A versatile method for simultaneous analysis of families of curves

We have developed a versatile new approach to the simultaneous analysis of families of curves, which combines the simplicity of empirical methods with several of the advantages of mathematical modeling, including objective comparison of curves and statistical hypothesis testing. The method uses weighted smoothing cubic splines; the degree of smoothing is adjusted automatically to satisfy constraints on curve chape (monotonicity, number of inflection points). By simultaneous analysis of a family of curves, one can extract the shape common to all the curves. Up to four linear scaling parameters are used to match the shape to each curve, and to provide optimal superimposition of the several curves. By applying constraints to these scaling factors, one can test a variety of hypotheses concerning comparisons of curves (e.g., identity, parallelism, or similarity of shape of two or more curves), and thus evaluate the effects of experimental manipulation. By optimal pooling of data one can avoid the need for arbitrary selection of a typical experiment, and can detect subtle but reproducible effects that might otherwise be overlooked. This approach can facilitate the development of an appropriate model. The method has been implemented in a Turbo-Pascal program for IBM-PC compatible microcomputers, and in FORTRAN-77 for the DEC-10 mainframe, and has been utilized successfully in a wide variety of applications.