The statistics of quantifiable homeostasis. I. Simple linear homeostasis

The statistical properties are explored of the least-squares estimators of the parameters of a deterministic linear model for homeostasis with random, normally distributed technical errors. The parameters are: the amplitude of perturbation (A), the homing value (H), the restoration constant (B), and the lag time (L). The former two, although important in their own right, tell us nothing about the processes of biologic restitution and are thus considered "nuisance" parameters; the latter two are the "business" parameters, which tell us about the processes of biologic adjustment. The properties studied include bias and precision and their relationships to the number of data points and to the size of the technical error. Efforts have been concentrated on data points equally spaced over six lag times (which corresponds roughly to the period of observation used in loading studies, such as glucose tolerance tests). The distributions of these four estimators are studied in sets of 100 Monte Carlo simulations, each with 15, 30, and 100 data points, respectively. The estimators are virtually unbiased and all satisfactorily close to Gaussian. The residual mean-square errors (the divisor being four less than the number of data points, since four parameters are estimated) seem to be unbiased estimators of the variances of the errors. Moreover, when suitably scaled, the residual sums of squares have distributions close to the chi-square distribution for the appropriate degrees of freedom and with the corresponding means and variances. The correlations among the estimators are modest and, except where A is involved, small.