Why Information-Based Plans Are Easy to Misread
An information fraction is often described as the proportion of the planned information available at an interim analysis. The phrase sounds simple, but errors arise when the quantities being compared are not actually on the same scale. Enrollment counts, event counts, effective sample sizes, variances, and standard errors are related in some designs, but they are not interchangeable by default.
A planning calculation is sound only if its numerator and denominator refer to the same treatment contrast, information measure, and assumptions about how observations contribute. This tutorial focuses on auditing those comparisons. It does not address how to choose sequential efficacy thresholds; the key question here is whether the stated information fraction means what the planner intends it to mean.
The definition is about precision for an estimator, not simply the number of people recruited. A participant can contribute less information than expected, for example if the outcome is missing, follow-up is incomplete, observations are clustered, or the design gives unequal precision across participants. Conversely, an analysis that adjusts for prognostic covariates can sometimes gain precision relative to a simpler analysis. The earlier material on information fractions and information-based sample-size calculations gives the broader planning context; here the emphasis is on spotting mismatched quantities.
Information, Variance, and Standard Error
A common error is to compare standard errors as if their ratio were the information ratio. Under the variance-information relationship in the definition, information is inversely proportional to variance. Since standard error is the square root of variance, converting a standard-error ratio to an information ratio requires squaring it.
This comparison requires the same estimand and a compatible variance calculation at both points. If the estimator, model, or variance method changes, the reported standard errors may not represent information on a directly comparable scale. The reciprocal-variance relation is exact for the stated definition; in practice, estimated or model-based standard errors may make the resulting information an estimate.
Worked Example: Correcting a Standard-Error Ratio
Suppose a hypothetical interim analysis estimates a treatment effect with standard error \(0.20\), while the planned final analysis is expected to have standard error \(0.15\). Assume the same treatment contrast and compatible variance model apply at both analyses. What is the planned information fraction?
Using the final analysis as the maximum-information reference,
The interim analysis has 56.25% of the planned final information under these assumptions. The standard-error ratio is \(0.20/0.15=1.333\), but it is not an information fraction. A common incorrect calculation is to use \(0.15/0.20=0.75\) directly. That mistakes a standard-error ratio for an information ratio and overstates the fraction.
The same relationship is useful as a diagnostic. If a claimed information fraction is \(0.56\), the corresponding interim standard error should be about \(1/\sqrt{0.56}=1.336\) times the final standard error, provided the estimand and variance assumptions match. If the reported standard errors do not roughly align with that relationship, check for a calculation error or a change in the analysis model.
Enrollment Is Not Information
Enrollment is a count of people assigned or recruited. Information describes precision for the target estimator. Equating the two assumes, at minimum, that each enrolled participant contributes the same expected amount of information and that the participants counted have usable outcome data. Those assumptions may be reasonable approximations in a simple design, but they must not be silently presumed.
For a simple illustration, consider a hypothetical two-arm comparison of means with equal allocation, independent observations, a common known outcome variance \(\sigma^2\), and complete outcome data. With total sample size \(N\), each group has \(N/2\) observations, and the variance of the difference in sample means is \(4\sigma^2/N\). Thus information for that difference is \(N/(4\sigma^2)\), so information is proportional to \(N\) under this particular model. Change the assumptions—for example, introduce missing outcomes, unequal variances, or clustering—and raw enrollment need not remain proportional to information.
Worked Example: Enrollment Versus Event Information
Consider a hypothetical, equally allocated time-to-event trial. At an interim review, 520 participants have been enrolled and 240 events have been observed. The plan targets 400 events by the maximum analysis, and enrollment is projected to reach 1,000 participants. For a rough illustration, assume the same event definition and equal allocation throughout, and use the usual approximation that information for a log hazard ratio is proportional to the number of events, with about \(0.25\) information units per event.
The event-based interim and maximum information estimates are
Therefore the approximate information fraction is
The enrollment fraction is \(520/1000=0.52\), which differs from the event-based information fraction. It would be wrong to report \(0.52\) as the information fraction merely because it is easy to calculate. The event-based approximation is appropriate only insofar as the allocation, event definition, and information contributed per event remain comparable. If those conditions are doubtful, use a direct information estimate from the planned or fitted model rather than treating event count as exact information.
A related error is to use an “effective sample size” without checking what it represents. Such a quantity can summarize losses from clustering or other design features, but it is not automatically the information for every estimand or analysis. It is a proxy whose interpretation depends on the design and assumptions used to construct it.
Keep the Maximum-Information Assumptions Compatible
The denominator \(I_{\max}\) is not just a convenient target number. It defines the scale on which the fraction is measured. A numerator based on observed events cannot be divided by a denominator based on raw enrollment unless the conversion between those quantities is specified and justified. Similarly, an interim estimate based on observed complete cases should not be compared uncritically with a maximum based on a different missingness or follow-up assumption.
Worked Example: Matching Usable Outcomes to the Planned Maximum
Suppose a hypothetical study plans to enroll 800 participants by the final analysis. For planning, 80% are expected to provide usable outcomes, giving a maximum of \(800(0.80)=640\) usable outcomes. At an interim review, 400 participants have enrolled, of whom 90% have usable outcomes, or \(400(0.90)=360\). Assume, for this illustration, that each usable outcome contributes the same information and that other design features are unchanged.
On the usable-outcome scale, the information fraction is
The raw enrollment fraction is \(400/800=0.50\); it is not the fraction implied by the stated usable-outcome assumptions. Another incorrect calculation would divide the interim usable count by the planned randomized count, \(360/800=0.45\). That mixes the numerator and denominator scales. The correction is to compare like with like: usable outcomes at interim with planned usable outcomes at maximum, or information estimates with information estimates.
The maximum can also become incompatible when planning assumptions change. For example, a revised final sample size, event target, allocation ratio, follow-up duration, or variance assumption can change \(I_{\max}\). A fraction calculated against the old maximum does not automatically retain its original interpretation after such a change. Record the assumptions behind the denominator, and recalculate the fraction if those assumptions are revised.
A Practical Audit of an Information Fraction
The following sequence is a compact way to locate common planning errors before relying on a reported fraction.
State the treatment contrast and analysis model. Information is tied to precision for an estimator, not to a study in the abstract.
Decide whether the calculation uses inverse variance, event-based information, or a justified proxy such as effective sample size. Do not mix scales without a conversion model.
Compare like quantities, including the same outcome definition, allocation, follow-up, missingness treatment, and variance assumptions where relevant.
If using standard errors, square the relevant ratio. If using counts, check that the count is a justified proxy for information and that both counts use the same definition.
Separate an observed information estimate from a planned expectation, and explain which assumptions make the fraction approximate.
Warning signs include an information fraction that is larger than 1 despite a stated maximum, a fraction that conflicts sharply with the standard errors, and a count-based fraction that changes substantially when missing outcomes or event timing are accounted for. These signs do not prove that the design is wrong: a revised maximum or a changed analysis model may explain them. They do require the planner to clarify the scale and assumptions before interpreting the number.
Choosing Among Information Measures
There is no single proxy that is best for every design. Direct variance-based information is closely tied to the estimator but may depend on model-based or estimated variances. Event counts can be convenient in event-driven designs when the information-per-event approximation is appropriate. Enrollment is easiest to track, but is least informative about precision unless equal contribution and complete ascertainment are plausible.
| Measure | When it can be useful | Main caution |
|---|---|---|
| Inverse variance | When a compatible variance estimate is available for the target estimator | Variance estimates or analysis models may differ across looks |
| Event count | When information is approximately proportional to events under a stable design | Event definitions, allocation, and information per event must remain comparable |
| Usable outcome count | When each usable outcome contributes approximately equal information | Unequal precision, clustering, or changes in outcome availability break the simple proportionality |
| Enrollment count | For operational tracking or a justified simple-design approximation | Participants may not have complete outcomes or equal information contributions |
All of these measures have limitations. Information may be estimated with uncertainty, and a planned maximum can be wrong if its assumptions about outcomes or follow-up are wrong. A mathematically precise ratio is not necessarily a credible description of the design if its underlying inputs are not credible. The most useful report states both the fraction and the definition and assumptions used to calculate it.
Common Mistakes and Corrections
- Wrong reasoning: “Half the planned participants are enrolled, so the information fraction is one half.” Correction: This is true only if information is proportional to enrollment under the design and outcome-availability assumptions. Check those assumptions or calculate information directly.
- Wrong reasoning: “The interim standard error is 20% larger, so information is 20% lower.” Correction: Information varies with the inverse square of standard error. Compute the squared standard-error ratio with numerator and denominator in the correct order.
- Wrong reasoning: “The interim count is usable outcomes, but the planned maximum is randomized participants; both are counts, so the ratio is valid.” Correction: Counts with different definitions are not comparable. Convert both to the same information scale.
- Wrong reasoning: “The assumptions behind the maximum can change without changing the interpretation of the fraction.” Correction: A changed maximum changes the denominator. Recalculate and state the revised assumptions.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- An interim standard error is \(0.30\), and the planned final standard error is \(0.20\). Under a compatible variance model, what is the interim information fraction?
- A trial has enrolled 300 of a planned 600 participants. Is the information fraction necessarily \(0.50\)? State the condition required for that conclusion.
- A planned study targets 500 events, and an interim review has observed 350 events. Under a justified assumption of information proportional to events, calculate the event-based information fraction.
- Explain why dividing 270 interim usable outcomes by a planned maximum of 600 randomized participants may give a misleading fraction.
Solutions. (1) The fraction is \((0.20/0.30)^2=0.4444\), or about 44.4%; the unsquared standard-error ratio is not the information fraction. (2) Not necessarily. It is \(0.50\) only if information is proportional to enrollment, such as when participants contribute equal expected information and outcome availability and other design features are compatible. (3) The event-based fraction is \(350/500=0.70\), or 70%, conditional on the stated proportionality assumption. (4) The numerator counts usable outcomes while the denominator counts randomized participants. They are on different scales; the planned maximum should be expressed as usable outcomes or converted to information using an explicit assumption.