Hypothesis framework and error rates
A trial tests $H_0: \delta = 0$ (no treatment effect) vs. $H_1: \delta = \delta_*$ (a clinically meaningful effect).
Type I error $\alpha = P(\text{reject }H_0 \mid H_0\text{ true})$ — the false positive rate. Conventionally $\alpha = 0.05$ (two-sided).
Type II error $\beta = P(\text{fail to reject }H_0 \mid H_1\text{ true})$ — the false negative rate. Power $= 1 - \beta$, typically $0.80$ or $0.90$.
Signal-to-noise intuition: the test statistic $Z = \hat{\delta} / \text{SE}(\hat{\delta})$ has a non-central normal distribution under $H_1$. The non-centrality parameter is $\lambda = \delta_* / \text{SE}$. Power increases as:
- The true effect $\delta_*$ grows
- The sample size $n$ grows (reducing $\text{SE}$)
- Outcome variability $\sigma^2$ shrinks
Sample size formulas
Two-sample comparison of means: equal group sizes $n$ per group, common variance $\sigma^2$, two-sided $\alpha$, power $1-\beta$:
\[n = \frac{(z_{\alpha/2} + z_\beta)^2 \cdot 2\sigma^2}{\delta^2}\]where $z_p$ is the $p$-th quantile of $N(0,1)$ (e.g., $z_{0.025} = 1.96$, $z_{0.10} = 1.28$ for $90\%$ power, $z_{0.20} = 0.84$ for $80\%$ power).
Survival endpoint (log-rank test): sample size is driven by the number of events $E$:
\[E = \frac{(z_{\alpha/2} + z_\beta)^2}{[\log(\text{HR})]^2 \cdot p_1 p_2}\]where $p_1, p_2$ are allocation proportions ($p_1 = p_2 = 0.5$ for 1:1). The total $n$ follows from the expected event probability over follow-up.
Binary endpoint (two proportions $\pi_1, \pi_2$):
\[n = \frac{(z_{\alpha/2} + z_\beta)^2 [\pi_1(1-\pi_1) + \pi_2(1-\pi_2)]}{(\pi_1 - \pi_2)^2}\]Determinants of sample size:
| Factor | Direction | Impact |
|---|---|---|
| Smaller $\alpha$ | Stricter | $\uparrow n$ |
| Higher power $(1-\beta)$ | More power | $\uparrow n$ |
| Smaller effect $\delta$ | Harder to detect | $\uparrow n$ |
| Larger $\sigma$ | Noisier outcome | $\uparrow n$ |
| Unequal allocation | Imbalanced | $\uparrow n$ |
Randomization methods
Simple randomization: each subject assigned independently with probability $p$. Imbalance likely in small trials.
Block randomization: within blocks of size $2k$, exactly $k$ subjects assigned to each treatment. Block sizes are concealed to prevent allocation prediction.
Stratified randomization: perform block randomization separately within strata defined by prognostic factors (e.g., disease stage, center). Ensures balance on key variables. Recommended when $< 200$ subjects per stratum.
Minimization (Pocock-Simon): new subject assigned to the group that minimizes imbalance on multiple factors simultaneously, using a deterministic or probabilistic rule. Produces superior covariate balance to stratified randomization but allocation is less truly random.
Cluster randomization: entire clusters (practices, schools, villages) randomized. Intra-cluster correlation (ICC) inflates required sample size by the design effect $\text{DEFF} = 1 + (m-1)\rho$, where $m$ is cluster size and $\rho$ is the ICC.
Group sequential designs
Sequential designs allow interim analyses with stopping rules for efficacy or futility, avoiding the need to wait until the full sample is enrolled.
The challenge: each interim look at cumulative data inflates type I error. If the trial is analyzed $K$ times at nominal $\alpha = 0.05$, the actual type I error is $> 0.05$.
O’Brien-Fleming boundaries: apply a stringent boundary at early looks (allowing stopping only for overwhelming evidence) and relax the boundary toward the final analysis:
\[b_k = z_{\alpha/2} \sqrt{K/k}\]| At interim $k$ of $K$, reject $H_0$ if $ | Z_k | > b_k$. The final boundary $b_K = z_{\alpha/2}$ (essentially unchanged), while early boundaries are very large. |
Pocock boundaries: constant threshold $b_k = c_\alpha$ for all $k$, where $c_\alpha$ is chosen to maintain overall type I error $\alpha$. More likely to stop early but the final $p$-value threshold is more conservative than $z_{0.025}$.
Information fraction: at interim $k$, the information fraction is $\mathcal{I}k / \mathcal{I}{\max}$, where $\mathcal{I}$ is Fisher information. For a fixed sample, $\mathcal{I}k / \mathcal{I}{\max} = n_k / N$.
Spending functions (Lan-DeMets): instead of pre-specifying look times, define an error-spending function $\alpha^*(t)$ that allocates the overall $\alpha$ continuously as a function of information fraction $t \in [0,1]$. The O’Brien-Fleming spending function: $\alpha_{\text{OF}}(t) = 2[1 - \Phi(z_{\alpha/2}/\sqrt{t})]$.
Adaptive designs
Sample size re-estimation (SSR): after an interim look, revise the sample size based on the observed variance (blinded) or effect size (unblinded), while controlling type I error via conditional power arguments.
Adaptive enrichment: modify eligibility criteria during the trial based on interim data — e.g., restrict to the subgroup showing larger treatment benefit.
Seamless phase II/III: a single trial starts with multiple doses (phase II) and adapts to carry forward only the promising doses into the confirmatory phase (III), with pre-specified rules and combined inference.
Estimands (ICH E9(R1)): the 2020 addendum requires explicit specification of:
- Population: which patients
- Variable: the outcome
- Intercurrent events: how are events like treatment discontinuation handled (treatment policy, hypothetical, composite, while on treatment, principal stratum)
- Summary measure: mean difference, HR, OR
This forces alignment between the scientific question, the design, and the analysis.
Blinding and CONSORT reporting
Blinding levels:
- Open-label: all parties know assignment
- Single-blind: subjects unaware
- Double-blind: subjects and investigators unaware (gold standard)
- Triple-blind: subjects, investigators, and outcome assessors unaware
CONSORT (Consolidated Standards of Reporting Trials) is the international standard for reporting parallel-group RCTs. Key elements:
- CONSORT flow diagram (enrollment → randomization → follow-up → analysis)
- Allocation concealment mechanism
- Randomization sequence generation
- Primary outcome pre-specification
- All enrolled, randomized, analyzed counts
ICH E9 requires the primary estimand and primary estimator (method) to be pre-specified in the statistical analysis plan (SAP) before database lock.