Information Fractions and Stage Sizes

Why Stage Size Is Not the Same as Information

In a seamless trial, a stage’s contribution to the final analysis depends not only on how many participants were recruited, but also on how precisely the treatment effect can be estimated from their outcomes. A stage with fewer participants can contribute substantial information if its outcome is measured precisely or if many participants experience the event of interest. Conversely, recruitment can increase without information rising as much as expected when outcomes are missing or event rates are low.

Information gives a common scale for comparing stages and planning their contributions. It is also useful when choosing inverse-normal combination-test weights: under standard large-sample conditions, a stage’s information determines the precision of its treatment-effect estimate. The calculations below concern planned information; realized information can differ from plan.

For \(K\) planned stages with information increments \(I_1,\ldots,I_K\), the planned total is \(I_{\mathrm{total}}=\sum_{j=1}^{K} I_j\). The fraction reached after stage \(k\) is \(f_k=(\sum_{j=1}^{k} I_j)/I_{\mathrm{total}}\). The fraction assigned to stage \(j\) is \(q_j=I_j/I_{\mathrm{total}}\). Thus \(f_k\) is cumulative, whereas \(q_j\) describes one stage; these quantities should not be interchanged.

Information and Inverse-Normal Weights

Suppose each stage produces a one-sided stagewise \(p\)-value \(p_j\), converted to a standard normal score \(Z_j=\Phi^{-1}(1-p_j)\) in the direction favoring treatment. If the stagewise statistics have independent increments and similar large-sample behavior, a common information-based choice of inverse-normal weight is \(w_j=\sqrt{I_j/I_{\mathrm{total}}}\). The combined statistic is \(Z_{\mathrm{comb}}=\sum_j w_j Z_j\), with \(\sum_jw_j^2=1\). The square root matters: inverse-normal weights are not the information fractions themselves.

This relationship has a simple precision justification. If a treatment-effect estimate has variance \(1/I_j\), its standard error is \(1/\sqrt{I_j}\). Standardizing a common effect \(\theta\) therefore gives an expected stage statistic approximately \(\theta\sqrt{I_j}\). A normalized weighted sum that reflects this precision assigns weights proportional to \(\sqrt{I_j}\), not to \(I_j\).

This is a design rationale, not a claim that every valid combination test must use these weights. Inverse-normal tests can use other prespecified weights, and validity depends on the stagewise \(p\)-values, dependence structure, and adaptation rules. When information is estimated from accruing data, the design must also specify how the weights and analysis will preserve the intended testing properties.

Binary Outcomes: Information From a Difference in Proportions

Consider a two-arm comparison in stage \(j\), with \(n_{Tj}\) participants assigned to treatment and \(n_{Cj}\) assigned to control. Let \(\hat{p}_{Tj}\) and \(\hat{p}_{Cj}\) be the observed response proportions, and estimate the treatment effect by \(\hat{p}_{Tj}-\hat{p}_{Cj}\). For independent Bernoulli outcomes, the variance is

For planning a score test under the null hypothesis of equal response probabilities, a common null response probability \(p_0\) gives the approximate information \(I_j=1/\{p_0(1-p_0)(1/n_{Tj}+1/n_{Cj})\}\). A different variance model, such as a Wald calculation under assumed alternative rates, can yield a different planned information. State which model is used; the formula is not independent of that choice.

Continuous Outcomes: Information From the Mean Difference

For a continuous endpoint, let the treatment-effect estimate in stage \(j\) be the difference in sample means. If outcomes are independent, have common variance \(\sigma^2\) in the two arms, and the analysis uses that variance, then

The common-variance assumption is a planning simplification, not a requirement for all analyses. With unequal variances, use the variance of the estimator under the specified model. If baseline adjustment or another model changes the residual variance, use the variance appropriate to that planned analysis rather than applying the unadjusted formula automatically.

Time-to-Event Outcomes: Information From Events

For a log-rank comparison, information is driven primarily by events, rather than by the number randomized alone. Under proportional hazards and a suitable large-sample approximation, if a proportion \(r\) of the risk set is assigned to treatment, information is approximately \(D r(1-r)\), where \(D\) is the number of observed events. With balanced allocation, \(r=0.5\), so the approximation is \(D/4\).

This event-count formula assumes that allocation remains balanced among participants at risk, treatment groups have comparable follow-up and censoring patterns for the approximation, and the proportional-hazards log-rank framework is suitable. More generally, log-rank information accumulates across event times according to the treatment and control composition of each risk set. If those compositions vary materially, use the planned risk-set information rather than treating every event as contributing exactly one quarter.

From Information Targets to Stage Recruitment

For binary and continuous endpoints, information can often be translated directly into arm-specific sample sizes by rearranging the relevant variance formula. With equal allocation and a common variance, both examples above have information proportional to the number per arm. For survival outcomes, first specify a target number of events, then estimate recruitment and follow-up needed to observe those events. In each case, distinguish the number randomized from the number contributing usable outcome information.

A practical planning sequence is:

Diagnostics, Alternatives, and Limitations

Before finalizing a plan, check whether the assumptions that make information additive are plausible. Stage increments should be based on nonoverlapping observations or otherwise have their covariance accounted for. The endpoint definition and measurement scale should be comparable across stages. Check that planned allocation is reflected in the variance calculation, that assumed response rates or standard deviations are reasonable, and that event accrual assumptions fit the expected risk-set pattern.

A frequent error is to use the fraction of participants recruited as the information fraction. This is correct only in special cases, such as equal allocation, equal outcome variance, complete ascertainment, and information proportional to sample size. Another error is to use \(q_j\) rather than \(\sqrt{q_j}\) as the inverse-normal weight. For example, a stage contributing 40% of total information receives weight \(\sqrt{0.40}\), not 0.40, under the information-based convention.

A group-sequential design may express interim timing through cumulative information fractions because precision, rather than recruitment alone, governs the sampling variation of the treatment estimate. A combination test instead combines stagewise evidence using its chosen weights; information-based weights are one natural option, not the only valid one. Neither approach eliminates the need to prespecify the analysis and justify the assumptions. Information calculations are planning approximations, and departures in variance, missingness, allocation, or event patterns can alter realized information.