The Critical Nutrient Deficiency Level and Economic Optimum Nutrient Level for the Mitscherlich Growth Curve

February 1988 Plant growth is modeled with the Mitscherlich response equation. Critical nutrient deficiency levels are defined in various ways including: the nutrient level associated with lOOp% of maximum yield for 0nutrient level associated with the point of maximum curvature on the response function (denoted xc); and, the economic optimum nutrient level under a linear cost function for the nutrient (denoted Xopt)· The Mitscherlich growth curve is reparameterized to include each of these definitions of a critical nutrient deficiency level. Such a reparameterization enables easy access to approximate standard errors of these critical nutrient levels as well as approximate oneor two-sided confidence intervals. An example is presented using the SAS® procedure for nonlinear least squares. 1 Assistant Professor, Biometrics Unit, Cornell University, Ithaca, NY 2In the Biometrics Unit technical report series, Cornell University, Ithaca, NY, 14853. Mitscherlich Growth BU-951-M -2INTRODUCTION M.P. Meredith March 1988 Critical nutrient deficiency levels associated with 90% of maximum yield have been determined by both free-hand graphical procedures and the fitting of the Mitscherlich growth equation (Ohki, 1977; Ware, Ohki, and Moon, 1982). Only point estimates have been reported for the nutrient level associated with 90% of maximum yield (denoted xgo) with no indication of how to determine a standard error associated with the reported critical nutrient deficiency level. It is shown that a reparameterization of the usual Mitscherlich equation enables determination of an asymptotically valid standard error for the parameter representing the critical nutrient deficiency level. Thus, a useful interval estimate may be calculated for a stated level of confidence. Current definitions of the critical nutrient deficiency level are arbitrary and subject to criticism. A new definition is proposed. The definition is based upon determining the nutrient level associated with the point of maximum curvature on the Mitscherlich growth curve and is denoted by Xc· This formulation provides an estimate distinct from that provided by the estimate of x90. Another potentially useful statistic for the Mitscherlich curve is the economic optimum nutrient level. This is determined by maximizing the response subject to a linear cost function for the nutrient. The resulting optimum, Xopt, is similar in form to Xc· The Mitscherlich model may again be reparameterized to include Xopt as a parameter in the model. In general, the practice of reparameterizing a nonlinear (or even linear) response equation to include a parameter of research interest may be a profitable one. Not only are asymptotically valid standard errors output by statistical computing software, but estimators are often (though not necessarily) less correlated after reparameterization. This results in faster, more stable convergence to the least Mitscherlich Growth BU-951-M -3M.P. Meredith March 1988 squares estimates. Further, the reparameterization results in a model whose parameters are of direct research interest to the investigator and are thus immediately interpretable.