Linear mixed effects models assume that a linear relationship exists between the predictors and the target variable. In many cases, this is a problematic assumption; whenever the target is expected to show any kind of saturation effect or have an exponential response with respect to any of the regressors, the linearity assumption needs to be removed. 

In medicine and biology, this is usually the case, as dose response studies almost always exhibit a certain kind of saturation effect. The same happens for marketing studies, because spending increasing amounts of resources in order to drive sales up might be effective, but it won’t be effective if that spend is too large. 

Fitting nonlinear mixed effects models is much harder than their linear counterpart. Here, we can’t rely on any matrix techniques and we need to attack the problem using numerical approximation techniques. 

Mixed effects models generate a likelihood that, has as many integrals as there are random effects for us. The problem is that, here, there are no OLS and matrix calculations that can help us. We need to calculate the likelihood along with the integrals.

Every software implementation uses quadrature formulas to calculate these integrals (if the random effects are generated according to a Gaussian distribution). This is usually very fast but numerical problems usually appear. 

Instead of following the quadrature approach, we will use simulation techniques to calculate the integrals. It will be much slower and less precise than quadrature formulas, but it will be much simpler to illustrate how the mechanics of nonlinear mixed effects models work (the other advantage is that this approach can be extended to nonlinear mixed effects involving other distributions, such as Student's t-distributions).