Information Fractions After Sample Size Changes

Why information fractions change

Interim analyses are often planned at specified fractions of the trial’s total statistical information. A sample-size re-estimation can change the planned final information, so a look that was previously scheduled at one fraction may occur at a different fraction under the revised plan. The same issue arises in an event-driven trial when the target number of events changes. The data already collected do not disappear; it is the planned final-information denominator that has changed.

This distinction matters because an information fraction describes the look’s position in the statistical information accumulated for the primary estimator, not merely the proportion of the original calendar time or enrollment completed. Planned interim boundaries and monitoring schedules are tied to the design’s information plan. After re-estimation, the fractions should therefore be calculated from the revised target, then checked against the intended monitoring plan.

Definition: Let $I_k$ denote cumulative statistical information available at interim look $k$, and let $I_K$ denote the planned information at the final analysis. The information fraction at look $k$ is $$ t_k=\frac{I_k}{I_K}. $$ It is a ratio of information, not necessarily a ratio of participants, events, or elapsed time. Under a valid plan with nondecreasing information, $0\leq t_k\leq 1$.

Information and sample size

Information is commonly defined as the reciprocal of the variance of an estimator, or as the corresponding expected Fisher information under the model. The exact calculation depends on the endpoint, estimator, allocation, and assumptions. A participant-count ratio is a convenient shortcut only when information is proportional to sample size across the relevant looks and the final target.

For example, suppose a continuous outcome is measured in two equally sized arms, with independent observations and a common known variance $\sigma^2$. If $N$ is the total number randomized, then each arm has $N/2$ observations. The variance of the difference in sample means is $$ \operatorname{Var}(\bar{x}_T-\bar{x}_C) =\frac{\sigma^2}{N/2}+\frac{\sigma^2}{N/2} =\frac{4\sigma^2}{N}. $$ Thus information for that difference is $I=N/(4\sigma^2)$. With the same variance and allocation assumptions at every look, the information fraction equals the fraction of total sample size enrolled. If the variance assumptions or allocation change, that shortcut need not hold.

In an event-driven survival trial, the number of observed events is often used as a practical information proxy. For a log-rank comparison under proportional hazards and stable allocation, information is approximately proportional to the number of events. This approximation can be useful for planning, but it depends on the design and event process; participants who have not yet had an event do not contribute the same information as participants with an event.

Recalculation after changing a target

At a look, recalculate the fraction using the cumulative information at that look and the revised planned final information. When the relevant information is proportional to total sample size, this reduces to dividing the cumulative sample size by the revised total. When it is proportional to the event count, use cumulative events divided by the revised event target. If the information measure is not proportional to either count, calculate information directly under the specified model.

Worked Example: Increasing the Total Sample Size

Consider a hypothetical two-arm trial with equal allocation and a continuous outcome. The original plan is to randomize 600 participants, with an interim look after 240 have been randomized. Assume independent observations and the same known outcome variance in both arms and at every look, so information is proportional to total sample size.

Under the original target, the look’s information fraction is $$ t_{\text{old}}=\frac{240}{600}=0.400. $$ Suppose a re-estimation increases the planned final total to 800 participants. The current cumulative sample size remains 240, so the revised fraction is $$ t_{\text{new}}=\frac{240}{800}=0.300. $$ The data from the 240 participants still provide the information they provided before the re-estimation; the fraction falls because planned final information has increased. For this design, reaching an information fraction of 0.500 would require 400 participants, since $0.500\times800=400$. That is 160 more than the current 240.

Worked Example: Changing an Event Target

In a hypothetical event-driven trial, assume the planned analysis uses event count as an information proxy under the trial’s specified survival model and stable allocation. At an interim look, 180 primary events have occurred. The original target was 400 events, but re-estimation raises it to 500.

Using the event-count approximation, the original fraction is $180/400=0.450$, while the revised fraction is $$ t_{\text{new}}=\frac{180}{500}=0.360. $$ A later look planned at 280 events would have fraction $280/500=0.560$ under the revised target, rather than $280/400=0.700$ under the original target. To place that later look at a fraction of 0.700 under the revised event target would require $0.700\times500=350$ events. From the current 180 events, that means 170 additional events. These calculations describe information timing; they do not determine how quickly events will accrue.

Why the fraction uses information

The use of information fractions follows from how uncertainty changes as data accumulate. Under a model in which information adds across independent increments, cumulative information at look $k$ is the sum of information from the observed increments. For an estimator with variance equal to the reciprocal of information, more information means less variance. If the final planned information changes while the information already accrued stays fixed, the ratio of accrued to planned information must change.

For the equal-allocation continuous-outcome example, let the current total sample size be $N_k$ and the revised target be $N^*$. With common known variance $\sigma^2$, $$ I_k=\frac{N_k}{4\sigma^2},\qquad I_K^*=\frac{N^*}{4\sigma^2}. $$ Therefore, $$ t_k^*=\frac{I_k}{I_K^*} =\frac{N_k/(4\sigma^2)}{N^*/(4\sigma^2)} =\frac{N_k}{N^*}. $$ The variance term cancels only because the same variance model applies to both quantities. If information is recalculated using a different variance estimate, or the design changes in another way, calculate the numerator and denominator under the revised information definition rather than assuming they cancel.

In group sequential monitoring, information fractions also describe the planned positions of looks. Under the usual independent-increments formulation, the correlation between standardized statistics at two looks is related to the square root of the ratio of their information, $\sqrt{I_j/I_k}$, for $j\leq k$. This is one reason a changed information schedule can matter to the monitoring design: the old look labels may no longer describe the same positions in the information process. The actual boundaries and error control must follow the design’s specified recalibration approach, not an informal relabeling of looks.

Worked Example: Translating a Planned Fraction to a Revised Look

A hypothetical equal-allocation trial originally plans a total of 600 participants and interim looks after 200 and 400 participants. Assume the same known variance throughout, so participant count is proportional to information. After the first look at 200 participants, the target is revised to 800.

The first look is now at fraction $200/800=0.250$. If the second look stays fixed at 400 participants, its revised fraction is $400/800=0.500$. Under the original target, that same second look was at $400/600=0.667$.

If the intention is instead to keep the second look at its original fraction of $0.667$, the revised participant count is approximately $$ 0.667\times800=533.6. $$ A practical plan might round to 534 participants, subject to the protocol’s allocation and rounding rules. At that count the fraction is $534/800=0.6675$, approximately 0.668. The look’s target has shifted from 400 to about 534 participants; its timing in calendar time will also depend on enrollment and follow-up.

Check the schedule and its assumptions

A reliable update distinguishes what has been observed from what is planned. Record the information available at each completed look, the revised final information, and the resulting fractions. For future looks, decide whether they remain at their original participant or event counts, or instead move to preserve particular information fractions. Either choice changes the schedule differently.

Useful diagnostics include the following:

Check the denominator. Confirm that every fraction uses the revised planned final information, not the superseded target.
Check the information model. Verify that a count ratio is justified by the endpoint, estimator, allocation, and variance assumptions.
Check ordering and range. Planned fractions should increase with cumulative information and ordinarily remain no greater than 1. A fraction above 1 signals that the target is below the information already accrued, or that quantities from inconsistent plans have been combined.
Check feasibility. Confirm that revised look targets respect sample-size floors, caps, allocation rules, event definitions, and operational constraints.
Check the monitoring plan. A changed fraction can affect how a look relates to the planned group sequential design. Do not assume that an old boundary remains appropriate merely because the look number is unchanged.

A simple participant-count schedule is the main alternative when information is not directly available or when it is a valid proxy. It is easier to communicate and may be adequate under stable assumptions, but can misstate the information fraction when information per participant changes. Direct information-based scheduling more closely tracks estimator precision, although it requires a defensible information calculation and reliable inputs. In event-driven trials, event counts often provide a practical middle ground; they are not a universal substitute for model-based information.

The limitation is important: recalculating fractions does not, by itself, establish that the revised design preserves the desired type I error rate or that its interim boundaries remain valid. That depends on the design and the adaptation procedure. Fractions are an input to interpreting and planning interim looks, not a stand-alone guarantee of statistical validity.

Common mistake

Common pitfall: “The interim occurred after 40% of the originally planned participants, so it is still at information fraction 0.40 after the target increases.” This is wrong when the revised final information is larger and the current information is unchanged. Under proportional information, 240 participants out of a revised target of 800 give $240/800=0.300$, not 0.400. Recalculate using the revised denominator, and use an information calculation rather than a count ratio when proportionality is not justified.

Check Your Understanding

Answer each question using the stated information model. Solutions follow the questions.

A trial has 300 participants at an interim look. Under an assumption of information proportional to sample size, the revised total target is 750. What is the revised information fraction?
A trial has accrued 240 of a planned 400 events. The target is increased to 480 events. What is the revised event-count information fraction?
Under the first question’s revised target, how many participants correspond to an information fraction of 0.600?
Why might the ratio of enrolled participants to the revised total fail to equal the information fraction?
A calculated fraction is 1.08. Give one design explanation and one data-check explanation for this result.

Solutions: (1) With proportional information, $t=300/750=0.400$. (2) Using the stated event-count proxy, $t=240/480=0.500$. (3) The target count is $0.600\times750=450$ participants. (4) Information per participant may vary with the endpoint, variance, allocation, or other design features, so a count ratio may not represent the ratio of estimator information. (5) The revised final target may imply less information than has already accrued, or the numerator may use a different information definition, units, or target than the denominator. Reconcile the plan and calculations before interpreting the fraction.