Statistical power
- Concept
Power is the probability that a study detects an effect that truly exists. An underpowered study can miss real effects.
What it is
Statistical power (1 − β) is the probability of rejecting the null hypothesis when it is false, that is, of finding a real difference of the expected size. It is set when the study is designed, usually at 80% or 90%. An underpowered study can give a false negative, and when it does come out significant by chance it tends to exaggerate the size of the effect.
Worked example
- Data: a trial compares two diets on systolic pressure; SD = 12 mmHg, two-sided α = 0.05, 40 people per group.
- SE of the difference = 12 × √(2 / 40) = 12 × 0.224 = 2.68 mmHg.
- If the true effect is 5 mmHg: 5 / 2.68 = 1.86; power ≈ normal probability of (1.86 − 1.96) ≈ 46%.
- With 91 people per group, the SE falls to 1.78 and power rises to about 80%.
- With 40 per group but a true effect of 8 mmHg: 8 / 2.68 = 2.98; power ≈ probability of (2.98 − 1.96) ≈ 85%.
- Reading: with 40 per group, a 5 mmHg effect has less than a 50% chance of being detected; more people or a bigger effect raises power.
What raises and what lowers power
| Factor | To raise power | Comment |
|---|---|---|
| Sample size | More participants | The most used lever |
| Effect size | Larger true effect | Not under control; choose the smallest relevant effect |
| α level | Higher α (0.10 instead of 0.05) | At the cost of more false positives |
| Variability | More precise measurements, paired design | Smaller SD, smaller SE |
| Type of test | One-sided (if justified in advance) | Two-sided is the standard |
Markers
- 1 − β
- 80%
- 90%
Common pitfalls
- Reading a non-significant result from a small study as proof of no effect.
- Planning the study around an unrealistically large expected effect so that a small sample comes out.
Test yourself
Think of the answer, then tap the question to see it.
What is statistical power?
The probability of rejecting the null hypothesis when it is false: detecting a real effect. It equals 1 − β.
Which four factors increase power?
A larger sample size, a larger effect, a higher α and less variability in the measurement.
Why does a paired design usually have more power?
Because each person is their own control, removing between-person variability and so reducing the standard error.
A 30-patient trial finds no difference. What should be checked before concluding?
Its power and the confidence interval of the effect: with few patients, important effects may still be compatible with the data.
Connections
Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.
Sources: Sample sizes required (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012); What are statistical tests? (NIST/SEMATECH e-Handbook of Statistical Methods, National Institute of Standards and Technology, 2012).
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Representative model, for educational purposes only.