Reassessment of biases in predicted nitrogen flows to the duodenum by NRC 2001

The appearance of numerous plots in recent literature from which the residuals are plotted against observed values (Y) to assess a model's potential bias raises this question: should residuals be regressed against Y or against predicted values (Y)? The answer requires knowing the expected relationship under the assumption of an unbiased model. The objectives of this paper are: 1) to derive the expected relationship between residuals, Y, and Y; 2) to determine whether Y or Y should be used for the assessment of bias; and 3) to reassess the extent of mean and linear bias in the prediction of N flows to the duodenum by the NRC (2001). In the simplest case, we can assume a true model of the form Y = Xbeta + epsilon. This model is estimated by Y = Xb + e, and Y = Xb. The correlation between the residual vector e and the vector of observations Y can easily be derived. The numerator of the correlation coefficient is shown to be equal to e'e, the residual sum of squares. The denominator of this correlation is equal to the square root of e'e multiplied by the total sum of squares. Algebraic simplifications show that the correlation between e and Y is equal to the square root of (1-R2). That is, under the assumption of an unbiased model, the residuals are correlated with the observed values and the slope of e regressed on Y is equal to (1-R2). Thus, a graph of e versus Y will show a positive slope between e and Y unless the model is a perfect predictor (i.e., R2 is equal to 1.0). Significant slopes linking e to Y have been erroneously interpreted as evidence of biased models in the NRC (2001). Conversely, the slope of e regressed on Y is expected to be zero under the assumption of an unbiased model. Therefore, residuals should be regressed against Y and not Y. When Y, as opposed to Y, was used to assess biases in the prediction of flows to the duodenum of microbial N, nonammonia-nonmicrobial N and nonammonia N in NRC (2001), mean biases became nonsignificant and linear biases over the range of predicted values are of the same magnitude or smaller than the standard errors of measurements reported in literature. Thus, although N flow predictions from NRC (2001) may not be precise, they appear to have insignificant and inconsequential biases.