Transforming signals into quantitative values and mathematical treatment of data

The purpose of this article is to review some mathematical models which are available to calibrate immunoassays and interpolate analyte concentration. Several curve fitting methods are available, but few of them are currently used. Point-to-point linear interpolation is prone to bias when the rate of curvature of the calibration curve or when the random error are high. Polygonal interpolation connects each neighboring pair of data points with multiple connected line segments. Neither smooths the data. Smoothing models are often preferred because they can correct some of the random error. Weighted least squares must be used when the calibration points have nonhomogeneous variance. The most used methods are empirical (spline function) or semi-empirical (logit-log, logistic). The logitlog straight line is often too rigid. Logistic equation can fit numerous assays. The best empirical model is the smoothed cubic spline. Model-based methods attempt to describe the data in terms of a given theoretical model for the assay chemistry. The choice of a method to efficiently control calculations must take into account the quality of the fit, the robustness with outliers, the proper computation of confidence limits and the user-friendliness. Errors associated with the different processing steps can also be due to the calibration point distribution, the axes in which the curve is shown, the model used, and, of course the measurement errors.