Multivariate modeling for retained protein and lipid

Energy efficiencies and maintenance parameters have been traditionally estimated with a linear regression model that treated metabolizable energy intake as the dependent variable and protein and lipid depositions as the independent variables. Several studies have described the statistical issues associated with this approach, such as the reverse role of dependent and independent variables and a potential multicollinearity issue due to the high correlation between protein and lipid depositions. Biased regression techniques have been proposed to minimize the harmful effects of multicollinearity on the estimates of energy efficiencies. These approaches, however, only partially addressed the issues described for the linear regression approach. A first multivariate approach was developed by L. J. Koong in the 1970s, who estimated the energy parameters using a set of simultaneous equations. This multivariate approach has been considerably extended in the past two decades with the complete characterization of model's biological interpretation under different feeding conditions, the simultaneous estimation of maintenance requirements, the extension of the model to a mixed-effects framework, and the implementation of a Bayesian framework for model fitting. The multivariate approach has been successfully applied to model energy deposition and partitioning by mice, pigs, salmon, and rainbow trout. However, multivariate models are, in general, harder to fit than linear regression models due to 1) larger number of parameters, 2) issues with parameter identifiability, and 3) overall lack of algorithm convergence. Therefore, with the recent availability of easy to use and efficient computer packages for model fitting, the use of a Bayesian framework seems to be an attractive approach for fitting multivariate models describing protein and lipid deposition.