From exact gradients to exact standard errors: third-order sensitivity equations for the FOCE and FOCEI population likelihood

Asymptotic standard errors for the parameters 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 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 less accurate and step-size dependent. Our objectives are to (i) derive the FOCE and FOCEI observed information in closed form within the same sensitivity-equation framework, and (ii) quantify the precision this recovers. Writing the objective as a data term plus the log-determinant of the first-order inner Hessian, the population Hessian splits so that the data term reuses the second-order sensitivities already needed for the gradient, whereas the log-determinant term requires third-order sensitivity equations -- confining the third-order dependence to a single term, where it enters in exactly two places. A finite-difference error analysis shows the differenced Hessian attains an accuracy no better than the square root of the objectives evaluation accuracy, whereas the analytic form is limited only by the sensitivity and differential-equation solutions, with no step size to tune. We illustrate this on a one-compartment oral model with first-order absorption fitted to warfarin data, where the differenced standard error is usable only over a narrow band of step sizes while the analytic value carries none. Implemented in the open-source R package nlmixr2, the method makes exact, reproducible standard errors routine, supporting more dependable confidence intervals, identifiability assessment, and uncertainty propagation.