Recomputing Information After Interim Data

Why Recompute Information at an Interim Analysis?

At an interim analysis, some observations have already been collected, and the current parameter distribution may differ from the one used to plan the original design. A useful comparison of candidate future samples therefore needs two pieces: the information contributed by the data already collected, and the expected information from each proposed future design under the updated parameter uncertainty. Comparing future samples alone can give the wrong impression of how much total information a design will provide.

The calculation is conditional on the interim data and on a specified model. It is not a new parameter update: the interim parameter distribution is an input. The task is to evaluate the information in each candidate future design across that distribution, then account for the accrued information once.

Information to Add, and What the Average Means

For a likelihood with score vector \(U(\theta)=\partial \ell(\theta)/\partial\theta\), the Fisher information is the expected outer product of the score, \(I(\theta)=E_\theta\{U(\theta)U(\theta)^\mathsf{T}\}\), under regularity conditions. Equivalently, when differentiation and expectation can be interchanged, it is the negative expected Hessian of the log likelihood. The information is a function of the parameter, even when the observations themselves have already been collected.

For fixed, conditionally independent observations, log likelihoods add, so their scores add. Each score has expectation zero under the model. Consequently, the expected cross-products between scores from distinct observations are zero, and Fisher information adds. If \(A\) denotes the accrued sampling plan and \(d\) the proposed future plan, then

The interim distribution is then used to average this total. Linearity of expectation gives an equivalent and often convenient calculation: average the accrued and future contributions separately, then add them. The accrued contribution is common to all candidates if every candidate continues from the same interim design. It still belongs in the reported total, although it does not change a ranking based only on a scalar criterion that increases with total information.

Worked Examples

Assumptions, Diagnostics, and Alternatives

The addition-and-average calculation requires a clearly specified model and design. In particular, the candidate future schedule is treated as fixed conditional on the interim data; its information is evaluated under the assumed observation model; and the likelihood contributions are conditionally independent, or have their dependence correctly represented. The posterior distribution used for averaging must correspond to the same parameterization as the information calculation. If the future sampling plan changes in response to future outcomes, its information must account for that outcome-dependent rule rather than treating the schedule as fixed.

Several checks can reveal errors before a design comparison is reported:

• Check the accounting. Each accrued observation should appear once, and future observations should not be included in the accrued component. Candidate totals should share the same accrued component when they start from the same interim design.
• Check dimensions and units. Information matrices for different candidates must use the same parameter order. Scalar information has units reciprocal to the squared parameter unit; matrix entries have units determined by the corresponding parameter pair.
• Check numerical behavior. Information matrices should be symmetric and positive semidefinite up to numerical rounding. Recalculate exponentials, weights, and matrix sums at adequate precision; posterior weights should be nonnegative and sum to one.
• Check the model and support. If the posterior places weight in parameter regions where the model is invalid or the information calculation is numerically unstable, the resulting average is not trustworthy. Confirm that every parameter value used is admissible.

A common mistake is to compute the future information at a single interim point estimate and then add the accrued information at that same point, calling the result an uncertainty-averaged total. That calculation is a plug-in evaluation, not an average over the interim parameter distribution. The correction is to evaluate \(I_A(\theta)+I_F(\theta;d)\) across the distribution and average those values, or equivalently average the two terms separately. Another mistake is to add prior precision as though it were accrued likelihood information. If a criterion uses posterior precision, prior curvature may be relevant, but it must be identified separately and not counted as data-derived Fisher information.

The main alternative summaries answer different questions. A nominal calculation evaluates information at one designated parameter value; it is simpler, but does not average over interim uncertainty. A worst-case calculation takes the least favorable information over a specified parameter set, emphasizing protection rather than posterior average performance. For matrix information, averaging the matrix first and then applying a criterion is generally not equivalent to applying that criterion at each parameter value and then averaging. The chosen operation should match the design objective rather than being treated as interchangeable.

Expected Fisher information is a model-based planning quantity, not realized information, a posterior covariance, or a direct measure of clinical benefit. Its reliability depends on the model, parameter distribution, and proposed schedule. A posterior average can also conceal a poor result in a low-probability but important region; when that matters, report additional summaries of performance across parameter values. For a longer adaptive sequence, future information may depend on interim outcomes and the policy that selects later samples, requiring an expectation over those outcomes as well as parameter uncertainty.