1. Definition
- Power is the probability of correctly rejecting the null hypothesis (H₀) when the alternative hypothesis (H₁) is true.
- Put simply:
- $\text{Power} = 1 – \beta$
- where β (beta) = probability of a Type II error (failing to reject H₀ when it is false).
2. Interpretation
- High power → the test is good at detecting true effects.
- Low power → the test often misses true effects (false negatives).
Typical target: Power ≥ 0.80 (80%)
- Means: if there’s a true effect, you have an 80% chance of detecting it.
3. Relationship with Errors
- Type I error (α): Rejecting H₀ when it’s true (false positive).
- Type II error (β): Failing to reject H₀ when it’s false (false negative).
- Power (1 – β): Correctly rejecting H₀ when it’s false (true positive).
4. Factors that Affect Power
- Effect Size (δ):
- Bigger true differences → easier to detect → higher power.
- Small effects → harder to detect → lower power.
- Sample Size (n):
- Larger n reduces standard error → higher power.
- Small n → noisy data, low power.
- Significance Level (α):
- Higher α (e.g., 0.10 instead of 0.05) → more liberal rejection → higher power.
- But risk of more false positives.
- Variance (σ²):
- Less variability in data → higher power.
- High variance → lower power.
5. Example
Drug Test Scenario
- H₀: New drug has no effect.
- H₁: Drug reduces blood pressure.
- Suppose:
- True effect size = medium (δ = 0.5).
- n = 30 patients.
- α = 0.05.
→ Power might be 0.60 (60%), meaning there’s a 40% chance you miss the effect (false negative).
If n increases to 100 patients → power rises to ~0.90 (very reliable).
6. Power Analysis (Sample Size Planning)
Researchers use power analysis before experiments to determine the minimum sample size needed.
General idea:
- Given α, desired power (usually 0.8), and expected effect size (δ), compute required n.
7. Summary Table
| Term | Meaning |
|---|---|
| α (Significance Level) | Risk of false positive (Type I error) |
| β (Beta) | Risk of false negative (Type II error) |
| 1 – β (Power) | Probability of detecting a true effect |
In short:
Power (1 – β) is the probability of correctly rejecting the null when the alternative is true. It depends on effect size, sample size, significance level, and variance. High power (≥80%) means your test is sensitive enough to detect real effects and avoid false negatives.