Physiological response curve analysis using nonlinear mixed models

Nonlinear response curves are often used to model the physiological responses of plants. These models are preferable to polynomials because the coefficients fit to the curves have biological meaning. The response curves are often generated by repeated measurements on one subject, over a range of values for the environmental variable of interest. However, the typical analysis of differences in coefficients between experimental groups does not include a repeated measures approach. This may lead to inappropriate estimation of error terms. Here, we show how to combine mixed model analysis, available in SAS, that allows for repeated observations on the same experimental unit, with nonlinear response curves. We illustrate the use of this nonlinear mixed model with a study in which two plant species were grown under contrasting light environments. We recorded light levels and net photosynthetic response on anywhere from 8 to 10 points per plant and fit a Mitscherlich model in which each plant has its own coefficients. The coefficients for the photosynthetic light-response curve for each plant were assumed to follow a multivariate normal distribution in which the mean was determined by the treatment. The approach yielded biologically relevant coefficients and unbiased standard error estimates for multiple treatment comparisons.