Two-Proportion Sample Size Calculator

This calculator determines the sample size required to detect a difference between two proportions (e.g., event rates in treatment vs. control groups) with specified statistical power and significance level.

Enter study parameters
Expected Proportion in Group 1 (%) e.g., event rate in treatment group
Expected Proportion in Group 2 (%) e.g., event rate in control group
Significance Level (α) Statistical Power (1 − β)
Test Type Allocation Ratio (n₂/n₁)

Formula

Sample Size for Two Proportions

\( n_1 = \frac{\left[ z_{\alpha} \sqrt{(1 + 1/k) \bar{p}(1-\bar{p})} + z_{\beta} \sqrt{p_1(1-p_1) + p_2(1-p_2)/k} \right]^2}{(p_1 - p_2)^2} \)
\( n_2 = k \cdot n_1 \)

where \( \bar{p} = \frac{p_1 + k \cdot p_2}{1 + k} \) is the pooled proportion and \( k \) is the allocation ratio.

Understanding the Calculation

This calculator uses the standard formula for comparing two independent proportions. The sample size depends on:

  • Effect size — the difference between the two proportions (larger differences require smaller samples)
  • Significance level (α) — the probability of a Type I error (false positive)
  • Power (1 − β) — the probability of detecting a true difference
  • Allocation ratio — the ratio of sample sizes between groups

Practical Considerations

When planning your study:

  • Add 10-20% to account for anticipated dropout or loss to follow-up
  • Consider using unequal allocation if one treatment is more expensive or has more side effects
  • Use conservative (higher) estimates of the control group proportion if uncertain
  • A two-sided test is generally preferred unless there is strong justification for one-sided

Specific Examples

Example 1: A clinical trial expects 30% event rate in the control group and 20% in the treatment group. With 80% power and α = 0.05 (two-sided), approximately 294 patients per group are needed.

Example 2: A vaccine trial expects 5% infection rate in the placebo group and 1% in the vaccine group. With 90% power and α = 0.05, approximately 435 patients per group are needed.

Assumptions & Limitations

  • Assumes independent samples from each group
  • Based on normal approximation to the binomial distribution
  • May be inaccurate for very small proportions (< 5%) or very large proportions (> 95%)
  • Does not account for clustering or repeated measures
  • Actual power may differ if assumptions are violated

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