Information About Fixed Effects

Why Isolate Information About Typical Values?

In a population model, a design is informative about typical values when the observations change predictably as those values change. In a nonlinear mixed-effects model, the observed data also vary because of random effects and residual error. The Gaussian population approximation summarizes those sources of variation in a subject-level mean and covariance. The fixed-effect information asks how precisely the typical-value parameters can be estimated from that approximate distribution, with the population covariance and residual-error parameters treated as known.

This qualification matters. Holding the parameters that define random-effect and residual variability fixed does not always make the resulting subject-level covariance matrix constant as a typical value changes. For example, the first-order approximation to the population covariance can depend on typical values through individual sensitivities. The information then has both a mean-sensitivity contribution and a covariance-sensitivity contribution. If the covariance matrix itself is frozen locally, only the mean contribution remains.

Assume subjects are independent, each \(V_i\) is positive definite, and the mean and covariance are differentiable in \(\theta\). The approximation is used at a specified parameter value, so all derivatives and matrices below are evaluated there. These assumptions make the Gaussian likelihood well-defined and permit the usual expected Fisher information calculation. The result is local: it describes information near the parameter value used to evaluate the model.

Deriving the Fixed-Effect Information

For one subject, suppress the subject index and write \(r=Y-m(\theta)\). The Gaussian log likelihood, omitting constants that do not depend on \(\theta\), is

For component \(\theta_a\), define \(m_a=\partial m/\partial\theta_a\) and \(V_a=\partial V/\partial\theta_a\). Differentiating the log determinant and inverse covariance gives the score component

The first term records how the mean changes with \(\theta_a\). The last two terms record how the covariance changes. Under the model, \(E(r)=0\) and \(E(rr^\mathsf{T})=V\). For a multivariate normal residual, the expected products of the linear and centered quadratic terms are zero, and the covariance of the centered quadratic terms yields the trace expression below. Thus the expected Fisher information between components \(\theta_a\) and \(\theta_b\) is

Applying this result to each independent subject and summing gives the fixed-effect information matrix:

where \(J_i=\partial m_i/\partial\theta^\mathsf{T}\) is the mean sensitivity matrix, and \(V_{i,a}=\partial V_i/\partial\theta_a\). In the matrix above, the entry for components \(a,b\) in the covariance term is \(\frac{1}{2}\operatorname{tr}(V_i^{-1}V_{i,a}V_i^{-1}V_{i,b})\). This notation makes explicit that each subject contributes information under their own sampling design.

A useful special case arises when \(V_i\) is treated as locally constant with respect to \(\theta\). Then \(V_{i,a}=0\) for every component, and

This is the mean-only fixed-effect information. It is also the appropriate expression for a Gaussian model whose covariance is parameter-independent in the fixed effects. It should not be confused with the full fixed-effect block when \(V_i(\theta)\) changes with typical values, even if the parameters defining random-effect and residual variability are held fixed.

When Typical Values Also Change the Covariance

In a population approximation, the covariance may depend on the typical values even when the random-effect covariance and residual-error parameters are held fixed. A local change in a typical value can change the individual sensitivities that map random effects to observations. The covariance contribution then reflects information carried by changes in the distribution’s spread or correlation, not just by changes in its mean.

The covariance term is positive semidefinite as a matrix, although individual off-diagonal entries can be positive or negative. It adds information when changes in the covariance distinguish nearby parameter values. It does not mean that greater variability automatically improves estimation: the mean term is weighted by \(V_i^{-1}\), so larger observational variability can reduce information from mean changes.

Design Interpretation, Diagnostics, and Limits

For fixed covariance, each subject contributes \(J_i^\mathsf{T}V_i^{-1}J_i\). A design is useful for typical values when its observation times, doses, or other design choices create sensitivities that are both sufficiently large and sufficiently distinct across parameters. If two columns of the sensitivity matrix are nearly proportional after accounting for \(V_i^{-1}\), the corresponding parameters may be difficult to distinguish. Increasing the number of independent subjects with the same design adds their information matrices, but it does not repair a structural inability of that design to separate parameter effects.

Several checks help identify a calculation or design that is going wrong:

• Check the covariance: each \(V_i\) must be symmetric positive definite for the inverse and Gaussian likelihood to be valid. A singular or nearly singular matrix may signal redundant observations or numerical instability.
• Check derivatives: compare analytic sensitivities with finite differences at the evaluation point. Poor scaling or a coding error can produce implausibly small, large, or discontinuous information.
• Check the matrix: expected Fisher information should be symmetric and positive semidefinite, apart from small numerical rounding errors. A materially negative eigenvalue suggests an implementation or approximation problem.
• Check identifiability: a very small eigenvalue, large condition number, or nearly singular information matrix indicates a weakly informed parameter combination, even when individual diagonal entries look substantial.
• Check the covariance dependence: determine whether \(V_i\) changes with \(\theta\) in the chosen approximation. Omitting \(V_{i,a}\) is justified only for a mean-only calculation or when those derivatives are zero by model structure.

The alternatives answer different questions. The mean-only expression is simpler and may be appropriate when covariance is fixed in the fixed effects or when deliberately assessing mean sensitivity alone. The full Gaussian fixed-effect block also uses covariance sensitivity. A likelihood-based analysis can instead use the original nonlinear mixed-effects likelihood, without relying on the Gaussian population approximation, but may require substantially more computation and is not the same local information calculation. The Gaussian approximation is most useful as a design tool when it adequately represents the subject-level distribution near the parameter values of interest.

Information is local and model-dependent. It does not guarantee unbiased estimates, adequate finite-sample precision, or robustness to misspecification. The approximation can be poor when nonlinearities are strong over the random-effect distribution, the subject-level distribution is far from Gaussian, or the design produces weakly identified parameters. Also, this fixed-effect calculation conditions on covariance and residual-error parameters as known. Estimating those parameters jointly can affect fixed-effect precision; accounting for that impact requires the joint information and nuisance-parameter analysis.