All comparisons were performed for a one‑arm, fixed‑design clinical trial with a binary primary endpoint. Such designs are often used in rare diseases or early‑phase settings where recruiting a concurrent control group is not feasible. The objective of this section is to compare sample size calculations across software implementations using two types of analyses:
Both approaches were evaluated over the same factorial design space used in the preceding two‑arm sections.
The same four parameters were systematically varied:
Type I error rate (\(\alpha\), two-sided):
\(\alpha \in {0.01,\ 0.05,\ 0.10,\ 0.20,\ 0.49}\)
Power (\(1 - \beta\)):
\(1 - \beta \in {0.51,\ 0.80,\ 0.90,\ 0.99}\)
Control (historical) response rate:
\(\pi_c \in {0.10,\ 0.30,\ 0.50,\ 0.80,\ 0.90}\)
Minimum clinically important improvement:
\(\Delta\pi \in {0.05,\ 0.15,\ 0.25,\ 0.49}\)
The experimental response rate target for design purposes was defined as:
\[ \pi_e = \pi_c + \Delta\pi, \]
and scenarios yielding \(\pi_e > 1\) were discarded. A total of 300 valid design scenarios were evaluated.
To ensure comparability between methods and isolate software‑level differences, the following assumptions were fixed across all scenarios:
No adjustments for stratification, covariates, or missingness were included. The focus is strictly on sample size determination under the assumed null response rate \(\pi_c\).
Relevancy was determined using the same rule‑based criteria applied in the two‑arm sections, relying only on α and power:
High relevance
Scenarios satisfying \[
0.02 < \alpha < 0.15 \quad\text{and}\quad \text{power} > 0.75.
\] These constitute realistic one‑arm designs often used in phase II or rare‑disease contexts.
Medium relevance
Scenarios not meeting criteria for high or low relevance, representing plausible but less standard combinations of α and power.
Low relevance
Scenarios with \[
\alpha > 0.20
\quad\text{or}\quad
\text{power} < 0.60
\quad\text{or}\quad
\text{power} \ge 0.99.
\] Such settings are operationally uncommon but included to stress‑test asymptotic and exact methods across extreme operating conditions.
The classification is used purely for interpretability and does not impact sample size computations.
rpact::getSampleSizeRates(group = 1)📌Overall: for one‑arm binary fixed designs using the Z‑test, rpact and nQuery are fully concordant with East, yielding identical sample sizes across the full design space.
nQuery and rpact produce identical sample sizes, with N‑ratios effectively equal to 1.00 for every design.| N-Ratio 1-Arms Binary, normal approximation | ||||||
| According to East sample sizes | ||||||
| Relevancy | Min | Q1 | Mean | Median | Q3 | Max |
|---|---|---|---|---|---|---|
| high | ||||||
| nquery | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| rpact | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| medium | ||||||
| nquery | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| rpact | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| low | ||||||
| nquery | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| rpact | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| N-Ratio 1-Arms Binary, normal approximation | ||||||
| According to East sample sizes | ||||||
| Relevancy | Min | Q1 | Mean | Median | Q3 | Max |
|---|---|---|---|---|---|---|
| high | ||||||
| nquery | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| rpact | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| medium | ||||||
| nquery | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| rpact | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| low | ||||||
| nquery | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
| rpact | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 |
📌Overall: A’Hern increasingly diverges from East as relevancy drops, shifting from mild underestimation to wider underestimation.
| N-Ratio 1-Arms Binary, exact computation | ||||||
| According to East sample sizes | ||||||
| Relevancy | Min | Q1 | Mean | Median | Q3 | Max |
|---|---|---|---|---|---|---|
| high | ||||||
| ahern | 0.80 | 0.94 | 0.97 | 0.96 | 1.00 | 1.19 |
| medium | ||||||
| ahern | 0.62 | 0.91 | 0.96 | 0.95 | 1.00 | 1.33 |
| low | ||||||
| ahern | 0.33 | 0.83 | 0.89 | 0.92 | 1.00 | 1.67 |
| Red : < 50% of N-ratios ±10% | ||||||
| N-Ratio 1-Arms Binary, exact computation | ||||||
| According to East sample sizes | ||||||
| Relevancy | Min | Q1 | Mean | Median | Q3 | Max |
|---|---|---|---|---|---|---|
| high | ||||||
| ahern | 0.80 | 0.94 | 0.97 | 0.96 | 1.00 | 1.19 |
| medium | ||||||
| ahern | 0.62 | 0.91 | 0.96 | 0.95 | 1.00 | 1.33 |
| low | ||||||
| ahern | 0.33 | 0.83 | 0.89 | 0.92 | 1.00 | 1.67 |
| Red : < 50% of N-ratios ±10% | ||||||
A’Hern’s design is based on a single‑stage exact binomial test with integer‑grid optimisation of \((n, r)\) pairs targeting feasibility and minimax or optimality criteria.
East, by contrast, uses an exact unconditional power calculation (not the A’Hern grid search) and optimises \(n\) directly to meet the power constraint.
These two formulations:
Because of these structural differences, A’Hern is not expected to match East, even though your A’Hern implementation is correct (it reproduces the original published tables).
Overall: the observed discrepancies reflect fundamental differences in how exact one‑arm binary designs are defined and optimised.
You can explore all results interactively, filter designs, compare methods, and inspect individual cases in the interactive section of the website.