What is statistical power, and how can you increase it?
Statistical power is the probability of correctly rejecting a false null hypothesis, equal to 1 minus the Type II error rate (beta). It rises with larger sample size, larger true effect size, higher alpha, or lower measurement variance.
How to think about it
Power is the sensitivity of your test — its ability to detect a real signal. Running an underpowered study is wasteful: you collect data, run the test, fail to reject H0, and cannot tell whether the effect is absent or whether you simply lacked the sensitivity to find it.
Formal definition
Power = P(reject H0 | H0 is false) = 1 - beta
Power depends on four quantities, and the relationship is exact enough for sample-size calculations:
- Effect size (delta): The magnitude of the true difference. A larger true effect is easier to detect.
- Sample size (n): More data reduces the standard error, pushing the two distributions further apart relative to their spread.
- Significance level (alpha): A more lenient alpha (e.g., 0.10 vs 0.05) moves the critical value closer to H0’s center, catching more of H1’s distribution.
- Population variance (sigma^2): Less noisy measurements mean less overlap between H0 and H1 distributions.
The standard error link
The test statistic scales as delta / (sigma / sqrt(n)). Quadrupling n halves the standard error, doubling the signal-to-noise ratio. Power climbs steeply with n at first, then flattens — a key reason why sample-size calculations must happen before data collection, not after.
Practical ways to increase power
| Lever | Mechanism | Practical note |
|---|---|---|
| Increase n | Shrinks SE | Pre-register sample size via power analysis (e.g., 80% power target) |
| Reduce variance | Shrinks SE | Stratified sampling, within-subject (paired) designs, covariate adjustment |
| Increase alpha | Relaxes threshold | Only defensible when Type I cost is genuinely low |
| Use one-tailed test | Concentrates the rejection region | Only valid when a directional effect is pre-specified |
| Increase effect size | Bigger signal | Often not controllable; but higher drug dose, stronger intervention |
Interpreting a power analysis
A power of 0.80 is a conventional minimum: you accept a 20% chance of missing a real effect of the specified size. If the effect could be clinically important even if small, set power higher (0.90 or 0.95).