Factors affecting power

Factors Affecting Power - The Core Quartet

• Statistical Power: The probability of correctly rejecting a false null hypothesis (H₀), i.e., detecting a true effect. It is the complement of a Type II error (Power = $1 - \beta$).

Power & Error Rates - A Balancing Act

• Power is the probability of correctly rejecting a false null hypothesis (detecting a true effect).

• It's directly related to the Type II error rate ($\beta$): $Power = 1 - \beta$.

  • $\beta$ is the probability of a false negative (failing to detect a true effect).
  • A lower $\beta$ leads to a higher power.

• The Trade-Off: Power is inversely affected by the Type I error rate ($\alpha$).

  • $\alpha$ is the probability of a false positive (rejecting a true null hypothesis), usually set at 0.05.
  • To be more confident in our result (↓ $\alpha$), we must accept a higher chance of missing a true effect (↑ $\beta$), which in turn ↓ Power.
  • ↓ $\alpha$ → ↑ $\beta$ → ↓ Power.

⭐ By convention, an acceptable $\beta$ is often set at 0.2, which corresponds to a power of 0.8 (or 80%). This means researchers accept a 20% chance of missing a real effect to achieve 80% power.

Sample Size Formula - Power by the Numbers

A simplified formula shows how different factors influence the required sample size ($n$):

$n \propto \frac{(\sigma^2)(Z_\alpha + Z_\beta)^2}{(\mu_1 - \mu_2)^2}$

• $n$ (Sample Size): Number of subjects needed.
• $\sigma$ (Standard Deviation): Data variability. As $\sigma$ ↑, $n$ ↑.
• $Z_\alpha$ (Significance): Z-score for the chosen $\alpha$ level (e.g., 0.05). As $\alpha$ ↓, $n$ ↑.
• $Z_\beta$ (Power): Z-score for the desired power (1 - $\beta$). As power ↑, $n$ ↑.
• $\mu_1 - \mu_2$ (Effect Size): Magnitude of the difference to be detected. As effect size ↓, $n$ ↑.

⭐ To detect an effect size that is half as large, you must quadruple the sample size.

High‑Yield Points - ⚡ Biggest Takeaways

• Power is the probability of detecting a true effect and avoiding a Type II error (Power = 1 - β).
• ↑ Sample size (n) is the most common way to increase power.
• ↑ Effect size (the magnitude of the difference) makes it easier to find a statistically significant result.
• ↑ Significance level (α) increases power, but also increases the risk of a Type I error.
• ↓ Variability (standard deviation) in the data increases power.