Nonlinear Regression with Variance Components: Modeling Effects of Ozone on Crop Yield

Split‐plot experimental designs are common in studies of the effects of air pollutants on crop yields. Nonlinear functions such as the Weibull function have been used extensively to model the effect of ozone (O 3 ) exposure on yield of several crop species. The usual nonlinear regression model, which assumes independent errors, is not appropriate for data from nested or split‐plot designs in which there is more than one source of random variation. The nonlinear model with variance components combines a nonlinear model for the mean with additive random effects to describe the covariance structure. We propose an estimated generalized least squares (EGLS) method of estimating the parameters for this model. This method is demonstrated and compared with results from ordinary nonlinear least squares for data from the National Crop Loss Assessment Network (NCLAN) program regarding the effects of O 3 on soybean [ Glycine max (L.) Merr.]. In this example, all methods give similar point estimates of the parameters of the Weibull function. The advantage of estimated generalized least squares is that it produces proper estimates of the variances of the parameters, estimated yields, and relative yield losses, which take the covariance structure into account. Model selection, hypothesis testing, and construction of confidence intervals are also demonstrated. A computer program that fits the nonlinear model with variance components by the EGLS method is available from the authors.