effective mating, gilts were on average 227-d-old. Average age at first farrowing was 342 d. Length of productive life was defined as the number of days between first farrowing and culling.
The culling date was defined as follows: 1) if the last weaning of the sow was recorded and the interval between the end of data recording in that herd (last event in the herd) and the last weaning of the sow was over 150 d (this period corresponds to the interval between 2 weanings), then the sow was considered as culled at the day after her last weaning; 2) if the interval was less than 150 d, the sow’s observation was censored at the last event in the herd; 3) if the last weaning of the sow was not recorded and the interval between the last event in the herd and the last farrowing of the sow was over 180 d (this period corresponds to the interval between 2 farrowings plus the average time between farrowing and weaning), then the sow was considered as culled at the day after her last farrowing; 4) if the interval was lower than 180 d, the sow was censored at the last event in the herd. After this, if the censored animals had a known reason of culling, then the culling date was the recorded culling date, and the animal was treated as uncensored. Sold animals were considered as censored, with date of sale as censoring date. Finally, editing constraints imposed were 1) only sows in herds with more than 50 sows with longevity records were retained, and 2) sows with age at first farrowing under 250 or over 480 d were excluded.
After editing the data set, 16,464 records of Large White sows with at least 1 farrowing were considered for survival analysis, including 13,321 (80.9%) and 3,143 (19.1%) uncensored and censored records, respectively. The time interval covered was sows born from 1989 to 2004. However, morphological characteristics had been recorded only for sows born from 1999 to 2004. After deleting sows without morphological records, the final data set consisted of only 5,077 records of Large White sows, including 2,827 (55.7%) and 2,250 (44.3%) uncensored and censored records, respectively. Individual records included animal, sire, and dam identification codes, date of first farrowing, culling or censoring date, censoring code, parity number, herd of origin, morphological characteristics, and records of all observed farrowings between March 2000 and December 2004. The culling reason was not available. These 5,077 sows were from 50 herds. The average number of sows per herd was 87, with a minimum of 29 sows and a maximum of 300 sows.
The file on morphological characteristics included 58,728 records. This file contained all animals scored for exterior traits at the end of the on-farm test; however, only a few were selected and remained in the nucleus herds. Two traits were considered: teat quality and feet and leg morphology. The first included the total number of teats (right and left), the number of inverse teats, and the number of intermediate teats. The number of good teats represents the total number of teats minus the number of inverse teats. The second group of traits consisted was an X-O rear leg (XORL) score, a side view angle rear leg (SVARL) score, an angle pastern rear leg (APRL) score, and a size of inner claws rear leg (SICRL) score.
Feet and leg characteristics were measured on a linear scale, which varied from 1 to 7 (SUISAG, 1999), for which the optimal value is 4. The distributions of these values in the data set changed from 1999 to 2003 (Table 1). After a preliminary study, extreme classes with too few uncensored records were grouped with their adjacent class. Eventually, 5 classes (13 to 17) were considered for the number of good teats. A variable number of classes were defined for feet and leg traits: 3 (2 to 4) for SICRL, 4 (2 to 5) for XORL and SVARL, and 5 (2 to 6) for APRL.
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The analysis began with the computation of a nonparametric estimate of the survival function over the entire productive life (Kaplan and Meier, 1958). However, previous work in dairy cattle (Ducrocq, 2002, 2005; Röxström et al., 2003) showed the benefit of modeling the baseline hazard function not over the entire productive life of the animals, but within parity. The observed hazard functions within parity were derived from the Kaplan-Meier (raw) estimate of the survivor curves stratified by parity. These empirical functions are the result of the influence of various factors involved in the culling process. To investigate the influence of these factors on the hazard function h(t;xm), the following piecewise Weibull mixed model was assumed for individual m, with covariate vector x'm(t) at time t:
 | [1] |
where h0,jn(
) is the baseline hazard function for the jth period of the nth parity, which follows a Weibull distribution (the parameters of the distribution are different for each jth period of the nth parity); and is the number of days since the previous farrowing date ( = t – tn, where tn is the value of t at the nth farrowing date).
Validation of the Baseline Hazard Function
The construction of the parametric model (Eq. [1]) began with an examination of a smoothed version of the nonparametric hazard function that led to the choice of a number of candidate periods (cj–1,cj, with c0 = 0), within which the Weibull assumption seems to hold. The limits of these periods were further refined by comparing the likelihood values obtained from piecewise baseline Weibull models, relying on the candidate periods or on slight variations (x2 d) of their bounds. Once the limits and periods were chosen, the baseline parametric hazard function of sows within the jth period of the nth parity was defined as
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with a different set of Weibull parameters, 
jnand 
jn, for each jth period of the nth parity.
This model can be simplified to a less demanding one if the proportionality assumption holds between parities:
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where h0,j() is the baseline hazard function for the jth period, which follows a Weibull distribution; and pn(t) is the time-dependent effect of the nth parity. To check whether this simplification is possible, from the raw data and for each parity (1, 2, ...9, 10, and above), a nonparametric estimate of the survivor function was obtained as a by-product of the Kaplan-Meier estimate, 
n(), of the survivor function (Klein and Moeschberger, 1997). A graphical test was performed plotting log (–logn()) vs. log. If the lines run parallel, the proportionality assumption can be accepted and the simplified model can be assumed.
Usually, such a plot is also used to check the validity of the Weibull assumption. If the baseline hazard is a Weibull hazard, a straight line is obtained. However this is the case for the piecewise Weibull model only for the first period. Indeed, if 
(0,c1), then:
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But if > c1, then (see Appendix)
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which is no longer a linear function of log . An alternative graphical validation test of the piecewise Weibull function was developed. Once the parametric survival function, S(), was computed for all times (see Appendix), the graphical validation test of the piecewise Weibull function consisted of comparing the shape of the plot of the nonparametric estimate of log(–logn()) vs. log with the one for the parametric distribution. The similarity between log–logn()) and its parametric equivalent was also measured as the squared Pearson correlation coefficient between the daily value of these functions.
Culling Factors
The second term, x'm(t)ß, of Equation [1] and [2] represents the effect on the hazard of a subset of (possibly time-dependent) explanatory variables. The time-dependent variables included in the model were the number of weaned piglets in each parity (NWw; 0 to 15) and herd-year-season effect (HYSx), with 4 seasons (calendar trimesters). The time-independent variable, birth year (BYy; 1999 to 2004), was also included to measure a potential phenotypic trend of the hazard function over years. All these variables were treated as categorical. The herd-year-season effect was included to permit a proper modeling of the changes in the culling process in a herd over time. Herd-year-season was treated as random (log-gamma distribution) because there were only a few uncensored records for most levels and integrated out to avoid having to estimate the effect of each level that was not of direct interest.
Exterior Traits. To analyze the phenotypic relationship between exterior traits and longevity, the model of Equation [2] was extended to
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where the exterior traits were included as time-independent categorical variables. Hypothesis testing was performed using likelihood ratio tests. A series of tests, hereafter called sequential tests, included the effects of exterior traits in the Model 1 at a time in sequential order. A second series of tests (last tests) compared the full model with the models excluding one effect at a time, for each effect separately. These and the subsequent computations were performed using the Survival Kit package (Ducrocq and Sölkner, 1998). The proportional hazards assumption was tested by extending the model with interaction terms between each exterior trait and a function of time (i.e., the parity). The proportional hazards assumption was checked via likelihood ratio tests (LRT), comparing the time independent model with models including one time dependent interaction factor at a time.
Phenotypic Index for Feet and Leg Scores. Model [2] was also extended to account for an index that combines the information of 3 feet and leg scores. This model was:
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where, instead of each exterior trait separately, the following index was included:
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i.e., the sum for the 3 traits of the squared deviations of the current value from the optimum 1 (level 4). Thus, a sow with an index value of 0 has optimal feet and legs. The larger the index value, the more severe the leg problems. For example, the index values were: 0 = optimal score; 1 = 1 slight defect; 2 = 2 slight defects; 3 = 3 slight defects; 4 = 1 severe defect; 5 = 1 severe defect and 1 slight defect, and so on.
Current Genetic Index for Exterior Traits.
A routine evaluation for exterior traits was implemented at SUI-SAG in 2001 (Hofer, 2001). The EBV used in this study were estimated by SUISAG in December 2004. Sows born between 1999 and 2001 in nucleus farms and with at least 1 litter were considered as base animals, and the index of base animals was transformed to have a mean value of 100 and an SD of 20. To check whether the current index for exterior traits (I_EXT) can be used to indirectly select for longevity, this index was used to define 6 classes with cutpoints at –2 SD, –SD, 0 SD, and 2 SD; i.e., the 6 classes were for EBV <60, 60 to 80, 80 to 100, 100 to 120, 120 to 140, and >140. Then, Kaplan-Meier estimates of the survival function stratified by the index class were obtained. An index class effect was also included in the following proportional hazards model:
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Measures of Culling Factors Effects
The effect of culling factors was measured by defining hazard ratios, HR = exp(x' m(t)ß), which compare the hazard of sows with the level of interest to the hazard for a reference level. However, to take into account the influence of the baseline hazard function, the effect of culling factors was also illustrated through predicted survival functions. This was done differently for time-dependent and time-independent effects. As the hazard ratio of time-dependent effects changes at each farrowing, it was illustrated by constructing the parametric survival function within parity, S() = [SREF()]HR, and predicting the percentage of sows that would farrow again. On the other hand, as the hazard ratio for time-independent effects remains the same over the entire productive life, their effect was illustrated using the predicted survival function of the sows for a level over the entire productive life. This was done elevating the Kaplan-Meier estimate of the survival function to the power of the hazard ratio for each level: (t) = [KM(t)]HR. This survival function estimate, (t), was used to estimate the average length of productive life: E(T) = 
0
S(t) dt (Kalbfleisch and Prentice, 1980). In swine, it is more sensible to estimate the average interval from first successful mating to culling; i.e., E(M) = E(T) + 115, which is the length of productive life plus the first gestation length. Then, the theoretical replacement rate for a stable herd is RR = 1/E(M) (Tarrés et al., 2004).
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RESULTS |
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The nonparametric survival function over the entire productive life is shown in Figure 1. The average length of productive life of sows was 602 d (1 yr and 7.75 mo). Then the average length from first mating to culling was 717 d, and the corresponding annual replacement rate is 50.9%.
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Culling Pattern Within Parity.
When the baseline hazard function was modeled within parity, the highest daily culling rate occurred during the very first day after each farrowing. The loss of animals during first day was relatively important (about 2% of farrowed sows for each parity). Excluding this first day, the observed hazard functions within parity derived from the Kaplan-Meier survivor curve are presented in Figure 2. The hazard was low during the first 3 wk after farrowing. Then, it strongly increased during the next 2 wk until it reached a maximum at 35 d. Then, it decreases at a similar rate during 2 wk, and after 49 d, it stayed at a very low level until the next farrowing. The pattern was very similar for all parities although the risk of culling was higher for the older ones. Figure 3 shows that this increase was proportional between parities because the plots of the logarithm of minus the logarithm of the Kaplan-Meier survivor curves against the logarithm of time are roughly parallel. Hence, there is no need to stratify the baseline per parity.
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shows the estimated parameters of this function for each period. This function fits the strong increase in hazard with a 3 (Weibull scale parameter for the third period) close to 4 between 20 and 40 d after farrowing. However, the strong decrease between 40 and 50 d cannot be appropriately modeled: it implicitly requires a very large hazard value at the starting point (farrowing date) and a 4 value corresponding to a drastic decrease of the hazard. This is not mathematically possible. As an alternative, a constant hazard 4 = 1 during this period was chosen as a compromise. Here it could be possible to use other (piecewise) parametric distributions with more flexibility. The most popular is the exponential one with many more intervals (periods). We tried a compromise with few periods. Despite this apparent lack of fit during the 41 to 50 d period, Figure 4a shows that the empirical estimate of the hazard function in first parity is well approximated: the R2 value of the regression of the empirical vs. the parametric hazard function is R2 = 0.849. The plot in Figure 4b confirm this and show an even closer fit of the survival function (R2 = 0.997) and of the logarithm of minus the cumulative hazard function log(–H(t)) = log(–logS(t)) (R2 = 0.994). Therefore, the piecewise Weibull model appears to adequately describe the culling pattern of sows within parity.
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The time dependent variables included in Model [2], parity and number of weaned piglets, were significantly related to risk of culling of sows. Results of LRT are presented in Table 3. The herd-year-season effect is not included in Table 3 because it has been treated as random and integrated out, but it also had a strong influence on the hazard function. The parameter was estimated at = 2.69 (95% confidence interval: [2.20, 3.29]), i.e., the variance of the log-gamma herd-year-season effect is equal to 
(1)(
) = 0.449 (95% confidence interval: [0.355, 0.573]), where (1)(.) is the trigamma function.
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). This increase is moderate (nonsignificant) for the first 3 parities and stronger later, especially for the last ones: a sow with 10 parities had over 4 times greater risk of being culled than first parity sows. When these hazard ratios are used to compute survival functions within parity, it was observed that on average 80% of sows with less than 3 farrowings were able to farrow again. This percentage decreased from 78 to 62% for the sows in parities 4 to 8 and was between 40 and 50% after the ninth parity.
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; P < 0.001). The risk of culling increased as fewer piglets were weaned (Table 4). If a sow in the first 3 parities weaned more than 10 piglets, there was a probability between 85 and 90% that she would farrow again. As the number of weaned piglets decreased below 10 piglets, these percentages decreased from 80% to less than 50%. If no piglet was weaned, there was a very low probability that the sow would farrow again. These percentages were lower for later parities. The other variables included in the Models [2], [3], [4], and [5] were considered as time-independent. For testing the proportional hazards assumption, interaction terms between the time-independent factors and parity were defined. The inclusion of these interaction terms did not significantly increase the likelihood for any of the models analyzed (results not shown), and thus, the proportionality hypothesis was not rejected. Then, it was possible to estimate the expected survival function over the entire productive life and to predict the effect of the factors on productive life expectancy (from first farrowing to culling) and replacement rates (from first successful mating to culling).
The time-independent effect in Model [2] was birth year (P < 0.001). Culling risk reached a maximum for sows born in 2000. From 2000 onward, the hazard ratio decreased (Table 4), and thus, the survival function increased. Figure 5 shows the phenotypic trend in survival of sows over time. Sows born in 2003 had a significantly higher probability of surviving than sows born in 2000 (P < 0.001). Productive life expectancy increased from 560 d in 2000 to nearly 710 d in 2003. Then, the annual replacement rate taking into account the first gestation was reduced from 54% in 2000 to 44% in 2003. Part of this improvement may be due to the better market situation for piglets that leads to an increase of sow numbers (and potentially to longer productive life).
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Number of good teats and feet and leg scores, the exterior traits added in Model [3], had a much lower influence on culling risk compared with the previous factors. However, some of them reached statistical significance, when they were entered sequentially in the model or excluded from the full model, one at a time (Table 3). In fact, the 2 types of tests gave very similar results, showing very little redundancy in the traits considered. Despite the moderate hazard ratios, their effect on life expectancy and replacement rates is not negligible.
The number of good teats had a significant effect on risk of culling when it was treated as a covariate with linear effect (Table 3; P < 0.05). The risk of culling increased as the number of good teats was reduced with a coefficient ß = –0.049 (95% confidence interval: –0.005 to –0.092). Sows with 13 or less good teats had 1.347 times greater risk of being culled (95% confidence interval: 1.003 to 1.809) and a lower survival curve than sows with 14 and more good teats. The predicted life expectancy of sows with 13 or less good teats is only 459 d, whereas for the other sows, it is close to 600 d. This means that the predicted annual replacement rate of these sows is 13% higher, going from 51 up to 64%.
Results for feet and leg scores agreed with what was expected intuitively: the extreme values seemed unfavorable to longevity (Table 5). For example, sows with an X-O rear leg score of 2 had 1.4 times greater risk of being culled than sows with an intermediate score (P < 0.05). Thus, their survival function is lower. The corresponding productive life expectancy is only 449 d compared with 602 d for the optimal score. The annual replacement rate of these sows is 14% higher than for other sows, rising from 51 to 65%. Sows at the optimum score 4 for the size of inner claws rear leg score had 0.83 times less risk of being culled (P < 0.01) than sows with scores 2 and 3. Although this effect is not large, resulting in a slightly higher survival function that decreases replacement rate from 51 to 48%, it is highly significant because the differences are found for levels with large number of records. Finally, the side view angle rear leg score and the angle pastern rear leg score had no significant effect on risk of being culled. However, there was an increase in the hazard of sows scored 5 for side view angle rear leg that approaches statistical significance (P = 0.08).
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The effect of the index for feet and legs in Model [4] was highly significant whether it is treated as categorical (LRT= 23. 0; 6 df; P < 0.001) or linear (LRT = 21. 3; 1 df; P < 0.001). The quadratic term was not significant (LRT = 1.01; 1 df). The hazard ratio increased from 0.936 for the optimal level to 1.762 for the worst 1 (Table 6). The linear regression coefficient of the index is 0.092 (95% confidence interval: 0.053 to 0.131). The Kaplan-Meier estimate of survival functions stratified according to the index value shows how survival decreased with this value (Figure 6). Productive life expectancy is reduced from 681 d for the optimal value to less than 1 yr in the worst value. The annual replacement rate increases from 45.8 to 76.4%.
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The effect of the current genetic index for exterior traits index for feet and legs in Model [5] was highly significant (LRT = 17.53; 5 df; P < 0.01). The instantaneous risk of culling decreased as the sow’s value for the current index for exteriors traits increased (Table 7). The hazard ratio decreased from 1 in the median category to 0.84 or 0.62 in categories over 1 or 2 genetic SD (value over 120 and 140). Figure 7 shows the phenotypic trend in sow survival for increasing index values. The productive life expectancy increased from 602 d in the median population to 699 or 771 d in categories over 1 or 2 genetic standard deviations. The annual replacement rate of sows was reduced from 51 to 45 or 41%, respectively.
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DISCUSSION |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONIMPLICATIONSAPPENDIXLITERATURE CITED |
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) and Large White x Landrace crossbred sows in France (Cozler et al., 1998), whereas Brandt et al. (1999) found a higher average length (880 d) for crossbred sows in weaner production herds in Germany. This difference may be due to the different type of herd. In selection herds, as is the case here, we expect shorter length of productive life than in production herds. The previous authors modeled the hazard function over the entire productive life. However, with our data, the modeling of the hazard function in this way leads to confounding: indeed, culling appears stronger during the first parity (Figure 1). This can be easily explained: in first parity, all sows started from the same point (1 d of productive life) and culling was concentrated on a short period. On the other hand, culling at farrowing in second parity and after was distributed over a longer period of time because the farrowing interval is a random variable. This explains why modeling the cyclic pattern of hazard function within parity is more precise, as in dairy cattle (Ducrocq, 2002, 2005; Röxström et al., 2003). The hazard functions of sows were proportional across parities. They increased with parity number but were not constant within parity. Culling rate was concentrated at the first day after each farrowing or at the first day after weaning. The weaning process itself was concentrated between d 21 and 49 after farrowing with an average weaning age of 35 d. Because of the chosen definition of culling date, risk of culling out of these periods was extremely low. The different culling rates at different periods explain why the simple (monotone) Weibull distribution fails the validation test of the baseline distribution and why a piecewise baseline distribution is needed. In this study, a piecewise Weibull function and a simple graphical method to validate its adequacy were proposed for sow longevity analysis.
Several culling factors were found to influence sow longevity (see for example Brandt et al., 1999; Yazdi et al., 2000a). In our study, the number of weaned piglets in each parity had the strongest influence. Another reproduction variable, the number of piglets born alive, also had a significant, yet lower effect on risk of culling (results not shown). These 2 litter size traits were not included together in the same model because they are strongly correlated. However, for both variables, the hazard ratio increased with decreasing litter size, as reported previously (Friendship et al., 1996; Brandt et al., 1999; Yazdi et al., 2000a). This is indicative that reproductive performance is a major reason for voluntary culling of the animal. Voluntary culling caused by poor performance has also been described for milk production in dairy cattle (Ducrocq et al., 1988; Rajala-Schultz and Gröhn, 2001). The same was found for low calf weaning weight in beef cattle (Díaz et al., 2002; Tarrés et al., 2004). On the other hand, other traits (e.g., growth rate and backfat thickness) that are also under selection within the Swiss program had smaller importance as voluntary reasons for culling and were discarded after a preliminary analysis (results not shown).
The exterior traits analyzed (number of good teats and feet and leg scores) had a moderate influence on risk of culling compared with other factors when they are analyzed separately. However, the combinations of nonoptimal levels for the different exterior traits considerably increase the hazard. Furthermore, an apparently small increase in culling risk related to a time independent factor such as an exterior trait can eventually have an important effect on overall survival, life expectancy, and annual replacement rate because its effect is cumulative and is amplified in the periods when the baseline hazard is very high. Therefore, a way to improve longevity is through phenotypic selection of replacement gilts based on exterior traits. For this reason, gilts with 13 or less good teats or with extreme feet and leg index should be culled. It should be remembered that sows with the optimum score 4 for all feet and leg scores had a longer productive life (681 d) than the average value in the whole population (602 d). The expected annual replacement rate decreased from 51% for the whole population to 46% for the optimum sow.
Clearly, culling animals with extreme feet and leg scores appears to work. From 1999 to 2003, the trend has been to concentrate most of the animals’ characteristics on exterior traits near their optimum level (Tables 1), denoting a more pronounced phenotypic selection especially on feet and legs. This probably can explain part of the improvement in sow longevity observed from 1999 to 2003. However, there were so few extreme animals left that it is not possible to carry out indirect selection further, except when the phenotypic score is replaced by a genetic index. In this study, we checked the efficiency of the latter through the inclusion in the model of the current index for exterior traits. The result of the use of such an index is very promising because the females with highly favorable index values have better survival (correlated response). Here, it is important to note that breeders may keep females with more favorable feet and leg breeding values longer to improve the genetic merit of their sow herd, although this type of selection is certainly not of relevance in the Swiss breeding program. However, the accuracy of such indirect selection still has to be confirmed on new groups of animals. Furthermore, the performance of indirect selection should also be compared with direct selection on longevity breeding values.
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IMPLICATIONS |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONIMPLICATIONSAPPENDIXLITERATURE CITED |
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APPENDIX |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONIMPLICATIONSAPPENDIXLITERATURE CITED |
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The piecewise Weibull hazard function of sows within the jth period is defined as:
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with a different set of Weibull parameters j and j for each period j. Then, the piecewise Weibull survival function S(), for the first period (0,c1), is
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and for the other periods (cj–1,cj) with j > 1, the piecewise Weibull survival function S() is
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where 
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Footnotes |
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2 Corresponding author: joaquim.tarres{at}dga.jouy.inra.fr
Received for publication December 9, 2005. Accepted for publication June 19, 2006.
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LITERATURE CITED |
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TopAbstractINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONIMPLICATIONSAPPENDIXLITERATURE CITED |
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