Continued progress has been made on the algorithm underlying the """"""""universal"""""""" curve fitting method, FLEXIFIT. This method combines advantages of empirical, nonparametric methods with those of traditional parametric modeling approaches. It is especially appropriate when quantitative estimates are required based on a family of curves with a common shape not easily described by a simple mathematical expression. The method uses cubic smoothing splines to describe the curve shape, together with four parameters to describe horizontal and vertical shifts and stretches required to superimpose members of the family. A major theoretical advance allows for the calculation of equivalent degrees of freedom for the residual sum-of-squares, and asymptotic standard errors of the shift and scale parameters. This advance is based on a definition of the problem as a penalized sum-of-squares, and a reexpression as a least-squares in a transformed variable. This expression makes possible a more efficient numerical algorithm, and provides a theoretical statistical basis for an otherwise apparently ad hoc method. We have applied the method to numerous data sets arising in immunoassay, bioassay, and pharmacology, using data from experiments in our own laboratory and from a number of others outside the NIH. The first phase of a study of optimal design of ligand binding was completed, resulting in a computer program for determining these designs, together with an extensive catalog of designs for a variety of commonly used experimental protocols. Use of an optimal design can result in a reduction in variance of the parameter estimates of up to 50% compared with some commonly used designs. This study also produced some general rules-of-thumb for designing binding experiments which are useful even in the absence of a computerized analysis. A second phase of this study has begun to provide optimal design of experiments involving multiple ligands simultaneously (""""""""blocking studies and dose response surfaces"""""""").
