Population based growth curve analysis: a comparison between models based on ordinary or stochastic differential equations implemented in a nonlinear mixed effect framework

Abstract

Growth curve analysis is frequently carried out in the study of growth of farm animals because measurements are taken repeatedly over time on the same individual within groups of animals. Growth functions like the Gompertz, Richards and Lopez, which are based on ordinary differential equations (ODE), can be used to model the functional relationship between size or mass and age. Recently, nonlinear mixed effect models have become very popular in the analysis of growth data because these models quantify both the population mean and its variation in structural parameters. The within animal variability is modelled as being independent and normally distributed and thus assumed to be uncorrelated. Systematic deviations from the individual growth curve can occur, which is the consequence of animal intrinsic variability in response to environmental impacts. This leads to correlated residuals. A way to model correlated residuals is to adopt a stochastic differential equation (SDE) model for the within animal variability. The objective of the present work was to compare two methods for population based growth curve analysis, where the individual growth curve was given as a solution to an ODE or a SDE. Data used to compare the two approaches was based on 40 pigs of three genders where body weight (BW) measurements were conducted weekly until 150 kg BW and then biweekly until slaughter. It was found that the population structural parameter estimates provided by the two methods were similar although the ODE models may underestimate parameter standard errors in the presence of serial correlation. Moreover, variance components may be severely overestimated if serial correlation is not addressed properly. In conclusion, application of SDEs model is biologically meaningful as there are many complex factors that affect growth in pigs and thus deviations from the time trend are mainly driven by the state of the pig.