Translating Precision Targets Into Data Requirements

From a Precision Goal to a Data Plan

A precision goal is useful only if it can guide a design decision. For example, a team may require a 95% confidence interval for a pathway rate constant to have a half-width no greater than a specified value. The design question is then concrete: how many independent observations are needed, how many replicates should be collected, which sampling times are most informative, or how precise must the assay be?

Fisher information provides a local, model-based translation from a design to expected parameter uncertainty. Its approximation is most useful when the model is locally identifiable, the likelihood is sufficiently regular, and the estimate is not strongly affected by a boundary or severe nonlinearity. Profile likelihood provides a complementary check by evaluating likelihood support over a range of parameter values rather than relying only on local curvature.

Translate an Interval Target Into Required Information

For a scalar parameter \(\theta\), let \(I(\theta)\) denote the Fisher information for the planned observations, evaluated at a plausible parameter value. Under appropriate regularity conditions, an asymptotically efficient estimator (such as the maximum likelihood estimator) is approximately normal with variance \(1/I(\theta)\). A two-sided approximate 95% confidence interval therefore has half-width \(1.96/\sqrt{I(\theta)}\). If the desired half-width is at most \(h\), the local information requirement is:

This calculation is for a marginal interval for one parameter under the stated normal approximation. A one-sided target, a joint confidence region, or a simultaneous guarantee across several parameters requires an appropriate critical value or joint criterion; the number 1.96 should not be used automatically for every precision statement. For a relative-precision target, it is often clearer to work with a transformed parameter such as \(\log(\theta)\), provided \(\theta>0\), and define the desired uncertainty on that scale.

Information depends on the model, the assumed parameter values, the observation error, and the design. Thus a calculation does not establish that a design will meet its target in every realized data set. It predicts precision under the assumptions used to compute the information.

Why Replicates and Independent Observations Can Be Counted

For independent observations with the same design and error model, log-likelihood contributions add, so their Fisher information adds. If one observation contributes information \(I_1\), then \(n\) such observations contribute \(nI_1\). In the scalar case, approximate standard error consequently decreases as \(1/\sqrt{n}\), not as \(1/n\).

With multiple estimated parameters, the information is a matrix. The variance for a parameter of interest depends not only on its own diagonal information but also on its correlation with nuisance parameters. For a two-parameter information matrix with target \(\theta\) and nuisance parameter \(\phi\), the local information for \(\theta\) after accounting for uncertainty in \(\phi\) is the Schur complement \(I_{\theta\theta}-I_{\theta\phi}^2/I_{\phi\phi}\). Ignoring that adjustment can overstate precision.

Replicate Counts, Assay Precision, and Sampling Times

When each replicate has the same independent contribution to information, the replicate count can be found by dividing required information by information per replicate and rounding up. This is not the same as simply multiplying the number of measurements if replicates share substantial within-subject or batch correlation. In that case, the joint likelihood and its dependence structure determine the information; treating correlated measurements as independent exaggerates the gain.

For independent normally distributed assay errors with known standard deviation \(\sigma\), a single observation with model mean \(f(\theta)\) contributes scalar information \(\{ \partial f(\theta)/\partial\theta\}^2/\sigma^2\). Thus, with a fixed number of observations and fixed sensitivities, information is proportional to \(1/\sigma^2\). This relationship makes it possible to translate a parameter target into an assay precision requirement.

Sampling times change the sensitivities of model outputs to parameters, and therefore change the information contributed at each time. For a single parameter, a simple design comparison can sum the expected information at the selected times. With several parameters, add the information matrices first and then assess the target parameter’s uncertainty; nuisance-parameter correlations can make the best schedule different from the one suggested by one sensitivity alone.

Check the Local Prediction With Profile Likelihood

A Fisher-information calculation uses local curvature near a parameter estimate or assumed design point. If the likelihood is asymmetric, flat in one direction, multimodal, or constrained by a parameter boundary, its normal-based interval may give a poor description of uncertainty. The profile likelihood addresses this by fixing the target parameter at candidate values and maximizing the likelihood over the remaining parameters. A profile-based interval consists of values whose likelihood-ratio statistic does not exceed the chosen cutoff.

For prospective planning, a practical approach is to specify a candidate design, generate or otherwise evaluate its expected data under plausible parameter values, fit the model, and calculate the target parameter’s profile interval. The design meets a maximum half-width target only if the profile interval satisfies the prespecified width criterion. Because a future data set is not known, repeated simulations can estimate how often the criterion is met. The likelihood-ratio cutoff and any small-sample calibration should match the intended inferential setting.

The two approaches answer related but different questions. Fisher information is efficient for screening many designs and showing how changes in sample size, replicate count, or error scale affect local precision. Profile likelihood checks the shape of likelihood support beyond the local approximation, but requires repeated fitting and profiling and can be computationally demanding. If a target must be met with a stated probability across future studies, simulation of the full design and analysis is more informative than either a single expected-information calculation or a single profile.

Diagnostics, Limitations, and a Common Pitfall

A calculated data requirement is suspect if the information matrix is singular or nearly singular, if estimated parameters approach bounds, if fitted profiles are highly asymmetric, or if small changes in the assumed parameter values substantially change the required design. Numerical instability in optimization or profiling is another warning: an apparently narrow interval is not credible if the fit has not converged reliably. Sensitivity to the assumed observation-error variance also deserves attention because information scales inversely with variance in the simple Gaussian setting.

A common mistake is to calculate the required information for each parameter separately and conclude that every target is met once its diagonal information is large enough. This ignores covariance with other parameters. The correct marginal variance comes from the inverse of the full information matrix, not from the reciprocal of a diagonal entry alone. Another mistake is to treat a nominal expected-information target as a guarantee: the realized data can be less informative than expected, and the local approximation itself can fail. Evaluate the full matrix, inspect profiles where needed, and make clear whether the goal is expected precision or a probability of meeting a target.

These calculations also rely on the assumed model and design being meaningful. More observations at an uninformative time may add little information; improved assay precision cannot resolve structural confounding by itself. A precision target may be infeasible for the proposed measurement types and parameterization. In that case, the result is a reason to reconsider the design, the estimand, or the target—not to report a rounded sample size as if it solved the underlying identifiability problem.