Why Combine the Updates?
For a continuous outcome, the planned number of participants depends on how variable the outcome is and how many randomized participants are expected to provide usable outcome data. If an interim blinded review suggests greater variability than planned, more participants with observed outcomes may be needed. If it also suggests more missing outcomes, enrollment must increase further to obtain that number of observed outcomes. These are distinct adjustments: the variance update changes the required number of complete outcomes, while the attrition update translates that requirement into randomized enrollment.
The calculation below concerns a two-arm, balanced parallel-group trial with a continuous endpoint. It uses a normal approximation for planning and assumes that the target effect, significance level, and power remain fixed. It is a nuisance-parameter update: it does not use an interim treatment-effect estimate to change the target effect.
Model, Assumptions, and Calculation
Suppose the estimand is the difference in population means, \(\Delta=\mu_1-\mu_0\), for a prespecified endpoint. For planning, assume independent participants, equal allocation, a common within-arm outcome variance \(\sigma^2\), and a normal approximation for the test statistic. The calculation also assumes that the target difference \(\Delta\), two-sided significance level \(\alpha\), and power \(1-\beta\) are specified in advance. It uses complete endpoint observations; the attrition adjustment is applied separately.
For \(n_C\) observed outcomes in each arm, the standard error of the difference in sample means is approximately \(\sigma\sqrt{2/n_C}\). To achieve the planned power for a two-sided test at level \(\alpha\), the standardized target difference is set to the sum of the two normal quantiles \(z_{1-\alpha/2}\) and \(z_{1-\beta}\). Solving for \(n_C\) gives:
The interim blinded variance estimate replaces \(\sigma^2\) in this planning formula. Denote that estimate by \(\hat{\sigma}^2\). If the blinded estimator is designed to estimate the variance relevant to the planning model, the updated complete-outcome requirement is
Next, under the stated equal-allocation and common-missingness assumptions, expected observed outcomes per arm are \(N(1-q)/2\). Equating that expectation to \(\hat n_C\) yields the unrounded randomized enrollment target:
This is an expectation-based planning calculation, not a guarantee that every arm will have at least \(\hat n_C\) observed outcomes. Final integer allocation and any minimums or maximums require operational rules; those should be specified separately rather than hidden in the algebra.
How the Two Updates Combine
If the original design used variance \(\sigma_0^2\) and missing-outcome proportion \(q_0\), while the update uses \(\hat{\sigma}^2\) and \(\hat q\), the other design inputs being unchanged, then the ratio of unrounded randomized enrollment targets is
The variance component is a ratio of variances, not a ratio of standard deviations. The attrition component compares the fractions expected to have observed outcomes. The factors multiply because the variance determines how many observed outcomes are required, and the observed-outcome fraction determines how many participants must be randomized to obtain them. If the original enrollment already included an attrition allowance, use the ratio formula or recalculate from the original complete-outcome requirement; do not apply the original allowance a second time.
Worked Example: Recalculating Enrollment From Updated Variance and Attrition
Consider a hypothetical equal-allocation trial with target mean difference \(\Delta=5\) points, two-sided \(\alpha=0.05\), power \(0.90\), interim blinded standard-deviation estimate \(\hat{\sigma}=12\) points, and updated missing-outcome proportion \(\hat q=0.15\). Assume the variance estimator is appropriate for the planning model and that the missingness proportion applies equally to both arms.
The relevant normal quantiles are \(z_{0.975}=1.959964\) and \(z_{0.90}=1.281552\). Their sum is \(3.241516\), and its square is approximately \(10.5074\). Thus the complete-outcome requirement per arm is
Inflating for the expected 15% missingness gives a total randomized target of
The continuous target is about 284.8 participants. For an equal-allocation illustration, rounding up to a balanced total gives 286 randomized participants, or 143 per arm. The expected observed outcomes per arm are \(143(0.85)=121.55\), above the unrounded requirement of 121.045. This rounding illustration does not imply that 121.55 observed outcomes are guaranteed.
Three Worked Applications
Worked Example: Comparing the Updated Target With the Original Plan
A hypothetical trial planned for \(\Delta=4\) units, two-sided \(\alpha=0.05\), and power \(0.80\), using \(\sigma_0=10\) units and \(q_0=0.10\). At the blinded interim review, the variance estimate corresponds to \(\hat{\sigma}=12\) units and the updated missing-outcome proportion is \(\hat q=0.20\). Assume the same model and allocation conditions as above.
Here \(z_{0.975}=1.959964\) and \(z_{0.80}=0.841621\); their sum is \(2.801585\), with square approximately \(7.84888\). The original complete-outcome requirement per arm was
The corresponding unrounded randomized target was \(N_0=2(98.111)/0.90=218.025\). With the updated variance, the complete-outcome requirement becomes
The updated randomized target is \(\hat N=2(141.280)/0.80=353.200\). As a check, the combined ratio is
The small difference between 353.200 and 353.201 is due to rounding intermediate values. The increase is substantial because the higher variance increases the complete-outcome requirement by 44%, while the lower expected observed fraction adds a further multiplicative increase.
Worked Example: Updating From Matured Blinded Interim Data
In a hypothetical trial, 100 randomized participants have reached the time when their endpoint status can be determined. Eighteen have no observed endpoint, so the interim missing-outcome proportion is \(18/100=0.18\). A prespecified blinded variance estimator, applied to the available masked outcome data under the trial’s variance model, returns \(\hat{\sigma}^2=196\) outcome-units squared. The target difference is 6 outcome units, with two-sided \(\alpha=0.05\) and power \(0.90\).
Because the estimator is specified to target the relevant planning variance, use \(\hat{\sigma}^2=196\) in the formula. The quantile sum squared is approximately \(10.5074\), as in the first example. The updated complete-outcome requirement per arm is
Accounting for the estimated missing-outcome proportion gives
An equal-allocation illustration rounds this continuous target up to 280 participants, 140 per arm. Under the assumed missingness fraction, the expected number with observed outcomes is \(140(0.82)=114.8\) per arm, slightly above the unrounded requirement.
This calculation is appropriate only if the 100 participants have matured endpoint status; participants whose follow-up is still pending should not be classified as missing merely because their endpoint has not yet been collected. It also relies on the estimator’s stated target. A raw masked variance can include between-treatment variation as well as within-arm variation, so it should not automatically be substituted unless the chosen blinded procedure accounts for that feature.
Assumptions, Diagnostics, and Failure Modes
The combined calculation is useful only when its inputs describe the future trial adequately. In particular, the variance update and missingness update need not be reliable just because they are computed while treatment assignments remain masked.
- Check the variance target. Confirm that the blinded estimator targets the variance required by the planned test. In a two-arm masked mixture, a pooled variance can reflect treatment-effect separation as well as within-arm variability. Use the prespecified method appropriate to the design; do not assume that any pooled sample variance estimates the desired \(\sigma^2\).
- Check outcome maturity. Calculate missingness using participants whose outcome status is known at the interim look. Counting participants with pending follow-up as missing can inflate \(\hat q\) and enrollment unnecessarily.
- Check whether missingness is plausibly common across arms. A pooled proportion is sufficient for this simple calculation only under the assumed common expected missingness and balanced allocation. If missingness differs by arm, a pooled \(q\) can hide the arm with fewer observed outcomes.
- Check the missing-data mechanism and estimand. Inflating enrollment addresses the expected number of observed endpoints; it does not by itself remove bias from outcome-dependent missingness or solve a mismatch between the endpoint analysis and the target estimand.
- Check whether the updated variance is stable. A small or unusual interim sample can produce a noisy variance estimate. Review the estimator, data quality, and whether the estimate is strongly affected by outliers or distributional features. The variance estimate should not be treated as exact.
- Check that inputs are not counted twice. If \(N_0\) already includes the original attrition allowance, do not inflate it again by \(1/(1-q_0)\) before applying the combined ratio.
A particularly common error is to change only one part of the calculation. A variance-only update may leave enrollment too low when missingness has increased. An attrition-only update may likewise leave too few observed outcomes if variability has increased. Another error is to multiply by \(\hat{\sigma}/\sigma_0\): the sample-size formula depends on variance, so the variance factor is \(\hat{\sigma}^2/\sigma_0^2\).
Comparison and Limits
| Approach | What it updates | What it can miss |
|---|---|---|
| Variance-only update | Required number of observed outcomes, using a revised variance | Enrollment needed to replace missing outcomes |
| Attrition-only update | Randomized enrollment, using a revised missing-outcome proportion | A changed complete-outcome requirement due to revised variance |
| Combined blinded update | Both the observed-outcome requirement and its enrollment inflation | Bias from informative missingness, unsuitable variance estimation, or arm-specific attrition hidden by pooling |
The combined method is a transparent plug-in calculation under its assumptions. It is not a substitute for a missing-data analysis, and it does not guarantee the desired realized number of outcomes or power. Nor does blinding alone establish that every adaptive procedure preserves the trial’s inferential properties. The estimator, timing, and update rule should be prespecified and evaluated for the design in which they will be used. This calculation changes enrollment through nuisance estimates while keeping the target effect fixed; using interim treatment-effect information is a different strategy.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A trial’s variance estimate changes from \(100\) to \(144\), while its missing-outcome proportion changes from \(0.10\) to \(0.20\). With all other design inputs fixed, what is the ratio of updated to original unrounded randomized enrollment?
- A trial requires 80 observed outcomes per arm and expects 20% missing outcomes, with equal allocation. What unrounded total enrollment follows from the expectation-based calculation?
- Why should participants with pending endpoint follow-up not automatically be counted as missing when estimating interim attrition?
- What can go wrong if a blinded pooled variance estimate is substituted without checking what variance it estimates?
Solutions. (1) The ratio is \((144/100)\{(1-0.10)/(1-0.20)\}=1.44(0.90/0.80)=1.62\). Updated randomized enrollment is 1.62 times the original unrounded target. (2) The total is \(2(80)/(1-0.20)=160/0.80=200\) participants. (3) Their outcome status is not yet known; treating pending outcomes as missing can overestimate the missing-outcome proportion and inflate enrollment. (4) A pooled masked variance may include between-arm treatment-effect variation rather than only the within-arm variance required by the planning formula. An unsuitable estimate can therefore distort the recalculated complete-outcome requirement.