Worked Michaelis-Menten Information Matrix

Why Work Through a Nonlinear Information Matrix?

For a nonlinear mean model, information depends on where the model is evaluated in parameter space. In Michaelis–Menten kinetics, the same concentration schedule can provide different amounts of information about the maximum rate and the half-saturation constant at different nominal parameter values. A useful calculation therefore needs more than a formula for an information matrix: it needs a stated model, a nominal parameter point, and an interpretation of the resulting matrix.

This tutorial derives and evaluates that calculation for a Michaelis–Menten mean with independent Gaussian errors. The emphasis is on how the matrix’s inverse and eigenstructure describe local parameter uncertainty and confounding. All numerical data below are hypothetical.

Deriving the Local Fisher Information

Write the parameter vector as \(\theta=(V_{\max},K_m)^\mathsf{T}\). The sensitivity vector is the gradient of the mean with respect to these parameters. It records the local change in the predicted response for a small change in each parameter:

The first component follows by differentiating with respect to \(V_{\max}\); the second follows by differentiating \(V_{\max}C(K_m+C)^{-1}\) with respect to \(K_m\). In particular, the second sensitivity is negative: increasing \(K_m\), with \(V_{\max}\) and concentration held fixed, lowers the predicted response.

For one observation, the log-likelihood, apart from a constant, is \(\ell(\theta)=-[Y-m(C;\theta)]^2/(2\sigma^2)\). Its score is \([Y-m(C;\theta)]g(C;\theta)/\sigma^2\). Since the model assumptions give \(E[Y-m(C;\theta)]=0\) and \(E[(Y-m(C;\theta))^2]=\sigma^2\), the expected outer product of the score is \(g(C;\theta)g(C;\theta)^\mathsf{T}/\sigma^2\). This is the Fisher information for one observation. Independence makes information additive across observations, so for a fixed design of \(n\) concentrations:

If concentration \(C_j\) is measured \(r_j\) times, its contribution is \(r_jg(C_j;\theta)g(C_j;\theta)^\mathsf{T}/\sigma^2\). The formula is local because the sensitivities are evaluated at the nominal \(\theta\). Under standard large-sample likelihood conditions and a nonsingular information matrix, \(\mathcal I(\theta)^{-1}\) approximates the covariance matrix of the parameter estimates. It is an approximation, not an exact finite-sample covariance identity.

A Four-Concentration Calculation

Consider a hypothetical design with one observation at each of \(C=1,2,4,8\) mg/L. Retain \(V_{\max}=12\) response units per hour, \(K_m=3\) mg/L, and \(\sigma=1.5\) response units per hour. The fixed design and common variance satisfy the stated calculation assumptions; the parameters are positive and the mean is differentiable at the nominal point.

Eigenstructure: Where Information Is Strong or Weak

For a symmetric information matrix, an eigenvector specifies a parameter direction and its eigenvalue measures curvature, or information, in that direction. A small eigenvalue indicates a direction in which relatively little information is available; a zero eigenvalue indicates a locally uninformative direction. These interpretations are for the parameter coordinates used to construct the matrix.

For a two-parameter matrix with entries \(a,b,b,d\), its eigenvalues are \(\{\operatorname{tr}(\mathcal I)\pm\sqrt{\operatorname{tr}(\mathcal I)^2-4\det(\mathcal I)}\}/2\). For the four-concentration matrix, the trace is approximately \(1.844943\) and the determinant is approximately \(0.0779316\). Substitution gives eigenvalues approximately \(1.8017\) and \(0.04325\), a ratio of about \(41.65\). Thus information is much stronger in one local direction than in the other. In the displayed parameter coordinates, a unit eigenvector for the smaller eigenvalue is approximately \((0.867,0.498)^\mathsf{T}\): increasing both numerical parameter coordinates together is relatively weakly informed. Because the parameters have different units, however, this raw-coordinate eigenvector should not be interpreted as a unit-independent biological trade-off.

The high positive parameter-estimate correlation in the example is compatible with a weak direction in which the parameter estimates move together. The negative off-diagonal in the information matrix does not imply a negative correlation: inversion changes the off-diagonal sign in this two-parameter case. More generally, use the inverse information to discuss approximate parameter correlations, and use the information eigenstructure to discuss weak directions; these are related but not interchangeable summaries.

Diagnostics, Alternatives, and Limits

A reliable calculation starts with analytic sensitivities and then checks the resulting matrix. Useful checks include symmetry, positive semidefiniteness, and the determinant or smallest eigenvalue. A negative eigenvalue beyond rounding error is inconsistent with a sum of sensitivity outer products and often signals an arithmetic or implementation error. A zero or very small eigenvalue signals rank deficiency or near-confounding. Compare the result with finite-difference derivatives at the nominal parameters, using a step size small enough to approximate the derivative but not so small that numerical round-off dominates.

Expected Fisher information is the expectation over possible responses at the assumed model and parameter values. An alternative is the observed information, the negative Hessian of the realized log-likelihood. In a nonlinear model the observed Hessian can depend on the actual residuals; it need not equal the expected information in a particular data set. The expected-information formula used here follows from the Gaussian model assumptions and is convenient for evaluating a design before collecting data. Numerical derivatives are an implementation check or alternative when analytic derivatives are difficult, not a replacement for checking the model and its assumptions.

Local information has important limits. It depends on the nominal parameter values, the chosen design, and the assumed error variance. A design that looks informative at one nominal point may be weak elsewhere. The matrix also depends on parameter units: rescaling a parameter changes matrix entries and eigenvalues. When comparing eigenstructure across parameterizations, first use a defensible scaling, such as expressing parameters relative to characteristic values; a raw eigenvalue ratio can otherwise reflect units as well as practical information. Finally, Gaussian independent errors with constant variance may be inadequate if measurement variability changes with concentration or if observations are correlated. In that case the likelihood and information calculation must reflect the appropriate error model.

A common mistake is to infer that two parameters are well identified because both diagonal information entries are large or positive. In the four-replicate example, both diagonal entries are positive, yet the determinant is zero and one eigenvalue vanishes. The correction is to inspect the matrix jointly—at minimum its determinant in the two-parameter case, and preferably its eigenstructure and the inverse when that inverse exists.