Checking Planned Information Fractions

Why Check a Planned Information Fraction?

A group sequential design schedules interim analyses by how much information is expected to be available, not simply by how much calendar time has passed. Before using a proposed schedule, check that its interim information fractions are consistent with the planned allocation, outcome variability, and number of outcomes expected to be usable at each analysis. A fraction that does not match these inputs may describe a different analysis schedule from the one the design assumes.

The central check is a comparison: calculate the information expected at an interim analysis and at the final analysis under the same planning assumptions, then divide the former by the latter. The result can be compared with the proposed fraction. Earlier material on Information Under Unequal Outcome Variances gives the variance structure that is especially important when treatment groups differ in variability.

Definition: Let \(I_k\) denote the planned information at interim analysis \(k\), and let \(I_K\) denote the planned information at the final analysis \(K\). The planned information fraction is \(t_k=I_k/I_K\). For an estimator of a scalar treatment contrast with planned variance \(V_k\), information is defined here as \(I_k=1/V_k\). Thus \(t_k=V_K/V_k\), when both variances refer to the same contrast and scale.

This definition makes the comparison specific to the estimator and its variance. It is not automatically equal to the fraction of participants randomized, the fraction of expected outcomes collected, or the fraction of calendar time elapsed. Those quantities may be useful approximations only when the assumptions connecting them to information are reasonable.

Model and Assumptions for the Check

For a transparent planning calculation, suppose the endpoint is a continuous outcome and the treatment effect is the difference between independent treatment- and control-group means. Let \(n_{Tk}\) and \(n_{Ck}\) be the numbers with usable outcomes at analysis \(k\), and let \(\sigma_T^2\) and \(\sigma_C^2\) be the corresponding outcome variances. Under independent observations within and between groups, the planned variance of the difference in means is

V_k=\operatorname{Var}(\bar{x}_{Tk}-\bar{x}_{Ck}) =\frac{\sigma_T^2}{n_{Tk}}+\frac{\sigma_C^2}{n_{Ck}}, \qquad I_k=\frac{1}{V_k}.

The calculation assumes the planned variances are suitable for the analysis population, that the observations counted in each group are independent under this model, and that the sample sizes are counts of outcomes actually available for the planned estimator. The variances may be external planning values or justified estimates; they need not be equal. This variance formula does not require normal outcomes, although a group sequential test may require additional distributional or approximation assumptions not examined here.

For a different estimator or dependence structure, the variance must be changed before taking its reciprocal. For example, paired outcomes, repeated measures, or clustered observations cannot be treated as independent observations merely by inserting their counts into this two-sample formula. Likewise, if the analysis adjusts for covariates, the information calculation should reflect the planned adjusted estimator rather than assume the unadjusted variance without justification.

Key check: Calculate \(V_k\) and \(V_K\) using the same contrast, variance assumptions, and definition of an analyzable outcome. Then compare \(t_k=V_K/V_k\) with the proposed interim fraction. A mismatch is a prompt to inspect the assumptions and inputs, not a reason to alter the fraction by guesswork.

Worked Information Checks

Worked Example: Equal Allocation and Equal Variances

Suppose a hypothetical two-group trial plans 80 usable outcomes per group at an interim analysis and 180 per group at the final analysis. Assume independent observations and a planning variance of \(100\) outcome-units squared in each group at both analyses. Check a proposed interim information fraction of \(0.4444\).

At the interim analysis, the variance of the difference in group means is

V_k=\frac{100}{80}+\frac{100}{80} =1.250+1.250=2.500, \qquad I_k=\frac{1}{2.500}=0.4000.

At the final analysis,

V_K=\frac{100}{180}+\frac{100}{180} =0.5556+0.5556=1.1111, \qquad I_K=\frac{1}{1.1111}=0.9000.

Therefore,

t_k=\frac{I_k}{I_K} =\frac{0.4000}{0.9000} =0.4444.

The proposed fraction agrees with the calculation to four significant figures. The check also shows why each arm's contribution must be calculated separately: \(100/80=1.250\), not \(2.500\); \(2.500\) is the sum of the two arm contributions. Under equal variances and equal allocation at both analyses, this information fraction matches the per-group sample-size ratio, \(80/180=0.4444\).

Worked Example: Unequal Allocation and Unequal Variances

Consider a hypothetical trial with treatment-group outcome variance \(144\) units squared and control-group variance \(64\) units squared. The interim plan has 60 usable treatment outcomes and 120 usable control outcomes. The final plan has 120 usable treatment outcomes and 240 usable control outcomes. Assume independent observations and unchanged planning variances. Check a proposed fraction of \(0.5000\).

At interim,

V_k=\frac{144}{60}+\frac{64}{120} =2.400+0.5333=2.9333, \qquad I_k=\frac{1}{2.9333}=0.3409.

At the final analysis,

V_K=\frac{144}{120}+\frac{64}{240} =1.200+0.2667=1.4667, \qquad I_K=\frac{1}{1.4667}=0.6818.

The information fraction is \(t_k=0.3409/0.6818=0.5000\), so the proposal is consistent. The raw total sample sizes are 180 and 360, also a ratio of \(0.5000\), but that agreement follows here because each group’s interim count is exactly half its final count. The unequal allocation and variances still matter to the absolute information values. If only the treatment count had grown by one half while the control count did not, the total sample-size ratio would not establish the information fraction.

Worked Example: Expected Completeness Changes the Check

Suppose a hypothetical trial plans to randomize 200 participants per group by the final analysis. At the interim analysis, 120 participants per group have been randomized, but expected outcome completeness by that analysis is 75% in treatment and 90% in control. By the final analysis, expected completeness is 90% in treatment and 96% in control. Assume independent outcomes, treatment variance \(225\) units squared, control variance \(100\) units squared, and that these expected complete-outcome counts are suitable planning inputs. Check a proposed interim fraction of \(0.5000\).

Use expected complete outcomes, not randomized counts: interim counts are \(0.75(120)=90\) in treatment and \(0.90(120)=108\) in control. Final counts are \(0.90(200)=180\) and \(0.96(200)=192\). Thus,

V_k=\frac{225}{90}+\frac{100}{108} =2.500+0.9259=3.4259, \qquad I_k=\frac{1}{3.4259}=0.2919,

V_K=\frac{225}{180}+\frac{100}{192} =1.250+0.5208=1.7708, \qquad I_K=\frac{1}{1.7708}=0.5647.

The resulting fraction is \(t_k=0.2919/0.5647=0.5169\), approximately \(0.517\), rather than \(0.500\). Under the stated model and completeness assumptions, the proposed fraction is not exactly supported by the expected information. The difference arises because the complete-outcome pattern is not a simple common proportion of final information in both groups; the groups also have different outcome variances. This is a planning discrepancy to investigate, not evidence that the actual future information is known exactly.

Why Proportional Counts Sometimes Work

A useful justification explains when a simple sample-size ratio is valid. Suppose each interim group count is the same proportion \(r\) of its corresponding final count, so \(n_{Tk}=r n_{TK}\) and \(n_{Ck}=r n_{CK}\), with \(0<r\leq1\). If the variances are unchanged, then

V_k=\frac{\sigma_T^2}{r n_{TK}}+\frac{\sigma_C^2}{r n_{CK}} =\frac{1}{r}\left(\frac{\sigma_T^2}{n_{TK}}+ \frac{\sigma_C^2}{n_{CK}}\right) =\frac{V_K}{r}.

Taking reciprocals gives \(I_k=rI_K\), and therefore \(t_k=r\). This result holds even with unequal variances and unequal allocation, provided both group counts grow by the same proportion and the same variance model applies at each analysis. If the growth proportions differ by group, the factor \(1/r\) cannot be taken out of the variance sum, so the information fraction generally is not a single raw sample-size ratio.

Practical Diagnostics and Alternatives

A careful check proceeds from the estimator to its variance, then from variance to information. If the proposed fraction and calculated fraction disagree, inspect the inputs before deciding that the schedule is wrong. In particular, verify the group-specific sample sizes, allocation ratio, variance inputs, and the distinction between randomized participants and outcomes expected to be available. Check that interim and final calculations use the same outcome definition, contrast, and analysis method.

Counts do not represent the analysis set. Replacing complete or otherwise usable outcomes with randomized counts can overstate planned information when outcomes are missing or not yet observed.
One common variance is imposed without support. When group variances differ, each variance contribution must use the count and variance for its own group.
Allocation changes over time. A ratio of total counts can conceal unequal growth across groups; calculate the variance at each look from its group-specific counts.
Expected completeness is treated as guaranteed. Expected counts are planning approximations. Actual completeness may differ, so a design should be checked under plausible departures when missingness is material.
Information is assumed to increase according to the schedule automatically. Added independent usable observations ordinarily reduce the variance in this model, but changes in the estimator, dependence, or outcome availability can invalidate a simple count-based expectation.

The main alternatives answer different questions. Calendar-time fractions measure elapsed time; they need not track recruitment or outcome availability. Event fractions can be useful for event-driven outcomes when event counts adequately represent information, but they are not interchangeable with variance-based information in every setting. Raw sample-size fractions are convenient when per-observation information is stable and counts grow proportionally. The variance-based check is more direct for the specified estimator, but it is only as reliable as its model and planning inputs.

The limitation is important: a planned calculation is not a guarantee about realized information. Variances may differ from their planning values, completeness may be uncertain, and dependence or estimator changes may require a more appropriate variance calculation. The check identifies whether the proposed schedule is coherent under stated assumptions; it does not remove uncertainty in those assumptions.

Common Mistake

Common mistake: “The interim plan has enrolled half of the final total sample, so its information fraction must be \(0.50\).” This reasoning is correct only under conditions that make information grow in proportion to that count, such as both groups reaching the same fraction of their final usable counts under a stable variance model. The correction is to calculate \(V_k\) and \(V_K\) from the group-specific usable counts and variances, then compare \(V_K/V_k\) with the proposed fraction.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In the two-group mean model, what quantity is inverted to define information, and what must remain consistent when comparing interim and final information?
Suppose both groups have interim counts equal to 60% of their respective final counts, with unchanged variances. What is the information fraction, and why?
Why can a total randomized-sample fraction misrepresent information when the treatment and control groups have different outcome variances?
A proposed fraction is \(0.40\), but a calculation based on expected complete outcomes gives \(0.47\). Name two planning inputs to check before revising the schedule.

Solutions. (1) Invert the planned variance of the estimator, \(I=1/V\); use the same contrast, analysis definition, and compatible variance assumptions at both analyses. (2) The fraction is \(0.60\): each group count is multiplied by \(r=0.60\), so the variance is \(V_k=V_K/r\), and \(I_k/I_K=r\). (3) The variance is a sum of group-specific contributions, each depending on its own count and variance; a total count alone does not preserve those contributions. (4) Check the group-specific expected complete counts and the variance assumptions, including whether the expected completeness rates and outcome variances are appropriate for each analysis.