Removing a Planned Interim Look in Group Sequential Trials

Why a Removed Look Requires Recalibration

A group sequential design is calibrated for a particular sequence of analyses. Each efficacy boundary is chosen with account taken of the earlier opportunities to stop. If a planned analysis is omitted, the set of possible crossing events changes: participants are not tested at the omitted look, and a later statistic may now be the first opportunity to cross. Simply keeping the remaining boundaries can therefore change the probability of rejecting $H_0$.

The goal here is to remove a planned look while preserving the design’s cumulative alpha-spending targets at the retained information times. This is a recalibration, not a claim that the trial has the same operating characteristics under every possible schedule. The target spending curve remains the reference, but the boundary sequence is solved again for the retained schedule.

Definition: Let $t_1<\cdots<t_K=1$ be the retained information fractions, and let $A(t)$ be a nondecreasing cumulative one-sided alpha-spending function with $A(0)=0$ and $A(1)=\alpha$. For upper efficacy boundaries $b_1,\ldots,b_K$, recalibration at the retained looks requires $$ P_{H_0}\!\left(\bigcup_{j=1}^{k}\{Z(t_j)\ge b_j\}\right)=A(t_k),\qquad k=1,\ldots,K. $$ The probability on the left is the cumulative null rejection probability through retained look $k$; it is not the nominal error allocated to look $k$ alone.

We use the canonical group sequential model under $H_0$: the standardized statistics have a joint normal distribution with mean zero, unit variances, and covariance $\operatorname{Cov}\{Z(s),Z(t)\}=\min(s,t)/\sqrt{st}$ for positive information fractions $s,t$. Thus, for $s\le t$, their correlation is $\sqrt{s/t}$. The model assumes the stated information fractions correctly represent the analysis information and that the planned statistic and spending function remain appropriate. For a two-sided design, both tails and the chosen error-allocation convention must be included; the examples below use an upper one-sided efficacy test.

Calibrating Only at Retained Looks

Suppose the original schedule included a look at $t_m$, but that analysis will not occur. Remove $t_m$ from the analysis sequence. Do not include a boundary or a crossing event at that time. At each retained time $t_k$, keep the cumulative target $A(t_k)$, then determine boundaries for the shorter sequence that satisfy those targets.

At the first retained look, there are no earlier retained crossing events, so the boundary has a direct expression. At later looks, a boundary must account for the joint distribution with the earlier retained statistics. A useful way to express the $k$th calibration equation is

P_{H_0}\!\left( Z(t_j)<b_j\ \text{for }j<k,\ Z(t_k)\ge b_k \right) =A(t_k)-A(t_{k-1}),

where the right side is the increase in cumulative target between consecutive retained looks. This identity follows by partitioning the event of any crossing through $t_k$ into (i) a crossing by $t_{k-1}$, and (ii) no earlier crossing followed by a crossing at $t_k$. It shows why a stagewise target is not generally the marginal tail probability $P\{Z(t_k)\ge b_k\}$: some paths crossing at $t_k$ would already have crossed an earlier retained boundary.

Key result: Calibrate sequentially on the retained schedule. Choose $b_1$ so that $P\{Z(t_1)\ge b_1\}=A(t_1)$. For $k\ge2$, choose $b_k$ so the probability of no crossing at retained looks $1,\ldots,k-1$ followed by crossing at $k$ equals $A(t_k)-A(t_{k-1})$. Under the canonical model, this is a joint normal probability calculation and usually requires numerical evaluation.

Worked Recalibrations

Worked Example: Removing the Only Interim Look

A hypothetical one-sided design has total $\alpha=0.025$, with a planned interim look at information fraction $t=0.50$ and a final analysis at $t=1$. The interim analysis is omitted, so only the final analysis remains. Assume the final information is still $t=1$, the test is standardized as specified by the canonical model, and the spending function has $A(1)=0.025$.

There is now one crossing event, at the final analysis. The calibration equation is $$ P\{Z(1)\ge b_1\}=A(1)=0.025,\qquad Z(1)\sim N(0,1). $$ Thus $b_1=\Phi^{-1}(0.975)=1.960$, where $\Phi$ is the standard normal cumulative distribution function. The recalibrated final rule is to reject $H_0$ if the final standardized statistic is at least 1.960.

The old interim boundary is not used, and the final boundary is not inherited from the former two-look design. With only one look, the final upper-tail probability itself must equal 0.025. Under the assumed null model, this gives cumulative Type I error 0.025.

Worked Example: Omitting the Middle Look

Consider a hypothetical one-sided design with $\alpha=0.025$, originally scheduled at $t=0.25,0.50,1$. The look at $t=0.50$ is omitted. Suppose the specified spending curve gives $A(0.25)=0.005$ and $A(1)=0.025$. Assume the retained information fractions and canonical joint normal model are valid.

At the first retained look, $$ P\{Z(0.25)\ge b_1\}=0.005, \qquad b_1=\Phi^{-1}(0.995)=2.576. $$ At the final look, the correlation is $\sqrt{0.25/1}=0.5$. Write $Z_1=Z(0.25)$ and $Z_2=Z(1)$. The final boundary must satisfy $$ P(Z_1\ge2.576)+P(Z_2\ge b_2)-P(Z_1\ge2.576,\ Z_2\ge b_2)=0.025. $$ The first probability is 0.005. Hence the required numerical equation is $$ 0.005+P(Z_2\ge b_2)-P(Z_1\ge2.576,\ Z_2\ge b_2)=0.025, \quad (Z_1,Z_2)\sim N_2\!\left( \begin{pmatrix}0\\0\end{pmatrix}, \begin{pmatrix}1&0.5\\0.5&1\end{pmatrix} \right). $$ A bivariate normal integration followed by one-dimensional root-finding gives $b_2$ near 2.03; the final digits depend on numerical integration and root-finding tolerance. The overlap probability is subtracted because some paths exceed both boundaries. The final marginal tail must therefore be slightly greater than 0.020 for the union probability to equal 0.025.

The omitted $t=0.50$ look contributes no crossing event. The target at the final look remains $A(1)=0.025$, but the recalibrated final boundary accounts only for the retained $t=0.25$ opportunity.

Worked Example: Removing an Early Look and Retaining Two Later Looks

A hypothetical design is planned at $t=0.20,0.60,1$, but the $t=0.20$ analysis is dropped. Let $\alpha=0.025$, with cumulative spending targets $A(0.60)=0.010$ and $A(1)=0.025$. Suppose the information fractions are correctly specified and the null joint distribution is canonical.

The first retained boundary is $$ b_1=\Phi^{-1}(1-0.010)=\Phi^{-1}(0.990)=2.326. $$ For the final boundary, the retained-look correlation is $\sqrt{0.60/1}=0.7746$. Set $Z_1=Z(0.60)$ and $Z_2=Z(1)$. Solve $$ P(Z_1\ge2.326)+P(Z_2\ge b_2) -P(Z_1\ge2.326,\ Z_2\ge b_2)=0.025, $$ with $(Z_1,Z_2)$ bivariate standard normal and correlation 0.7746. Since the first term is 0.010, the equation requires $$ P(Z_2\ge b_2)-P(Z_1\ge2.326,\ Z_2\ge b_2)=0.015. $$ A numerical bivariate normal solver can find $b_2$ from this equation. This is a complete specification of the recalibration: the final crossing probability among paths not already rejected at $t=0.60$ must be 0.015. It would be incorrect to set $b_2=\Phi^{-1}(0.975)$, because that would assign a final marginal tail probability of 0.025 in addition to the earlier crossing opportunity.

In practice, report the solver’s boundary with the integration tolerance and verify that substituting it into the union probability reproduces 0.025 to the stated numerical precision. The equation also makes the required target and the role of the overlap explicit even before the final numerical root is obtained.

Implementation, Diagnostics, and Limitations

A practical recalibration has four steps. First, list only the analyses that will actually occur, with their information fractions in increasing order. Second, evaluate the spending function at each retained fraction; do not replace $A(t_k)$ with a nominal per-look alpha. Third, solve the cumulative crossing equations sequentially under the joint null model. Finally, verify the resulting boundaries by evaluating the cumulative crossing probability at every retained look.

Fix the retained schedule.
Record the actual retained information fractions and confirm that the final analysis remains at the planned terminal information.

Carry forward cumulative targets.
Evaluate the same spending function $A(t)$ at each retained fraction. The omitted analysis is not assigned a boundary.

Re-solve the boundaries.
Use the canonical joint distribution for the retained statistics, including their correlations, and solve each cumulative crossing equation.

Verify the design.
Recalculate cumulative crossing probabilities at all retained looks and check that they match $A(t_k)$ within numerical tolerance.

Several diagnostics are especially useful. The targets must be nondecreasing, and each computed cumulative crossing probability should match its target. Boundaries should be finite when the corresponding target is positive and should not be treated as ordinary finite cutoffs when the increment in target is zero. Check that the correlation matrix uses the retained information fractions, not the original look indices. A solver that fails to converge, or a substituted boundary that misses the target, is a reason to inspect integration accuracy, root brackets, and input times rather than to accept the output.

Recalibration preserves the specified null rejection probabilities at the retained looks under the assumed model. It does not necessarily preserve power, expected sample size, or the interpretation of a design that is changed for reasons related to emerging unblinded results. It also depends on accurate information times and a valid joint distribution. If information differs from plan, or if the reason for changing the schedule creates additional concerns, the statistical and operational consequences require separate consideration.

The main alternative is to leave the old boundaries in place. That is simple but generally does not preserve the cumulative spending targets, because removing an earlier crossing opportunity alters the probability of reaching and crossing later boundaries. A conditional-error approach is another way to assess modifications: it asks how much rejection probability remains conditional on the observed history. It is not the same as recalibrating to $A(t_k)$ at retained information times, and the two approaches answer different design questions.

Common Mistakes

Keeping the old final boundary automatically. The original boundary was calibrated with the original sequence of looks. Recompute it for the retained sequence.
Using the spending increment as a marginal tail probability. $A(t_k)-A(t_{k-1})$ is the required probability of a new crossing after no earlier crossing, not generally $P\{Z(t_k)\ge b_k\}$.
Including the omitted look in the probability calculation. If no test occurs at that time, it is not part of the union of rejection events.
Using the wrong correlation. For retained times $s\le t$, the canonical correlation is $\sqrt{s/t}$, based on the information fractions.
Confusing alpha control with unchanged power. Matching cumulative null targets does not guarantee the same power or stopping behavior as the original design.

Key takeaway: When a planned look is omitted, remove its crossing event and recalibrate every remaining boundary against the original cumulative spending targets at the retained information times. Verify the resulting cumulative crossing probabilities under the joint null model.

Check Your Understanding

Use the definitions and calibration equations above. Solutions are included after the questions.

A one-sided trial has total $\alpha=0.025$ and only its final analysis at $t=1$ remains. What is the upper efficacy boundary?
Why is the later boundary not generally obtained by setting its marginal upper-tail probability equal to $A(t_k)-A(t_{k-1})$?
Two retained looks occur at $t=0.36$ and $t=0.81$. Under the canonical model, what is their correlation?
What probability should be checked after calibrating a final boundary in a two-look design?

Solutions. (1) The boundary is $\Phi^{-1}(0.975)=1.960$. (2) Some paths exceed the later boundary after already crossing an earlier boundary; the spending increment is the probability of a new crossing among paths with no previous crossing. (3) Since $0.36\le0.81$, the correlation is $\sqrt{0.36/0.81}=\sqrt{4/9}=2/3=0.6667$. (4) Check the joint null probability of crossing at either retained look, $P\{Z(t_1)\ge b_1\ \text{or}\ Z(t_2)\ge b_2\}$, and confirm it equals $A(t_2)$ within the chosen numerical tolerance.