Comparison of linear, skewed-linear, and proportional hazard models for the analysis of lambing interval in Ripollesa ewes1

Abstract

Lambing interval is a relevant reproductive indicator for sheep populations under continuous mating systems, although there is a shortage of selection programs accounting for this trait in the sheep industry. Both the historical assumption of small genetic background and its unorthodox distribution pattern have limited its implementation as a breeding objective. In this manuscript, statistical performances of 3 alternative parametrizations [i.e., symmetric Gaussian mixed linear (GML) model, skew-Gaussian mixed linear (SGML) model, and piecewise Weibull proportional hazard (PWPH) model] have been compared to elucidate the preferred methodology to handle lambing interval data. More specifically, flock-by-flock analyses were performed on 31,986 lambing interval records (257.3 ± 0.2 d) from 6 purebred Ripollesa flocks. Model performances were compared in terms of deviance information criterion (DIC) and Bayes factor (BF). For all flocks, PWPH models were clearly preferred; they generated a reduction of 1,900 or more DIC units and provided BF estimates larger than 100 (i.e., PWPH models against linear models). These differences were reduced when comparing PWPH models with different number of change points for the baseline hazard function. In 4 flocks, only 2 change points were required to minimize the DIC, whereas 4 and 6 change points were needed for the 2 remaining flocks. These differences demonstrated a remarkable degree of heterogeneity across sheep flocks that must be properly accounted for in genetic evaluation models to avoid statistical biases and suboptimal genetic trends. Within this context, all 6 Ripollesa flocks revealed substantial genetic background for lambing interval with heritabilities ranging between 0.13 and 0.19. This study provides the first evidence of the suitability of PWPH models for lambing interval analysis, clearly discarding previous parametrizations focused on mixed linear models.

Comparison between linear and proportional hazard models for the analysis of age at first lambing in the Ripollesa breed

“…Our estimates about goodness of fit suggested that linear models must be ruled out from AFL data analysis, whereas PWPH models were an appealing alternative with preferable statistical performances. This conclusion agreed with previous studies that advocated the superiority of survival analysis techniques when handling time interval data such as lambing interval in sheep (Casellas et al, 2012), age at first calving (García et al, 2014), or fertility-and longevity-related traits in cattle (Caraviello et al, 2004;Schneider et al, 2005Schneider et al, , 2006MacNeil and Vukasinovic, 2011). Within this context, the implementation of proportional hazards models seems a reasonable alternative for AFL data.…”

“…Note that both approaches provide similar estimates and departures must be partially attributable to the absence of systematic, permanent and genetic factors when computing the baseline hazard function under model PWPH3; the Kaplan-Meier estimate is computed from raw AFL data and all additional sources of variation are implicitly included. The need for this kind of additional modeling of the standard Weibull baseline function has been previously demonstrated on longevity (Tarrés et al, 2005;Casellas et al, 2008) and fertility data (Casellas and Bach, 2012), although this should not be viewed as a weakness of the survival analysis techniques but a remarkable advantage in terms of flexibility of the PWPH (Casellas, 2007) model. Systematic and genetic influences on AFL Although three systematic effects were included in the analytical model, only the season of birth of the ewe had relevant influences on the AFL under PWPH3 parameterization.…”

“…Both approaches are able to fit data under unsymmetrical distributions, although the realized goodness of fit can differ substantially (Casellas and Bach, 2012). Without loss of generality, all analyses accounted for three independent sources of variation, that is systematic (b), permanent environmental (p), and additive genetic (a) effects.…”

“…Both the generalization of Gaussian models to non-symmetrical patterns and the implementation of non-linear models on the basis of skew parametric distributions have been suggested in previous studies Casellas and Bach, 2012;García et al, 2014). The main objective of this research was to evaluate two main parameterizations for the analysis of AFL in the Ripollesa sheep breed, the skew-Gaussian mixed linear model (sGML, Sahu et al, 2003) and the piecewise Weibull proportional hazards model (Casellas, 2007).…”

“…Nevertheless, heritabilities form linear and proportional hazard models cannot be comparable at all. The advantage of proportional hazard models when capturing additive genetic variance from time-interval traits was previously reported by Caraviello et al (2004) in dairy cattle and Casellas and Bach (2012) in sheep, this potentially leading to more rapid genetic progress (Korsgaard et al, 2002). It is important to note that predicted breeding values from mixed linear and proportional hazard models cannot be directly compared because they are diametrically opposed.…”