Standard error of inverse prediction for dose-response relationship: approximate and exact statistical inference

This paper develops a new metric, the standard error of inverse prediction (SEIP), for a dose-response relationship (calibration curve) when dose is estimated from response via inverse regression. SEIP can be viewed as a generalization of the coefficient of variation to regression problem when x is predicted using y-value. We employ nonstandard statistical methods to treat the inverse prediction, which has an infinite mean and variance due to the presence of a normally distributed variable in the denominator. We develop confidence intervals and hypothesis testing for SEIP on the basis of the normal approximation and using the exact statistical inference based on the noncentral t-distribution. We derive the power functions for both approaches and test them via statistical simulations. The theoretical SEIP, as the ratio of the regression standard error to the slope, is viewed as reciprocal of the signal-to-noise ratio, a popular measure of signal processing. The SEIP, as a figure of merit for inverse prediction, can be used for comparison of calibration curves with different dependent variables and slopes. We illustrate our theory with electron paramagnetic resonance tooth dosimetry for a rapid estimation of the radiation dose received in the event of nuclear terrorism.