70 Curvilinear regression.

T. W. Popham. USDA, ARS, SPA, Biometrics, Stillwater, OK 74075.

Many times measured responses to changes in an independent “causative” variable are not linear. A choice must be made concerning which mathematical functions might be used as a model. If a linear function, which produces a curved response, is chosen it can be fitted with linear least squares. Two other groups of functions (intrinsically linear and non-linear) are available which will usually require fitting with non-linear least squares. Incorrect assumptions concerning distribution of errors can affect confidence limits about the regression. Various methods of fitting do not produce the same results. Choice of starting values for parameters to be estimated will affect the result. Poorly chosen initial values may cause the fitting process to find a local minimum but not a global minimum or simply not converge at all. The power function, logarithmic function and some probability functions (log-normal, 2- and 3-parameter Weibull, exponential) illustrate simple curve forms and how they might be used. When a response repeats the response pattern over time, periodic regression (trigonometric polynomials) may be required. Curved surfaces enable prediction of response to two or more “causative” variables.