I've been using nlme and more recently lmer to fit multi-level models of time course data using orthogonal polynomials. My colleagues and I originally chose polynomials because we believed that "nonlinear" functions such as the logistic could not be used for multi-level modeling because they are not dynamically consistent. In at least one case this constraint is articulated very explicitly (Willett, 1997, p. 238-239):
In general, the individual growth modeling approach can accommodate any level-1 model that is linear in the individual growth parameters...Many common growth functions are dynamically consistent, including the quadratic model cited above and all other polynomial models, regardless of their order. Other potentially important individual growth models such as the logistic model (which provides an important theoretical representation of human development from the perspective of some psychological theories - see Fischer & Pipp, 1984) is not linear in the individual growth parameters in its usual formulation.
However, I recently discovered that, as I understand it, both nlme and lmer can use SSfpl to fit 4-parameter logistic functions in a multi-level modeling context. Did we misunderstand the dynamic consistency constraint? Perhaps lmer and/or SSfpl implements the 4-parameter logistic in a dynamically consistent way? If so, does anyone know how it is constrained to be dynamically consistent?
Thanks in advance.
