Local Fisher Information At Nominal Parameters

Why Information Is Local in a Nonlinear Model

In a nonlinear model, the response to a change in a parameter can depend on the parameter values themselves. Fisher information is built from these response sensitivities, so it too depends on the parameter values at which it is evaluated. A design may therefore have different information matrices under different plausible parameter values, even when the observation locations and measurement precision are unchanged.

This tutorial develops that dependence for a two-parameter response model. The goal is to calculate information at explicitly stated nominal parameter values: values treated as the working parameter specification for a local calculation. The result is conditional on those values; it is not automatically a global summary of information across all plausible parameter values.

Model, Assumptions, and the Information Formula

Consider the two-parameter Emax mean response \(m(C;E_{\max},K)=E_{\max}C/(K+C)\), where \(C\) is a nonnegative concentration, \(E_{\max}>0\) is the maximum response, and \(K>0\) is the concentration producing half of \(E_{\max}\). Suppose the observed response at condition \(C_i\) is \(Y_i=m(C_i;\theta)+\varepsilon_i\), with independent errors \(\varepsilon_i\sim N(0,\sigma^2)\). Assume \(\sigma^2\) is known, is the same for all observations, and does not depend on \(\theta\). Also assume the mean is differentiable in the parameters and the nominal values are in the interior of the allowed parameter region.

These assumptions matter. Independence lets the information contributions add. The normal model with constant known variance makes the mean-sensitivity contribution the full information expression used here. If the variance also depends on parameters, its derivatives contribute additional information; the formula below would then be incomplete.

For this model, differentiating the mean gives

The first component describes the change in mean response per unit change in \(E_{\max}\); the second describes the change per unit change in \(K\). The second derivative is negative because increasing \(K\), at fixed positive concentration and \(E_{\max}\), lowers the predicted response. Both components depend on the model settings, and the second also depends directly on \(E_{\max}\).

Why the Local Information Formula Holds

For one observation \(Y_i\), the normal log likelihood, apart from constants not involving \(\theta\), is \(-[Y_i-m(C_i;\theta)]^2/(2\sigma^2)\). Differentiating with respect to \(\theta\) gives the score contribution \([Y_i-m(C_i;\theta)]g(C_i;\theta)/\sigma^2\). The expected outer product of this score contribution is

because the residual has mean zero and variance \(\sigma^2\). Independence makes cross-products between different observations have expectation zero, so summing the individual contributions gives the information matrix. Substituting \(\theta_0\) into every sensitivity gives the local matrix \(I(\theta_0)\). This calculation also shows why one must state the nominal values before reporting a numerical matrix.

Changing the Nominal Values Changes the Answer

Keep the same observation condition and measurement variance, but change the nominal \(K\) from 2 to 4 concentration units while retaining \(E_{\max}=100\) response units. The sensitivity becomes

Combining Contributions at Specified Nominal Values

For several independent observations, evaluate each sensitivity at the same nominal parameter vector, form its outer product, divide by the variance, and add the resulting matrices. The following example makes the calculation explicit. It uses one hypothetical observation at each of two concentrations; no population-level or study claim is intended.

Interpreting and Checking a Local Calculation

The matrix \(I(\theta_0)\) describes expected information under the stated model when the true parameter value is \(\theta_0\), or serves as a local approximation when \(\theta_0\) is a working value. It is not an observed-data quantity: the realized responses do not enter this expected-information calculation. It also does not establish that the nominal values are correct. It answers a conditional question: given these values, model, variance, and observation conditions, what information does the specified design provide locally?

Several checks help catch calculation or modeling failures:

• Check the gradient analytically and numerically. A small finite-difference change in each parameter should produce a mean change close to the corresponding derivative times that parameter change.
• Check units. Each matrix entry must have reciprocal units corresponding to the two parameters indexing its row and column.
• Check the variance factor. With the stated model, multiplying the residual variance by a constant divides every information entry by that constant.
• Check nominal-value sensitivity. Recalculate at other scientifically plausible values. Large changes indicate that a single local calculation is a fragile summary.
• Check the assumptions. Dependence between observations, unequal variances, or a variance that depends on parameters changes the calculation. Do not use the constant-variance formula without justification.

A common mistake is to compute derivatives at the nominal value but describe the resulting matrix as if it applied throughout the parameter space. The correction is to report the value explicitly, for example, “\(I(E_{\max}=100,K=2)\), under known \(\sigma=5\),” and explain that another nominal point may yield another matrix.

Local Information and Other Summaries

Local expected information differs from observed information, which is calculated from the realized data and the likelihood curvature at the fitted or specified parameter values. The two may be close in some settings, but they are not interchangeable. It also differs from a summary that averages performance across a range or distribution of parameter values: a single nominal calculation does not perform that averaging.

When uncertainty about the parameter values is material, a practical check is to repeat the local calculation at several plausible nominal values and report how the matrices change. This is a sensitivity analysis, not a replacement for stating the assumed model. Its purpose is to reveal whether conclusions depend strongly on one chosen parameter point. Local information remains useful because it gives a precise, interpretable calculation at a specified point; its limitation is precisely that locality.

Do not infer from one entry alone that the entire design is informative or uninformative. The matrix summarizes joint local sensitivity, and its entries must be read together with the model, parameter units, nominal values, and variance assumptions. Assessing structural problems in the matrix or translating it into particular design criteria requires additional tools beyond this calculation.