Sample size calculation for different study designs

Core Concepts - The Power Players

• Sample Size (n) is a function of four key variables:
  • α (Alpha/Type I Error): Probability of a false positive. Conventionally set at 0.05.
  • β (Beta/Type II Error): Probability of a false negative. Conventionally set at 0.20.
  • Power (1-β): Probability of detecting a true effect. Desired level is typically ≥80%.
  • Effect Size (d): Magnitude of the difference to be detected. Smaller effects need larger samples.

⭐ Halving the effect size you want to detect requires quadrupling the sample size. The relationship is an inverse square: $n \propto 1/d^2$.

Continuous Outcomes - Mean Feats

• Objective: To find the sample size needed to detect a specific difference between two means.
• Core Formula: For two independent sample means: $$ n \approx \frac{2\sigma^2(Z_{\alpha/2} + Z_\beta)^2}{d^2} $$
  • n: Sample size per group.
  • d: Smallest difference in means you want to detect.
  • σ: Population standard deviation (estimated from prior studies).
  • Zα/2: Z-score for desired significance level (e.g., 1.96 for α = 0.05).
  • Zβ: Z-score for desired power (e.g., 0.84 for 80% power).

⭐ To detect a difference half as small (d/2), you must quadruple (4x) the sample size.

Categorical Outcomes - Proportion Power

• Objective: Calculate sample size (n) per group to detect a specified difference between two proportions ($p_1$, $p_2$) with desired power.

• Core Components:

  • α (alpha): Significance level (e.g., 0.05).
  • Power (1-β): Probability of detecting a true effect (e.g., 0.80).
  • $p_1$, $p_2$: Estimated proportions in the two groups.
  • Effect Size: The clinically significant difference ($|p_1 - p_2|$).

• Sample Size Formula (per group): $$ n = \frac{2\bar{p}(1-\bar{p})(Z_{\alpha/2} + Z_{\beta})^2}{(p_1 - p_2)^2} $$

  • Where $\bar{p} = (p_1+p_2)/2$ is the average proportion.

• Key Relationships:

  • Sample size (n) ↑ as power ↑, significance ↓ (α ↓), or effect size ↓.

⭐ High-Yield Pearl: The required sample size is maximal for a given difference when the average proportion ($\bar{p}$) is 0.5 (50%), as this value represents maximum variance. Detecting small differences requires much larger samples.

Observational Studies - Cohort & Case Calculations

• Goal: Determine the minimum sample size to detect a statistically significant association (e.g., a specific Relative Risk or Odds Ratio).

• Core Inputs for Calculation:

  • α (alpha): Probability of Type I error (false positive). Conventionally set at 0.05.
  • β (beta): Probability of Type II error (false negative). Set at 0.20 or 0.10.
  • Power (1-β): Probability of detecting a true effect. Usually 80% or 90%.
  • Effect Size: The minimum expected difference (e.g., RR, OR) you want to detect.
  • Prevalence/Incidence: Expected frequency of the outcome (cohort) or exposure (case-control) in the comparison group.

⭐ In case-control studies, increasing the number of controls per case (e.g., 2:1, 3:1) boosts study power. This benefit significantly diminishes beyond a 4:1 ratio.

High‑Yield Points - ⚡ Biggest Takeaways

• Sample size is directly proportional to the desired power (1-β) and population variance (σ²).
• It is inversely proportional to the effect size and the significance level (α).
• To ↑ power or detect a smaller effect size, you must ↑ sample size.
• Halving the effect size requires quadrupling the sample size to maintain the same power.
• Case-control sample size is influenced by the prevalence of exposure.
• Cohort/RCT sample size is driven by the expected incidence of the outcome.