Power (1 – β)

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).

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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.

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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).

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4. Factors that Affect Power

1. Effect Size (δ):
   • Bigger true differences → easier to detect → higher power.
   • Small effects → harder to detect → lower power.
2. Sample Size (n):
   • Larger n reduces standard error → higher power.
   • Small n → noisy data, low power.
3. Significance Level (α):
   • Higher α (e.g., 0.10 instead of 0.05) → more liberal rejection → higher power.
   • But risk of more false positives.
4. Variance (σ²):
   • Less variability in data → higher power.
   • High variance → lower power.

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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).

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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.

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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

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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.