From an Information Target to an Interim Sample Size
A group sequential design specifies when interim analyses are planned. One way to describe a planned look is by its information fraction: the proportion of the design's maximum planned information expected to be available at that look. When information increases in direct proportion to the number of participants, this fraction can be translated into a sample-size target by multiplying it by the maximum sample size.
That translation is useful, but it is conditional. A fixed variance by itself does not guarantee that every participant contributes the same information. The proportionality calculation requires a model and an accrual plan under which information per participant is constant. The examples below use independent observations from a normal-mean model with known variance, or an equal-allocation comparison of two such normal means with a common known variance.
Why the Simple Multiplication Works
Suppose each observation contributes \(c\) units of information under the specified model, and the observations contribute independently. With \(N\) observations, planned information is \(I(N)=cN\). Therefore, for an interim sample size \(N_t\) and maximum sample size \(N_{\max}\),
Thus a target information fraction \(t\) gives \(N_t=tN_{\max}\). The equality follows because the same constant \(c\) applies to the observations included at each stage. It is not a general consequence of independence or fixed variance alone: if the information contributions differ by participant or change with the design, the cancellation in this ratio need not hold.
For a single normal mean with known variance \(\sigma^2\), each independent observation contributes \(1/\sigma^2\), so \(I(N)=N/\sigma^2\). For a two-group comparison, let \(n_1\) and \(n_0\) be the group sample sizes. Under independent normal outcomes with the same known variance \(\sigma^2\), the variance of the estimated difference in means is \(\sigma^2(1/n_1+1/n_0)\), and its information is the reciprocal of that variance. With equal allocation, \(n_1=n_0=N/2\), where \(N\) is the total sample size. Hence the information is \(N/(4\sigma^2)\), which is also proportional to total sample size.
The corresponding interim-to-maximum information ratio is \(N_t/N_{\max}\), provided the interim allocation is also equal and the same variance model applies. If a design uses a different allocation, the sample-size conversion must reflect that design rather than assuming this equal-allocation formula.
Planning and Checking the Target
Use the conversion only after specifying the outcome model, the planned variance, the maximum sample size, the information fraction, and the allocation rule. In practice, calculate the unrounded target first. Then choose integer group sizes that meet the allocation plan and report the information fraction those integers actually deliver. If the target is intended to be at least a stated fraction, rounding upward is generally safer than rounding to the nearest participant, because rounding down can leave the planned information short.
Specify the outcome model, variance, maximum sample size, and allocation. Confirm that the assumptions make planned information proportional to the relevant sample size.
Multiply the target information fraction by the maximum total sample size, or derive the target from the information formula if the design is not proportional in total sample size.
Choose feasible group counts. For an equal-allocation design, a feasible total is typically even; do not silently leave one group with an extra participant while claiming exact balance.
Calculate the information at the rounded sample sizes and divide by the planned maximum information. Report this achieved fraction, especially if rounding changed the target.
Worked Example: A Single-Mean Interim Target
A hypothetical study plans a maximum of 240 independent observations to estimate a mean under a normal model with known standard deviation \(12\) units. It plans an interim look at information fraction \(t=0.45\). Find the target sample size and verify the fraction using information.
For this model, each observation contributes \(1/\sigma^2=1/12^2=1/144\) information units per squared unit of outcome. The target based on proportional information is
The maximum planned information is \(I_{\max}=240/144=1.6667\), and the interim information is \(I_t=108/144=0.7500\). Thus
The target is exactly 108 observations, with no rounding required. This conclusion depends on the specified normal-mean, known-variance model and the assumption that all observations contribute the same information.
Worked Example: Equal-Allocation Interim Target
A hypothetical two-group trial plans 360 participants in total, allocated equally, to compare group means. Outcomes are independent and normally distributed with a common known standard deviation of \(10\) mm. The planned interim information fraction is \(0.40\). Find the total and per-group interim targets and verify the information ratio.
The continuous total-sample target is \(0.40(360)=144\) participants. Because the design is equal-allocation, this gives \(144/2=72\) participants per group. For the treatment-minus-control mean contrast, the interim information is
At the maximum sample size, there are 180 participants per group, so
The ratio is \(0.3600/0.9000=0.4000\). Thus 144 total participants, 72 per group, meet the target under the stated model and allocation.
Worked Example: Rounding a Balanced Target
A hypothetical equal-allocation study has a maximum total sample size of 202, or 101 participants per group. The requested interim information fraction is \(0.33\). Determine a feasible balanced interim target that does not fall below the requested fraction, then calculate the achieved fraction.
The unrounded total target is
A balanced allocation requires an even total. The smallest even total that is at least 66.66 is 68, or 34 participants per group. Under the fixed-variance equal-allocation model, the achieved information fraction is
The feasible target is therefore 68 participants, and it delivers about 33.66% of the maximum planned information, slightly above the requested 33%. Reporting only “33%” would conceal the small effect of integer and balance constraints.
What Can Go Wrong
The central diagnostic is to check whether the model and accrual plan support constant information per participant. For the examples above, that means independent outcomes, the stated normal model, a fixed known variance, and the specified allocation pattern. If the variance differs from the planned value, the calculated fraction may not describe the information actually observed. If participants have differing information contributions, multiplying the maximum sample size by the target fraction may also be wrong.
A second check is whether “sample size” means the number enrolled or the number with usable outcomes at the interim. The information formula concerns observations contributing outcome information. If some enrolled participants have not yet provided usable outcomes, enrollment count alone can overstate information. Likewise, the fraction of calendar time elapsed does not by itself determine the information fraction.
The calculation is a planning conversion, not a rule for deciding whether to stop a trial or how to change its maximum sample size. Those decisions require additional design methods. The target also does not guarantee that realized information will equal planned information: finite samples and departures from the planning assumptions can alter the information available at the look.
| Planning quantity | What it represents | When it matches information fraction |
|---|---|---|
| Interim sample size divided by maximum sample size | Share of planned participants observed | When information is proportional to sample size under the model and allocation |
| Events observed divided by planned events | Share of planned outcome events | Only when the event count is an appropriate proxy for information in the design |
| Elapsed calendar time divided by planned duration | Share of planned study time | Not generally; accrual and outcome availability need not be uniform over time |
A common mistake is to reason, “The variance is fixed and participants are independent, so every participant must add the same information; therefore the interim sample size is always the target fraction times the maximum.” Independence and a fixed variance alone do not establish equal information contributions. The correction is to identify a model under which the contribution is constant—for example, independent observations from a normal-mean model with known variance—and verify that the interim allocation preserves the design assumptions. If those conditions do not hold, calculate information under the appropriate model rather than applying the simple sample-size ratio.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A one-sample normal-mean design has maximum sample size 300 and plans a look at information fraction 0.25. What is the interim sample-size target if the variance is fixed and each observation has the same information contribution?
- A two-group equal-allocation design plans 280 participants total and a look at information fraction 0.50. What are the total interim target and the target per group?
- An equal-allocation design has maximum total sample size 198 and requests information fraction 0.20. What is the smallest balanced interim total that meets or exceeds the target, and what fraction does it achieve?
- Why does a fixed variance, without a specified model or equal-contribution assumption, not automatically justify multiplying the maximum sample size by the target information fraction?
- Does enrolling 40% of the planned participants necessarily mean that 40% of the planned information is available? State the condition needed for that conclusion.
Solutions. (1) \(0.25(300)=75\) observations. (2) The total target is \(0.50(280)=140\), giving 70 participants per group. (3) The unrounded target is \(0.20(198)=39.6\); the next feasible even total is 40, so the achieved fraction is \(40/198\approx0.2020\), or 20.20%. (4) Equal information contributions follow from the specified model and design, not merely from independence and a fixed variance; different observation models or contributions can break proportionality. (5) Not necessarily. It does so when information is proportional to sample size under the model and allocation and the participants counted have usable outcomes contributing information.