It is well known that there exist inter-individual variations of pharmacokinetic parameters. We can analyze pharmacokinetic data by Bayesian models expressing these variations by a prior distribution. Some programs, such as MULTI(ELS), are developed on this method and population distribution of pharmacokinetic parameters are estimated by maximizing log likelihood. In this program, however, the linear approximation by Taylor expansion is used in calculation of log likelihood and this linear approximation brings significant error in calculation of maximum log likelihood. We need to calculate maximum log likelihood unbiassedly for model selection by AIC. In this paper, we employ Monte Carlo method to calculate the likelihood of Bayesian models. We find that the estimate of maximum log likelihood by MULTI(ELS) has bias caused by the linear approximation in calculation of log likelihood. We propose the use of Monte Carlo method for unbiassed calculation of the log likelihood of Bayesian models.
Data of 44 lactations for Holstein cows sampled weekly in 8 prefecture experimental farms were used to compare two models of the lactation curves: the Wood model of y = a tb e-ct and the McMillan model of y = d (1—e-g (t-h) ) e-ft (t>h). The mathematical models of Wood and McMillan were fitted to the results for individual cows by a non-linear least square method. The results obtained were summarized as follows: (1) The standard deviations for all parameters of the Wood model were less than those of the McMillan model. (2) The McMillan model fitted the data better than the Wood model except four cows with respect to AIC (Akaike's Information Criterion). (3) Highly significant correlations were found between several parameters of the Wood model and the McMillan model: a, b, c of th Wood model and d of the McMillan model, c of the Wood model and f of the McMillan model, and, b, c of the Wood model and g of the McMillan model.
A fundamental process is characterized by the evidence that the among-person variability of its systemic parameter is small. The so-called homeostatis is necessary condition but not sufficient condition for this stability. This enables us to apply a linear model with respect to systemic parameter to biological field. The aim of this short notes is to notify several biological examples, which support our model [15, 16] but almost unknown among biometricians, since they were published mostly in a home journal. Among examples, 30, 50 and 60 are new.
A set of quadratic forms are proposed, which contain an approximate solution vector and the right-hand side vector in the mixed model equations after absorption of fixed effects on one and another side, respectively. The approximate solutions are based on utilizing information included in the offdiagonal elements as well as in the diagonal elements of the coefficient matrix. The quadratics are translation invariant and whose matrices are not symmetric. Using the quadratics, a method for unbiased estimation of variance components in the mixed linear model is described as a simple approximation to the minimum variance quadratic unbiased estimation. The current approach does not require the calculation of a generalized inverse of the coefficient matrix as do not other approximate approaches. The relative ability of the present method to eliminate bias due to a culling tyqe of selection is examined by Monte Cario simulation.