Two-stage nonlinear mixed model, analysis using expected value transformations

The standard two-stage method is a simple procedure for estimating parameters of the population mean curve for the nonlinear random coefficient regression model, but it produces biased estimates with unknown distributions. This simple method can be improved by first reparameterizing the model using the expected value transformation. We propose an adaptation of the standard two-stage method in which the first step is to reparameterize the model, then apply the standard two-stage method to the expected value parameters rather tl;an to the original parameters, and finally transform back to the original parameterization. The resulting estimates of the expected value parameters are approximately unbiased and normally distributed, consequently hypothesis tests and confidence intervals can be constructed using standard methods. Profile likelihood ellipses for the expected value parameters can be inverted to construct profile likelihood plots and confidence intervals for the original parameters. The final parameter estimates have smaller bias and variance than the standard two-stage estimates, and the confidence intervals are asymmetric, reflecting the actual distributions of parameter estimates more accurately. We demonstrate fitting a Mitscherlich function to the effect of ambient CO2 on photosynthesis of grasses (Potvin, Lechowicz, and Tardif 1990), comparing curves among treatment groups, and constructing confidence intervals. The coverage probabilities of confidence intervals for the standard two-stage method and the proposed method are compared in a Monte Carlo simulation.