First-Order and FOCE Information Approximations

Why the Linearization Point Matters

In a nonlinear mixed-effects model, an individual’s response depends on both population parameters and that individual’s random effects. Approximating the likelihood requires approximating how the response changes with those random effects. A first-order (FO) approximation linearizes at zero random effects; a first-order conditional estimation (FOCE) approximation linearizes at an individual-specific conditional mode. When the response is nonlinear, these points can have different slopes. Because the slope enters the curvature of the approximate conditional likelihood, the two methods can imply different amounts of local information.

This comparison concerns the location at which the response is linearized and the resulting curvature approximation. It does not make conditional curvature interchangeable with the population Fisher information matrix. The latter concerns sensitivity of the marginal subject likelihood to population parameters; the introductory material on exact marginal information distinguishes that target from conditional calculations.

Conditional Curvature Under FO and FOCE

For clarity, first consider one observation and one random effect, with residual variance \(\sigma^2\) and random-effect variance \(\omega\). The negative log conditional density, omitting constants that do not depend on \(\eta\), is

A first-order Taylor expansion of the response at a point \(a\) is \(f(\eta)\approx f(a)+f'(a)(\eta-a)\). Under this linearized response, the second derivative of \(Q\) with respect to \(\eta\) is constant:

The first term is the response contribution to conditional precision; the second is the prior precision from the random-effects distribution. FO uses \(a=0\). FOCE uses \(a=\hat{\eta}\), the conditional mode found using the nonlinear model. In multiple dimensions, the corresponding expression is \(H(a)=J(a)^\mathsf{T}R^{-1}J(a)+\Omega^{-1}\), where \(J(a)\) is the response Jacobian with respect to the random effects.

The FOCE mode is not generally the mode under the FO linearized model. It is obtained by minimizing the nonlinear \(Q(\eta)\), then the response is locally approximated there. In contrast, FO linearizes first at zero and uses the resulting approximation for its subsequent conditional calculation. This order of operations is consequential when the response curve bends over the range of plausible random effects.

Worked Example: A Nonlinear Response with Different Slopes

Why the Curvatures Differ: A Short Justification

The response contribution to curvature is the squared local slope divided by residual variance. Thus, in the scalar case,

The prior-precision terms cancel because both calculations use the same random-effects variance. In the example, the numerator is positive because \(14.820216^2>10^2\), so FOCE has greater conditional curvature. If the slopes have equal magnitudes, the two conditional curvatures are equal, even if the conditional modes differ. This derivation also shows why a difference in curvature is not guaranteed: it depends on the local derivatives, not simply on how far the mode lies from zero.

This is curvature of the linearized conditional objective. The exact nonlinear objective has an additional residual-times-second-derivative contribution to its second derivative. FOCE’s local linearization does not retain that term in \(J^\mathsf{T}R^{-1}J+\Omega^{-1}\). Consequently, this FOCE curvature should not be described as the exact observed Hessian of the nonlinear objective.

When the Approximations Agree, and When Design Rankings Can Shift

Practical Interpretation, Diagnostics, and Limitations

FO is often attractive because the expansion point is fixed and its linearized marginal model has a simple form. With \(\eta\sim N(0,\Omega)\), the FO response approximation is \(f(0;\theta)+J(0)\eta\), so the resulting approximate marginal distribution is normal with mean \(f(0;\theta)\) and covariance \(R+J(0)\Omega J(0)^\mathsf{T}\). FOCE can represent subject-specific local slopes more directly when the conditional modes differ appreciably from zero, but its expansion point depends on the observed subject data and on the model parameters.

Several practical checks help reveal when a comparison deserves caution. Examine the conditional modes relative to the random-effects scale, compare the response slopes at zero and at those modes, and check whether the approximate conditional curvature remains positive and numerically stable. Strong changes in slope across plausible random effects, extreme modes, or near-flat local responses can make a local approximation sensitive to small data or parameter changes. These checks flag potential problems; they do not establish that either approximation is adequate.

A frequent mistake is to treat the larger of \(H_{\mathrm{FO}}\) and \(H_{\mathrm{FOCE}}\) as proof that the corresponding method has more population information. The curvature here concerns uncertainty in an individual random effect under a local conditional approximation. Population information instead concerns estimation of \(\theta\), and depends on how the approximate marginal likelihood changes with \(\theta\), across subjects and observations. A second mistake is to assume FOCE always improves the approximation merely because it expands at a conditional mode. It uses a different local expansion, not an exact likelihood; its accuracy and design implications still depend on model curvature, data, and the approximation’s assumptions.

Both methods rely on an adequately specified nonlinear mixed-effects model, the stated random-effects and residual distributions, and a local Taylor approximation that is useful near its expansion point. Neither local curvature nor a single subject’s conditional mode captures all uncertainty in a population design. The main alternative for a particular comparison is to evaluate a more direct marginal-likelihood calculation, such as numerical integration where feasible; that can be computationally more demanding and is still subject to model assumptions. The choice should be guided by the target quantity and by whether the approximation is credible in the intended design region.