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Additional file 3: Figure S2. Total additive ( $${a}_{0}+{a}_{1}$$ a 0 + a 1 ) variance for daily feed intake (DFI) for different probabilities of the occurrence of an environmental challenge. Figure S3. Level ( $${a}_{0}$$ a 0 ) variance for daily feed intake (DFI) for different probabilities of the occurrence of an environmental challenge. Figure S4. Environmental sensitivity ( $${a}_{1}$$ a 1 ), variance for daily feed intake (DFI) for different probabilities of the occurrence of an environmental challenge. In all cases $$p$$ p is the environmental descriptor with $$p$$ p = 0 indicating non-challenging environmental conditions and $$p$$ p = 1 indicating highly challenging conditions. The interquartile range is shown in grey in all cases.

Additional file 1: Table S1. Parameter estimates obtained using the reaction norm animal model (RNAM) with homogeneous residual variance (column in grey) and heterogeneous residual variances (last nine columns). One residual variance was set for normal days and one for highly variable days. Nine values of p, going from 0.10 to 0.90 were set as cutting points to differentiate between normal days and highly variable days.

Additional file 2: Figure S1. Heritability estimates for daily feed intake (DFI) for different probabilities of the occurrence of an environmental challenge ( $$p$$ p is the environmental descriptor with $$p$$ p = 0 indicating non-challenging environmental conditions and $$p$$ p = 1 indicating highly challenging conditions). The interquartile range is shown in grey.

García Baccino, Carolina Andrea. Universidad de Buenos Aires. Facultad de Agronomía. Buenos Aires, Argentina. - Université de Toulouse. GenPhySE. INRAE, ENVT. Tolosan, France. Etancelin, Christel Marie. Université de Toulouse. GenPhySE. INRAE, ENVT. Tolosan, France. Tortereau, Flavie. Université de Toulouse. GenPhySE. INRAE, ENVT. Tolosan, France. Marcon, Didier. Unité Expérimentale INRAE, Domaine de La Sapinière, INRAE. France. Weisbecker, Jean Louis. Université de Toulouse. GenPhySE. INRAE, ENVT. Tolosan, France. Legarra, Andres. Université de Toulouse. GenPhySE. INRAE, ENVT. Tolosan, France. Background: Resilient animals can remain productive under different environmental conditions. Rearing in increas - ingly heterogeneous environmental conditions increases the need of selecting resilient animals. Detection of environmental challenges that affect an entire population can provide a unique opportunity to select animals that are more resilient to these events.\nThe objective of this study was two-fold: (1) to present a simple and practical data-driven approach to estimate the probability that, at a given date, an unrecorded environmental challenge occurred; and (2) to evaluate the genetic determinism of resilience to such events.\nMethods: Our method consists of inferring the existence of highly variable days (indicator of environmental challenges) via mixture models applied to frequently recorded phenotypic measures and then using the inferred probabilities of the occurrence of an environmental challenge in a reaction norm model to evaluate the genetic determinism of resilience to these events.\nThese probabilities are estimated for each day (or other time frame). We illustrate the method by using an ovine dataset with daily feed intake (DFI) records.\nResults: Using the proposed method, we estimated the probability of the occurrence of an unrecorded environmental challenge, which proved to be informative and useful for inclusion as a covariate in a reaction norm animal model.\nWe estimated the breeding values for sensitivity of the genetic potential for DFI of animals to environmental challenges.\nThe level and slope of the reaction norm were negatively correlated (−0.46±0.21).\nConclusions: Our method is promising and appears to be viable to identify unrecorded events of environmental challenges, which is useful when selecting resilient animals and only productive data are available. It can be generalized to a wide variety of phenotypic records from different species and used with large datasets.\nThe negative correlation between level and slope indicates that a hypothetical selection for increased DFI may not be optimal depending on the presence or absence of stress. We observed a reranking of individuals along the environmental gradient and low genetic correlations between extreme environmental conditions. These results confirm the existence of a G×E\ninteraction and show that the best animals in one environmental condition are not the best in another one. grafs.

The relationship between pairs of individuals is an important topic in many areas of population and quantitative genetics. It is usually measured as the proportion of thegenome identical by descent shared by the pair and it can be inferred from pedigree information. But there is a variance in actual relationships as a consequence of Mendelian sampling, whose general formula has not been developed. The goal of this work is to develop this general formula for the one-locus situation,. We provide simple expressions for the variances and covariances of all actual relationships in an arbitrary complex pedigree. The proposed method relies on the use of the nine identity coefficients and the generalized relationship coefficients; formulas have been checked by computer simulation. Finally two examples for a short pedigree of dogs and a long pedigree of sheep are given.