Power (1 – β)#
The probability a test correctly detects a real effect (rejects a false null).
Important
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What it is#
Power is the probability of correctly rejecting the null hypothesis \(H_0\) when the alternative is true — of detecting a real effect. Formally,
where \(\beta\) is the probability of a Type II error (missing a true effect). The usual target is power \(\ge 0.80\): an 80% chance of catching an effect that is really there.
The error triad#
Three quantities partition the possibilities when \(H_0\) is actually false or true: \(\alpha\) is the Type I error (a false positive — rejecting a true \(H_0\)), \(\beta\) the Type II error (a false negative), and \(1 - \beta\) the power (a true positive).
What raises power#
Four levers. A larger effect size \(\delta\) is easier to detect; a larger sample size \(n\) shrinks the standard error and lifts power; a more lenient significance level \(\alpha\) (say 0.10 rather than 0.05) raises power but admits more false positives; and lower variance \(\sigma^2\) sharpens detection.
Example#
Testing whether a drug lowers blood pressure, with a medium effect (\(\delta = 0.5\)), \(n = 30\) and \(\alpha = 0.05\), power might be only 0.60 — a 40% chance of missing the effect. Raising \(n\) to 100 lifts power to about 0.90.
Where it’s used#
Power is the target of a-priori power analysis: fixing \(\alpha\), a desired power (commonly 0.80) and an expected \(\delta\), one solves for the minimum sample size needed — so that a true effect is very likely to register rather than slip away as a false negative.
Theme: Statistical Inference & Power · All terminology
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Mind map — connected ideas
Type I Error · Statistical Power · A Priori Power Analysis · Effect Size (δ) · Sample size · Statistical Significance
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More in Statistical Inference & Power
A Priori Power Analysis · Chi-square (χ²) Test · Clopper–Pearson Interval · Compromise Power Analysis · Confidence Intervals (CIs) · Effect Size (δ) · Hypothesis Testing · Kolmogorov–Smirnov (KS) Test · Minimum Detectable Lift (MDL) · P-Value (probability value) · Post Hoc Power Analysis · Power Analysis · Sample size · Significance Level (α)
See also
Source article Adapted (context, re-expressed) in our own words from: Power (1 – β) (insightful-data-lab.com).