α Allocation Tables in seamless, adaptive, group sequential design
This example reflects a registrational Phase II/III seamless trial
incorporating
seamless + adaptive (dose/population selection) + group sequential
elements.
Trial Assumptions (for context)
- Overall type I error: Two-sided α = 0.05
- Primary endpoint: Progression-Free Survival (PFS)
- Populations:
- F = Full (ITT) population
- S = Biomarker-positive subgroup
(e.g., PD-L1 high)
- Design:
- Stage 1: Multi-dose exploration
+ adaptive decision
- Stage 2: Confirmatory testing
with selected dose (and possibly enriched population)
- Interim analyses:
- One adaptive interim (no
hypothesis testing)
- Two formal group sequential
looks (including final)
- Combination method: Inverse Normal Combination Test
Table 1. Family-wise α Allocation
and Gatekeeping Strategy
SAP-style α allocation across hypothesis families
|
Hypothesis Family |
Description |
Type |
Two-sided α |
Gatekeeping / Notes |
|
H1 (Primary) |
PFS in S population |
Confirmatory |
0.04 |
Primary confirmatory path if
enrichment is selected |
|
H2 (Supportive / Co-primary) |
PFS in F population |
Confirmatory |
0.01 |
Used if no enrichment, or as
supportive evidence |
|
H3 (Key Secondary) |
Overall Survival (OS) |
Confirmatory (conditional) |
0.00* |
*α may be recycled from H1/H2
upon success |
SAP Notes (typical language):
- Strong control of the family-wise
type I error rate at 0.05 (two-sided) will be maintained.
- A hierarchical gatekeeping
strategy will be applied: confirmatory testing of OS is permitted only if
the primary PFS hypothesis is rejected.
- The allocation reflects the
higher prior plausibility of treatment benefit in the biomarker-positive
population.
Table 2. Group Sequential α
Spending for Primary Hypothesis H1 (PFS in S)
Two-sided α = 0.04, O’Brien–Fleming–type boundary (Lan–DeMets)
|
Look |
Information Fraction |
Timing (Illustrative) |
Cumulative α |
Incremental α |
Notes |
|
1 |
0.50 |
~50% of planned events |
0.0005 |
0.0005 |
Very conservative early boundary |
|
2 |
0.75 |
~75% of events |
0.0060 |
0.0055 |
Moderate α release |
|
3 |
1.00 |
Final analysis |
0.0400 |
0.0340 |
Majority of α preserved for
final |
Interpretation:
- Early stopping for efficacy
requires very strong evidence.
- The final analysis boundary is
close to the nominal 0.05 test, minimizing loss of power.
(An analogous table would be specified for H2 using α = 0.01, if
applicable.)
Table 3. Combination Test Parameters
(Inverse Normal Method)
Pre-specified combination of Stage 1 and Stage 2 evidence (H1)
|
Component |
Stage 1 |
Stage 2 |
Notes |
|
p-value |
|
|
Each computed using pre-defined
methods |
|
Weight |
|
|
Based on information fractions |
|
Combined statistic |
\multicolumn{2}{c}{ |
Single final test |
|
|
Significance level |
\multicolumn{2}{c}{Two-sided α
= 0.04 (subject to sequential boundary)} |
SAP Clarification:
- Stage 1 data may inform adaptive
decisions but contribute to inference only through the pre-specified
combination test.
- Type I error is preserved
regardless of dose or population selection.
Table 4. Adaptive Decision Rules (Dose
and Population Selection)
Illustrative SAP decision framework (executed by the independent DMC)
|
Decision Step |
Data Used (Unblinded) |
Pre-specified Rule |
Outcome |
|
Dose elimination (safety) |
SAE/AESI rates |
Eliminate doses exceeding safety
threshold |
Reduced candidate set |
|
Dose selection (efficacy) |
PFS effect estimates |
Select dose meeting minimum
clinically relevant effect with acceptable safety |
Selected dose D* |
|
Population strategy |
Subgroup consistency / interaction
metrics |
If predefined criteria met →
enrich S |
Enrich S or remain F |
|
Confirmatory path |
Selected strategy |
Route hypothesis testing through H1
or H2 |
Final testing pathway |
Example SAP Text (Commonly Used)
“The family-wise two-sided type I error rate will be strongly controlled
at 0.05. Interim efficacy analyses will follow a group sequential design using
a Lan–DeMets O’Brien–Fleming–type alpha-spending function. Adaptive dose and
population selection at the interim analysis will be conducted by the
independent DMC. Confirmatory inference will be based on a pre-specified
inverse normal combination test to preserve the overall type I error rate.”
Key Takeaway (One Sentence)
In adaptive seamless trials, α is allocated at the family level,
spent only at formal group sequential looks, and preserved across adaptive
decisions through pre-specified combination testing.
This is basically a healthy design:
Stage 1 (Adaptation): ~35–40% of events
Look #1 (Sequential): ~60% of events
Final: 100% of events
O’Brien–Fleming–Type Alpha-Spending Function
3. Mathematical Form (Lan–DeMets
Implementation)
Under the Lan–DeMets framework, an OBF-type alpha-spending
function is commonly defined as:
where:
is the cumulative alpha spent
at information fraction 
- is the information fraction
(0–1)
is the standard normal cumulative
distribution function
is the standard normal quantile
corresponding to the nominal two-sided alpha
Key behavior:
- When is small →
is extremely small - When
→ 
4. Interpretation at Interim Analyses
At each formal group sequential look:
- The current information
fraction is calculated
- The cumulative alpha
is obtained from the spending
function - The corresponding critical
boundary (Z-value) is determined
Early stopping for efficacy is allowed only if the observed test
statistic exceeds this boundary.
5. Typical Boundary Behavior
(Illustrative)
For a two-sided α = 0.05 with one interim analysis:
|
Analysis |
Information Fraction |
Cumulative α |
Approximate Z Boundary |
|
Interim |
0.60 |
~0.005 |
±2.8 |
|
Final |
1.00 |
0.05 |
±1.96 |
Interpretation:
- At the interim look, the required
Z-value is very high, making early stopping rare
- At the final analysis, the
boundary is essentially identical to a standard 0.05 test