Inverse nonlinear estimation in the presence of measurement error.

Abstract

The inverse estimation problem consists of a calibration stage and a prediction stage. In the calibration stage a response variable is measured at predetermined values of a control variable from which the relationship between the control and response variables is determined. In the prediction stage, an independent sample of the response variable is obtained at an unknown value of the control variable. The inverse estimation problem is to make inferences about the unknown value. Two extensions to the nonlinear inverse regression model are developed: first, measurement error is included in the prediction stage only and, next, measurement error is included in both the calibration and prediction stages. Point and interval estimates are developed for these cases. These are applied to three nonlinear model functions: the Michaelis-Menten, and the two- and four-parameter logistic models. We show that ignoring the measurement error in the nonlinear inverse estimation problem produces a naive estimator whose mean and variance are both biased. Although the bias in the estimator is small for small model error variance in the three models considered, there can be large bias in the variance, even for relatively small measurement error variance. The underestimation of the variance causes confidence interval estimates whose widths are too small, producing noncoverage rates much greater than the nominal level. The proposed point estimators, in general, produced similar point estimates and had variances close to that of their empirical distribution. A one-step estimator, of use when there is error in both the calibration and prediction stages, can be computed explicitly from the naive estimator and performed well in simulation studies. The proposed interval estimators had noncoverage properties superior to the naive interval estimator. The one-step estimator produced similar results for the three models studied but can yield intervals whose noncoverage rates are unequally distributed on the left and right. Alternate estimators proposed to correct this problem, by accounting for the curvature in the model, performed with mixed results.