Bayesian, Likelihood-Free Modelling of Phenotypic Plasticity and Variability in Individuals and Populations

Abstract

There is a paradigm shift from the traditional focus on the “average” individual towards the definition and analysis of trait variation within individual life-history and among individuals in populations. This is a result of increasing availability of individual phenotypic data. The shift allows the use of genetic and environment-driven variations to assess robustness to challenge, gain greater understanding of organismal biological processes, or deliver individual-targeted treatments or genetic selection. These consequences apply, in particular, to variation in ontogenetic growth. We propose an approach to parameterise mathematical models of individual traits (e.g., reaction norms, growth curves) that address two challenges: 1) Estimation of individual traits while making minimal assumptions about data distribution and correlation, addressed via Approximate Bayesian Computation (a form of nonparametric inference). We are motivated by the fact that available information on distribution of biological data is often less precise than assumed by conventional likelihood functions. 2) Scaling-up to population phenotype distributions while facilitating unbiased use of individual data; this is addressed via a probabilistic framework where population distributions build on separately-inferred individual distributions and individual-trait interpretability is preserved. The approach is tested against Bayesian likelihood-based inference, by fitting weight and energy intake growth models to animal data and normal- and skewed-distributed simulated data. i) Individual inferences were accurate and robust to changes in data distribution and sample size; in particular, median-based predictions were more robust than maximum- likelihood-based curves. These results suggest that the approach gives reliable inferences using few observations and monitoring resources. ii) At the population level, each individual contributed via a specific data distribution, and population phenotype estimates were not disproportionally influenced by outlier individuals. Indices measuring population phenotype variation can be derived for study comparisons. The approach offers an alternative for estimating trait variability in biological systems that may be reliable for various applications, for example, in genetics, health, and individualised nutrition, while using fewer assumptions and fewer empirical observations. In livestock breeding, the potentially greater accuracy of trait estimation (without specification of multitrait variance-covariance parameters) could lead to improved selection and to more decisive estimates of trait heritability.

A Novel Estimation of Unobserved Pig Growth Traits for the Purposes of Precision Feeding Methods

“…In the context of pig production data, population and individual level trait estimation is typically carried out within a framework based on hierarchical regression models ( 19 , 73 ). Under this framework, the overall quality of inferences could be negatively impacted by having to directly estimate multiple variance-covariance parameters ( 27 ), which can be challenging due to data limitations ( 9 ). The technical difficulties associated with this estimation procedure are the main reason why several studies make various working assumptions that neglect trait correlations ( 20 , 54 , 61 ).…”

“…In an attempt to alleviate these concerns, the developed approach to estimate population and individual level traits described in this paper, shifted away from hierarchical regression modelling in favour of an alternative framework based on separately inferring individual level trait distributions, which were then scaled up to obtain population level traits. This alternative framework does not necessitate an explicit specification of the aforementioned variance-covariance parameters ( 27 ). Thus, reducing the number of assumptions and the number of parameters that need to be estimated should increase the ability to adequately characterise the traits of individual pigs, which should lead to a greater understanding of the impact of such differences on the estimation of population averages ( 74 ).…”

“…There was considerable variation in the estimated traits among pigs, but very few individuals were identified as potential outliers. However, it is difficult to ascertain if these potential outliers are a result of genetics, environment, feeding and management practises, a combination of some of these factors ( 76 ), or data limitations ( 27 ). Moreover, it is also important to note that since the parameters on body composition were estimated conditional on the bodyweight parameter estimates and without any additional data, some of these estimates could carry increased uncertainty and have limited biological interpretability ( 29 ).…”

“…The pig whose set of estimates Ŷ j had the lowest value of D was chosen to characterise the average pig as its traits were regarded as central in the population. This specific approach to multidimensional estimation of the average pig was chosen because it preserves the individual level correlations between traits which were estimated jointly for each animal in the population ( 27 ).…”

“…Within this setting, it is necessary to either assume or estimate multiple mean and variance-covariance parameters from data, which carries uncertainty ( 27 ) and can be challenging in practise ( 9 ). To avoid these challenges, a potential alternative approach to model trait variation, based on separately inferring individual level distributions within a Bayesian framework, has been recently suggested by Filipe and Kyriazakis ( 27 ). This framework is yet to be comprehensively tested in the context of the estimation of traits that are typically required for precision feeding purposes.…”