Comparison of growth curves of two genotypes of dairy goats using nonlinear mixed models

Abstract

SUMMARYThe objective of the current study was to assess the use of nonlinear mixed model methodology to fit the growth curves (weight v. time) of two dairy goat genotypes (Alpine, +A and Saanen, +S). The nonlinear functions evaluated included Brody, Von Bertalanffy, Richards, Logistic and Gompertz. The growth curve adjustment was performed using two steps. First, random effects u1, u2 and u3 were linked to the asymptotic body weight (β1), constant of integration (β2) and rate constant of growth (β3) parameters, respectively. In addition to a traditional fixed-effects model, four combinations of models were evaluated using random variables: all parameters associated with random effects (u1, u2 and u3), only β1 and β2 (u1 and u2), only β1 and β3 (u1 and u3) and only β1 (u1). Second, the fit of the best adjusted model was refined by using the power variance and modelling the error structure. Residual variance ($\sigma _e^2 $) and the Akaike information criterion were used to evaluate the models. After the best fitting model was chosen, the genotype curve parameters were compared. The residual variance was reduced in all scenarios for which random effects were considered. The Richards (u1 and u3) function had the best fit to the data. This model was reparameterized using two isotropic error structures for unequally spaced data, and the structure known in the literature as SP(MATERN) proved to be a better fit. The growth curve parameters differed between the two genotypes, with the exception of the constant that determines the proportion of the final size at which the inflection point occurs (β4). The nonlinear mixed model methodology is an efficient tool for evaluating growth curve features, and it is advisable to assign biologically significant parameters with random effects. Moreover, evaluating error structure modelling is recommended to account for possible correlated errors that may be present even when using random effects. Different Richard growth curve parameters should be used for the predominantly Alpine and Saanen genotypes because there are differences in their growth patterns.

Modelación de curvas de crecimiento de novillas lecheras de reemplazo criadas en condiciones tropicales de pastoreo

“…Usually, the adjustment of weight-age data through nonlinear models has been performed using ordinary least squares regressions, where errors are assumed to be homoscedastic and normally distributed (Wang and Zuidhof, 2004). Nevertheless, the unbalanced structure of the data, which is naturally generated by death or discard of animals during growth and the correlation of mesurements of an individual over time, tend to generate correlated errors and heterogeneous variances between observations from one age to another, violating some of the mentioned assumption and therefore, the parameters estimates may not be the most appropriate (Regadas-Filho et al, 2014;Tedeschi et al, 2000). Another approach to model longitudinal growth data is the use of nonlinear mixed models.…”

“…Although different approximations to the integral are available, in the present study, the approximation used in those models associated with random effects, was the first-order method of Sheiner (1982, 1988) available via METHOD=FIRO option. However, in the estimation of the fixed effects models, the approximation used was the adaptive Gaussian quadrature (Pinheiro and Bates 1995), since FIRO method is not appropriate if random effects are not considered in the model (Regadas-Filho et al, 2014). The estimated models were used to constructs Table II.…”

“…Gompertz (Laird, 1965) Von Bertalanffy (Von Bertalanffy, 1957) Verhulst (Macciotta et al, 2004) Logistic (Nelder, 1961) The variance partition is important because parameters estimates can vary from animal to animal even within each racial group, especially those related with asymptotic body weight and maturation rate (Regadas-Filho et al, 2014). In the case of VB-012, the error variation was partitioned in within animal variation and the between animal variation was further divided into variance due to asymptotic weigth (β0), constant of integration (β1), maturation rate (β2) and the covariance among them (Aggrey, 2009).…”

“…The estimated parameters by the evaluated mathematical functions can be found in Table V. In the nonlinear mixed model methodology, the estimation of fixed effects is considered the mean value of each parameter (β0, β1 or β2) within the studied population, meanwhile, as individual growth curve parameters are adjusted for each animal at the same time, the random effects represent differences between the individually fitted parameter for each animal and the population average (fixed effect). The random effects are modeled assuming a normal distribution with zero mean and unstructured covariance matrix G (Aggrey, 2009;Regadas-Filho et al, 2014). Figure 2 presents the general adjusted growth curve and an example of individual adjustment of a random animal obtained with VB-012 model.…”

“…The variance components significance yields important information suggesting variability in growth patterns within the racial groups. A special emphasis must be placed on variability related to asymptotic body weight (β0) and maturation rate (β2) parameters since it can be used in genetic programs to select the individuals with the best perfomance associated with these traits (Regadas-Filho et al, 2014;Menchaca et al, 1996).…”