title: Estimation of response curves in closed-loop physiological control.
abstract: Several recent reports have addressed the problem of estimating the response slope from repeated measurements of paired data when both stimulus and response variables are subject to biological variability. These earlier approaches suffer from several drawbacks: useful information about the relationships between the error components in a closed-loop system is not fully utilized; the response intercept cannot be directly estimated; and the normalization procedure required in some methods may fail under certain circumstances. This paper proposes a new, general method of simultaneously estimating the response slope and intercept from corrupted stimulus-response data when the errors in both variables are specifically related by the system structure. A direct extension of the least-squares approach, this method [directed least squares (DLS)] reduces to ordinary least-squares methods when either of the measured variables is error free and to the reduced-major-axis (RMA) method of Kermack and Haldane (Biometrics 37: 30-41, 1950) when the magnitudes of the normalized errors are equal. The DLS estimators are scale invariant, statistically unbiased and always assume the minimum variance. With simple modifications, the method is also applicable to paired data. If, however, the relation between error components is uncertain, then the RMA method is optimal, i.e., having the least possible asymptotic bias and variance. These results are illustrated by using various types of closed-loop respiratory response data.
mesh_terms: D000818:Animals; D008433:Mathematics; D008954:Models, Biological; D005082:Physical Exertion; D010827:Physiology; D013223:Statistics as Topic