Sample size for equivalence trials

Equivalence Trials - Same, But Not Inferior

• Goal: To demonstrate that a new treatment is therapeutically similar to a standard one (i.e., not clinically superior or inferior).
• Core Concept: Relies on a pre-defined equivalence margin (Δ), the maximum allowable difference that is clinically acceptable.
• Hypothesis Flip: Unlike superiority trials, the null hypothesis ($H_0$) is that the treatments are different, and the alternative ($H_A$) is that they are equivalent.
  • $H_0$: $|Effect_{new} - Effect_{standard}| ext{ ≥ } Δ$
  • $H_A$: $|Effect_{new} - Effect_{standard}| ext{ < } Δ$
• Conclusion: Equivalence is established if the 95% Confidence Interval for the treatment difference falls entirely within the equivalence margin (-Δ to +Δ).

⭐ A major pitfall is failing to pre-specify the equivalence margin (Δ) before the trial begins. This must be a clinically justified, not statistically derived, value.

Sample Size Formula - The Nitty-Gritty

• The sample size ($n$ per group) for an equivalence trial is determined by a specific formula that accounts for the unique goal of proving similarity, not difference.

• Core Formula:

  • $n \ge \frac{2(Z_{\alpha/2} + Z_{\beta})^2 \sigma^2}{(\delta)^2}$

• Key Inputs:

  • $Z_{\alpha/2}$: Critical value for significance level $\alpha$ (e.g., 1.96 for $\alpha = 0.05$).
  • $Z_{\beta}$: Critical value for statistical power (1-$eta$) (e.g., 0.84 for 80% power).
  • $\sigma^2$: Pooled variance of the outcome measure.
  • $\delta$: The pre-defined equivalence margin. This is the maximum difference between treatments that is considered clinically meaningless.

⭐ The most influential factor is the equivalence margin ($\delta$). A narrower, more stringent margin requires a substantially larger sample size to demonstrate equivalence with adequate power.

Sample Size Levers - Driving N Up or Down

Key factors that influence the required sample size (N) in equivalence trials:

• Factors that ↑ N (Larger Sample Needed):

  • ↓ Smaller equivalence margin (δ)
  • ↑ Higher desired power (1-β) (e.g., 90% vs. 80%)
  • ↓ Lower significance level (α) (e.g., 0.01 vs. 0.05)
  • ↑ Higher data variability (σ²)

• Factors that ↓ N (Smaller Sample Needed):

  • ↑ Larger equivalence margin (δ)
  • ↓ Lower power
  • ↑ Higher significance level (α)
  • ↓ Lower data variability (σ²)

⭐ The choice of the equivalence margin (δ) is paramount. It must be small enough to be clinically meaningful but large enough to make the trial feasible. It's set before the trial begins and represents the largest difference that is clinically unimportant.

High‑Yield Points - ⚡ Biggest Takeaways

• Equivalence trials test if a new treatment is "not unacceptably worse" than the standard one.
• The null hypothesis (H₀) assumes a difference, while the alternative (H₁) assumes equivalence.
• A pre-specified equivalence margin (Δ) is crucial; it defines the acceptable difference.
• The confidence interval of the treatment effect must fall entirely within this margin.
• Smaller (stricter) equivalence margins demand larger sample sizes.
• Generally, equivalence trials require larger sample sizes than superiority trials.