van de Beek, H.; Beldjenna, M.; Fidler, M. L.; Zwep, L. B.; van Hasselt, J. G. C.

Asymptotic standard errors for the parameter estimates of a nonlinear mixed-effects model fitted by first-order conditional estimation (FOCE) or FOCE with interaction (FOCEI) require the observed (Fisher) information, the negative second derivative of the population objective with respect to the parameters, evaluated at the optimum. The gradient of this objective can be computed exactly from sensitivity equations, but the observed information is conventionally still formed by finite differencing, which is intrinsically less accurate and dependent on a step size. We derive the FOCE and FOCEI observed information in closed form within the same sensitivity-equation framework. Writing the objective as a data term plus the log-determinant of the first-order inner Hessian, the population Hessian splits into two contributions: the data term reuses the second-order sensitivities already needed for the gradient, whereas the log-determinant term requires third-order sensitivity equations. The third-order dependence is therefore confined to a single term, where it enters in exactly two places. A finite-difference error analysis shows that the differenced Hessian attains an accuracy no better than the square root of the objective's evaluation accuracy, whereas the analytic form is limited only by the accuracy of the sensitivity and differential-equation solutions, with no step size to tune. The method is implemented in the open-source R package nlmixr2, making analytic standard errors available in routine model fitting.